Entropic Measure and Wasserstein Diffusion
Max-K. von Renesse, Karl-Theodor Sturm
Abstract
We construct a new random probability measure on the sphere and on the unit interval
which in both cases has a Gibbs structure with the relative entropy functional as Hamiltonian.
It satisfies a quasi-invariance formula with respect to the action of smooth diffeomorphism
of the sphere and the interval respectively. The associated integration by parts formula is
used to construct two classes of diffusion processes on probability measures (on the sphere or
the unit interval) by Dirichlet form methods. The first one is closely related to Malliavin’s
Brownian motion on the homeomorphism group. The second one is a probability valued
stochastic perturbation of the heat flow, whose intrinsic metric is the quadratic Wasserstein
distance. It may be regarded as the canonical diffusion process on the Wasserstein space.
1 Introduction
(a) Equipped with the L2-Wasserstein distance dW (cf. (2.1)), the space P(M) of probability
measures on an Euclidean or Riemannian space M is itself a rich object of geometric interest.
Due to the fundamental works of Y. Brenier, R. McCann, F. Otto, C. Villani and many others
(see e.g. [Bre91, McC97, CEMS01, Ott01, OV00, Vil03]) there are well understood and pow-
erful concepts of geodesics, exponential maps, tangent spaces TµP(M) and gradients Du(µ) of
functions on this space. In a certain sense, P(M) can be regarded as an infinite dimensional
Riemannian manifold, or at least as an infinite dimensional Alexandrov space with nonnegative
lower curvature bound if the base manifold (M,d) has nonnegative sectional curvature.
A central role is played by the relative entropy : P(M) → R ∪ {+∞} with respect to the
Riemannian volume measure dx on M
Ent(µ) =
ρ log ρ dx, if dµ(x) ≪ dx with ρ(x) = dµ(x)
+∞, else.
The relative entropy as a function on the geodesic space (P(M), dW ) is K-convex for a given
number K ∈ R if and only if the Ricci curvature of the underlying manifold M is bounded from
below by K, [vRS05, Stu06]. The gradient flow for the relative entropy in the geodesic space
(P(M), dW ) is given by the heat equation ∂∂tµ = ∆µ on M , [JKO98]. More generally, a large
class of evolution equations can be treated as gradient flows for suitable free energy functionals
S : P(M) → R, [Vil03].
What is missing until now, is a natural ’Riemannian volume measure’ P on P(M). The basic re-
quirement will be an integration by parts formula for the gradient. This will imply the closability
of the pre-Dirichlet form
E(u, v) =
〈Du(µ),Dv(µ)〉Tµ dP(µ)
in L2(P(M),P), – which in turn will be the key tool in order to develop an analytic and stochastic
calculus on P(M). In particular, it will allow us to construct a kind of Laplacian and a kind
of Brownian motion on P(M). Among others, we intend to use the powerful machinery of
Dirichlet forms to study stochastically perturbed gradient flows on P(M) which – on the level
http://arxiv.org/abs/0704.0704v1
of the underlying spaces M – will lead to a new concept of SPDEs (preserving probability by
construction).
Instead of constructing a ’uniform distribution’ P on P(M), for various reasons, we prefer to
construct a probability measure Pβ on P(M) formally given as
dPβ(µ) =
e−β·Ent(µ) dP(µ) (1.1)
for β > 0 and some normalization constant Zβ. (In the language of statistical mechanics, β is
the ’inverse temperature’ and Zβ the ’partition function’ whereas the entropy plays the role of
a Hamiltonian.)
(b) One of the basic results of this paper is the rigorous construction of such a entropic measure
Pβ in the one-dimensional case, i.e. M = S1 or M = [0, 1]. We will essentially make use of the
representation of probability measures by their inverse distributions function gµ. It allows to
transfer the problem of constructing a measure Pβ on the space of probability measures P([0, 1])
(or P(S1)) into the problem of constructing a measure Qβ0 (or Qβ) on the space G0 (or G, resp.)
of nondecreasing functions from [0, 1] (or S1, resp.) into itself.
In terms of the measure Q
0 on G0, for instance, the formal characterization (1.1) then reads as
follows
0 (g) =
e−β·S(g) dQ0(g). (1.2)
Here Q0 denotes some ’uniform distribution’ on G0 ⊂ L2([0, 1]) and S : G0 → [0,∞] is the
entropy functional
S(g) := Ent(g∗Leb) = −
log g′(t) dt.
This representation is reminiscent of Feynman’s heuristic picture of the Wiener measure, — now
with the energy
H(g) =
g′(t)2dt
of a path replaced by its entropy. Q
0 will turn out to be (the law of) the Dirichlet process or
normalized Gamma process.
(c) The key result here is the quasi-invariance – or in other words a change of variable formula –
for the measure Pβ (or P
0 ) under push-forwards µ 7→ h∗µ by means of smooth diffeomorphisms
h of S1 (or [0, 1], resp.). This is equivalent to the quasi-invariance of the measure Qβ under
translations g 7→ h ◦ g of the semigroup G by smooth h ∈ G. The density
dPβ(h∗µ)
dPβ(µ)
· Y 0h (µ)
consists of two terms. The first one
h (µ) = exp
log h′(t)dµ(t)
can be interpreted as exp(−βEnt(h∗µ))/ exp(−βEnt(µ)) in accordance with our formal inter-
pretation (1.1). The second one
Y 0h (µ) =
I∈gaps(µ)
h′(I−) · h′(I+)
|h(I)|/|I|
can be interpreted as the change of variable formula for the (non-existing) measure P. Here
gaps(µ) denotes the set of intervals I =]I−, I+[⊂ S1 of maximal length with µ(I) = 0. Note that
Pβ is concentrated on the set of µ which have no atoms and not absolutely continuous parts and
whose supports have Lebesgue measure 0.
(d) The tangent space at a given point µ in P = P(S1) (or in P0 = P([0, 1])) will be an
appropriate completion of the space C∞(S1,R) (or C∞([0, 1],R), resp.). The action of a tangent
vector ϕ on µ (’exponential map’) is given by the push forward ϕ∗µ. This leads to the notion
of the directional derivative
Dϕu(µ) = lim
[u((Id+ tϕ)∗µ)− u(µ)]
for functions u : P → R. The quasi-invariance of the measure Pβ implies an integration by parts
formula (and thus the closability)
D∗ϕu = −Dϕu− Vϕ · u
with drift Vϕ = limt→0
Id+tϕ
− 1).
The subsequent construction will strongly depend on the choice of the norm on the tangent
spaces TµP. Basically, we will encounter two important cases.
(e) Choosing TµP = Hs(S1,Leb) for some s > 1/2 — independent of µ — leads to a regular,
local, recurrent Dirichlet form E on L2(P,Pβ) by
E(u, u) =
|Dϕku(µ)|2 dPβ(µ).
where {ϕk}k∈N denotes some complete orthonormal system in the Sobolev space Hs(S1). Ac-
cording to the theory of Dirichlet forms on locally compact spaces [FOT94], this form is associ-
ated with a continuous Markov process on P(S1) which is reversible with respect to the measure
Pβ. Its generator is given by
DϕkDϕk +
Vϕk ·Dϕk . (1.3)
This process (gt)t≥0 is closely related to the stochastic processes on the diffeomorphism group
of S1 and to the ’Brownian motion’ on the homeomorphism group of S1, studied by Airault,
Fang, Malliavin, Ren, Thalmaier and others [AMT04, AM06, AR02, Fan02, Fan04, Mal99].
These are processes with generator 1
kDϕkDϕk . Hence, one advantage of our approach is to
identify a probability measure Pβ such that these processes — after adding a suitable drift —
are reversible.
Moreover, previous approaches are restricted to s ≥ 3/2 whereas our construction applies to all
cases s > 1/2.
(f) Choosing TµG = L2([0, 1], µ) leads to the Wasserstein Dirichlet form
E(u, u) =
‖Du(µ)‖2L2(µ) dP
0 (µ)
on L2(P0,Pβ0 ). Its square field operator is the squared norm of the Wasserstein gradient and its
intrinsic distance (which governs the short time asymptotic of the process) coincides with the
L2-Wasserstein metric. The associated continuous Markov process (µt)t≥0 on P([0, 1]), which we
shall call Wasserstein diffusion, is reversible w.r.t. the entropic measure P
0 . It can be regarded
as a stochastic perturbation of the Neumann heat flow on P([0, 1]) with small time Gaussian
behaviour measured in terms of kinetic energy.
2 Spaces of Probability Measures and Monotone Maps
The goal of this paper is to study stochastic dynamics on spaces P(M) in case M is the unit
interval [0, 1] or the unit circle S1.
2.1 The Spaces P0 = P([0, 1]) and G0
Let us collect some basic facts for the space P0 = P([0, 1]) of probability measures on the unit
interval [0, 1] the proofs of which can be found in the monograph [Vil03]. Equipped with the
L2-Wasserstein distance dW , it is a compact metric space. Recall that
dW (µ, ν) := inf
[0,1]2
|x− y|2γ(dx, dy)
, (2.1)
where the infimum is taken over all probability measures γ ∈ P([0, 1]2) having marginals µ and
ν (i.e. γ(A×M) = µ(A) and γ(M ×B) = ν(B) for all A,B ⊂M).
Let G0 denote the space of all right continuous nondecreasing maps g : [0, 1[→ [0, 1] equipped
with the L2-distance
‖g1 − g2‖L2 =
|g1(t)− g2(t)|2dt
Moreover, for notational convenience each g ∈ G0 is extended to the full interval [0, 1] by g(1) :=
1. The map
χ : G0 → P0, g 7→ g∗Leb
(= push forward of the Lebesgue measure on [0, 1] under the map g) establishes an isometry
between (G0, ‖.‖L2) and (P0, dW ). The inverse map χ−1 : P0 → G0, µ 7→ gµ assigns to each
probability measure µ ∈ P0 its inverse distribution function defined by
gµ(t) := inf{s ∈ [0, 1] : µ[0, s] > t} (2.2)
with inf ∅ := 1. In particular, for all µ, ν ∈ P0
dW (µ, ν) = ‖gµ − gν‖L2 . (2.3)
For each g ∈ G0 the generalized inverse g−1 ∈ G0 is defined by g−1(t) = inf{s ≥ 0 : g(s) > t}.
Obviously,
‖g1 − g2‖L1 = ‖g−11 − g
2 ‖L1 (2.4)
(being simply the area between the graphs) and (g−1)−1 = g. Moreover, g−1(g(t)) = t for all
t provided g−1 is continuous. (Note that under the measure Q
0 to be constructed below the
latter will be satisfied for a.e. g ∈ G0.)
On G0, there exist various canonical topologies: the L2-topology of G0 regarded as subset of
L2([0, 1],R); the image of the weak topology on P0 under the map χ−1 : µ 7→ gµ (= inverse
distribution function); the image of the weak topology on P0 under the map µ 7→ g−1µ (=
distribution function). All these – and several other – topologies coincide.
Proposition 2.1. For each sequence (gn)n ⊂ G0, each g ∈ G0 and each p ∈ [1,∞[ the following
are equivalent:
(i) gn(t) → g(t) for each t ∈ [0, 1] in which g is continuous;
(ii) gn → g in Lp([0, 1]);
(iii) g−1n → g−1 in Lp([0, 1]);
(iv) µgn → µg weakly;
(v) µgn → µg in dW .
In particular, G0 is compact.
Let us briefly sketch the main arguments of the
Proof. Since all the functions gn and g
n are bounded, properties (ii) and (iii) obviously are
independent of p. The equivalence of (ii) and (iii) for p = 1 was already stated in (2.4) and the
equivalence between (ii) for p = 2 and (v) was stated in (2.3). The equivalence of (iv) and (v)
is the well known fact that the Wasserstein distance metrizes the weak topology. Another well
known characterization of weak convergence states that (iv) is equivalent to (i’): g−1n (t) → g−1(t)
for each t ∈ [0, 1] in which g−1 is continuous. Finally, (i′) ⇔ (i) according to the equivalence
(ii) ⇔ (iii) which allows to pass from convergence of distribution functions g−1n to convergence
of inverse distribution functions gn. The last assertion follows from the compactness of P0 in
the weak topology.
2.2 The Spaces G, G1 and P = P(S1)
Throughout this paper, S1 = R/Z will always denote the circle of length 1. It inherits the group
operation + from R with neutral element 0. For each x, y ∈ S1 the positively oriented segment
from x to y will be denoted by [x, y] and its length by |[x, y]|. If no ambiguity is possible, the
latter will also be denoted by y − x. In contrast to that, |x − y| will denote the S1-distance
between x and y. Hence, in particular, |[y, x]| = 1 − |[x, y]| and |x − y| = min{|[y, x]|, |[x, y]|}.
A family of points t1, . . . , tN ∈ S1 is called an ’ordered family’ if
i=1 |[ti, ti+1]| = 1 with
tN+1 := t1 (or in other words if all the open segments ]ti, ti+1[ are disjoint).
G(R) = {g : R → R right continuous nondecreasing with g(x+ 1) = g(x) + 1 for all x ∈ R}.
Due to the required equivariance with respect to the group action of Z, each map g ∈ G(R)
induces uniquely a map π(g) : S1 → S1. Put G := π(G(R)). The monotonicity of the functions
in G(R) induces also a kind of monotonicity of maps in G: each continuous g ∈ G will be
order preserving and homotopic to the identity map. In the sequel, however, we often will have
to deal with discontinuous g ∈ G. The elements g ∈ G will be called monotone maps of S1.
G is a compact subspace of the L2-space of maps from S1 to S1 with metric ‖g1 − g2‖L2 =
|g1(t)− g2(t)|2dt
With the composition ◦ of maps, G is a semigroup. Its neutral element e is the identity map.
Of particular interest in the sequel will be the semigroup G1 = G/S1 where functions g, h ∈ G
will be identified if g(.) = h(.+ a) for some a ∈ S1.
Proposition 2.2. The map
χ : G1 → P, g 7→ g∗Leb
(= push forward of the Lebesgue measure on S1 under the map g) and its inverse χ−1 : P →
G1, µ 7→ gµ (with gµ as defined in (2.2)) establish an isometry between the space G1 equipped
with the induced L2-distance
‖g1 − g2‖G1 =
|g1(t)− g2(t+ s)|2dt
and the space P of probability measures on S1 equipped with the L2-Wasserstein distance. In
particular, G1 is compact.
Proof. The bijectivity of χ and χ−1 is clear. It remains to prove that
dW (µ, ν) = ‖gµ − gν‖G1 (2.5)
for all µ, ν ∈ P. Obviously, it suffices to prove this for all absolutely continuous µ, ν (or
equivalently for strictly increasing gµ, gν) since the latter are dense in P (or in G1, resp.). For
such a pair of measures, there exists a map F : S1 → S1 (’transport map’) which minimizes the
transportation costs [Vil03]. Fix any point in S1, say 0, and put s = F (0). Then the map F is
a transport map for the mass µ on the segment ]0, 1[ onto the mass ν on the segment ]s, s+ 1[.
Since these segments are isometric to the interval ]0, 1[, the results from the previous subsection
imply that the minimal cost for such a transport is given by
|g1(t) − g2(t + s)|2dt. Varying
over s finally proves the claim.
3 Dirichlet Process and Entropic Measure
3.1 Gibbsean Interpretation and Heuristic Derivation of the Entropic Mea-
One of the basic results of this paper is the rigorous construction of a measure Pβ formally given
as (1.1) in the one-dimensional case, i.e. M = S1 or M = [0, 1]. We will essentially make use
of the isometries χ : G1 → P = P(S1), g 7→ g∗Leb and χ : G0 → P0 = P([0, 1]). They allow to
transfer the problem of constructing measures Pβ on spaces of probability measures P (or P0)
into the problem of constructing measures Qβ (or Q
0 ) on spaces of functions G1 (or G0, resp.).
In terms of the measure Q
0 on G0, for instance, the formal characterization (1.1) then reads as
follows
0 (dg) =
e−β·S(g)Q0(dg). (3.1)
Here Q0 denotes some ’uniform distribution’ on G0 ⊂ L2([0, 1]) and S : G0 → [0,∞] is the
entropy functional S(g) := Ent(g∗Leb). If g is absolutely continuous then S(g) can be expressed
explicitly as
S(g) = −
log g′(t) dt.
The representation (3.1) is reminiscent of Feynman’s heuristic picture of the Wiener measure.
Let us briefly recall the latter and try to use it as a guideline for our construction of the measure
According to this heuristic picture, the Wiener measure Pβ with diffusion constant σ2 = 1/β
should be interpreted (and could be constructed) as
Pβ(dg) =
e−β·H(g)P(dg) (3.2)
with the energy functional H(g) = 1
g′(t)2dt. Here P(dg) is assumed to be the ’uniform
distribution’ on the space G∗ of all continuous paths g : [0, 1] → R with g(0) = 0. Even if
such a uniform distribution existed, typically almost all paths g would have infinite energy.
Nevertheless, one can overcome this difficulty as follows.
Given any finite partition {0 = t0 < t1 < · · · < tN = 1} of [0, 1], one should replace the energy
H(g) of the path g by the energy of the piecewise linear interpolation of g
HN (g) = inf {H(g̃) : g̃ ∈ G∗, g̃(ti) = g(ti) ∀i} =
|g(ti)− g(ti−1)|2
2(ti − ti−1)
Then (3.2) leads to the following explicit representation for the finite dimensional distributions
Pβ (gt1 ∈ dx1, . . . , gtN ∈ dxN ) =
|xi − xi−1|2
ti − ti−1
pN (dx1, . . . , xN ). (3.3)
Here pN (dx1, . . . , xN ) = P (gt1 ∈ dx1, . . . , gtN ∈ dxN ) should be a ’uniform distribution’ on RN
and Zβ,N a normalization constant. Choosing pN to be the N -dimensional Lebesgue measure
makes the RHS of (3.3) a projective family of probability measures. According to Kolmogorov’s
extension theorem this family has a unique projective limit, the Wiener measure Pβ on G∗ with
diffusion constant σ2 = 1/β.
Now let us try to follow this procedure with the entropy functional S(g) replacing the energy
functional H(g). Given any finite partition {0 = t0 < t1 < · · · < tN < tN+1 = 1} of [0, 1], we
will replace the entropy S(g) of the path g by the entropy of the piecewise linear interpolation
SN (g) = inf {S(g̃) : g̃ ∈ G0, g̃(ti) = g(ti) ∀i} = −
g(ti)− g(ti−1)
ti − ti−1
· (ti − ti−1).
This leads to the following expression for the finite dimensional distributions
0 (gt1 ∈ dx1, . . . , gtN ∈ dxN )
xi − xi−1
ti − ti−1
· (ti − ti−1)
qN(dx1 . . . dxN ) (3.4)
where qN (dx1, . . . , xN ) = Q0 (gt1 ∈ dx1, . . . , gtN ∈ dxN ) is a ’uniform distribution’ on the simplex
(x1, . . . , xN ) ∈ [0, 1]N : 0 < x1 < x2 . . . < xN < 1
and x0 := 0, xN+1 := 1.
What is a ’canonical’ candidate for qN? A natural requirement will be the invariance property
qN (dx1, . . . , dxN ) = [(Ξ
xi−1,xi+k)∗ qk(dxi, . . . , dxi+k−1)]
dqN−k(dx1, . . . , dxi−1, dxi+k, . . . , dxN ) (3.5)
for all 1 ≤ k ≤ N and all 1 ≤ i ≤ N − k + 1 with the convention x0 = 0, xN+1 = 1 and the
rescaling map Ξa,b : ]0, 1[k→ Rk, yj 7→ yj(b− a) + a for j = 1, · · · , k.
If the qN , N ∈ N, were probability measures then the invariance property admits the following
interpretation: under qN , the distribution of the (N − k)-tuple (x1, . . . , xi−1, xi+k, . . . , xN ) is
nothing but qN−k; and under qN , the distribution of the k-tuple (xi, . . . , xi+k−1) of points in
the interval ]xi−1, xk[ coincides — after rescaling of this interval — with qk. Unfortunately, no
family of probability measures qN , N ∈ N with property (3.5) exists. However, there is a family
of measures with this property.
By iteration of the invariance property (3.5), the choice of the measure q1 on the interval
Σ1 = ]0, 1[ will determine all the measures qN , N ∈ N. Moreover, applying (3.5) for N = 2,
k = 1 and both choices of i yields
(Ξ0,x1)∗ q1(dx2)
dq1(dx1) =
(Ξx2,1)∗ q1(dx1)
dq1(dx2) (3.6)
for all 0 < x1 < x2 < 1. This reflects the intuitive requirement that there should be no difference
whether we first choose randomly x1 ∈ ]0, 1[ and then x2 ∈ ]x1, 1[ or the other way round, first
x2 ∈ ]0, 1[ and then x1 ∈ ]0, x2[.
Lemma 3.1. A family of measures qN , N ∈ N, with continuous densities satisfies property (3.5)
if and only if
qN (dx1, . . . , dxN ) = C
N dx1 . . . dxN
x1 · (x2 − x1) · . . . · (xN − xN−1) · (1− xN )
(3.7)
for some constant C ∈ R+.
Proof. If q1(dx) = ρ(x)dx then (3.6) is equivalent to
ρ(y) · ρ
= ρ(x) · ρ
y − x
for all 0 < x < y < 1. For continuous ρ this implies that there exists a constant C ∈ R+ such
that ρ(x) = C
x(1−x)
for all 0 < x < 1. Iterated inserting this into (3.5) yields the claim.
Let us come back to our attempt to give a meaning to the heuristic formula (3.1). Combining
(3.4) with the choice (3.7) of the measure qN finally yields
0 (gt1 ∈ dx1, . . . , gtN ∈ dxN )
(xi − xi−1)β(ti−ti1 )
dx1 . . . dxN
x1 · (x2 − x1) · . . . · (1− xN )
(3.8)
with appropriate normalization constants Zβ,N . Now the RHS of this formula indeed turns out
to define a consistent family of probability measures. Hence, by Kolmogorov’s extension theorem
it admits a projective limit Q
0 on the space G0. The push forward of this measure under the
canonical identification χ : G0 → P0, g 7→ g∗Leb will be the entropic measure Pβ0 which we were
looking for.
The details of the rigorous construction of this measure as well as various properties of it will
be presented in the following sections.
3.2 The Measures Qβ and Pβ
The basic object to be studied in this section is the probability measure Qβ on the space G.
Proposition 3.2. For each real number β > 0 there exists a unique probability measure Qβ on
G, called Dirichlet process, with the property that for each N ∈ N and for each ordered family
of points t1, t2, . . . , tN ∈ S1
β (gt1 ∈ dx1, . . . , gtN ∈ dxN ) =
i=1 Γ(β(ti+1 − ti))
(xi+1 − xi)β(ti+1−ti)−1dx1 . . . dxN .
(3.9)
The precise meaning of (3.9) is that for all bounded measurable u : (S1)N → R
u (gt1 , . . . , gtN ) dQ
i=1 Γ(β · |[ti, ti+1]|)
u(x1, . . . , xN )
|[xi, xi+1]|β·|[ti,ti+1]|−1dx1 . . . dxN .
with ΣN =
(x1, . . . , xN ) ∈ (S1)N :
i=1 |[xi, xi+1]| = 1
and xN+1 := x1, tN+1 := t1. In
particular, with N = 1 this means
u(gt)dQ
β(g) =
u(x)dx for each t ∈ S1.
Proof. It suffices to prove that (3.9) defines a consistent family of finite dimensional distributions.
The existence of Qβ (as a ’projective limit’) then follows from Kolmogorov’s extension theorem.
The required consistency means that
i=1 Γ(β · |[ti, ti+1]|)
|[xi, xi+1]|β·|[ti,ti+1]|−1u(x1, . . . , xN ) dx1 . . . dxN
Γ(β · |[t1, t2]|) · . . . · Γ(β · |[tk−1, tk+1]|) · . . . · Γ(β · |[tN , t1]|)
|[x1, x2]|β·|[t1,t2]|−1 · . . . · |[xk−1, xk+1]|β·|[tk−1,tk+1]|−1 · . . . · |[xN , x1]|β·|[tN ,t1]|−1
·v(x1, . . . , xk−1, xk . . . , xN ) dx1 . . . dxk−1dxk+1 . . . dxN
whenever u(x1, . . . , xN ) = v(x1, . . . , xk−1, xk . . . , xN ) for all (x1, . . . xN ) ∈ ΣN . The latter is an
immediate consequence of the well-known fact (Euler’s beta integral) that
[xk−1,xk+1]
|[xk−1, xk]|β·|[tk−1,tk]|−1 · |[xk, xk+1]|β·|[tk,tk+1]|−1 dxk
Γ(β · |[tk−1, tk]|)Γ(β · |[tk, tk+1]|)
Γ(β · |[tk−1, tk+1]|)
|[xk−1, xk+1]|β·|[tk−1,tk+1]|−1.
For s ∈ S1 let θ̂s : G → G, g 7→ g ◦ θs be the isomorphism of G induced by the rotation
θs : S
1 → S1, t 7→ t+ s. Obviously, the measure Qβ on G is invariant under each of the maps θ̂s.
Hence, Qβ induces a probability measure Q
1 on the quotient spaces G1 = G/S1.
Recall the definition of the map χ : G → P, g 7→ g∗Leb. Since (g ◦ θs)∗Leb = g∗Leb this
canonically extends to a map χ : G1 → P. (As mentioned before, the latter is even an isometry.)
Definition 3.3. The entropic measure Pβ on P is defined as the push forward of the Dirichlet
process Qβ on G (or equivalently, of the measure Qβ1 on G1) under the map χ. That is, for all
bounded measurable u : P → R
u(µ) dPβ(µ) =
u(g∗Leb) dQ
β(g).
3.3 The Measures Q
0 and P
The subspaces {g ∈ G : g(0) = 0} and {g ∈ G0 : g(0) = 0} can obviously be identified.
Conditioning the probability measure Qβ onto this event thus will define a probability measure
0 on G0. However, we prefer to give the direct construction of Q
Proposition 3.4. For each real number β > 0 there exists a unique probability measure Q
G0, called Dirichlet process, with the property that for each N ∈ N and each family 0 = t0 <
t1 < t2 < . . . < tN < tN+1 = 1
0 (gt1 ∈ dx1, . . . , gtN ∈ dxN ) =
i Γ(β · (ti+1 − ti))
(xi+1 − xi)β·(ti+1−ti)−1dx1 . . . dxN .
(3.10)
The precise meaning of (3.10) is that for all bounded measurable u : [0, 1]N → R
u (gt1 , . . . , gtN ) dQ
0 (g)
i=1 Γ(β · (ti+1 − ti))
u(x1, . . . , xN )
(xi+1 − xi)β·(ti+1−ti)−1dx1 . . . dxN .
with ΣN =
(x1, . . . , xN ) ∈ [0, 1]N : 0 < x1 < x2 . . . < xn < 1
and xN+1 := x1, tN+1 := t1.
Remark 3.5. According to these explicit formulae, it is easy to calculate the moments of the
Dirichlet process. For instance,
0 (gt) :=
gt dQ
0 (g) = t
0 (gt) :=
(gt − t)2 dQβ0 (g) =
1 + β
t(1− t)
for all β > 0 and all t ∈ [0, 1].
Definition 3.6. The entropic measure P
0 on P0 = P([0, 1]) is defined as the push forward of
the Dirichlet process Q
0 on G0 under the map χ. That is, for all bounded measurable u : P0 → R
u(µ) dP
0 (µ) =
u(g∗Leb) dQ
0 (g).
Remark 3.7. (i) According to the above construction Q
0 ( . ) = Q
β( . |g(0) = 0) and
u(g) dQ
0 (g) =
u(g − g(0)) dQβ(g),
u(g) dQβ(g) =
u(g + x) dQ
0 (g) dx.
(ii) Analogously, the entropic measures on the sphere and on the are linked as follows
u(µ) dPβ(µ) =
u((θx)∗µ)dP
0 (µ) dx
or briefly
dPβ =
(θ̂x)∗dP
where θx : S
1 → S1, y 7→ x + y and θ̂x : P → P : µ 7→ (θx)∗µ. We would like to emphasize,
however, that Pβ 6= Pβ0 . For instance, consider u(µ) :=
f dµ for some f : S1 → R (which may
be identified with f : [0, 1] → R). Then
P(S1)
u(µ) dPβ(µ) =
f(x) dx
whereas
P([0,1])
u(µ) dP
0 (µ) =
[0,1]
f(x)ρβ(x) dx
with ρβ(x) =
Γ(βt)Γ(β(1−t))
xβt−1(1− x)β(1−t)−1 dt.
According to the last remark, it suffices to study in detail one of the four measures Qβ, Q
Pβ, and P
0 . We will concentrate in the rest of this chapter on the measure Q
0 which seems to
admit the most easy interpretations.
3.4 The Dirichlet Process as Normalized Gamma Process
We start recalling some basic facts about the real valued Gamma processes. For α > 0 denote
by G(α) the absolutely continuous probability measure on R+ with density
xα−1e−x.
Definition 3.8. A real valued Markov process (γt)t≥0 starting in zero is called standard Gamma
process if its increments γt − γs are independent and distributed according to G(t − s) for 0 ≤
s < t. Without loss of generality we may assume that almost surely the function t→ γt is right
continuous and nondecreasing.
Alternatively the Gamma-Process may be defined as the unique pure jump Levy process with
Levy measure Λ(dx) = 1‖‖x>0
dx. The connection between pure jump Levy and Poisson point
processes gives rise to several other equivalent representations of the Gamma process [Kin93,
Ber99]. For instance, let Π = {p = (px, py) ∈ R2} be the Poisson point process on R+×R+ with
intensity measure dx× Λ(dy) with Λ as above, then a Gamma process is obtained by
γt :=
p∈Π:px≤t
py. (3.11)
For β > 0 the process γt·β is a Levy process with Levy measure Λβ(dx) = β · 1‖‖x>0 e
dx. Its
increments are distributed according to
P (γβ·t − γβ·s ∈ dx) =
Γ(β · (t− s))x
β·(t−s)−1e−xdx. �
Proposition 3.9. For each β > 0, the law of the process (
)t∈[0,1] is the Dirichlet process Q
Proof. This well-known fact is easily obtained from Lukacs’ characterization of the Gamma
distribution [ÉY04].
3.5 Support Properties
Proposition 3.10. (i) For each β > 0, the measure Q
0 has full support on G0.
(ii) Q
0 -almost surely the function t 7→ g(t) is strictly increasing but increases only by jumps
(that is, the jumps heights add up to 1 and the jump locations are dense in [0, 1]).
(iii) For each fixed t0 ∈ [0, 1], Qβ0 -almost surely the function t 7→ g(t) is continuous at t0.
Proof. (i) Let g ∈ G ⊂ L2([0, 1], dx) and ǫ > 0 then we have to show Qβ(Bǫ(g)) > 0 where
Bǫ(g) = {h ∈ G0 : ‖h − g‖L2([0,1]) < ǫ}. For this choose finitely many points ti ∈ [0, 1] together
with δi > 0 such that the set S := {f ∈ G
∣ |f(ti)−g(ti)| ≤ δi ∀i} is contained in Bǫ(g). Clearly,
from (3.10) Qβ(S) > 0 which proves the claim.
(ii) (3.10) implies that Q
0 -almost surely g(s) < g(t) for each given pair s < t. Varying over all
such rational pairs s < t, it follows that a.e. g is strictly increasing on R+.
In terms of the probabilistic representation (3.9), it is obvious that g increases only by jumps.
(iii) This also follows easily from the representation as a normalized gamma process (3.9).
Restating the previous property (ii) in terms of the entropic measure yields that P
0 -a.e. µ ∈ P0
is ’Cantor like’. More precisely,
Corollary 3.11. P
0 -almost surely the measure µ ∈ P0 has no absolutely continuous part and
no discrete part. The topological support of µ has Lebesgue measure 0. Moreover,
Ent(µ) = +∞. (3.12)
Proof. The assertion on the entropy of µ is an immediate consequence of the statement on the
support of µ. The second claim follows from the fact that the jump heights of g add up to 1.
In terms of the measure Q
0 , the last assertion of the corollary states that S(g) = +∞ for Q
0 -a.e.
g ∈ G0.
3.6 Scaling and Invariance Properties
The Dirichlet process Q
0 on G0 has the following Markov property: the distribution of g|[s,t]
depends on g[0,1]\[s,t] only via g(s), g(t).
And the Dirichlet process Q
0 on G0 has a remarkable self-similarity property: if we restrict the
functions g onto a given interval [s, t] and then linearly rescale their domain and range in order
to make them again elements of G0 then this new process is distributed according to Qβ
0 with
β′ = β · |t− s|.
Proposition 3.12. For each β > 0, and each s, t ∈ [0, 1], s < t
g|[s,t] ∈ .
∣ g[0,1]\[s,t]
g|[s,t] ∈ .
∣ g(s), g(t)
(3.13)
(Ξs,t)∗Q
0 = Q
β·|t−s|
0 (3.14)
where Ξs,t : G0 → G0 with Ξs,t(g)(r) = g((1−r)s+rt)−g(s)g(t)−g(s) for r ∈ [0, 1].
Proof. Both properties follow immediately from the representation in Proposition 3.10.
Corollary 3.13. The probability measures Q
0 , β > 0 on G0 are uniquely characterized by the
self-similarity property (3.14) and the distributions of g1/2:
0 (g1/2 ∈ dx) =
Γ(β/2)2
· [x(1− x)]β/2−1dx.
Proposition 3.14. (i) For β → 0 the measures Qβ0 weakly converge to a measure Q00 defined
as the uniform distribution on the set {1[t,1] : t ∈ ]0, 1]} ⊂ G0.
Analogously, the measures Qβ weakly converge for β → 0 to a measure Q0 defined as the uniform
distribution on the set of constant maps {t : t ∈ S1} ⊂ G.
(ii) For β → ∞ the measures Qβ0 (or Qβ) weakly converge to the Dirac mass δe on the identity
map e of [0, 1] (or S1, resp.).
Proof. (i) Since the space G0 (equipped with the L2-topology) is compact, so is P(G0) (equipped
with the weak topology). Hence the family Q
0 , β > 0 is pre-compact. Let Q
0 denote the limit of
any converging subsequence of Q
0 for β → 0. According to the formula for the one-dimensional
distributions, for each t ∈ ]0, 1[
0 (gt ∈ dx) =
Γ(βt)Γ(β(1 − t)) · x
βt−1(1− x)β(1−t)−1dx
−→ (1− t)δ{0}(dx) + tδ{1}(dx)
as β → 0. Hence, Q00 is the uniform distribution on the set {1[t,1] : t ∈ ]0, 1]} ⊂ G0.
(ii) Similarly, Q
0 (gt ∈ dx) → δt(dx) as β → ∞. Hence, δe with e : t 7→ t will be the unique
accumulation point of Q
0 for β → ∞.
Restating the previous results in terms of the entropic measures, yields that the entropic mea-
sures P
0 converge weakly to the uniform distribution P
0 on the set {(1 − t)δ{0} + tδ{1} : t ∈
[0, 1]} ⊂ P0; and the measures Pβ converge weakly to the uniform distribution P0 on the set
{δ{t} : t ∈ S1} ⊂ P whereas for β → ∞ both, Pβ0 and Pβ, will converge to δLeb, the Dirac mass
on the uniform distribution of [0, 1] or S1, resp.
The assertions of Proposition 3.12 imply the following Markov property and self-similarity prop-
erty of the entropic measure.
Proposition 3.15. For each each x, y ∈ [0, 1], x < y
µ|[x,y] ∈ .
∣µ|[0,1]\[x,y]
µ|[x,y] ∈ .
∣µ([x, y]
µ|[x,y] ∈ .
∣µ([x, y]) = α
0 (µx,y ∈ . )
with µx,y ∈ P0 (’rescaling of µ|[x,y]’) defined by µx,y(A) = 1µ([x,y])µ(x+(y−x) ·A) for A ⊂ [0, 1].
3.7 Dirichlet Processes on General Measurable Spaces
Recall Ferguson’s notion of a Dirichlet process on a general measurable spaceM with parameter
measure m on M . This is a probability measure Qm
on P(M), uniquely defined by the fact
that for any finite measurable partition M = ˙
i=1 Mi and σi := m(Mi).
P(M) (µ : µ(M1) ∈ dx1, . . . , µ(MN ) ∈ dxN )
Γ(m(M))
i=1 Γ(σi)
1 · · · x
)σN+1−1dx1 · · · dxN ,
If a map h :M →M leaves the parameter measure m invariant then obviously the induced map
ĥ : P(M) → P(M), µ 7→ h∗µ leaves the Dirichlet process QmP(M) invariant.
In the particular case M = [0, 1] and m = β · Leb, the Dirichlet process Qm
can be obtained
as push forward of the measure Q
0 (introduced before) under the isomorphism ζ : G0 → P([0, 1])
which assigns to each g the induced Lebesgue-Stieltjes measure dg (the inverse ζ−1 assigns to
each probability measure its distribution function):
P([0,1]) = ζ∗Q
0 . (3.15)
Note that the support properties of the measure Qm
P([0,1])
are completely different from those of
the measure P
0 . In particular, Q
P([0,1])
-almost every µ ∈ P([0, 1]) is discrete and has full topo-
logical support, cf. Corollary 3.11. The invariance properties of Qm
P([0,1])
under push forwards by
means of measure preserving transformations of [0, 1] seems to have no intrinsic interpretation
in terms of Q
4 The Change of Variable Formula for the Dirichlet Process and
for the Entropic Measure
Our main result in this chapter will be a change of variable formula for the Dirichlet process.
To motivate this formula, let us first present an heuristic derivation based on the formal repre-
sentation (3.1).
4.1 Heuristic Approaches to Change of Variable Formulae
Let us have a look on the change of variable formula for the Wiener measure. On a formal level,
it easily follows from Feynman’s heuristic interpretation
dPβ(g) =
g′(t)2dt dP(g)
with the (non-existing) ’uniform distribution’ P. Assuming that the latter is ’translation invari-
ant’ (i.e. invariant under additive changes of variables, – at least in ’smooth’ directions h) we
immediately obtain
dPβ(h+ g) =
(h+g)′(t)2dt dP(h + g)
h′(t)2dt−β
h′(t)g′(t)dt · e−
g′(t)2dt dP(g)
h′(t)2dt−β
h′(t)dg(t) dPβ(g).
If we interpret
h′(t)dg(t) as the Ito integral of h′ with respect to the Brownian path g then
this is indeed the famous Cameron-Martin-Girsanov-Maruyama theorem.
In the case of the entropic measure, the starting point for a similar argumentation is the heuristic
interpretation
0 (g) =
log g′(t)dt dQ0(g),
again with a (non-existing) ’uniform distribution’ Q0 on G0. The natural concept of ’change
of variables’, of course, will be based on the semigroup structure of the underlying space G0;
that is, we will study transformations of G0 of the form g 7→ h ◦ g for some (smooth) element
h ∈ G0. It turns out that Q0 should not be assumed to be invariant under translations but
merely quasi-invariant:
dQ0(h ◦ g) = Y 0h (g) dQ0(g)
with some density Yh. This immediately implies the following change of variable formula for Q
0 (h ◦ g) =
log(h◦g)′(t)dt dQ0(h ◦ g)
log h′(g(t))dt · eβ
log g′(t)dt · Y 0h (g) dQ0(g)
log g′(t)dt · Y 0h (g) dQ
0 (g).
This is the heuristic derivation of the change of variables formula. Its rigorous derivation (and
the identification of the density Yh) is the main result of this chapter.
4.2 The Change of Variables Formula on the Sphere
For g, h ∈ G with h ∈ C2 we put
Y 0h (g) :=
h′ (g(a−)) · h′ (g(a+))
δ(h◦g)
, (4.1)
where Jg ⊂ S1 denotes the set of jump locations of g and
δ(h ◦ g)
(a) :=
h (g(a+)) − h (g(a−))
g(a+)− g(a−) .
To simplify notation, here and in the sequel (if no ambiguity seems possible), we write y − x
instead of |[x, y]| to denote the length of the positively oriented segment from x to y in S1. We
will see below that the infinite product in the definition of Y 0h (g) converges for Q
β-a.e. g ∈ G.
Moreover, for β > 0 we put
h (g) := exp
log h′ (g(s)) ds
h (g) := X
h (g) · Y
h (g). (4.2)
Theorem 4.1. Each C2-diffeomorphism h ∈ G induces a bijective map τh : G → G, g 7→ h ◦ g
which leaves the measure Qβ quasi-invariant:
dQβ(h ◦ g) = Y βh (g) dQ
β(g).
In other words, the push forward of Qβ under the map τ−1h = τh−1 is absolutely continuous w.r.t.
Qβ with density Y
d(τh−1)∗Q
dQβ(g)
h (g).
The function Y
h is bounded from above and below (away from 0) on G.
By means of the canonical isometry χ : G → P, g 7→ g∗Leb, Theorem 4.1 immediately implies
Corollary 4.2. For each C2-diffeomorphism h ∈ G the entropic measure Pβ is quasi-invariant
under the transformation µ 7→ h∗µ of the space P:
dPβ(h∗µ) = Y
−1(µ)) dPβ(µ).
The density Y
−1(µ)) introduced in (4.2) can be expressed as follows
−1(µ)) = exp
log h′(s)µ(ds)
I∈gaps(µ)
h′(I−) · h′(I+)
|h(I)|/|I|
where gaps(µ) denotes the set of segments I = ]I−, I+[⊂ S1 of maximal length with µ(I) = 0
and |I| denotes the length of such a segment.
4.3 The Change of Variables Formula on the Interval
From the representation of Qβ as a product of Q
0 and Leb (see Remark 3.7) and the change of
variable formulae for Qβ and Leb, one can deduce a change of variable formula for Q
0 similar to
that of Theorem 4.1 but containing an additional factor 1
h′(0)
. In this case, one has to restrict
to translations by means of C2-diffeomorphisms h ∈ G with h(0) = 0.
More generally, one might be interested in translations of G0 by means of C2-diffeomorphisms
h ∈ G0. In contrast to the previous situation, it now may happen that h′(0) 6= h′(1).
For g ∈ G0 and C2-ismorphism h : [0, 1] → [0, 1] we put
h,0(g) := X
h (g) · Yh,0(g) (4.3)
Yh,0(g) =
h′(0) · h′(1)
· Y 0h (g)
and X
(g) and Y 0h (g) defined as before in (4.1), (4.2). Note that here and in the sequel by a
C2-isomorphism h ∈ G0 we understand an increasing homeomorphism h : [0, 1] → [0, 1] such that
h and h−1 are bounded in C2([0, 1]), which in particular implies h′ > 0.
Theorem 4.3. Each translation τh : G0 → G0, g 7→ h ◦ g by means of a C2-isomorphism h ∈ G0
leaves the measure Q
0 quasi-invariant:
0 (h ◦ g) = Y
(g) dQ
0 (g)
or, in other words,
d(τh−1)∗Q
0 (g)
0 (g)
The function Y
h,0 is bounded from above and below (away from 0) on G0.
Corollary 4.4. For each C2-isomorphism h ∈ G0 the entropic measure Pβ0 is quasi-invariant
under the transformation µ 7→ h∗µ of the space P0:
0 (h∗µ)
0 (µ)
= exp
log h′(s)µ(ds)
h′(0) · h′(1)
I∈gaps(µ)
h′(I−) · h′(I+)
|h(I)|/|I|
where gaps(µ) denotes the set of intervals I = ]I−, I+[⊂ [0, 1] of maximal length with µ(I) = 0
and |I| denotes the length of such an interval.
Remark 4.5. Theorem 4.3 seems to be unrelated to the quasi-invariance of the measure Qm
P([0,1])
under the transformation dg → h · dg/〈h, dg〉 shown in [Han02]. Nor is it anyhow implied by
the quasi-ivarariance formula for the general measure valued gamma process as in [TVY01] with
respect to a similar transformation. In our present case the latter would correspond to the
mapping dγ → h · dγ of the (measure valued) Gamma process dγ.
4.4 Proofs for the Sphere Case
Lemma 4.6. For each C2-diffeomorphism h ∈ G
h (g) = lim
h (g(ti+1))− h (g(ti))
g(ti+1)− g(ti)
]β(ti+1−ti)
(4.4)
Here ti =
for i = 0, 1, . . . , k − 1 and tk = 0. Thus ti+1 − ti := |[ti, ti+1]| = 1k for all i.
Proof. Without restriction, we may assume β = 1. According to Taylor’s formula
h (g(ti+1)) = h (g(ti)) + h
′ (g(ti)) · (g(ti+1)− g(ti)) + 12h
′′(γi) · (g(ti+1)− g(ti))2
for some γi ∈ [g(ti), g(ti+1)]. Hence,
h (g(ti+1))− h (g(ti))
g(ti+1)− g(ti)
]ti+1−ti
= lim
h′ (g(ti)) +
h′′(γi) · (g(ti+1)− g(ti))
]ti+1−ti
= lim
log h′ (g(ti)) + log
1 + 1
h′′(γi)
h′ (g(ti))
(g(ti+1)− g(ti))
· (ti+1 − ti)
= exp
log h′ (g(ti)) · (ti+1 − ti)
= exp
log h′ (g(t)) dt
= X1h(g).
Here (⋆) follows from the fact that
1 + 1
h′′(γi)
h′ (g(ti))
· (g(ti+1)− g(ti)) =
h (g(ti+1))− h (g(ti))
g(ti+1)− g(ti)
h′ (g(ti))
= h′(ηi) ·
h′ (g(ti))
≥ ε > 0
for some ηi ∈ [g(ti), g(ti+1)] and some ε > 0, independent of i and k. Thus
1 + 1
h′′(γi)
h′ (g(ti))
· (g(ti+1)− g(ti))
· (ti+1 − ti)
≤ C1 ·
h′′(γi)
h′ (g(ti))
· (g(ti+1)− g(ti)) · (ti+1 − ti)
≤ C2 ·
(g(ti+1)− g(ti)) · (ti+1 − ti)
≤ C3 · 1k .
Lemma 4.7. For each C3-diffeomorphism h ∈ G
Y 0h (g) := lim
h′ (g(ti)) ·
g(ti+1)− g(ti)
h (g(ti+1))− h (g(ti))
(4.5)
where ti =
for i = 0, 1, . . . , k − 1 and tk = 0.
Proof. Let h and g be given. Depending on some ε > 0 let us choose l ∈ N large enough (to be
specified in the sequel) and let a1, . . . , al denote the l largest jumps of g. Put J
g = Jg\{a1, . . . , al}
and for simplicity al+1 := a1. For k very large (compared with l) and j = 1, . . . , l let kj denote
the index i ∈ {0, 1, . . . , k − 1}, for which aj ∈ [ti, ti+1[. Then again by Taylor’s formula
kj+1−1
i=kj+1
h′ (g(ti)) ·
g(ti+1)− g(ti)
h (g(ti+1))− h (g(ti))
kj+1−1
i=kj+1
1 + 1
h′′ (g(ti))
h′ (g(ti))
· (g(ti+1)− g(ti)) + 16
h′′′(ηi)
h′ (g(ti))
· (g(ti+1)− g(ti))2
≤ exp
kj+1−1
i=kj+1
log h′
(g(ti)) +
· (g(ti+1)− g(ti))
≤ eε/l · exp
kj+1−1
i=kj+1
log h′
(g(ti)) · (g(ti+1)− g(ti))
provided l and k are chosen so large that
|g(ti+1)− g(ti)| ≤
C1 · l
for all i ∈ {0, . . . , k − 1} \ {k1, . . . , kl}, where C1 = sup
|h′′′(x)|
6·h′(y)
On the other hand,
g(tkj+1)
g(tkj+1)
) = exp
∫ g(tkj+1 )
g(tkj+1)
log h′
(s) ds
= exp
kj+1−1
i=kj+1
log h′
(g(ti)) · (g(ti+1)− g(ti)) +
log h′
(γi) · 12 (g(ti+1)− g(ti))
≥ e−ε/l · exp
kj+1−1
i=kj+1
log h′
(g(ti)) · (g(ti+1)− g(ti))
provided l and k are chosen so large that
|g(ti+1)− g(ti)| ≤
C2 · l
for all i ∈ {0, 1, . . . , k − 1} \ {k1, . . . , kl}, where C2 = sup
log h′
Therefore,
i∈{0,1,...,k−1}\{k1,...,kl}
h′ (g(ti)) ·
g(ti+1)− g(ti)
h (g(ti+1))− h (g(ti))
≤ e2ε ·
g(tkj+1)
g(tkj+1)
) = (I).
In order to derive the corresponding lower estimate, we can proceed as before in (1a) and (2)
(replacing ε by −ε and ≤ by ≥ and vice versa). To proceed as in (1b) we have to argue as
follows
kj+1−1
i=kj+1
log h′
(g(ti))− εl
· (g(ti+1)− g(ti))
≥ e−ε/l · exp
kj+1−1
i=kj+1
(1− ε) ·
log h′
(g(ti)) · (g(ti+1)− g(ti))
provided l and k are chosen so large that
log (1 + C3 · (g(ti+1)− g(ti))) ≥ (1− ε) · C3 · (g(ti+1)− g(ti))
for all i ∈ {0, 1, . . . , k − 1} \ {k1, . . . , kl}, where C3 = sup
log h′
Thus we obtain the following lower estimate
i∈{0,1,...,k−1}\{k1,...,kl}
h′ (g(ti)) ·
g(ti+1)− g(ti)
h (g(ti+1))− h (g(ti))
≥ e−2ε ·
g(tkj+1)
g(tkj+1)
≥ e−2ε · C−ε/23 ·
g(tkj+1)
g(tkj+1)
) = (II),
since
g(tkj+1)
g(tkj+1)
= exp
log h′
g(tkj+1)
− log h′
g(tkj+1)
≤ exp
g(tkj+1)− g(tkj+1)
≤ exp
where C3 = sup
∣(log h′)
Now for fixed l as k → ∞ the bound (I) converges to
(I′) = e2ε ·
h′ (g(aj+1−))
h′ (g(aj+))
and the bound (II) to
(II′) = e−2ε · C−ε/23 ·
h′ (g(aj+1−))
h′ (g(aj+))
Finally, it remains to consider
i∈{k1,...,kl}
h′ (g(ti)) ·
g(ti+1)− g(ti)
h (g(ti+1))− h (g(ti))
= (III).
Again for fixed l and k → ∞ this obviously converges to
(III′) =
h′ (g(aj−))
· δ(h ◦ g)
Putting together these estimates and letting l → ∞, we obtain the claim.
Lemma 4.8. (i) For all g, h ∈ G with h ∈ C2 strictly increasing, the infinite product in the
definition of Y 0h (g) converges. There exists a constant C = C(β, h) such that ∀g ∈ G
≤ Y β
(g) ≤ C.
(ii) If hn → h in C2 then Y 0hn(g) → Y
h (g).
(iii) Let Y 0h,k,X
denote the sequences used in Lemma 4.6 and 4.7 to approximate Y 0h ,X
Then there exists a constant C = C(β, h) such that ∀g ∈ G, ∀k ∈ N
≤ Y βh,k(g) ≤ C.
Proof. (i) Put C = sup |(log h′)′|. Given g ∈ G and ǫ > 0, we choose k large enough such that
a∈Jg(k)
|g(a+)−g(a−)| ≤ ǫ where Jg(k) = Jg \{a1, a2, . . . , ak} denotes the ’set of small jumps’
of g. Here we enumerate the jump locations a1, a2, . . . ∈ Jg according to the size of the respective
jumps. Then with suitable ξa ∈ [g(a−), g(a+)]
a∈Jg(k)
h′(g(a−))
h′(g(a+))
δ(h◦g)
a∈Jg(k)
log h′(g(a−)) + 1
log h′(g(a−)) − log h′(ξ(a))
a∈Jg(k)
|C · (g(a+)− g(a−))| = C · ǫ.
Hence, the infinite sum
h′(g(a−))
h′(g(a+))
δ(h◦g)
= lim
a∈Jg(k)
h′(g(a−))
h′(g(a+))
δ(h◦g)
is absolutely convergent and thus also infinite product in the definition of Y 0h (g) converges. The
same arguments immediately yield
∣log Y 0h (g)
log h′(g(a−)) + 1
log h′(g(a−)) − log h′(ξ(a))
≤ C. (4.6)
(ii) In order to prove the convergence Y 0hn(g) → Y
h (g), for given g ∈ G we split the product over
all jumps into a finite product over the big jumps and an infinite product over all small jumps.
Obviously, the finite products will converge (for any choice of k)
a∈{a1,...,ak}
h′n(g(a−))
h′n(g(a+))
δ(hn◦g)
a∈{a1,...,ak}
h′(g(a−))
h′(g(a+))
δ(h◦g)
as n → ∞ provided hn → h in C2. Now let C = supn supx |(log h′n)′(x)| and choose k as before.
Then uniformly in n
a∈Jg\{a1,...,ak}
h′n(g(a−))
h′n(g(a+))
δ(hn◦g)
≤ C · ǫ.
(iii) Let C1 = sup
|h′(x)| and C2 = sup
∣(log h′)
∣. Then for all g and k:
Xh,k(g) =
h′(ηi)
ti+1−ti ≤ C1
Y 0h,k(g) =
h′ (g(ti))
h′(γi)
= exp
log h′
(ζi) · (g(ti)− γi)
≤ exp
|g(ti)− γi|
≤ exp(C2)
(with suitable γi, ηi ∈ [g(ti), g(ti+1)] and ζi ∈ [g(ti), γi]). Analogously, the lower estimates
follow.
Proof of Theorem 4.1. In order to prove the equality of the two measures under consideration, it
suffices to prove that all of their finite dimensional distributions coincide. That is, for each m ∈
N, each ordered family t1, . . . , tm of points in S
1 and each bounded continuous u : (S1)m −→ R
one has to verify that
h−1 (g(t1)) , h
−1 (g(t2)) , . . . , h
−1 (g(tm))
dQβ(g)
u (g(t1), g(t2), . . . , g(tm)) · Y βh (g) dQ
β(g).
Without restriction, we may restrict ourselves to equidistant partitions, i.e. ti =
for i =
1, . . . ,m. Let us fix m ∈ N, u and h. For simplicity, we first assume that h is C3. Then by
Lemmas 4.6 - 4.8 and Lebesgue’s theorem
, . . . , g (1)
· Y βh (g) dQ
, . . . , g (1)
· lim
(g) dQβ(g)
= lim
, . . . , g (1)
dQβ(g)
= lim
[Γ(β/km)]km
u(xk, x2k, . . . , xmk)
h′(xi) ·
[h(xi+1)− h(xi)]
dx1 . . . dxmk
= lim
[Γ(β/km)]km
u(xk, x2k, . . . , xmk) ·
[h(xi+1)− h(xi)]
dh(x1) . . . dh(xmk)
= lim
[Γ(β/km)]km
h−1(yk), h
−1(y2k), . . . , h
−1(ymk)
[yi+1 − yi]
dy1 . . . dymk
, h−1
, . . . , h−1 (g (1))
dQβ(g).
Now we treat the general case h ∈ C2. We choose a sequence of C3-functions hn ∈ G with hn → h
in C2. Then
h−1 (g(t1)) , h
−1 (g(t2)) , . . . , h
−1 (g(tm))
dQβ(g)
= lim
h−1n (g(t1)) , h
n (g(t2)) , . . . , h
n (g(tm))
dQβ(g)
= lim
u (g(t1), g(t2), . . . , g(tm)) · Y βhn(g) dQ
u (g(t1), g(t2), . . . , g(tm)) · Y βh (g) dQ
β(g).
For the last equality, we have used the dominated convergence Y
(g) → Y βh (g) (due to Lemma
4.8).
4.5 Proof for the Interval Case
The proof of Theorem 4.3 uses completely analogous arguments as in the previous section. To
simplify notation, for h ∈ C1([0, 1]), k ∈ N let Xh,k, Y 0h,k : G0 → R be defined by
Xh,k(g) :=
h (g(ti+1))− h (g(ti))
g(ti+1)− g(ti)
]ti+1−ti
Y 0h,k(g) :=
g(t1)− g(t0)
h (g(t1))− h (g(t0))
] k−1
h′ (g(ti)) ·
g(ti+1)− g(ti)
h (g(ti+1))− h (g(ti))
where ti =
with i = 0, 1, . . . , k. Similar to the proof of theorem 4.1 the measure Q
0 satisfies
the following finite dimensional quasi-invariance formula.
For any u : [0, 1]m−1 → R, m, l ∈ N and C1-isomorphism h : [0, 1] → [0, 1]
h−1 (g(t1)) , h
−1 (g(t2)) , . . . , h
−1 (g(tm−1))
0 (g)
u (g(t1), g(t2), . . . , g(tm−1)) ·Xβh,l·m(g) · Y
h,l·m(g) dQ
0 (g),
where ti =
, i = 1, · · · ,m−1. The passage to the limit for letting first l and then m to infinity
is based on the following assertions.
Lemma 4.9. (i) For each C2-isomorphism h ∈ G0 and g ∈ G0
Xh(g) = lim
Xh,k(g).
(ii) For each C3-isomorphism h ∈ G0 and g ∈ G0
Y 0h,k(g) =
h′(g(a+)) · h′(g(a−))
δ(h◦g)
h′(g(0)) · h′(g(1−))
1 if g(1−) = g(1)
h′(g(1−))
δ(h◦g)
else,
where Jg ⊂]0, 1[ is the set of jump locations of g on ]0, 1[. In particular,
Y 0h,k(g) = Yh,0(g) for Q
0 -a.e.g.
(iii) For all g ∈ G0 and C2-isomorphism h ∈ G0, the infinite product in the definition of Yh,0(g)
converges. There exists a constant C = C(β, h) such that ∀g ∈ G0
≤ Y β
(g) ≤ C.
(iv) If hn → h in C2([0, 1], [0, 1]) with h as above, then Y 0,hn(g) → Y0,h(g).
(v) For each C3-isomorphism h ∈ G0 there exists a constant C = C(β, h) such that ∀g ∈ G,
∀k ∈ N
≤ Xβh,k(g) · Y
h,k(g) ≤ C.
Proof. The proofs of (i) and (iii)-(iv) carry over from their respective counterparts on the sphere,
lemmas 4.6 and 4.8 above. We sketch the proof of statement (ii) which needs most modification.
For ε > 0 choose l ∈ N large enough and let a2, . . . , al−1 denote the l − 2 largest jumps of g
on ]0, 1[. For k very large (compared with l) we may assume that a2, . . . , al−2 ∈] 2k , 1 −
[. Put
a1 :=
, al := 1 − 1k . For j = 1, . . . , l let kj denote the index i ∈ {1, . . . , k − 1}, for which
aj ∈ [ti, ti+1[. In particular, k1 = 1 and kl = k−1. Then using the same arguments as in lemma
4.7 one obtains, for k and l sufficiently large, the two sided bounds
(I) = e2ε ·
g(tkj+1)
g(tkj+1)
i∈{1,...,k−1}\{k1,...,kl}
h′ (g(ti)) ·
g(ti+1)− g(ti)
h (g(ti+1))− h (g(ti))
≥ e−2ε · C−ε/23 ·
g(tkj+1)
g(tkj+1)
) = (II)
For fixed l and k → ∞ the bounds (I) and (II) converge to
(I′) = e2ε
h′(g(a2−))
h′(g(0))
h′ (g(aj+1−))
h′ (g(aj+))
h′(g(1−))
h′(g(al−1+))
(II′) = e−2ε · C−ε/23 ·
h′(g(a2−))
h′(g(0))
h′ (g(aj+1−))
h′ (g(aj+))
h′(g(1−))
h′(g(al−1+))
It remains to consider the three remaining terms
(III) =
i∈{k2,...,kl−1}
h′ (g(ti)) ·
g(ti+1)− g(ti)
h (g(ti+1))− h (g(ti))
which for fixed l and k → ∞ converges to
(III′) =
h′ (g(aj−))
· δ(h ◦ g)
(IV) =
)− g(0)
− h (g(0))
)− g( 1
converging by right continuity of g to
(IV′) = h′(g(0))
(V) =
k − 1
g(1) − g(k−1
h (g(1)) − h
g(k−1
which tends, also for k → ∞, to
(V′) =
1 if g continuous in 1
δ(h◦g)
(1) 1
h′(g(1−))
else.
Combining these estimates and letting l → ∞, we obtain the first claim. The second claim in
statement (ii) follows from the fact that g is continuous in t = 1 Q
0 -almost surely.
5 The Integration by Parts Formula
In order to construct Dirichlet forms and Markov processes on G, we will consider it as an infinite
dimensional manifold. For each g ∈ G, the tangent space TgG will be an appropriate completion
of the space C∞(S1,R). The whole construction will strongly depend on the choice of the norm
on the tangent spaces TgG. Basically, we will encounter two important cases:
• in Chapter 6 we will study the case TgG = Hs(S1,Leb) for some s > 1/2, independent
of g; this approach is closely related to the construction of stochastic processes on the
diffeomorphism group of S1 and Malliavin’s Brownian motion on the homeomorphism
group on S1, cf. [Mal99].
• in Chapters 7-9 we will assume TgG = L2(S1, g∗Leb); in terms of the dynamics on the
space P(S1) of probability measures, this will lead to a Dirichlet form and a stochastic
process associated with the Wasserstein gradient and with intrinsic metric given by the
Wasserstein distance.
In this chapter, we develop the basic tools for the differential calculus on G. The main result
will be an integration by parts formula. These results will be independent of the choice of the
norm on the tangent space.
5.1 The Drift Term
For each ϕ ∈ C∞(S1,R), the flow generated by ϕ is the map eϕ : R × S1 → S1 where for each
x ∈ S1 the function eϕ(., x) : R → S1, t 7→ eϕ(t, x) denotes the unique solution to the ODE
= ϕ(xt) (5.1)
with initial condition x0 = x. Since eϕ(t, x) = etϕ(1, x) for all ϕ, t, x under consideration, we
may simplify notation and write etϕ(x) instead of eϕ(t, x).
Obviously, for each ϕ ∈ C∞(S1,R) the family etϕ, t ∈ R is a group of orientation preserving,
C∞-diffeomorphism of S1. (In particular, e0 is the identity map e on S1, etϕ ◦ esϕ = e(t+s)ϕ for
all s, t ∈ R and (eϕ)−1 = e−ϕ.)
Since ∂
etϕ(x)|t=0 = ϕ(x) we obtain as a linearization for small t
etϕ(x) ≈ x+ tϕ(x). (5.2)
More precisely,
|etϕ(x)− (x+ tϕ(x))| ≤ C · t2
as well as
etϕ(x)− (1 + t
ϕ(x))| ≤ C · t2
uniformly in x and |t| ≤ 1.
For ϕ ∈ C∞(S1,R) and β > 0 we define functions V βϕ : G → R by
V βϕ (g) := V
ϕ (g) + β
ϕ′(g(x))dx
where
V 0ϕ (g) :=
ϕ′(g(a+)) + ϕ′(g(a−))
− ϕ(g(a+)) − ϕ(g(a−))
g(a+)− g(a−)
. (5.3)
Lemma 5.1. (i) The sum in (5.3) is absolutely convergent. More precisely,
|V 0ϕ (g)| ≤
ϕ′(g(a+)) + ϕ′(g(a−))
− ϕ(g(a+)) − ϕ(g(a−))
g(a+)− g(a−)
|ϕ′′(x)|dx
|V βϕ (g)| ≤ (1/2 + β) ·
|ϕ′′(x)|dx.
(ii) For each β ≥ 0
V βϕ (g) =
Y βetϕ(g)
e+tϕ(g)
. (5.4)
Proof. (i) According to Taylor’s formula, for each a ∈ Jg
ϕ′(g(a+)) + ϕ′(g(a−))
− δ(ϕ ◦ g)
(a) =
2(g(a+) − g(a−))
∫ g(a+)
g(a−)
∫ g(a+)
g(a−)
sgn(y−x) ·ϕ′′(y)dydx.
Hence,
ϕ′(g(a+)) + ϕ′(g(a−))
− δ(ϕ ◦ g)
(g(a+)− g(a−))
∫ g(a+)
g(a−)
∫ g(a+)
g(a−)
sgn(y − x) · ϕ′′(y)dydx
∫ g(a+)
g(a−)
|ϕ′′(y)|dy = 1
|ϕ′′(y)|dy.
Finally,
ϕ′(g(x))dx| ≤ sup
|ϕ′(y)| ≤
|ϕ′′(y)|dy.
(ii) Let us first consider the case β = 0.
log Y 0etϕ(g)
etϕ)(g(a+)) +
etϕ)(g(a−)) − log
δ(etϕ ◦ g)
etϕ)(g(a+)) +
etϕ)(g(a−)) − log
δ(etϕ ◦ g)
In order to justify that we may interchange differentiation and summation, we decompose (as
we did several times before) the infinite sum over all jumps in Jg into a finite sum over big
jumps a1, . . . , ak and an infinite sum over small jumps in Jg(k) = Jg \ {a1, . . . , ak}. Of course,
the finite sum will make no problem. We are going to prove that the contribution of the small
jumps is arbitrarily small. Recall from Lemma 4.8 that
a∈Jg(k)
etϕ)(g(a+)) +
etϕ)(g(a−)) − log
δ(etϕ ◦ g)
≤ Ct·
a∈Jg(k)
[g(a+)− g(a−)]
where Ct := supx
log( ∂
etϕ)(x)
∣. Now Ct ≤ C · |t| for all |t| ≤ 1 and an appropriate constant
C. Thus for any given ǫ > 0
a∈Jg(k)
etϕ)(g(a+)) +
etϕ)(g(a−)) − log
δ(etϕ ◦ g)
provided k is chosen large enough (i.e. such that C ·
a∈Jg(k)
|g(a+)−g(a−)| ≤ ǫ). This justifies
the above interchange of differentiation and summation.
Now for each x ∈ S1
etϕ(x)
= ϕ′(x)
since the linearization of etϕ for small t yields
etϕ(x) ≈ x+ tϕ(x),
etϕ(x) ≈ 1 + tϕ′(x).
Similarly, for small t we obtain
δ(etϕ ◦ g)
(a) ≈ 1 + t · δ(ϕ ◦ g)
and thus
δ(etϕ ◦ g)
δ(ϕ ◦ g)
Therefore,
log Y 0etϕ(g)
= V 0ϕ (g).
On the other hand, obviously
log Y 0etϕ(g)
Y 0etϕ(g)
since Y 0e0(g) = 1.
Finally, we have to consider the derivative of Xetϕ . Based on the previous arguments and using
the fact that ∂
(x) is uniformly bounded in t ∈ [−1, 1] and x ∈ S1 we immediately
logXetϕ(g)
(g(y))dy
(g(y)) dy =
ϕ′(g(y))dy.
Again Xe0(g) = 1. Therefore,
= β ·
ϕ′(g(y))dy
and thus
Y βetϕ(g)
= V βϕ (g).
this proves the first identity in (5.4). The proof of the second one V
ϕ (g) =
e+tϕ(g)
similar (even slightly easier).
5.2 Directional Derivatives
For functions u : G → R we will define the directional derivative along ϕ ∈ C∞(S1,R) by
Dϕu(g) := lim
[u(etϕ ◦ g)− u(g)] (5.5)
provided this limit exists. In particular, this will be the case for the following ’cylinder functions’.
Definition 5.2. We say that u : G → R belongs to the class Sk(G) if it can be written as
u(g) = U(g(x1), . . . , g(xm)) (5.6)
for some m ∈ N, some x1, . . . , xm ∈ S1 and some Ck-function U : (S1)m → R.
It should be mentioned that functions u ∈ Sk(G) are in general not continuous on G.
Lemma 5.3. The directional derivative exists for all u ∈ S1(G). In particular, for u as above
Dϕu(g) = lim
[u(g + t · ϕ ◦ g) − u(g)]
∂iU(g(x1), . . . , g(xm)) · ϕ(g(xi))
with ∂iU :=
U . Moreover, Dϕ : S
k(G) → Sk−1(G) for all k ∈ N ∪ {∞} and
‖Dϕu‖L2(Qβ) ≤
m · ‖∇U‖∞ · ‖ϕ‖L2(S1).
Proof. The first claim follows from
Dϕu(g) =
U(etϕ(g(x1)), . . . , etϕ(g(xm)))
∂iU(etϕ(g(x1)), . . . , etϕ(g(xm))) ·
etϕ(g(xi))
∂iU(g(x1), . . . , g(xm)) · ϕ(g(xi))
U(g(x1) + tϕ(g(x1)), . . . , g(xm) + tϕ(g(xm)))
= lim
[u(g + t · ϕ ◦ g)− u(g)] .
For the second claim,
‖Dϕu‖2L2(Qβ) =
∂iU(g(x1), . . . , g(xm)) · ϕ(g(xi))
dQβ(g)
(∂iU)
2(g(x1), . . . , g(xm)) ·
ϕ2(g(xi))
dQβ(g)
≤ ‖∇U‖2∞ ·
ϕ2(g(xi)) dQ
= m · ‖∇U‖2∞ ·
ϕ2(y) dy.
5.3 Integration by Parts Formula on P(S1)
For ϕ ∈ C∞(S1,R) let D∗ϕ denote the operator in L2(G,Qβ) adjoint to Dϕ with domain S1(G).
Proposition 5.4. Dom(D∗ϕ) ⊃ S1(G) and for all u ∈ S1(G)
D∗ϕu = −Dϕu− V βϕ · u. (5.7)
Proof. Let u, v ∈ S1(G). Then
Dϕu · v dQβ = lim
[u(etϕ ◦ g) − u(g)] · v(g) dQβ(g)
= lim
u(g) · v(e−tϕ ◦ g) · Y βe−tϕ − u(g) · v(g)
dQβ(g)
= lim
u(g) · [v(e−tϕ ◦ g)− v(g)] dQβ(g)
+ lim
u(g) · v(g) ·
Y βe−tϕ − 1
dQβ(g)
+ lim
u(g) · [v(e−tϕ ◦ g) − v(g)] ·
Y βe−tϕ − 1
dQβ(g)
u ·Dϕv dQβ(g)−
u · v · V βϕ dQβ(g) + 0.
To justify the last equality, note that according to Lemma 4.8 | log Y βetϕ | ≤ C · |t| for |t| ≤ 1.
Hence, the claim follows with dominated convergence and Lemma 5.4.
Corollary 5.5. The operator (Dϕ,S
1(G)) is closable in L2(Qβ). Its closure will be denoted by
(Dϕ,Dom(Dϕ)).
In other words, Dom(Dϕ) is the closure (or completion) of S
1(G) with respect to the norm
[u2 + (Dϕu)
2] dQβ
Of course, the space Dom(Dϕ) will depend on β but we assume β > 0 to be fixed for the sequel.
Remark 5.6. The bilinear form
Eϕ(u, v) :=
Dϕu ·Dϕv dQβ , Dom(Eϕ) := Dom(Dϕ) (5.8)
is a Dirichlet form on L2(G,Qβ) with form core S∞(G). Its generator (Lϕ,Dom(Lϕ)) is the
Friedrichs extension of the symmetric operator
(−D∗ϕ ◦Dϕ, S2(G)).
5.4 Derivatives and Integration by Parts Formula on P([0, 1])
Now let us have a look on flows on [0, 1]. To do so, let a function ϕ ∈ C∞([0, 1],R) with
ϕ(0) = ϕ(1) = 0 be given. (Note that each such function can be regarded as ϕ ∈ C∞(S1,R)
with ϕ(0) = 0.) The flow equation (5.1) now defines a flow etϕ, t ∈ R, of order preserving C∞
diffeomorphisms of [0, 1]. In particular, etϕ(0) = 0 and etϕ(1) = 1 for all t ∈ R.
Lemma 5.1 together with Theorem 4.3 immediately yields
Lemma 5.7. For ϕ ∈ C∞([0, 1],R) with ϕ(0) = ϕ(1) = 0 and each β ≥ 0
etϕ,0
= V βϕ (g)−
ϕ′(0) + ϕ′(1)
ϕ,0(g). (5.9)
For functions u : G0 → R we will define the directional derivative along ϕ ∈ C∞([0, 1],R) with
ϕ(0) = ϕ(1) = 0 as before by
Dϕu(g) := lim
[u(etϕ ◦ g)− u(g)] (5.10)
provided this limit exists. We will consider three classes of ’cylinder functions’ for which the
existence of this limit is guaranteed.
Definition 5.8. (i) We say that a function u : G0 → R belongs to the class Ck(G0) (for k ∈
N ∪ {0,∞}) if it can be written as
u(g) = U
~f(t)g(t)dt
(5.11)
for some m ∈ N, some ~f = (f1, . . . , fm) with fi ∈ L2([0, 1],Leb) and some Ck-function U : Rm →
R. Here and in the sequel, we write
~f(t)g(t)dt =
f1(t)g(t)dt, . . . ,
fm(t)g(t)dt
(ii) We say that u : G0 → R belongs to the class Sk(G0) if it can be written as
u(g) = U (g(x1), . . . , g(xm)) (5.12)
for some m ∈ N, some x1, . . . , xm ∈ [0, 1] and some Ck-function U : Rm → R.
(iii) We say that u : G0 → R belongs to the class Zk(G0) if it can be written as
u(g) = U
~α(gs)ds
(5.13)
with U as above, ~α = (α1, . . . , αm) ∈ Ck([0, 1],Rm) and
~α(gs)ds =
α1(gs)ds, . . . ,
αm(gs)ds
Remark 5.9. For each ϕ ∈ C∞(S1,R) with ϕ(0) = 0 (which can be regarded as ϕ ∈ C∞([0, 1],R)
with ϕ(0) = ϕ(1) = 0), the definitions of Dϕ in (5.5) and (5.10) are consistent in the following
sense. Each cylinder function u ∈ S1(G0) defines by v(g) := u(g − g0) (∀g ∈ G) a cylinder
function v ∈ S1(G) with Dϕv = Dϕu on G0. Conversely, each cylinder function v ∈ S1(G)
defines by u(g) := v(g) (∀g ∈ G0) a cylinder function u ∈ S1(G0) with Dϕv = Dϕu on G0.
Lemma 5.10. (i) The directional derivative Dϕu(g) exists for all u ∈ C1(G0)∪S1(G0)∪Z1(G0)
(in each point g ∈ G0 and in each direction ϕ ∈ C∞([0, 1],R) with ϕ(0) = ϕ(1) = 0) and
Dϕu(g) = limt→0
[u(g + t · ϕ ◦ g)− u(g)]. Moreover,
Dϕu(g) =
~f(t)g(t)dt
fi(t)ϕ(g(t))dt
for each u ∈ C1(G0) as in (5.11),
Dϕu(g) =
∂iU(g(x1), . . . , g(xm)) · ϕ(g(xi))
for each u ∈ S1(G0) as in (5.12), and
Dϕu(g) =
~α(gs)ds
α′i(gs)ϕ(gs)ds
for each u ∈ Z1(G0) as in (5.13).
(ii) For ϕ ∈ C∞([0, 1],R) with ϕ(0) = ϕ(1) = 0 let D∗ϕ,0 denote the operator in L2(G0,Q
adjoint to Dϕ. Then for all u ∈ C1(G0) ∪S1(G0) ∪ Z1(G0)
D∗ϕ,0u = −Dϕu− V
ϕ,0 · u. (5.14)
Proof. See the proof of the analogous results in Lemma 5.3 and Proposition 5.4.
Remark 5.11. The operators (Dϕ,C
1(G0)), (Dϕ,S1(G0)), and (Dϕ,Z1(G0)) are closable in
0 ). The closures of (Dϕ,C
1(G0)), (Dϕ,Z1(G0)) and (Dϕ,S1(G0)) coincide. They will be
denoted by (Dϕ,Dom(Dϕ)). See (proof of) Corollary 6.11.
6 Dirichlet Form and Stochastic Dynamics on on G
At each point g ∈ G, the directional derivative Dϕu(g) of any ’nice’ function u on G defines
a linear form ϕ 7→ Dϕu(g) on C∞(S1). If we specify a pre-Hilbert norm ‖.‖g on C∞(S1) for
which this linear form is continuous then there exists a unique element Du(g) ∈ TgG with
Dϕu(g) = 〈Du(g), ϕ〉g for all ϕ ∈ C∞(S1). Here TgG denotes the completion of C∞(S1) w.r.t.
the norm ‖.‖g.
The canonical choice of a Dirichlet form on G will then be (the closure of)
E(u, v) =
〈Du(g),Dv(g)〉g dQβ(g), u, v ∈ S1(G). (6.1)
Given such a Dirichlet form, there is a straightforward procedure to construct an operator (’gen-
eralized Laplacian’) and a Markov process (’generalized Brownian motion’). Different choices of
‖.‖g in general will lead to completely different Dirichlet forms, operators and Markov processes.
We will discuss in detail two choices: in this chapter we will choose ‖.‖g (independent of g) to
be the Sobolev norm ‖.‖Hs for some s > 1/2; in the remaining chapters, ‖.‖g will always be the
L2-norm ϕ 7→ (
ϕ(gt)
2dt)1/2 of L2(S1, g∗Leb).
For the sequel, fix – once for ever – the number β > 0 and drop it from the notations, i.e.
Q := Qβ, Vϕ := V
ϕ etc.
6.1 The Dirichlet Form on G
Let (ψk)k∈N denote the standard Fourier basis of L
2(S1). That is,
ψ2k(x) =
2 · sin(2πkx), ψ2k+1(x) =
2 · cos(2πkx)
for k = 1, 2, . . . and ψ1(x) = 1. It constitutes a complete orthonormal system in L
2(S1): each
ϕ ∈ L2(S1) can uniquely be written as ϕ(x) =
k=1 ck · ψk(x) with Fourier coefficients of ϕ
given by ck :=
ϕ(y)ψk(y)dy. In terms of these Fourier coefficients we define for each s ≥ 0
the norm
‖ϕ‖Hs :=
c21 +
k2s · (c22k + c22k+1)
(6.2)
on C∞(S1). The Sobolev space Hs(S1) is the completion of C∞(S1) with respect to the norm
‖.‖Hs . It has a complete orthonormal system consisting of smooth functions (ϕk)k∈N. For
instance, one may choose
ϕ2k(x) =
2 · k−s · sin(2πkx), ϕ2k+1(x) =
2 · k−s · cos(2πkx) (6.3)
for k = 1, 2, . . . and ϕ1(x) = 1.
A linear form A : C∞(S1) → R is continuous w.r.t. ‖.‖Hs — and thus can be represented as
A(ϕ) = 〈ψ,ϕ〉Hs for some ψ ∈ Hs(S1) with ‖ψ‖Hs = ‖A‖Hs — if and only if
‖A‖Hs :=
|A(ψ1)|2 +
k2s · (|A(ψ2k)|2 + |A(ψ2k+1)|2)
<∞. (6.4)
Proposition 6.1. Fix a number s > 1/2. Then for each cylinder function u ∈ S(G) and each
g ∈ G, the directional derivative defines a continuous linear form ϕ 7→ Dϕu(g) on C∞(S1) ⊂
Hs(S1). There exists a unique tangent vector Du(g) ∈ Hs(S1) such that Dϕu(g) = 〈Du(g), ϕ〉Hs
for all ϕ ∈ C∞(S1).
In terms of the family Φ = (ϕk)k∈N from (6.3)
Du(g) =
Dϕku(g) · ϕk(.)
‖Du(g)‖2Hs =
|Dϕku(g)|2. (6.5)
Proof. It remains to prove that the RHS of (6.5) is finite for each u and g under consideration.
According to Lemma 5.3, for any u ∈ S(G) represented as in (5.12)
|Dϕku(g)|2 =
∂iU(g(x1), . . . , g(xm)) · ϕk(g(xi))
≤ m · ‖∇U‖2∞ · ‖
ϕ2k‖∞ = m · ‖∇U‖2∞ · (1 + 4
k−2s).
And, indeed, the latter is finite for each s > 1/2.
For the sequel, let us now fix a number s > 1/2 and define
E(u, v) =
〈Du(g),Dv(g)〉Hs dQ(g) (6.6)
for u, v ∈ S1(G). Equivalently, in terms of the family Φ = (ϕk)k∈N from (6.3)
E(u, v) =
Dϕku(g) ·Dϕkv(g) dQ(g). (6.7)
Theorem 6.2. (i) (E ,S1(G)) is closable. Its closure (E ,Dom(E)) is a regular Dirichlet form
on L2(G,Q) which is strongly local and recurrent (hence, in particular, conservative).
(ii) For u ∈ S1(G) with representation (5.6)
E(u, u) =
∂iU(g(x1), . . . , g(xm)) · ϕk(g(xi))
dQ(g).
The generator of the Dirichlet form is the Friedrichs extension of the operator L given on S2(G)
Lu(g) =
i,j=1
∂i∂jU (g(x1), . . . , g(xm))ϕk(g(xi))ϕk(g(xj))
∂iU (g(x1), . . . , g(xm)) [ϕ
k(g(xi)) + Vϕk(g)]ϕk(g(xi)).
(iii) Z1(G) is a core for Dom(E) (i.e. it is contained in the latter as a dense subset). For
u ∈ Z1(G) with representation (5.13)
E(u, u) =
~α(gt)dt) ·
α′i(gt)ϕk(gt)dt
dQ(g).
The generator of the Dirichlet form is the Friedrichs extension of the operator L given on Z2(G)
Lu(g) =
i,j=1
∂i∂jU
~α(gt)dt
α′i(gt)ϕk(gt)dt ·
α′j(gt)ϕk(gt)dt
~α(gt)dt
{Vϕk(g) +
[α′′i (gt)ϕ
k(gt) + α
i(gt)ϕ
k(gt)ϕk(gt)]dt}.
(iv) The intrinsic metric ρ can be estimated from below in terms of the L2-metric:
ρ(g, h) ≥ 1√
‖g − h‖L2 .
Remark 6.3. All assertions of the above Theorem remain valid for any E defined as in (6.7)
with any choice of a sequence Φ = (ϕk)k∈N of smooth functions on S
1 with
C := ‖
ϕ2k‖∞ <∞. (6.8)
(This condition is satisfied for the sequence from (6.3) if and only if s > 1/2.)
The proof of the Theorem will make use of the following
Lemma 6.4. (i) Dom(E) contains all functions u which can be represented as
u(g) = U(‖g − f1‖L2 , . . . , ‖g − fm‖L2) (6.9)
with some m ∈ N, some f1, . . . , fm ∈ G and some U ∈ C1(Rm,R).
For each u as above, each ϕ ∈ C∞(S1) and Q-a.e. g ∈ G
Dϕu(g) =
∂iU(‖g − f1‖L2 , . . . , ‖g − fm‖L2) ·
sign(g(t)− fi(t))
|g(t) − fi(t)|
‖g − fi‖L2
ϕ(g(t))dt
where sign(z) := +1 for z ∈ S1 with |[0, z]| ≤ 1/2 and sign(z) := −1 for z ∈ S1 with |[z, 0]| < 1/2.
(ii) Moreover, Dom(E) contains all functions u which can be represented as
u(g) = U(gǫ1(x1), . . . , gǫm(xm)) (6.10)
with some m ∈ N, some x1, . . . , xm ∈ S1, some ǫ1, . . . , ǫm ∈ ]0, 1[ and some U ∈ C1((S1)m,R).
Here gǫ(x) :=
∫ x+ǫ
g(t)dt ∈ S1 for x ∈ S1 and 0 < ǫ < 1. More precisely,
gǫ(x) := π(
∫ x+ǫ
π−1g(t)dt)
where π : G(R) → G (cf. section 2.2) denotes the projection and π−1 : G → G(R) the canonical
lift with π−1(g)(t) ∈ [g(x), g(x) + 1] ⊂ R for t ∈ [x, x+ 1] ⊂ R.
For each u as above, each ϕ ∈ C∞(S1) and each g ∈ G
Dϕu(g) =
∂iU(gǫ1(x1), . . . , gǫm(xm)) ·
∫ xi+ǫi
ϕ(g(t))dt.
(iii) The set of all u of the form (6.10) is dense in Dom(E).
Proof. (i) Let us first prove that for each f ∈ G, the map u(g) = ‖g − f‖L2 lies in Dom(E). For
n ∈ N, let πn : G → G be the map which replaces each g by the piecewise constant map:
πn(g)(t) := g(
) for t ∈ [ i
Then by right continuity πn(g) → g as n→ ∞ and thus
|g( i
)− f( i
)|2 −→
|g(t) − f(t)|2dt.
Therefore, for each g ∈ G as n→ ∞
un(g) := Un(g(0), g(
), . . . , g(
n − 1
)) −→ u(g) (6.11)
where Un(x1, . . . , xn) :=
i=0 dn(xi+1 − f( in))
and dn is a smooth approximation of
the distance function x 7→ |x| on S1 (which itself is non-differentiable at x = 0 and x = 1
) with
|d′n| ≤ 1 and dn(x) → |x| as n→ ∞. Obviously, un ∈ S1(G).
By dominated convergence, (6.11) also implies that un → u in L2(G,Q). Hence, u ∈ Dom(E) if
(and only if) we can prove that
E(un) <∞.
E(un) =
∂iUn(g(0), g(
), . . . , g(
n − 1
)) · ϕ(g( i − 1
dQ(g)
ϕ2k(g(
i − 1
)) dQ(g) =
‖ϕk‖2L2 <∞,
uniformly in n ∈ N. This proves the claim for the function u(g) = ‖g − f‖L2 .
From this, the general claim follows immediately: if vn, n ∈ N, is a sequence of S1(G) approx-
imations of g 7→ ‖g − 0‖L2 then un(g) := U(vn(g − f1), . . . , vn(g − fm)) defines a sequence of
S1(G) approximations of u(g) = U(‖g − f1‖L2 , . . . , ‖g − fm‖L2).
(ii) Again it suffices to treat the particular case m = 1 and U = id, that is, u(g) = gǫ(x)
for some x ∈ S1 and some 0 < ǫ < 1. Let g̃ ∈ G(R) be the lifting of g and recall that
u(g) = π(1
∫ x+ǫ
g̃(t)dt). Define un ∈ S1(G) for n ∈ N by un(g) = π( 1n
i=0 g̃(x +
ǫ)). Right
continuity of g̃ implies un → u as n → ∞ pointwise on G and thus also in L2(G,Q). To see the
boundedness of E(un) note that Dϕun(g) = 1n
i=0 ϕ(g(x +
ǫ)). Thus
E(un) ≤
ϕ2k(g(x +
ǫ))dQ(g) =
‖ϕk‖2L2 <∞.
(iii) We have to prove that each u ∈ S1(G) can be approximated in the norm (‖.‖2 + E(.))1/2 by
functions un of type (6.10). Again it suffices to treat the particular case u(g) = g(x) for some
x ∈ S1. Choose un(g) = g1/n(x). Then by right continuity of g, un → u pointwise on G and
thus also in L2(G,Q). Moreover, Dϕun(g) = n
∫ x+1/n
ϕ(g(t))dt (for all ϕ and g) and therefore
E(un) ≤
∫ x+1/n
ϕ2k(g(t))dtdQ(g) =
‖ϕk‖2L2 <∞.
Proof of the Theorem. (a) The sum E of closable bilinear forms with common domain S1(G)
is closable, provided it is still finite on this domain. The latter will follow by means of Lemma
5.3 which implies for all u ∈ S1(G) with representation (5.11)
E(u, u) =
∂iU(g(x1), . . . , g(xm)) · ϕk(g(xi))
dQ(g)
≤ m · ‖∇U‖2∞ ·
‖ϕk‖2L2(S1) <∞.
Hence, indeed E is finite on S1(G).
(b) The Markov property for E follows from that of the Eϕk(u, v) =
Dϕku ·Dϕkv dQ.
(c) According to the previous Lemma, the class of continuous functions of type (6.10) is dense
in Dom(E). Moreover, the class of finite energy functions of type (6.9) is dense in C(G) (with
the L2 topology of G ⊂ L2(S1), cf. Proposition 2.1). Therefore, the Dirichlet form E is regular.
(e) The estimate for the intrinsic metric is an immediate consequence of the following estimate
for the norm of the gradient of the function u(g) = ‖g − f‖L2 (which holds for each f ∈ G
uniformly in g ∈ G):
‖Du(g)‖2 =
sign(g(t)− fi(t))
|g(t) − fi(t)|
‖g − fi‖L2
ϕk(g(t))dt
ϕ2k(g(t))dt ≤ ‖
ϕ2k‖∞ =: C.
(f) The locality is an immediate consequence of the previous estimate: Given functions u, v ∈
Dom(E) with disjoint supports, one has to prove that E(u, v) = 0. Without restriction, one may
assume that supp[u] ⊂ Br(g) and supp[v] ⊂ Br(h) with ‖g − h‖L2 > 2r + 2δ. (The general
case will follow by a simple covering argument.) Without restriction, u, v can be assumed to be
bounded. Then |u| ≤ Cwδ,g and |v| ≤ Cwδ,h for some constant C where
wδ,g(f) =
(r + δ − ‖f − g‖L2) ∧ 1
Given un ∈ S1(G) with un → u in Dom(E) put
un = (un ∧ wδ,g) ∨ (−wδ,g).
Then un → u in Dom(E). Analogously, vn → v in Dom(E) for vn = (vn ∧ wδ,h) ∨ (−wδ,h). But
obviously, E(un, vn) = 0 since un · vn = 0. Hence, E(u, v) = 0.
(g) In order to prove that Z1(G) is contained in Dom(E) it suffices to prove that each u ∈ Z1(G)
of the form u(g) =
α(gt)dt can be approximated in Dom(E) by un ∈ S1(G). Given u as above
with α ∈ C1(S1,R) put un(g) = 1n
i=1 α(gi/n). Then un ∈ S1(G), un → u on G and
Dϕun(g) =
α′(gi/n)ϕ(gi/n) →
α′(gt)ϕ(gt)dt = Dϕu(g).
Moreover,
E(un, un) =
α′(gi/n)ϕ(gi/n)
dQ(g)
≤ C ·
α′(gi/n)
2 dQ(g) = C ·
α′(t)2dt
uniformly in n ∈ N. Hence, u ∈ Dom(E) and
E(u, u) = lim
E(un, un) =
α′(gt)ϕk(gt)dt
dQ(g).
(h) The set Z1(G) is dense in Dom(E) since according to assertion (ii) of the previous Lemma
already the subset of all u of the form (6.10) is dense in Dom(E).
Finally, one easily verifies that Z2(G) is dense in Z1(G) and (using the integration by parts
formula) that L is a symmetric operator on Z2(G) with the given representation.
Corollary 6.5. There exists a strong Markov process (gt)t≥0 on G, associated with the Dirichlet
form E. It has continuous trajectories and it is reversible w.r.t. the measure Q. Its generator
has the form
DϕkDϕk +
Vϕk ·Dϕk
with {ϕk}k∈N being the Fourier basis of Hs(S1).
Remark 6.6. This process (gt)t≥0 is closely related to the stochastic processes on the diffeo-
morphism group of S1 and to the ’Brownian motion’ on the homeomorphism group of S1,
studied by Airault, Fang, Malliavin, Ren, Thalmaier and others [AMT04, AM06, AR02, Fan02,
Fan04, Mal99]. These are processes with generator 1
kDϕkDϕk . For instance, in the
case s = 3/2 our process from the previous Corollary may be regarded as ’Brownian motion
plus drift’. All the previous approaches are restricted to s ≥ 3/2. The main improvements of
our approach are:
• identification of a probability measure Q such that these processes — after adding a
suitable drift — are reversible;
• construction of such processes in all cases s > 1/2.
6.2 Finite Dimensional Noise Approximations
In the previous section, we have seen the construction of the diffusion process on G under minimal
assumptions. However, the construction of the process is rather abstract. In this section, we try
to construct explicitly a diffusion process associated with the generator of the Dirichlet form E
from Theorem 6.2. Here we do not aim for greatest generality.
Let a finite family Φ = (ϕk)k=1,...,n of smooth functions on S
1 be given and let (Wt)t≥0 with
Wt = (W
t , . . . ,W
t ) be a n-dimensional Brownian motion, defined on some probability space
(Ω,F ,P). For each x ∈ S1 we define a stochastic processes (ηt(x))t≥0 with values in S1 as the
strong solution of the Ito differential equation
dηt(x) =
ϕk(ηt(x))dW
ϕ′k(ηt(x))ϕk(ηt(x))dt (6.12)
with initial condition η0(x) = x. Equation (6.12) can be rewritten in Stratonovich form as
follows
dηt(x) =
ϕk(ηt(x)) ⋄ dW kt . (6.13)
Obviously, for every t and for P-a.e. ω ∈ Ω, the function x 7→ ηt(x, ω) is an element of the
semigroup G. (Indeed, it is a C∞-diffeomorphism.) Thus (6.13) may also be interpreted as a
Stratonovich SDE on the semigroup G:
dηt =
ϕk(ηt) ⋄ dW kt , η0 = e. (6.14)
This process on G is right invariant: if gt denotes the solution to (6.14) with initial condition
g0 = g for some initial condition g ∈ G then gt = ηt ◦ g. One easily verifies that the generator
of this process (gt)t≥0 is given on S
2(G) by 1
k=1DϕkDϕk . What we aim for, however, is a
process with generator
D∗ϕkDϕk =
DϕkDϕk +
Vϕk ·Dϕk .
Define a new probability measure Pg on (Ω,F), given on Ft by
dPg = exp
Vϕk(ηs ◦ g)dW ks −
|Vϕk(ηs ◦ g)|2ds
dP (6.15)
and a semigroup (Pt)t≥0 acting on bounded measurable functions u on G as follows
Ptu(g) =
u(ηt(g(.), ω)) dP
g(ω).
Proposition 6.7. (Pt)t≥0 is a strongly continuous Markov semigroup on G. Its generator is an
extension of the operator 1
L = −1
Dϕk with domain S
2(G). That is, for all u ∈ S2(G)
and all g ∈ G
(Ptu(g)− u(g)) =
Lu(g). (6.16)
Proof. The strong continuity follows easily from the fact that ηt(x, .) → x a.s. as t → 0 which
implies by dominated convergence
Ptu(g) =
u(ηt ◦ g) dPg → u(g)
for each continuous u : G → R.
Now we aim for identifying the generator. According to Girsanov’s theorem, under the measure
Pg the processes
W̃ kt =W
Vϕk(ηs ◦ g)ds
for k = 1, . . . , n will define n independent Brownian motions. In terms of these driving processes,
(6.12) can be reformulated as
dgt(x) =
ϕk(gt(x))dW̃
[ϕ′k(gt(x)) + Vϕk(gt)]ϕk(gt(x))dt (6.17)
(recall that gs = ηs ◦ g). The chain rule applied to a smooth function U on (S1)m, therefore,
yields
dU (gt(y1), . . . , gt(ym))
U (gt(y1), . . . , gt(ym)) dgt(yi)
i,j=1
∂xi∂xj
U (gt(y1), . . . , gt(ym)) d〈g.(yi), g.(yj)〉t
U (gt(y1), . . . , gt(ym))ϕk(gt(yi))dW̃
U (gt(y1), . . . , gt(ym)) [ϕ
k(gt(yi)) + Vϕk(gt)]ϕk(gt(yi))dt
i,j=1
∂xi∂xj
U (gt(y1), . . . , gt(ym))ϕk(gt(yi))ϕk(gt(yj))dt.
Hence, for a cylinder function of the form u(g) = U(g(y1), . . . , g(ym)) we obtain
(Ptu(g) − u(g))
= lim
[U (gt(y1), . . . , gt(ym))− U (g0(y1), . . . , g0(ym))] dPg
= lim
U (gs(y1), . . . , gs(ym)) [ϕ
k(gs(yi)) + Vϕk(gs)]ϕk(gs(yi))
i,j=1
∂xi∂xj
U (gs(y1), . . . , gs(ym))ϕk(gs(yi))ϕk(gs(yj))
 ds dPg
U (g(y1), . . . , g(ym)) [ϕ
k(g(yi)) + Vϕk(g)]ϕk(g(yi))
i,j=1
∂xi∂xj
U (g(y1), . . . , g(ym))ϕk(g(yi))ϕk(g(yj))
[DϕkDϕku(g) + Vϕk(g) ·Dϕku(g)] = −
D∗ϕkDϕku(g).
In order to justify (∗), we have to verify continuity in s in all the expressions preceding (∗). The
only term for which this is not obvious is Vϕk(gs). But gs = ηs ◦g with a function ηs(x, ω) which
is continuous in x and in s. Thus Vϕk(ηs(., ω) ◦ g) is continuous in s.
Remark 6.8. All the previous argumentations in principle also apply to infinite families of
(ϕk)k=1,2,..., provided they have sufficiently good integrability properties. For instance, the
family (6.3) with s > 5
will do the job. There are three key steps which require a careful
verification:
• the solvability of the Ito equation (6.12) and the fact that the solutions are homeomor-
phisms of S1; here s ≥ 3
suffices, cf. [Mal99];
• the boundedness of the quadratic variation of the drift to justify Girsanov’s transformation
in (6.15); for s > 5
this will be satisfied since Lemma 5.1 implies (uniformly in g)
|Vϕk(g)|
2 ≤ (β + 1)2
|ϕ′′k(x)|2dx ≤ 4(β + 1)2
k4−2s;
• the finiteness of the generator and Ito’s chain rule for C2-cylinder functions; here s > 3
will be sufficient.
Remark 6.9. Another completely different approximation of the process (gt)t≥0 in terms of
finite dimensional SDEs is obtained as follows. For N ∈ N, let S1N denote the set of cylinder
functions u : G → R which can be represented as u(g) = U(g(1/N), g(2/N), . . . , g(1)) for some
U ∈ C1((S1)N ). Denote the closure of (E ,S1N ) by (EN ,Dom(EN )). It is the image of the
Dirichlet form (EN ,Dom(EN )) on ΣN ⊂ (S1)N given by
EN (U) =
i,j=1
∂iU(x)∂jU(x) aij(x)ρ(x) dx (6.18)
aij(x) =
ϕk(xi)ϕk(xj), ρ(x) =
Γ(β/N)N
(xi+1 − xi)β/N−1dx.
and (as before) ΣN =
(x1, . . . , xN ) ∈ (S1)N :
i=1 |[xi, xi+1]| = 1
. That is,
EN (u) = EN (U)
for cylinder functions u ∈ S1N as above. Let (Xt,Px)t≥0,x∈ΣN be the Markov process on ΣN
associated with EN . Then the semigroup associated with EN is given by
TNt u(g) = Eg(1/N),...,g(1) [U(Xt)] .
Now let (gt,Pg)t≥0,g∈G and (Tt)t≥0 denote the Markov process and the L
2-semigroup associated
with E . Then as N → ∞
t → Tt strongly in L2
since
E2N ց E
in the sense of quadratic forms, [RS80], Theorem S.16. (Note that ∪N∈NS12N is dense inDom(E).)
6.3 Dirichlet Form and Stochastic Dynamics on G1 and P
In order to define the derivative of a function u : G1 → R we regard it as a function ũ on G with
the property ũ(g) = ũ(g ◦θz) for all z ∈ S1. This implies that Dϕũ(g) = (Dϕũ)(g ◦θz) whenever
one of these expressions is well-defined. In other words, Dϕũ defines a function on G1 which will
be denoted by Dϕu and called the directional derivative of u along ϕ.
Corollary 6.10. (i) Under assumption (6.8), with the notations from above,
E(u, u) =
|Dϕku|
2 dQ.
defines a regular, strongly local, recurrent Dirichlet form on L2(G1,Q).
(ii) The Markov process on G analyzed in the previous section extends to a (continuous, re-
versible) Markov process on G1.
In order to see the second claim, let g, g̃ ∈ G with g̃ = g ◦ θz for some z ∈ S1. Then obviously,
g̃t(., ω) = ηt(g̃(.), ω) = ηt(g(. + z), ω) = gt(., ω) ◦ θz.
Moreover,
Pg̃ = Pg
since Vϕ(g ◦ θz) = Vϕ(g) for all ϕ under consideration and all z ∈ S1.
The objects considered previously – derivative, Dirichlet form and Markov process on G1 – have
canonical counterparts on P. The key to these new objects is the bijective map χ : G1 → P.
The flow generated by a smooth ’tangent vector’ ϕ : S1 → R through the point µ ∈ P will be
given by ((etϕ)∗µ)t∈R. In these terms, the directional derivative of a function u : P → R at the
point µ ∈ P in direction ϕ ∈ C∞(S1,R) can be expressed as
Dϕu(µ) = lim
[u((etϕ)∗µ)− u(µ)] ,
provided this limit exists. The adjoint operator to Dϕ in L
2(P,P) is given (on a suitable dense
subspace) by
D∗ϕu(µ) = −Dϕ(µ)− Vϕ(χ−1(µ)) · u(µ).
The drift term can be represented as
−1(µ)) = β
ϕ′(s)µ(ds) +
I∈gaps(µ)
ϕ′(I−) + ϕ
′(I+)
− ϕ(I+)− ϕ(I−)|I|
Given a sequence Φ = (ϕk)k∈N of smooth functions on S
1 satisfying (6.8), we obtain a (regular,
strongly local, recurrent) Dirichlet form E on L2(P,P) by
E(u, u) =
|Dϕku(µ)|2dP(µ). (6.19)
It is the image of the Dirichlet form defined in (6.7) under the map χ. The generator of E is
given on an appropriate dense subspace of L2(P,P) by
L = −
D∗ϕkDϕk . (6.20)
For P-a.e. µ0 ∈ P, the associated Markov process (µt)t≥0 on P starting in µ0 is given as
µt(ω) = gt(ω)∗Leb
where (gt)t≥0 is the process on G, starting in g0 := χ−1(µ0). (As mentioned before, (gt)t≥0 admits
a more direct construction provided we restrict ourselves to a finite sequence Φ = (ϕk)k=1,...,n.)
6.4 Dirichlet Form and Stochastic Dynamics on G0 and P0
For s > 0 and ϕ : [0, 1] → R let the Sobolev norm ‖ϕ‖Hs be defined as in (6.2) and let Hs0([0, 1])
denote the closure of C∞c (]0, 1[), the space of smooth ϕ : [0, 1] → R with compact support
in ]0, 1[. If s ≥ 1/2 (which is the only case we are interested in) Hs0([0, 1]) can be identified
with {ϕ ∈ Hs([0, 1]) : ϕ(0) = ϕ(1) = 0} or equivalently with {ϕ ∈ Hs(S1) : ϕ(0) = 0}.
For the sequel, fix s > 1/2 and a complete orthonormal basis Φ = {ϕk}k∈N of Hs0([0, 1]) with
C := ‖
k‖∞ <∞, and define
E0(u, u) =
|Dϕk ,0u(g)|2 dQ0(g).
Corollary 6.11. (E0,S1(G0)), (E0,Z1(G0)) and (E0,C1(G0)) are closable. Their closures coincide
and define a regular, strongly local, recurrent Dirichlet form (E0,Dom(E0)) on L2(G0,Q0).
Proof. For the closability (and the equivalence of the respective closures) of (E0,S1(G0)) and
(E0,Z1(G0)), see the proof of Theorem 6.2. Also all the assertions on the closure are deduced in
the same manner. For the closability of (E0,C1(G0)) (and the equivalence of its closure with the
previously defined closures), see the proof of Theorem 7.8 below.
As explained in the previous subsection, these objects (invariant measure, derivative, Dirichlet
form and Markov process) on G0 have canonical counterparts on P0 defined by means of the
bijective map χ : G0 → P0.
7 The Canonical Dirichlet Form on the Wasserstein Space
7.1 Tangent Spaces and Gradients
The aim of this chapter is to construct a canonical Dirichlet form on the L2-Wasserstein space
P0. Due to the isometry χ : G0 → P0 this is equivalent to construct a canonical Dirichlet form
on the metric space (G0, ‖.‖L2). This can be realized in two geometric settings which seem to be
completely different:
• Like in the preceding two chapters, G0 can be considered as a group, with composition of
functions as group operation. The tangent space TgG0 is the closure (w.r.t. some norm)
of the space of smooth functions ϕ : [0, 1] → R with ϕ(0) = ϕ(1) = 0. Such a function ϕ
induces a flow on G0 by (g, t) 7→ etϕ ◦ g ≈ g + t ϕ ◦ g and it defines a directional derivative
by Dϕu(g) = limt→0
[u(etϕ ◦ g)−u(g)] for u : G0 → R. The norm on TgG0 we now choose
to be ‖ϕ‖Tg := (
ϕ(gs)
2ds)1/2. That is,
TgG0 := L2([0, 1], g∗Leb).
For given u and g as above, a gradient Du(g) ∈ TgG0 exists with
Dϕu(g) = 〈Du(g), ϕ〉Tg (∀ϕ ∈ Tg)
if and only if supϕ
Dϕu(g)
‖ϕ◦g‖
• Alternatively, we can regard G0 as a closed subset of the space L2([0, 1],Leb). The lin-
ear structure of the latter (with the pointwise addition of functions as group operation)
suggests to choose as tangent space
TgG0 := L2([0, 1],Leb).
An element f ∈ TgG0 induces a flow by (g, t) 7→ g+tf and it defines a directional derivative
(’Frechet derivative’) by Dfu(g) = limt→0
[u(g + tf) − u(g)] for u : G0 → R, provided u
extends to a neighborhood of G0 in L2([0, 1],Leb) or the flow (induced by f) stays within
G0. A gradient Du(g) ∈ TgG0 exists with
Dfu(g) = 〈Du(g), f〉L2 (∀ϕ ∈ L2)
if and only if supf
Dfu(g)
<∞. In this case, Du(g) is the usual L2-gradient.
Fortunately, both geometric settings lead to the same result.
Lemma 7.1. (i) For each g ∈ G0, the map ιg : ϕ 7→ ϕ ◦ g defines an isometric embedding
of TgG0 = L2([0, 1], g∗Leb) into TgG0 = L2([0, 1],Leb). For each (smooth) cylinder function
u : G0 → R
Dϕu(g) = Dϕ◦gu(g).
If Du ∈ L2(Leb) exists then Du ∈ L2(g∗Leb) also exists.
(ii) For Q0-a.e. g ∈ G0, the above map ιg : TgG0 → TgG0 is even bijective. For each u as above
Du(g) = Du(g) ◦ g−1 and
‖Du(g)‖Tg = ‖Du(g)‖Tg .
Proof. (i) is obvious, (ii) follows from the fact that for Q0-a.e. g ∈ G0 the generalized inverse
g−1 is continuous and thus g−1(gt) = t for all t (see sections 3.5 and 2.1). Hence, the map
ιg : TgG0 → TgG0 is surjective: for each f ∈ TgG0
ιg(f ◦ g−1) = f ◦ g−1 ◦ g = f.
Example 7.2. (i) For each u ∈ Z1(G0) of the form u(g) = U(
~α(gt)dt) with U ∈ C1(Rm,R)
and ~α = (α1, . . . , αm) ∈ C1([0, 1],Rm), the gradients Du(g) ∈ TgG0 = L2([0, 1], g∗Leb) and
Du(g) ∈ TgG0 = L2([0, 1],Leb) exist:
Du(g) =
~α(gt)dt) · α′i(g(.)), Du(g) =
~α(gt)dt) · α′i(.)
and their norms coincide:
‖Du(g)‖2Tg = ‖Du(g)‖
~α(gt)dt) · α′i(g(s))
(ii) For each u ∈ C1(G0) of the form u(g) = U(
~f(t)g(t)dt) with U ∈ C1(Rm,R) and ~f =
(f1, . . . , fm) ∈ L2([0, 1],Rm), the gradient
Du(g) =
~f(t)g(t)dt) · αi(.) ∈ L2([0, 1],Leb)
exists and
‖Du(g)‖2Tg =
~f(t)g(t)dt) · fi(s)
For u ∈ C1(G0) ∪ Z1(G0), the gradient Du can be regarded as a map G0 × [0, 1] → R, (g, t) 7→
Du(g)(t). More precisely,
D : C1(G0) ∪ Z1(G0) → L2(G0 × [0, 1],Q0 ⊗ Leb).
Proposition 7.3. The operator D : Z1(G0) → L2(G0×[0, 1],Q0⊗Leb) is closable in L2(G0,Q0).
Proof. LetW ∈ L2(G0×[0, 1],Q0⊗Leb) be of the formW (g) = w(g)·ϕ(gt) with some w ∈ Z1(G0)
and some ϕ ∈ C∞([0, 1]) satisfying ϕ(0) = ϕ(1) = 0. Then according to the integration by parts
formula for each u ∈ Z1(G0) with u(g) = U(
~α(gs)ds)
G0×[0,1]
Du ·W d(Q0 ⊗ Leb) =
~α(gs)ds)α
i(gt)w(g)ϕ(gt)dtdQ0(g)
Dϕu(g)w(g) dQ0(g) =
u(g)D∗ϕw(g) dQ0(g).
To prove the closability of D, consider a sequence (un)n in Z
1(G0) with un → 0 in L2(Q0) and
Dun → V in L2(Q0 ⊗ Leb). Then
V ·W d(Q0 ⊗ Leb) = lim
Dun ·W d(Q0 ⊗ Leb) = lim
ϕw dQ0 = 0 (7.1)
for all W as above. The linear hull of the latter is dense in L2(Q0 ⊗ Leb). Hence, (7.1) implies
V = 0 which proves the closability of D.
The closure of (D,Z1(G0)) will be denoted by (D,Dom(D). Note that a priori it is not clear
whether D coincides with D on C1(G0). (See, however, Theorem 7.8 below.)
7.2 The Dirichlet Form
Definition 7.4. For u, v ∈ Z1(G0) ∪ C1(G0) we define the ’Wasserstein Dirichlet integral’
E(u, v) =
〈Du(g),Dv(g)〉L2 dQ0(g). (7.2)
Theorem 7.5. (i) (E,Z1(G0)) is closable. Its closure (E,Dom(E)) is a regular, recurrent
Dirichlet form on L2(G0,Q0).
Dom(E) = Dom(D) and for all u, v ∈ Dom(D)
E(u, v) =
G0×[0,1]
Du · Dv d(Q0 ⊗ Leb).
(ii) The set Z∞0 (G0) of all cylinder functions u ∈ Z∞(G0) of the form u(g) = U(
~α(gs)ds) with
U ∈ C∞(Rm,R) and ~α = (α1, . . . , αm) ∈ C∞([0, 1],Rm) satisfying α′i(0) = α′i(1) = 0 is a core
for (E,Dom(E)).
(iii) The generator (L,Dom(L) of (E,Dom(E)) is the Friedrichs extension of the operator
(L,Z∞0 (G0) given by
Lu(g) = −
D∗αiui(g)
i,j=1
∂i∂jU
~α(gs)ds
α′i(gs)α
j(gs)ds +
~α(gs)ds
· V β
where ui(g) := ∂iU(
~α(gs)ds) and V
(g) denotes the drift term defined in section 5.1 with
ϕ = α′i; β > 0 is the parameter of the entropic measure fixed throughout the whole chapter.
(iv) The Dirichlet form (E,Dom(E)) has a square field operator given by
Γ(u, v) := 〈Du,Dv〉L2(Leb) ∈ L1(G0,Q0)
with Dom(Γ) = Dom(E) ∩ L∞(G0,Q0). That is, for all u, v, w ∈ Dom(E) ∩ L∞(G0,Q0)
w · Γ(u, v) dQ0 = E(u, vw) + E(uw, v) − E(uv,w). (7.3)
Proof. (a) The closability of the form (E,Z1(G0)) follows immediately from the previous Propo-
sition 7.3. Alternatively, we can deduce it from assertion (iii) which we are going to prove first.
(b) Our first claim is that E(u,w) = −
u · Lw dQ0 for all u,w ∈ Z∞0 (G0). Let u(g) =
~α(gs)ds) and w(g) = W (
~γ(gs)ds) with U,W ∈ C∞(Rm,R) and ~α = (α1, . . . , αm), ~γ =
(γ1, . . . , γm) ∈ C∞([0, 1],Rm) satisfying α′i(0) = α′i(1) = γ′i(0) = γ′i(1) = 0. Observe that
〈Du(g),Dw(g)〉L2 =
i,j=1
~α(gs)ds) · ∂jW (
~γ(gs)ds) ·
α′i(gs)γ
j(gs)ds
ui(g) ·Dα′
w(g).
Hence, according to the integration by parts formula from Proposition 5.10
E(u,w) =
〈Du(g),Dw(g)〉L2 dQ0(g)
ui(g) ·Dα′
w(g) dQ0(g)
ui(g) · w(g) dQ0(g)
Lu(g) · w(g) dQ0(g).
This proves our first claim. In particular, (L,Z∞0 (G0)) is a symmetric operator. Therefore, the
form (E,Z∞0 (G0)) is closable and its generator coincides with the Friedrichs extension of L.
(c) Now let us prove that Z∞0 (G0) is dense in Z1(G0). That is, let us prove that each function
u ∈ Z1(G0) can be approximated by functions uǫ ∈ Z∞0 (G0). For simplicity, assume that u is of the
form u(g) = U(
α(gs)ds) with U ∈ C1(R) and α ∈ C1([0, 1]). (That is, for simplicity, m = 1.)
Let Uǫ ∈ C∞(R) for ǫ > 0 be smooth approximations of U with ‖U − Uǫ‖∞ + ‖U ′ − U ′ǫ‖∞ → 0
as ǫ → 0 and let αǫ ∈ C∞(R) with α′ǫ(0) = α′ǫ(1) = 0 be smooth approximations of α with
‖α−αǫ‖∞ → 0 and α′ǫ(t) → α′(t) for all t ∈]0, 1[ as ǫ → 0. Moreover, assume that supǫ ‖α′‖∞ <
Define uǫ ∈ Z∞0 (G0) as uǫ(g) = Uǫ(
αǫ(gs)ds). Then uǫ → u in L2(G0,Q0) by dominated
convergence relative Q0.
Since
U ′ǫ(
αǫ(g(s))ds)
[0,1]
α′ǫ(gs)
2ds ≤ C,
(α′ǫ)
2(g(s))
ǫ→0−→ α′(gs)2 ∀s ∈ [0, 1] \
{g = 0} ∩ {g = 1}
[0, 1] \
{g = 0} ∩ {g = 1}
=]0, 1[ for Q0-almost all g ∈ G0
one finds by dominated convergence in L2([0, 1],Leb), for Q0-almost all g ∈ G0
U ′ǫ(
αǫ(gs)ds)
[0,1]
α′ǫ(gs)
ǫ→0−→
α(gs)ds)
[0,1]
α′(gs)
Hence also with
E(uǫ, uǫ) =
U ′ǫ(
αǫ(gs)ds)
α′ǫ(gs)
2dsQ0(dg)
ǫ→0−→
α(gs)ds)
α′(gs)
2dsQ0(dg)
by dominated convergence in L2(G0,Q0). In particular, {uǫ}ǫ constitutes a Cauchy sequence
relative to the norm ‖v‖2
E,1 := ‖v‖2L2(G,Q) + E(v, v). In fact, since the sequence uǫ is uniformly
bounded w.r.t. to ‖.‖E,1, by weak compactness there is a weakly converging subsequence in
(Dom(E), ‖.‖E,1). Since the associated norms converge, the convergence is actually strong in
(Dom(E), ‖.‖E,1). Moreover, since uǫ → u in L2(G0,Q0), this limit is unique. Hence the entire
sequence converges to u ∈ (Dom(E), ‖.‖E,1), such that in particular E(u, u) = limǫ→0E(uǫ, uǫ).
This proves our second claim. In particular, it implies that also (E,Z1(G0)) is closable and that
the closures of Z∞0 (G0) and Z1(G0) coincide.
(d) Obviously, (E,Dom(E)) has the Markovian property. Hence, it is a Dirichlet form. Since
the constant functions belong to Dom(E), the form is recurrent. Finally, the set Z1(G0) is dense
in (C(G0), ‖.‖∞) according to the theorem of Stone-Weierstrass since it separates the points in
the compact metric space G0. Hence, (E,Dom(E)) is regular.
(e) According to Leibniz’ rule, (7.3) holds true for all u, v, w ∈ Z1(G0). Arbitrary u, v, w ∈
Dom(E)∩L∞(G0,Q0) can be approximated in (E(.) + ‖.‖2)1/2 by un, vn, wn ∈ Z1(G0) which are
uniformly bounded on G0. Then unvn → uv, unwn → uw and vnwn → vw in (E(.) + ‖.‖2)1/2.
Moreover, we may assume that wn → w Q0-a.e. on G0 and thus
|wΓ(u, v) − wnΓ(un, vn)| dQ0 ≤
|w−wn|Γ(u, v)dQ0 +
|wn| · |Γ(u, v)−Γ(un, vn)|dQ0 → 0
by dominated convergence. Hence, (7.3) carries over from Z1(G0) to Dom(E) ∩L∞(G0,Q0).
Lemma 7.6. For each f ∈ G0 the function u : g 7→ 〈f, g〉L2 belongs to Dom(E).
Proof. (a) For f, g ∈ G0 put µf = f∗Leb and µg = g∗Leb. Recall that by Kantorovich duality
‖f − g‖2L2 =
d2W (µf , µg)
= sup
ϕdµf +
= sup
ϕ(ft)dt+
ψ(gt)dt
where the supϕ,ψ is taken over all (smooth, bounded) ϕ ∈ L1([0, 1], µf ), ψ ∈ L1([0, 1], µg)
satisfying ϕ(x) + ψ(y) ≤ 1
|x − y|2 for µf -a.e. x and µg-a.e. y in [0, 1]. Replacing ϕ(x) by
|x|2/2 − ϕ(x) (and ψ(y) by . . .) this can be restated as
〈f, g〉L2 = inf
ϕ(ft)dt+
ψ(gt)dt
(7.4)
where the infϕ,ψ now is taken over all (smooth, bounded) ϕ ∈ L1([0, 1], µf ), ψ ∈ L1([0, 1], µg)
satisfying ϕ(x) + ψ(y) ≥ 〈x, y〉 for µf -a.e. x and µg-a.e. y in [0, 1]. If g is strictly increasing
then ψ can be chosen as
ψ′ = f ◦ g−1,
cf. [Vil03], sect. 2.1 and 2.2.
(b) Now fix a countable dense set {gn}n∈N of strictly increasing functions in G0 and an arbitrary
function f ∈ G0. Let (ϕn, ψn) denote a minimizing pair for (f, gn) in (7.4) and define un : G0 → R
un(g) := min
i=1,...,n
ϕ(fi(t))dt+
ψi(g(t))dt
Note that ψ′i = f ◦ g
i and thus un(gi) = 〈f, gi〉 for all i = 1, . . . , n. Therefore,
|un(g)− un(g̃)| ≤ max
|ψi(g(t))dt − ψi(g̃(t))|dt ≤ max
‖ψ′i‖∞ ·
|g(t)− g̃(t)|dt ≤ ‖g − g̃‖L1
for all g, g̃ ∈ G0. Hence, un → u pointwise on G0 and in L2(G0,Q0) where u(g) := 〈f, g〉.
(c) The function un is in the class Z
0(G0):
un(g) = Un
~α(gt)dt
with Un(x1, . . . , xn) = min{c1 + x1, . . . , cn + xn}, ci =
ϕi(f(t))dt and αi = ψi. The function
Un can be easily approximated by C1 functions in order to verify that un ∈ Dom(E) and
Dun(g) =
1Ai(g) · ψ′i(g(.))
with a suitable disjoint decomposition G0 = ∪iAi. (More precisely, Ai denotes the set of all
g ∈ G0 satisfying
ϕ(fi(t))dt +
ψi(g(t))dt <
ϕ(fj(t))dt +
ψj(g(t))dt for all j < i and
ϕ(fi(t))dt+
ψi(g(t))dt ≤
ϕ(fi(t))dt+
ψi(g(t))dt for all j > i.) Thus
‖Dun(g)‖2 =
1Ai(g) ·
ψ′i(g(t))
E(un) ≤ max
‖ψ′i ◦ g‖2L2dQ0(g).
In particular, since |ψ′i| ≤ 1,
E(un) ≤ 1
and thus u ∈ Dom(E).
Lemma 7.7. For all u ∈ Z1(G0) and all w ∈ C1(G0) ∩Dom(E)
E(u,w) =
〈Du(g),Dw(g)〉L2dQ0(g) (7.5)
(with Du(g) and Dw(g) given explicitly as in Example 7.2).
Proof. Recall that for u ∈ Z∞0 (G0) of the form u(g) = U(
~α(gt)dt)
Lu(g) =
ui(g)
with ui(g) = ∂iU(
~α(gt)dt). Hence, for w ∈ C1(G0) of the form w(g) =W (〈~h, g〉)
E(u,w) = −
Lu(g)w(g) dQ0(g)
ui(g)w(g) dQ0(g) =
ui(g)Dα′
ui(g)w(g) dQ0(g)
i,j=1
~α(gt)dt) · ∂jW (
~h(t)g(t)dt) ·
α′i(g(t))hj(t)dt dQ0(g)
〈Du(g),Dw(g)〉dQ0(g).
This proves the claim provided u ∈ Z∞0 (G0). By density this extends to all u ∈ Z1(G0).
Theorem 7.8. (i) (E,C1(G0)) is closable and its closure coincides with (E,Dom(E)). Similarly,
(D,C1(G0)) is closable and its closure coincides with (D,Dom(D)).
(ii) For all u,w ∈ Z1(G0) ∪ C1(G0)
Γ(u,w)(g) = 〈Du(g),Dw(g)〉L2 , (7.6)
in particular, E(u,w) =
〈Du(g),Dw(g)〉L2dQ0(g) (with Du(g) and Dw(g) given explicitly as
in Example 7.2).
(iii) For each f ∈ G0 the function uf : g 7→ ‖f − g‖L2 belongs to Dom(E) and Γ(uf , uf ) ≤ 1
Q0-a.e. on G0.
(iv) (E,Dom(E)) is strongly local.
Proof. (a) Claim: For each f ∈ L2([0, 1],Leb) the function uf : g 7→ 〈f, g〉L2 belongs to Dom(E)
and E(uf , uf ) = ‖f‖2L2 .
Indeed, if f ∈ L2 ∩ C1 then f = c0 + c1f1 + c2f2 with f1, f2 ∈ G0 and c0, c1, c2 ∈ R. Hence,
uf ∈ Dom(E) according to Lemma 7.6 and E(uf , uf ) =
‖Duf‖2dQ0 = ‖f‖2 according to
Lemma 7.7. Finally, each f ∈ L2 can be approximated by fn ∈ L2 ∩ C1 with ‖f − fn‖ → 0.
Hence, uf ∈ Dom(E) and E(uf , uf ) = ‖f‖2.
(b) Claim: C1(G0) ⊂ Dom(E).
Let u ∈ C1(G0) be given with u(g) = U(〈~f , g〉), U ∈ C1(Rm,R), ~f = (f1, . . . , fm) ∈ L2([0, 1],Rm).
For each i = 1, . . . ,m let (wi,n)n∈N be an approximating sequence in (Z
1(G0), (E + ‖.‖2)1/2) for
wi : g 7→ 〈fi, g〉. Put un(g) = U(w1,n(g), . . . , wm,n(g)). Then un ∈ Z1(G0), un → u pointwise on
G0 and in L2(G0,Q0). Moreover,
E(un, un) =
∂iU(w1,n(g), . . . , wm,n(g))Dwi,n(g)‖2L2 dQ0(g)
∂iU(〈f1, g〉, . . . , 〈fm, g〉)Dwi(g)‖2L2 dQ0(g)
‖Du(g)‖2dQ0(g).
Hence, u ∈ Dom(E) and E(u, u) =
‖Du(g)‖2dQ0(g).
(c) Assertion (ii) then follows via polarization and bi-linearity. Assertion (iii) is an immediate
consequence of assertion (ii). Assertion (iii) allows to prove the locality of the Dirichlet form
(E,Dom(E)) in the same manner as in the proof of Theorem 6.2.
(d) Claim: C1(G0) is dense in Dom(E).
We have to prove that each u ∈ Z1(G0) can be approximated by un ∈ C1(G0). As usual, it suffices
to treat the particular case u(g) =
α(gt)dt for some α ∈ C1([0, 1]). Put Un(x1, . . . , xn) =
i=1 α(xi) and fn,i(t) = n · 1[ i−1
(t). Then
un(g) := Un(〈fn,1, g〉, . . . 〈fn,n, g〉) =
defines a sequence in C1(G0) with un(g) → u(g) pointwise on G0 and in L2(G0,Q0).
Moreover,
Dun(g) =
· 1[ i−1
[(.) (7.7)
and therefore
E(un) =
dQ0(g) −→
α′(gt)
2dtdQ0(g) = E(u). (7.8)
Thus (un)n is Cauchy in Dom(E) and un → u in Dom(E).
7.3 Rademacher Property and Intrinsic Metric
We say that a function u : G0 → R is 1-Lipschitz if
|u(g)− u(h)| ≤ ‖g − h‖L2 (∀g, h ∈ G0).
Theorem 7.9. Every 1-Lipschitz function u on G0 belongs to Dom(E) and Γ(u, u) ≤ 1 Q0-a.e.
on G0.
Before proving the theorem in full generality, let us first consider the following particular case.
Lemma 7.10. Given n ∈ N, let {h1, . . . , hn} be a orthonormal system in L2([0, 1],Leb) and let
U be a 1-Lipschitz function on Rn. Then the function u(g) = U(〈h1, g〉, . . . , 〈hn, g〉) belongs to
Dom(E) and Γ(u, u) ≤ 1 Q0-a.e. on G0.
Proof. Let us first assume that in addition U is C1. Then according to Theorem 7.8, u is in
Dom(E) and Du(g) =
i=1 ∂iU(〈~h, g〉) · hi. Thus
Γ(u, u)(g) = ‖Du(g)‖L2 =
|∂iU(〈~h, g〉)|2 ≤ 1.
In the case of a general 1-Lipschitz continuous U on Rn we choose an approximating sequence
of 1-Lipschitz functions Uk, k ∈ N, in C1(Rn) with Uk → U uniformly on Rn and put uk(g) =
Uk((〈~h, g〉) for g ∈ G0. Then uk → u pointwise and in L2(G0,Q0). Hence, u ∈ Dom(E) and
Γ(u, u) ≤ 1 Q0-a.e. on G0.
Proof of Theorem 7.9. Every 1-Lipschitz function u on G0 can be extended to a 1-Lipschitz
function ũ on L2([0, 1],Leb) (’Kirszbraun extension’). Hence, without restriction, assume that
u is a 1-Lipschitz function on L2([0, 1],Leb). Choose a complete orthonormal system {hi}i∈N of
the separable Hilbert space L2([0, 1],Leb) and define for each n ∈ N the function Un : Rn → R
Un(x1, . . . , xn) = u
for x = (x1, . . . , xn) ∈ Rn. This function Un is 1-Lipschitz on Rn:
|Un(x)− Un(y)| ≤
xihi −
≤ |x− y|.
Hence, according to the previous Lemma the function
un(g) = Un(〈h1, g〉, . . . , 〈hn, g〉)
belongs belongs to Dom(E) and Γ(un, un) ≤ 1 Q0-a.e. on G0.
Note that
un(g) = u
〈hi, g〉hi
for each g ∈ L2([0, 1],Leb). Therefore, un → u on L2([0, 1],Leb) since
i=1〈hi, g〉hi → g
on L2([0, 1],Leb) and since u is continuous on L2([0, 1],Leb). Thus, finally, u ∈ Dom(E) and
Γ(u, u) ≤ 1 Q0-a.e. on G0.
Our next goal is the converse to the previous Theorem.
Theorem 7.11. Every continuous function u ∈ Dom(E) with Γ(u, u) ≤ 1 Q0-a.e. on G0 is
1-Lipschitz on G0.
Lemma 7.12. For each u ∈ C1(G0) ∪ Z1(G0) and all g0, g1 ∈ G0
u(g1)− u(g0) =
〈Du ((1− t)g0 + tg1) , g1 − g0〉L2dt. (7.9)
Proof. Put gt = (1− t)g0 + tg1 and consider the C1 function η : [0, 1] → R defined by ηt = u(gt).
η̇t = Dg1−g0u(gt) = 〈Du(gt), g1 − g0〉
and thus
η1 − η0 =
η̇tdt =
〈Du(gt), g1 − g0〉dt.
Lemma 7.13. Let g0, g1 ∈ G0 ∩C3 and put gt = (1− t)g0 + tg1. Then for each u ∈ Dom(E) and
each bounded measurable Ψ : G0 → R
[u(g1 ◦ h)− u(g0 ◦ h)]Ψ(h) dQ0(h) =
〈Du(gt ◦ h, (g1 − g0) ◦ h〉Ψ(h)Q0(h)dt. (7.10)
Proof. Given g0, g1, Ψ and u ∈ Dom(E) as above, choose an approximating sequence in Z1(G0)∪
C1(G0) with un → u in Dom(E) as n→ ∞. According to the previous Lemma for each n
[un(g1◦h)−un(g0◦h)]Ψ(h) dQ0(h) =
〈Dun (gt ◦ h) , (g1−g0)◦h〉Ψ(h) dQ0(h)dt. (7.11)
By assumption un → u in L2(G0,Q0) and Dun → Du in L2(G0 × [0, 1],Q0 ⊗ Leb) as n → ∞.
Using the quasi-invariance of Q0 (Theorem 4.3) this implies
|u(gt ◦ h)− un(gt ◦ h)|Ψ(h) dQ0(h) =
[u(h)− un(h)|Ψ(g−1t ◦ h) · Y
(h) dQ0(h) → 0
as n→ ∞ as well as
‖Du(gt ◦ h)− Dun(gt ◦ h)|2L2Ψ(h)Q0(h)
‖Du(h) − Dun(h)|2L2Ψ(g
t ◦ h) · Y
(h)Q0(h) → 0
Hence, we may pass to the limit n→ ∞ in (7.11) which yields the claim.
Proof of Theorem 7.11. Let a continuous u ∈ Dom(E) be given with Γ(u, u) ≤ 1 Q0-a.e. on G0.
We want to prove that u(g1)− u(g0) ≤ ‖g1 − g0‖L2 for all g0, g1 ∈ G0. By density of G0 ∩ C3 in
G0 and by continuity of u it suffices to prove the claim for g0, g1 ∈ G0 ∩ C3.
Choose a sequence of bounded measurable Ψk : G0 → R+ such that the probability measures
ΨkdQ0 on G0 converge weakly to δe, the Dirac mass in the identity map e ∈ G0. Then according
to the previous Lemma and the assumption ‖Du‖ ≤ 1
[u(g1 ◦ h)− u(g0 ◦ h)]Ψk((h)dQ0(h)
〈Du(gt ◦ h, (g1 − g0) ◦ h〉Ψk(h) dQ0(h)dt
‖Du(gt ◦ h)‖L2 · ‖(g1 − g0) ◦ h‖L2 ·Ψk(h) dQ0(h)dt
‖(g1 − g0) ◦ h‖L2 ·Ψk(h) dQ0(h).
Now the integrands on both sides, h 7→ u(g1 ◦h)−u(g0 ◦h) as well as h 7→ ‖(g1 − g0) ◦h‖L2 , are
continuous in h ∈ G0. Hence, as k → ∞ by weak convergence ΨkdQ0 → δe we obtain
u(g1)− u(g0) ≤ ‖g1 − g0‖L2 .
Corollary 7.14. The intrinsic metric for the Dirichlet form (E,Dom(E)) is the L2-metric:
‖g1 − g0‖L2 = sup {u(g1)− u(g0) : u ∈ C(G0) ∩Dom(E), Γ(u, u) ≤ 1Q0-a.e. on G0}
for all g0, g1 ∈ G0.
7.4 Finite Dimensional Noise Approximations
The goal of this section is to present representations – and finite dimensional approximations –
of the Dirichlet form
E(u, v) =
〈Du(g),Dv(g)〉L2 dQ0(g)
in terms of globally defined vector fields.
If (ϕi)i∈N is a complete orthonormal system in Tg = L
2([0, 1], g∗Leb) for a given g ∈ G0 then
obviously
〈Du(g),Dv(g)〉L2 =
Dϕiu(g)Dϕiv(g). (7.12)
Unfortunately, however, there exists no family (ϕi)i∈N which is simultaneously orthonormal in
all Tg = L
2([0, 1], g∗Leb), g ∈ G0. For a general family, the representation (7.12) should be
replaced by
〈Du(g),Dv(g)〉L2 =
i,j=1
Dϕiu(g) · aij(g) ·Dϕjv(g) (7.13)
where a(g) = (aij(g))i,j∈N is the ’generalized inverse’ to Φ(g) = (Φij(g))i,j∈N with
Φij(g) := 〈ϕi, ϕj〉Tg =
ϕi(gt)ϕj(gt)dt.
In order to make these concepts rigorous, we have to introduce some notations.
For fixed n ∈ N let S+(n) ⊂ Rn×n denote the set of symmetric nonnegative definite real (n×n)-
matrices. For each A ∈ S+(n) a unique element A−1 ∈ S+(n), called generalized inverse to A,
is defined by
A−1x :=
0 if x ∈ Ker(A),
y if x ∈ Ran(A) with x = Ay
This definition makes sense since (by the symmetry of A) we have an orthogonal decomposition
Rn = Ker(A)⊕ Ran(A). Obviously,
A−1 ·A = A · A−1 = πA
where πA denotes the projection onto Ran(A).
Moreover, for each A ∈ S+(n) there exists a unique element A1/2 ∈ S+(n), called nonnegative
square root of A, satisfying
A1/2 · A1/2 = A.
Let Ψ(n) denote the map A 7→ A−1, regarded as a map from S+(n) ⊂ Rn×n to Rn×n, with
ij (A) = (A
−1)ij for i, j = 1, . . . , n. Similarly, put
Ξ(n) : S+(n) → S+(n), A 7→ (A1/2)−1 = (A−1)1/2.
Note that Ψ(n)(A) = Ξ(n)(A) · Ξ(n)(A) for all A ∈ S+(n).
The maps Ψ(n) and Ξ(n) are smooth on the subset of positive definite matrices A ∈ S+(n) but
unfortunately not on the whole set S+(n). However, they can be approximated from below
(in the sense of quadratic forms) by smooth maps: there exists a sequence of C∞ maps Ξ(n,l) :
Rn×n → Rn×n with
ξ · Ξ(n,k)(A) · ξ ≤ ξ · Ξ(n,l)(A) · ξ
for all A ∈ S+(n), ξ ∈ Rn and all k, l ∈ N with k ≤ l and
(n,l)
ij (A) → Ξ
ij (A) = (A
−1/2)ij
for all A ∈ S+(n), i, j ∈ {1, . . . , n} as l → ∞. Put Ψ(n,l)(A) = Ξ(n,l)(A) ·Ξ(n,l)(A) for A ∈ Rn×n.
Then the sequence (Ψ(n,l))l∈N approximates Ψ
(n) from below in the sense of quadratic forms.
Now let us choose a family {ϕi}i∈N of smooth functions ϕi : [0, 1] → R which is total in C0([0, 1])
w.r.t. uniform convergence (i.e. its linear hull is dense). Put
Φij(g) := 〈ϕi, ϕj〉Tg =
ϕi(gx)ϕj(gx)dx
(n,l)
ij (g) = Ψ
(n,l)
ij (Φ(g)) , σ
(n,l)
ij (g) = Ξ
(n,l)
ij (Φ(g)).
Note that the maps g 7→ a(n,l)ij (g) and g 7→ σ
(n,l)
ij (g) (for each choice of n, l, i, j) belong to the
class Z∞(G0). Moreover, put
ij (g) = Ψ
ij (Φ(g)) .
Then obviously the orthogonal projection πn onto the linear span of {ϕ1, . . . , ϕn} ⊂ Tg =
L2([0, 1], g∗Leb) is given by
πnu =
i,j=1
ij (g) · 〈u, ϕi〉Tg · ϕj
〈πnu, πnv〉Tg =
i,j=1
〈u, ϕi〉Tg · a
ij (g) · 〈v, ϕj〉Tg
for all u, v ∈ Tg.
Theorem 7.15. (i) For each n, l ∈ N the form (E(n,l),Z1(G0)) with
(n,l)(u, v) =
i,j=1
Dϕiu(g) · a
(n,l)
ij (g) ·Dϕjv(g) dQ0(g)
is closable. Its closure is a Dirichlet form with generator being the Friedrichs extension of the
symmetric operator (L(n,l),Z2(G0)) given by
(n,l) =
i,j=1
(n,l)
ij ·DϕiDϕj +
i,j=1
(n,l)
ij + a
(n,l)
ij · V
Dϕj . (7.14)
(ii) As l → ∞
(n,l) ր E(n)
where
(n)(u, v) =
i,j=1
Dϕiu(g) · a
ij (g) ·Dϕjv(g) dQ0(g).
for u, v ∈ Z1(G0). Hence, in particular, E(n) is a Dirichlet form.
(iii) As n→ ∞
(n) ր E
(which provides an alternative proof for the closability of the form (E,Z1(G0))).
Proof. (i) The function a
(n,l)
i,j on G0 is a cylinder function in the class Z1(G0). The integration
by parts formula for the Dϕi , therefore, implies that for all u, v ∈ Z2(G0)
(n,l)(u, v) =
Dϕiu(g)Dϕjv(g)a
(n,l)
ij (g)dQ0(g)
u(g) ·D∗ϕi
(n,l)
ij Dϕjv
(g) dQ0(g) = −
u(g) · L(n,l)v(g) dQ0(g).
(n,l) = −
i,j=1
(n,l)
ij Dϕj
Hence, (E(n,l),Z2(G0)) is closable and the generator of its closure is the Friedrichs extension of
(L(n,l),Z2(G0)).
(ii) The monotone convergence E(n,l) ր E(n) of the quadratic forms is an immediate consequence
of the fact that a(n,l)(g) ր a(n)(g) (in the sense of symmetric matrices) for each g ∈ G0 which in
turn follows from the defining properties of the approximations Ψ(n,l) of the generalized inverse
Ψ(n).
The limit of an increasing sequence of Dirichlet forms is itself again a Dirichlet form provided it
is densely defined which in our case is guaranteed since it is finite on Z2(G0).
(iii) Obviously, the En, n ∈ N constitute an increasing sequence of Dirichlet forms with En ≤ E
for all n. Moreover, Z1(G0) is a core for all the forms under consideration. Hence, it suffices to
prove that for each u ∈ Z1(G0) and each ǫ > 0 there exists an n ∈ N such that
(n)(u, u)− E(u, u)
To simplify notation, assume that u is of the form u(g) = U(
α(gt)dt) for some U ∈ C1c (R)
and some α ∈ C1([0, 1]). By assumption, the set {ϕi, i ∈ N} is total in C0([0, 1]) w.r.t. uniform
convergence. Hence, for each δ > 0 there exist n ∈ N and ϕ ∈ span(ϕ1, . . . , ϕn) with ‖α′−ϕ‖sup ≤
δ which implies
〈α′, ϕ〉Tg
‖ϕ‖Tg
≥ ‖ϕ‖Tg − δ ≥ ‖α′‖Tg − 2δ.
E(u, u) ≥ E(n)(u, u) ≥
α(gt)dt)
2 · 〈α′, ϕ〉2Tg ·
‖ϕ‖2Tg
dQ0(g)
α(gt)dt)
‖α′‖Tg − 2δ
dQ0(g)
α(gt)dt)
1 + δ
‖α′‖2Tg − 4δ
dQ0(g)
1 + δ
E(u, u)− 4δ‖U ′‖2sup.
Hence, for δ sufficiently small, E(u, u) and E(n)(u, u) are arbitrarily close to each other.
Remark 7.16. For any given g0 ∈ G0, let (gt)t≥0 with gt : (x, ω) 7→ gxt (ω) be the solution to the
dgxt =
i,j=1
(n,l)
ij (gt) · ϕj(g
t ) dW
i,j=1
(n,l)
ij (gt) · ϕj(g
t ) ·
ϕ′i(g
t ) + V
i,j=1
k,m=1
(n,l)
ij (Φ(gt)) · 〈(ϕkϕm)
′, ϕi〉Tg · ϕj(gxt )dt
where ∂kmΨ
(n,l)
ij for (k,m) ∈ {1, . . . , n}2 denotes the 1st order partial derivative of the function
(n,l)
ij : R
n×n → R with respect to the coordinate xkm. Then the generator of the process coincides
on Z2(G0) with the operator 12L
(n,l) from (7.14), the generator of the Dirichlet form E(n,l).
Let us briefly comment on the various terms in the SDE from above:
• The first one,
i,j=1 σ
(n,l)
ij (gt) · ϕj(gxt ) dW it is the diffusion term, written in Ito form;
• the second one, 1
i,j=1 a
(n,l)
ij (gt) ·ϕj(gxt ) ·ϕ′i(gxt )dt is a drift which comes from the trans-
formation between Stratonovich and Ito form (it would disappear if we wrote the diffusion
term in Stratonovich form).
• The next one, 1
i,j=1 a
(n,l)
ij (gt) ·ϕj(gxt ) · V
ϕi(gt)dt is a drift which arises from our change
of variable formula. Actually, since
V βϕi(g) = β
ϕ′i(g(y))dy +
ϕ′i(g(a+)) + ϕ
i(g(a−))
− ϕi(g(a+)) − ϕi(g(a−))
g(a+)− g(a−)
it consists of two parts, one originates in the logarithmic derivative of the entropy of the
g’s (which finally will force the process to evolve as a stochastic perturbation of the heat
equation), the other one is created by the jumps of the g’s.
• The last term, 1
i,j=1
k,m=1 ∂kmΨ
(n,l)
ij (Φ(gt)) · 〈(ϕkϕm)′, ϕi〉Tg · ϕj(gxt )dt involves the
derivative of the diffusion matrix. It arises from the fact that the generator is originally
given in divergence form.
7.5 The Wasserstein Diffusion (µt) on P0
The objects considered previously – derivative, Dirichlet form and Markov process on G0 – have
canonical counterparts on P0. The key to these objects is the bijective map χ : G0 → P0,
g 7→ g∗Leb.
We denote by Zk(P0) the set of all (’cylinder’) functions u : P0 → R which can be written as
u(µ) = U
α1dµ, . . . ,
(7.15)
with some m ∈ N, some U ∈ Ck(Rm) and some ~α = (α1, . . . , αm) ∈ Ck([0, 1],Rm) . The subset
of u ∈ Zk(P0) with α′i(0) = α′i(1) = 0 for all i = 1, . . . ,m will be denoted by Zk0(P0). For
u ∈ Z1(P0) represented as above we define its gradient Du(µ) ∈ L2([0, 1], µ) by
Du(µ) =
~αdµ) · α′i(.)
with norm
‖Du(µ)‖L2(µ) =
~αdµ) · α′i
The tangent space at a given point µ ∈ P0 can be identified with L2([0, 1], µ). The action of a
tangent vector ϕ ∈ L2([0, 1], µ) on µ (’exponential map’) is given by the push forward ϕ∗µ.
Theorem 7.17. (i) The image of the Dirichlet form defined in (7.2) under the map χ is the
regular, strongly local, recurrent Wasserstein Dirichlet form E on L2(P0,P0) defined on its core
Z1(P0) by
E(u, v) =
〈Du(µ),Dv(µ)〉2L2(µ)dP0(µ). (7.16)
The Dirichlet form has a square field operator, defined on Dom(E) ∩ L∞, and given on Z1(P0)
Γ(u, v)(µ) = 〈Du(µ),Dv(µ)〉2L2(µ).
The intrinsic metric for the Dirichlet form is the L2-Wasserstein distance dW . More precisely,
a continuous function u : P0 → R is 1-Lipschitz w.r.t. the L2-Wasserstein distance if and only
if it belongs to Dom(E) and Γ(u, u)(µ) ≤ 1 for P0-a.e. µ ∈ P0.
(ii) The generator of the Dirichlet form is the Friedrichs extension of the symmetric operator
(L,Z20(P0) on L2(P0,P0) given as L = L1 + L2 + β · L3 with
L1u(µ) =
i,j=1
∂i∂jU(
~αdµ) ·
L2u(µ) =
~αdµ) ·
I∈gaps(µ)
α′′i (I−) + α
i (I+)
i(I+)− α′i(I−)
i (0) + α
i (1)
L3u(µ) =
~αdµ) ·
α′′i dµ.
Recall that gaps(µ) denotes the set of intervals I = ]I−, I+[⊂ [0, 1] of maximal length with
µ(I) = 0 and |I| denotes the length of such an interval.
(iii) For P0-a.e. µ0 ∈ P0, the associated Markov process (µt)t≥0 on P0 starting in µ0, called
Wasserstein diffusion, with generator 1
L is given as
µt(ω) = gt(ω)∗Leb
where (gt)t≥0 is the Markov process on G0 associated with the Dirichlet form of Theorem 7.5,
starting in g0 := χ
−1(µ0).
For each u ∈ Z20(P0) the process
u(µt)− u(µ0)−
Lu(µs)ds
is a martingale whenever the distribution of µ0 is chosen to be absolutely continuous w.r.t. the
entropic measure P0. Its quadratic variation process is
Γ(u, u)(µs)ds.
Remark 7.18. L1 is the second order part (’diffusion part’) of the generator L, L2 and L3
are first order operators (’drift parts’). The operator L1 describes the diffusion on P0 in all
directions of the respective tangent spaces. This means that the process (µt) at each time t ≥ 0
experiences the full ’tangential’ L2([0, 1], µt)-noise.
L3 is the generator of the deterministic semigroup (’Neumann heat flow’) (Ht)t≥0 on L
2(P0,P0)
given by
Htu(µ) = u(htµ).
Here ht is the heat kernel on [0, 1] with reflecting (’Neumann’) boundary conditions and htµ(.) =
ht(., y)µ(dy). Indeed, for each u ∈ Z10(P0) given as u(g) = U(
~αdµ) we obtain Htu(µ) =
~α(x)ht(x, y)µ(dy)dx
and thus
∂tHtu(µ) =
∂iU(htµ) · ∂t
αi(x)ht(x, y)µ(dy)dx
∂iU(htµ) ·
αi(x)h
t (x, y)µ(dy)dx
∂iU(htµ) ·
α′′i (x)ht(x, y)µ(dy)dx = L3Htu(µ).
Note that L depends on β only via the drift term L3 and
L → L3 as β → ∞.
The following statement, which in the finite dimensional case is known as Varadhan’s formula,
exhibits another close relationship between (µt) and the geometry of (P([0, 1]), dW ). The Gaus-
sian short time asymptotics of the process (µt)t≥0 are governed by the L
2-Wasserstein distance.
Corollary 7.19. For measurable sets A,B ∈ P0 with positive P0-measure, let dW (A,B) =
inf{dW (ν, ν̃) | ν ∈ A, ν̃ ∈ B} and pt(A,B) =
pt(ν, dν̃)P0(dν) where pt(ν, dν̃) denotes the
transition semigroup for the process (µt)t≥0.
t log pt(A,B) = −
dW (A,B)
. (7.17)
Proof. This type of result is known as Varadhan’s formula. Its respective form for (E,Dom(E) on
L2(P0,P0) holds true by the very general results of [HR03] for conservative symmetric diffusions,
and the identification of the intrinsic metric as dW in our previous Theorem.
Due to the sample path continuity of (µt) the Wasserstein diffusion is equivalently characterized
by the following martingale problem. Here we use the notation 〈α, µt〉 =
α(x)µt(dx).
Corollary 7.20. For each α ∈ C2([0, 1]) with α′(0) = α′(1) = 0 the process
Mt = 〈α, µt〉 −
〈α′′, µs〉ds
I∈gaps(µs)
α′′(I−) + α
′′(I+)
′(I+)− α′(I−)
′′(0) + α′′(1)
is a continuous martingale with quadratic variation process
[M ]t =
〈(α′)2, µs〉ds.
Remark 7.21. For illustration one may compare corollary 7.20 for (µt) in the case β = 1 to
the respective martingale problems for four other well-known measure valued process, say on
the real line, namely the so-called super-Brownian motion or Dawson-Watanabe process (µDWt ),
the Fleming-Viot process (µFW ), both of which we can consider with the Laplacian as drift, the
Dobrushin-Doob process (µDDt ) which is the empirical measure of independent Brownian motions
with locally finite Poissonian starting distribution, cf. [AKR98], and finally simply the empirical
measure process of a single Brownian motion (µBMt = δXt). For each i ∈ {DW,FV,DD,BM}
and sufficiently regular α : R → R the process M it := 〈α, µit〉 − 12
〈α′′, µis〉ds is a continuous
martingale with quadratic variation process
[MDW ]t =
〈α2, µDWs 〉ds,
[MFV ]t =
[〈α2, µFVs 〉 − (〈α, µFVs 〉)2]ds,
[MDD]t =
〈(α′)2, µDDs 〉ds,
[MBM ]t =
〈(α′)2, µBMs 〉ds.
In view of corollary 7.19 the apparent similarity of µDD and µBM to the Wasserstein diffusion
µ is no suprise. However, the effective state spaces of µDD, µBM and µt are as much different
as their invariant measures.
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	Introduction
	Spaces of Probability Measures and Monotone Maps
	The Spaces P0=P([0,1]) and G0
	The Spaces G, G1 and P=P(S1)
	Dirichlet Process and Entropic Measure
	Gibbsean Interpretation and Heuristic Derivation of the Entropic Measure
	The Measures Q and P 
	The Measures Q0 and P0 
	The Dirichlet Process as Normalized Gamma Process
	Support Properties
	Scaling and Invariance Properties
	Dirichlet Processes on General Measurable Spaces
	The Change of Variable Formula for the Dirichlet Process and for the Entropic Measure
	Heuristic Approaches to Change of Variable Formulae
	The Change of Variables Formula on the Sphere
	The Change of Variables Formula on the Interval
	Proofs for the Sphere Case
	Proof for the Interval Case
	The Integration by Parts Formula
	The Drift Term
	Directional Derivatives
	Integration by Parts Formula on P(S1)
	Derivatives and Integration by Parts Formula on P([0,1])
	Dirichlet Form and Stochastic Dynamics on on G
	The Dirichlet Form on G
	Finite Dimensional Noise Approximations
	Dirichlet Form and Stochastic Dynamics on G1 and P
	Dirichlet Form and Stochastic Dynamics on G0 and P0
	The Canonical Dirichlet Form on the Wasserstein Space
	Tangent Spaces and Gradients
	The Dirichlet Form
	Rademacher Property and Intrinsic Metric
	Finite Dimensional Noise Approximations
	The Wasserstein Diffusion (t) on P0

  We construct a new random probability measure on the sphere and on the unit
interval which in both cases has a Gibbs structure with the relative entropy
functional as Hamiltonian. It satisfies a quasi-invariance formula with respect
to the action of smooth diffeomorphism of the sphere and the interval
respectively. The associated integration by parts formula is used to construct
two classes of diffusion processes on probability measures (on the sphere or
the unit interval) by Dirichlet form methods. The first one is closely related
to Malliavin's Brownian motion on the homeomorphism group. The second one is a
probability valued stochastic perturbation of the heat flow, whose intrinsic
metric is the quadratic Wasserstein distance. It may be regarded as the
canonical diffusion process on the Wasserstein space.

Introduction
(a) Equipped with the L2-Wasserstein distance dW (cf. (2.1)), the space P(M) of probability
measures on an Euclidean or Riemannian space M is itself a rich object of geometric interest.
Due to the fundamental works of Y. Brenier, R. McCann, F. Otto, C. Villani and many others
(see e.g. [Bre91, McC97, CEMS01, Ott01, OV00, Vil03]) there are well understood and pow-
erful concepts of geodesics, exponential maps, tangent spaces TµP(M) and gradients Du(µ) of
functions on this space. In a certain sense, P(M) can be regarded as an infinite dimensional
Riemannian manifold, or at least as an infinite dimensional Alexandrov space with nonnegative
lower curvature bound if the base manifold (M,d) has nonnegative sectional curvature.
A central role is played by the relative entropy : P(M) → R ∪ {+∞} with respect to the
Riemannian volume measure dx on M
Ent(µ) =
ρ log ρ dx, if dµ(x) ≪ dx with ρ(x) = dµ(x)
+∞, else.
The relative entropy as a function on the geodesic space (P(M), dW ) is K-convex for a given
number K ∈ R if and only if the Ricci curvature of the underlying manifold M is bounded from
below by K, [vRS05, Stu06]. The gradient flow for the relative entropy in the geodesic space
(P(M), dW ) is given by the heat equation ∂∂tµ = ∆µ on M , [JKO98]. More generally, a large
class of evolution equations can be treated as gradient flows for suitable free energy functionals
S : P(M) → R, [Vil03].
What is missing until now, is a natural ’Riemannian volume measure’ P on P(M). The basic re-
quirement will be an integration by parts formula for the gradient. This will imply the closability
of the pre-Dirichlet form
E(u, v) =
〈Du(µ),Dv(µ)〉Tµ dP(µ)
in L2(P(M),P), – which in turn will be the key tool in order to develop an analytic and stochastic
calculus on P(M). In particular, it will allow us to construct a kind of Laplacian and a kind
of Brownian motion on P(M). Among others, we intend to use the powerful machinery of
Dirichlet forms to study stochastically perturbed gradient flows on P(M) which – on the level
http://arxiv.org/abs/0704.0704v1
of the underlying spaces M – will lead to a new concept of SPDEs (preserving probability by
construction).
Instead of constructing a ’uniform distribution’ P on P(M), for various reasons, we prefer to
construct a probability measure Pβ on P(M) formally given as
dPβ(µ) =
e−β·Ent(µ) dP(µ) (1.1)
for β > 0 and some normalization constant Zβ. (In the language of statistical mechanics, β is
the ’inverse temperature’ and Zβ the ’partition function’ whereas the entropy plays the role of
a Hamiltonian.)
(b) One of the basic results of this paper is the rigorous construction of such a entropic measure
Pβ in the one-dimensional case, i.e. M = S1 or M = [0, 1]. We will essentially make use of the
representation of probability measures by their inverse distributions function gµ. It allows to
transfer the problem of constructing a measure Pβ on the space of probability measures P([0, 1])
(or P(S1)) into the problem of constructing a measure Qβ0 (or Qβ) on the space G0 (or G, resp.)
of nondecreasing functions from [0, 1] (or S1, resp.) into itself.
In terms of the measure Q
0 on G0, for instance, the formal characterization (1.1) then reads as
follows
0 (g) =
e−β·S(g) dQ0(g). (1.2)
Here Q0 denotes some ’uniform distribution’ on G0 ⊂ L2([0, 1]) and S : G0 → [0,∞] is the
entropy functional
S(g) := Ent(g∗Leb) = −
log g′(t) dt.
This representation is reminiscent of Feynman’s heuristic picture of the Wiener measure, — now
with the energy
H(g) =
g′(t)2dt
of a path replaced by its entropy. Q
0 will turn out to be (the law of) the Dirichlet process or
normalized Gamma process.
(c) The key result here is the quasi-invariance – or in other words a change of variable formula –
for the measure Pβ (or P
0 ) under push-forwards µ 7→ h∗µ by means of smooth diffeomorphisms
h of S1 (or [0, 1], resp.). This is equivalent to the quasi-invariance of the measure Qβ under
translations g 7→ h ◦ g of the semigroup G by smooth h ∈ G. The density
dPβ(h∗µ)
dPβ(µ)
· Y 0h (µ)
consists of two terms. The first one
h (µ) = exp
log h′(t)dµ(t)
can be interpreted as exp(−βEnt(h∗µ))/ exp(−βEnt(µ)) in accordance with our formal inter-
pretation (1.1). The second one
Y 0h (µ) =
I∈gaps(µ)
h′(I−) · h′(I+)
|h(I)|/|I|
can be interpreted as the change of variable formula for the (non-existing) measure P. Here
gaps(µ) denotes the set of intervals I =]I−, I+[⊂ S1 of maximal length with µ(I) = 0. Note that
Pβ is concentrated on the set of µ which have no atoms and not absolutely continuous parts and
whose supports have Lebesgue measure 0.
(d) The tangent space at a given point µ in P = P(S1) (or in P0 = P([0, 1])) will be an
appropriate completion of the space C∞(S1,R) (or C∞([0, 1],R), resp.). The action of a tangent
vector ϕ on µ (’exponential map’) is given by the push forward ϕ∗µ. This leads to the notion
of the directional derivative
Dϕu(µ) = lim
[u((Id+ tϕ)∗µ)− u(µ)]
for functions u : P → R. The quasi-invariance of the measure Pβ implies an integration by parts
formula (and thus the closability)
D∗ϕu = −Dϕu− Vϕ · u
with drift Vϕ = limt→0
Id+tϕ
− 1).
The subsequent construction will strongly depend on the choice of the norm on the tangent
spaces TµP. Basically, we will encounter two important cases.
(e) Choosing TµP = Hs(S1,Leb) for some s > 1/2 — independent of µ — leads to a regular,
local, recurrent Dirichlet form E on L2(P,Pβ) by
E(u, u) =
|Dϕku(µ)|2 dPβ(µ).
where {ϕk}k∈N denotes some complete orthonormal system in the Sobolev space Hs(S1). Ac-
cording to the theory of Dirichlet forms on locally compact spaces [FOT94], this form is associ-
ated with a continuous Markov process on P(S1) which is reversible with respect to the measure
Pβ. Its generator is given by
DϕkDϕk +
Vϕk ·Dϕk . (1.3)
This process (gt)t≥0 is closely related to the stochastic processes on the diffeomorphism group
of S1 and to the ’Brownian motion’ on the homeomorphism group of S1, studied by Airault,
Fang, Malliavin, Ren, Thalmaier and others [AMT04, AM06, AR02, Fan02, Fan04, Mal99].
These are processes with generator 1
kDϕkDϕk . Hence, one advantage of our approach is to
identify a probability measure Pβ such that these processes — after adding a suitable drift —
are reversible.
Moreover, previous approaches are restricted to s ≥ 3/2 whereas our construction applies to all
cases s > 1/2.
(f) Choosing TµG = L2([0, 1], µ) leads to the Wasserstein Dirichlet form
E(u, u) =
‖Du(µ)‖2L2(µ) dP
0 (µ)
on L2(P0,Pβ0 ). Its square field operator is the squared norm of the Wasserstein gradient and its
intrinsic distance (which governs the short time asymptotic of the process) coincides with the
L2-Wasserstein metric. The associated continuous Markov process (µt)t≥0 on P([0, 1]), which we
shall call Wasserstein diffusion, is reversible w.r.t. the entropic measure P
0 . It can be regarded
as a stochastic perturbation of the Neumann heat flow on P([0, 1]) with small time Gaussian
behaviour measured in terms of kinetic energy.
2 Spaces of Probability Measures and Monotone Maps
The goal of this paper is to study stochastic dynamics on spaces P(M) in case M is the unit
interval [0, 1] or the unit circle S1.
2.1 The Spaces P0 = P([0, 1]) and G0
Let us collect some basic facts for the space P0 = P([0, 1]) of probability measures on the unit
interval [0, 1] the proofs of which can be found in the monograph [Vil03]. Equipped with the
L2-Wasserstein distance dW , it is a compact metric space. Recall that
dW (µ, ν) := inf
[0,1]2
|x− y|2γ(dx, dy)
, (2.1)
where the infimum is taken over all probability measures γ ∈ P([0, 1]2) having marginals µ and
ν (i.e. γ(A×M) = µ(A) and γ(M ×B) = ν(B) for all A,B ⊂M).
Let G0 denote the space of all right continuous nondecreasing maps g : [0, 1[→ [0, 1] equipped
with the L2-distance
‖g1 − g2‖L2 =
|g1(t)− g2(t)|2dt
Moreover, for notational convenience each g ∈ G0 is extended to the full interval [0, 1] by g(1) :=
1. The map
χ : G0 → P0, g 7→ g∗Leb
(= push forward of the Lebesgue measure on [0, 1] under the map g) establishes an isometry
between (G0, ‖.‖L2) and (P0, dW ). The inverse map χ−1 : P0 → G0, µ 7→ gµ assigns to each
probability measure µ ∈ P0 its inverse distribution function defined by
gµ(t) := inf{s ∈ [0, 1] : µ[0, s] > t} (2.2)
with inf ∅ := 1. In particular, for all µ, ν ∈ P0
dW (µ, ν) = ‖gµ − gν‖L2 . (2.3)
For each g ∈ G0 the generalized inverse g−1 ∈ G0 is defined by g−1(t) = inf{s ≥ 0 : g(s) > t}.
Obviously,
‖g1 − g2‖L1 = ‖g−11 − g
2 ‖L1 (2.4)
(being simply the area between the graphs) and (g−1)−1 = g. Moreover, g−1(g(t)) = t for all
t provided g−1 is continuous. (Note that under the measure Q
0 to be constructed below the
latter will be satisfied for a.e. g ∈ G0.)
On G0, there exist various canonical topologies: the L2-topology of G0 regarded as subset of
L2([0, 1],R); the image of the weak topology on P0 under the map χ−1 : µ 7→ gµ (= inverse
distribution function); the image of the weak topology on P0 under the map µ 7→ g−1µ (=
distribution function). All these – and several other – topologies coincide.
Proposition 2.1. For each sequence (gn)n ⊂ G0, each g ∈ G0 and each p ∈ [1,∞[ the following
are equivalent:
(i) gn(t) → g(t) for each t ∈ [0, 1] in which g is continuous;
(ii) gn → g in Lp([0, 1]);
(iii) g−1n → g−1 in Lp([0, 1]);
(iv) µgn → µg weakly;
(v) µgn → µg in dW .
In particular, G0 is compact.
Let us briefly sketch the main arguments of the
Proof. Since all the functions gn and g
n are bounded, properties (ii) and (iii) obviously are
independent of p. The equivalence of (ii) and (iii) for p = 1 was already stated in (2.4) and the
equivalence between (ii) for p = 2 and (v) was stated in (2.3). The equivalence of (iv) and (v)
is the well known fact that the Wasserstein distance metrizes the weak topology. Another well
known characterization of weak convergence states that (iv) is equivalent to (i’): g−1n (t) → g−1(t)
for each t ∈ [0, 1] in which g−1 is continuous. Finally, (i′) ⇔ (i) according to the equivalence
(ii) ⇔ (iii) which allows to pass from convergence of distribution functions g−1n to convergence
of inverse distribution functions gn. The last assertion follows from the compactness of P0 in
the weak topology.
2.2 The Spaces G, G1 and P = P(S1)
Throughout this paper, S1 = R/Z will always denote the circle of length 1. It inherits the group
operation + from R with neutral element 0. For each x, y ∈ S1 the positively oriented segment
from x to y will be denoted by [x, y] and its length by |[x, y]|. If no ambiguity is possible, the
latter will also be denoted by y − x. In contrast to that, |x − y| will denote the S1-distance
between x and y. Hence, in particular, |[y, x]| = 1 − |[x, y]| and |x − y| = min{|[y, x]|, |[x, y]|}.
A family of points t1, . . . , tN ∈ S1 is called an ’ordered family’ if
i=1 |[ti, ti+1]| = 1 with
tN+1 := t1 (or in other words if all the open segments ]ti, ti+1[ are disjoint).
G(R) = {g : R → R right continuous nondecreasing with g(x+ 1) = g(x) + 1 for all x ∈ R}.
Due to the required equivariance with respect to the group action of Z, each map g ∈ G(R)
induces uniquely a map π(g) : S1 → S1. Put G := π(G(R)). The monotonicity of the functions
in G(R) induces also a kind of monotonicity of maps in G: each continuous g ∈ G will be
order preserving and homotopic to the identity map. In the sequel, however, we often will have
to deal with discontinuous g ∈ G. The elements g ∈ G will be called monotone maps of S1.
G is a compact subspace of the L2-space of maps from S1 to S1 with metric ‖g1 − g2‖L2 =
|g1(t)− g2(t)|2dt
With the composition ◦ of maps, G is a semigroup. Its neutral element e is the identity map.
Of particular interest in the sequel will be the semigroup G1 = G/S1 where functions g, h ∈ G
will be identified if g(.) = h(.+ a) for some a ∈ S1.
Proposition 2.2. The map
χ : G1 → P, g 7→ g∗Leb
(= push forward of the Lebesgue measure on S1 under the map g) and its inverse χ−1 : P →
G1, µ 7→ gµ (with gµ as defined in (2.2)) establish an isometry between the space G1 equipped
with the induced L2-distance
‖g1 − g2‖G1 =
|g1(t)− g2(t+ s)|2dt
and the space P of probability measures on S1 equipped with the L2-Wasserstein distance. In
particular, G1 is compact.
Proof. The bijectivity of χ and χ−1 is clear. It remains to prove that
dW (µ, ν) = ‖gµ − gν‖G1 (2.5)
for all µ, ν ∈ P. Obviously, it suffices to prove this for all absolutely continuous µ, ν (or
equivalently for strictly increasing gµ, gν) since the latter are dense in P (or in G1, resp.). For
such a pair of measures, there exists a map F : S1 → S1 (’transport map’) which minimizes the
transportation costs [Vil03]. Fix any point in S1, say 0, and put s = F (0). Then the map F is
a transport map for the mass µ on the segment ]0, 1[ onto the mass ν on the segment ]s, s+ 1[.
Since these segments are isometric to the interval ]0, 1[, the results from the previous subsection
imply that the minimal cost for such a transport is given by
|g1(t) − g2(t + s)|2dt. Varying
over s finally proves the claim.
3 Dirichlet Process and Entropic Measure
3.1 Gibbsean Interpretation and Heuristic Derivation of the Entropic Mea-
One of the basic results of this paper is the rigorous construction of a measure Pβ formally given
as (1.1) in the one-dimensional case, i.e. M = S1 or M = [0, 1]. We will essentially make use
of the isometries χ : G1 → P = P(S1), g 7→ g∗Leb and χ : G0 → P0 = P([0, 1]). They allow to
transfer the problem of constructing measures Pβ on spaces of probability measures P (or P0)
into the problem of constructing measures Qβ (or Q
0 ) on spaces of functions G1 (or G0, resp.).
In terms of the measure Q
0 on G0, for instance, the formal characterization (1.1) then reads as
follows
0 (dg) =
e−β·S(g)Q0(dg). (3.1)
Here Q0 denotes some ’uniform distribution’ on G0 ⊂ L2([0, 1]) and S : G0 → [0,∞] is the
entropy functional S(g) := Ent(g∗Leb). If g is absolutely continuous then S(g) can be expressed
explicitly as
S(g) = −
log g′(t) dt.
The representation (3.1) is reminiscent of Feynman’s heuristic picture of the Wiener measure.
Let us briefly recall the latter and try to use it as a guideline for our construction of the measure
According to this heuristic picture, the Wiener measure Pβ with diffusion constant σ2 = 1/β
should be interpreted (and could be constructed) as
Pβ(dg) =
e−β·H(g)P(dg) (3.2)
with the energy functional H(g) = 1
g′(t)2dt. Here P(dg) is assumed to be the ’uniform
distribution’ on the space G∗ of all continuous paths g : [0, 1] → R with g(0) = 0. Even if
such a uniform distribution existed, typically almost all paths g would have infinite energy.
Nevertheless, one can overcome this difficulty as follows.
Given any finite partition {0 = t0 < t1 < · · · < tN = 1} of [0, 1], one should replace the energy
H(g) of the path g by the energy of the piecewise linear interpolation of g
HN (g) = inf {H(g̃) : g̃ ∈ G∗, g̃(ti) = g(ti) ∀i} =
|g(ti)− g(ti−1)|2
2(ti − ti−1)
Then (3.2) leads to the following explicit representation for the finite dimensional distributions
Pβ (gt1 ∈ dx1, . . . , gtN ∈ dxN ) =
|xi − xi−1|2
ti − ti−1
pN (dx1, . . . , xN ). (3.3)
Here pN (dx1, . . . , xN ) = P (gt1 ∈ dx1, . . . , gtN ∈ dxN ) should be a ’uniform distribution’ on RN
and Zβ,N a normalization constant. Choosing pN to be the N -dimensional Lebesgue measure
makes the RHS of (3.3) a projective family of probability measures. According to Kolmogorov’s
extension theorem this family has a unique projective limit, the Wiener measure Pβ on G∗ with
diffusion constant σ2 = 1/β.
Now let us try to follow this procedure with the entropy functional S(g) replacing the energy
functional H(g). Given any finite partition {0 = t0 < t1 < · · · < tN < tN+1 = 1} of [0, 1], we
will replace the entropy S(g) of the path g by the entropy of the piecewise linear interpolation
SN (g) = inf {S(g̃) : g̃ ∈ G0, g̃(ti) = g(ti) ∀i} = −
g(ti)− g(ti−1)
ti − ti−1
· (ti − ti−1).
This leads to the following expression for the finite dimensional distributions
0 (gt1 ∈ dx1, . . . , gtN ∈ dxN )
xi − xi−1
ti − ti−1
· (ti − ti−1)
qN(dx1 . . . dxN ) (3.4)
where qN (dx1, . . . , xN ) = Q0 (gt1 ∈ dx1, . . . , gtN ∈ dxN ) is a ’uniform distribution’ on the simplex
(x1, . . . , xN ) ∈ [0, 1]N : 0 < x1 < x2 . . . < xN < 1
and x0 := 0, xN+1 := 1.
What is a ’canonical’ candidate for qN? A natural requirement will be the invariance property
qN (dx1, . . . , dxN ) = [(Ξ
xi−1,xi+k)∗ qk(dxi, . . . , dxi+k−1)]
dqN−k(dx1, . . . , dxi−1, dxi+k, . . . , dxN ) (3.5)
for all 1 ≤ k ≤ N and all 1 ≤ i ≤ N − k + 1 with the convention x0 = 0, xN+1 = 1 and the
rescaling map Ξa,b : ]0, 1[k→ Rk, yj 7→ yj(b− a) + a for j = 1, · · · , k.
If the qN , N ∈ N, were probability measures then the invariance property admits the following
interpretation: under qN , the distribution of the (N − k)-tuple (x1, . . . , xi−1, xi+k, . . . , xN ) is
nothing but qN−k; and under qN , the distribution of the k-tuple (xi, . . . , xi+k−1) of points in
the interval ]xi−1, xk[ coincides — after rescaling of this interval — with qk. Unfortunately, no
family of probability measures qN , N ∈ N with property (3.5) exists. However, there is a family
of measures with this property.
By iteration of the invariance property (3.5), the choice of the measure q1 on the interval
Σ1 = ]0, 1[ will determine all the measures qN , N ∈ N. Moreover, applying (3.5) for N = 2,
k = 1 and both choices of i yields
(Ξ0,x1)∗ q1(dx2)
dq1(dx1) =
(Ξx2,1)∗ q1(dx1)
dq1(dx2) (3.6)
for all 0 < x1 < x2 < 1. This reflects the intuitive requirement that there should be no difference
whether we first choose randomly x1 ∈ ]0, 1[ and then x2 ∈ ]x1, 1[ or the other way round, first
x2 ∈ ]0, 1[ and then x1 ∈ ]0, x2[.
Lemma 3.1. A family of measures qN , N ∈ N, with continuous densities satisfies property (3.5)
if and only if
qN (dx1, . . . , dxN ) = C
N dx1 . . . dxN
x1 · (x2 − x1) · . . . · (xN − xN−1) · (1− xN )
(3.7)
for some constant C ∈ R+.
Proof. If q1(dx) = ρ(x)dx then (3.6) is equivalent to
ρ(y) · ρ
= ρ(x) · ρ
y − x
for all 0 < x < y < 1. For continuous ρ this implies that there exists a constant C ∈ R+ such
that ρ(x) = C
x(1−x)
for all 0 < x < 1. Iterated inserting this into (3.5) yields the claim.
Let us come back to our attempt to give a meaning to the heuristic formula (3.1). Combining
(3.4) with the choice (3.7) of the measure qN finally yields
0 (gt1 ∈ dx1, . . . , gtN ∈ dxN )
(xi − xi−1)β(ti−ti1 )
dx1 . . . dxN
x1 · (x2 − x1) · . . . · (1− xN )
(3.8)
with appropriate normalization constants Zβ,N . Now the RHS of this formula indeed turns out
to define a consistent family of probability measures. Hence, by Kolmogorov’s extension theorem
it admits a projective limit Q
0 on the space G0. The push forward of this measure under the
canonical identification χ : G0 → P0, g 7→ g∗Leb will be the entropic measure Pβ0 which we were
looking for.
The details of the rigorous construction of this measure as well as various properties of it will
be presented in the following sections.
3.2 The Measures Qβ and Pβ
The basic object to be studied in this section is the probability measure Qβ on the space G.
Proposition 3.2. For each real number β > 0 there exists a unique probability measure Qβ on
G, called Dirichlet process, with the property that for each N ∈ N and for each ordered family
of points t1, t2, . . . , tN ∈ S1
β (gt1 ∈ dx1, . . . , gtN ∈ dxN ) =
i=1 Γ(β(ti+1 − ti))
(xi+1 − xi)β(ti+1−ti)−1dx1 . . . dxN .
(3.9)
The precise meaning of (3.9) is that for all bounded measurable u : (S1)N → R
u (gt1 , . . . , gtN ) dQ
i=1 Γ(β · |[ti, ti+1]|)
u(x1, . . . , xN )
|[xi, xi+1]|β·|[ti,ti+1]|−1dx1 . . . dxN .
with ΣN =
(x1, . . . , xN ) ∈ (S1)N :
i=1 |[xi, xi+1]| = 1
and xN+1 := x1, tN+1 := t1. In
particular, with N = 1 this means
u(gt)dQ
β(g) =
u(x)dx for each t ∈ S1.
Proof. It suffices to prove that (3.9) defines a consistent family of finite dimensional distributions.
The existence of Qβ (as a ’projective limit’) then follows from Kolmogorov’s extension theorem.
The required consistency means that
i=1 Γ(β · |[ti, ti+1]|)
|[xi, xi+1]|β·|[ti,ti+1]|−1u(x1, . . . , xN ) dx1 . . . dxN
Γ(β · |[t1, t2]|) · . . . · Γ(β · |[tk−1, tk+1]|) · . . . · Γ(β · |[tN , t1]|)
|[x1, x2]|β·|[t1,t2]|−1 · . . . · |[xk−1, xk+1]|β·|[tk−1,tk+1]|−1 · . . . · |[xN , x1]|β·|[tN ,t1]|−1
·v(x1, . . . , xk−1, xk . . . , xN ) dx1 . . . dxk−1dxk+1 . . . dxN
whenever u(x1, . . . , xN ) = v(x1, . . . , xk−1, xk . . . , xN ) for all (x1, . . . xN ) ∈ ΣN . The latter is an
immediate consequence of the well-known fact (Euler’s beta integral) that
[xk−1,xk+1]
|[xk−1, xk]|β·|[tk−1,tk]|−1 · |[xk, xk+1]|β·|[tk,tk+1]|−1 dxk
Γ(β · |[tk−1, tk]|)Γ(β · |[tk, tk+1]|)
Γ(β · |[tk−1, tk+1]|)
|[xk−1, xk+1]|β·|[tk−1,tk+1]|−1.
For s ∈ S1 let θ̂s : G → G, g 7→ g ◦ θs be the isomorphism of G induced by the rotation
θs : S
1 → S1, t 7→ t+ s. Obviously, the measure Qβ on G is invariant under each of the maps θ̂s.
Hence, Qβ induces a probability measure Q
1 on the quotient spaces G1 = G/S1.
Recall the definition of the map χ : G → P, g 7→ g∗Leb. Since (g ◦ θs)∗Leb = g∗Leb this
canonically extends to a map χ : G1 → P. (As mentioned before, the latter is even an isometry.)
Definition 3.3. The entropic measure Pβ on P is defined as the push forward of the Dirichlet
process Qβ on G (or equivalently, of the measure Qβ1 on G1) under the map χ. That is, for all
bounded measurable u : P → R
u(µ) dPβ(µ) =
u(g∗Leb) dQ
β(g).
3.3 The Measures Q
0 and P
The subspaces {g ∈ G : g(0) = 0} and {g ∈ G0 : g(0) = 0} can obviously be identified.
Conditioning the probability measure Qβ onto this event thus will define a probability measure
0 on G0. However, we prefer to give the direct construction of Q
Proposition 3.4. For each real number β > 0 there exists a unique probability measure Q
G0, called Dirichlet process, with the property that for each N ∈ N and each family 0 = t0 <
t1 < t2 < . . . < tN < tN+1 = 1
0 (gt1 ∈ dx1, . . . , gtN ∈ dxN ) =
i Γ(β · (ti+1 − ti))
(xi+1 − xi)β·(ti+1−ti)−1dx1 . . . dxN .
(3.10)
The precise meaning of (3.10) is that for all bounded measurable u : [0, 1]N → R
u (gt1 , . . . , gtN ) dQ
0 (g)
i=1 Γ(β · (ti+1 − ti))
u(x1, . . . , xN )
(xi+1 − xi)β·(ti+1−ti)−1dx1 . . . dxN .
with ΣN =
(x1, . . . , xN ) ∈ [0, 1]N : 0 < x1 < x2 . . . < xn < 1
and xN+1 := x1, tN+1 := t1.
Remark 3.5. According to these explicit formulae, it is easy to calculate the moments of the
Dirichlet process. For instance,
0 (gt) :=
gt dQ
0 (g) = t
0 (gt) :=
(gt − t)2 dQβ0 (g) =
1 + β
t(1− t)
for all β > 0 and all t ∈ [0, 1].
Definition 3.6. The entropic measure P
0 on P0 = P([0, 1]) is defined as the push forward of
the Dirichlet process Q
0 on G0 under the map χ. That is, for all bounded measurable u : P0 → R
u(µ) dP
0 (µ) =
u(g∗Leb) dQ
0 (g).
Remark 3.7. (i) According to the above construction Q
0 ( . ) = Q
β( . |g(0) = 0) and
u(g) dQ
0 (g) =
u(g − g(0)) dQβ(g),
u(g) dQβ(g) =
u(g + x) dQ
0 (g) dx.
(ii) Analogously, the entropic measures on the sphere and on the are linked as follows
u(µ) dPβ(µ) =
u((θx)∗µ)dP
0 (µ) dx
or briefly
dPβ =
(θ̂x)∗dP
where θx : S
1 → S1, y 7→ x + y and θ̂x : P → P : µ 7→ (θx)∗µ. We would like to emphasize,
however, that Pβ 6= Pβ0 . For instance, consider u(µ) :=
f dµ for some f : S1 → R (which may
be identified with f : [0, 1] → R). Then
P(S1)
u(µ) dPβ(µ) =
f(x) dx
whereas
P([0,1])
u(µ) dP
0 (µ) =
[0,1]
f(x)ρβ(x) dx
with ρβ(x) =
Γ(βt)Γ(β(1−t))
xβt−1(1− x)β(1−t)−1 dt.
According to the last remark, it suffices to study in detail one of the four measures Qβ, Q
Pβ, and P
0 . We will concentrate in the rest of this chapter on the measure Q
0 which seems to
admit the most easy interpretations.
3.4 The Dirichlet Process as Normalized Gamma Process
We start recalling some basic facts about the real valued Gamma processes. For α > 0 denote
by G(α) the absolutely continuous probability measure on R+ with density
xα−1e−x.
Definition 3.8. A real valued Markov process (γt)t≥0 starting in zero is called standard Gamma
process if its increments γt − γs are independent and distributed according to G(t − s) for 0 ≤
s < t. Without loss of generality we may assume that almost surely the function t→ γt is right
continuous and nondecreasing.
Alternatively the Gamma-Process may be defined as the unique pure jump Levy process with
Levy measure Λ(dx) = 1‖‖x>0
dx. The connection between pure jump Levy and Poisson point
processes gives rise to several other equivalent representations of the Gamma process [Kin93,
Ber99]. For instance, let Π = {p = (px, py) ∈ R2} be the Poisson point process on R+×R+ with
intensity measure dx× Λ(dy) with Λ as above, then a Gamma process is obtained by
γt :=
p∈Π:px≤t
py. (3.11)
For β > 0 the process γt·β is a Levy process with Levy measure Λβ(dx) = β · 1‖‖x>0 e
dx. Its
increments are distributed according to
P (γβ·t − γβ·s ∈ dx) =
Γ(β · (t− s))x
β·(t−s)−1e−xdx. �
Proposition 3.9. For each β > 0, the law of the process (
)t∈[0,1] is the Dirichlet process Q
Proof. This well-known fact is easily obtained from Lukacs’ characterization of the Gamma
distribution [ÉY04].
3.5 Support Properties
Proposition 3.10. (i) For each β > 0, the measure Q
0 has full support on G0.
(ii) Q
0 -almost surely the function t 7→ g(t) is strictly increasing but increases only by jumps
(that is, the jumps heights add up to 1 and the jump locations are dense in [0, 1]).
(iii) For each fixed t0 ∈ [0, 1], Qβ0 -almost surely the function t 7→ g(t) is continuous at t0.
Proof. (i) Let g ∈ G ⊂ L2([0, 1], dx) and ǫ > 0 then we have to show Qβ(Bǫ(g)) > 0 where
Bǫ(g) = {h ∈ G0 : ‖h − g‖L2([0,1]) < ǫ}. For this choose finitely many points ti ∈ [0, 1] together
with δi > 0 such that the set S := {f ∈ G
∣ |f(ti)−g(ti)| ≤ δi ∀i} is contained in Bǫ(g). Clearly,
from (3.10) Qβ(S) > 0 which proves the claim.
(ii) (3.10) implies that Q
0 -almost surely g(s) < g(t) for each given pair s < t. Varying over all
such rational pairs s < t, it follows that a.e. g is strictly increasing on R+.
In terms of the probabilistic representation (3.9), it is obvious that g increases only by jumps.
(iii) This also follows easily from the representation as a normalized gamma process (3.9).
Restating the previous property (ii) in terms of the entropic measure yields that P
0 -a.e. µ ∈ P0
is ’Cantor like’. More precisely,
Corollary 3.11. P
0 -almost surely the measure µ ∈ P0 has no absolutely continuous part and
no discrete part. The topological support of µ has Lebesgue measure 0. Moreover,
Ent(µ) = +∞. (3.12)
Proof. The assertion on the entropy of µ is an immediate consequence of the statement on the
support of µ. The second claim follows from the fact that the jump heights of g add up to 1.
In terms of the measure Q
0 , the last assertion of the corollary states that S(g) = +∞ for Q
0 -a.e.
g ∈ G0.
3.6 Scaling and Invariance Properties
The Dirichlet process Q
0 on G0 has the following Markov property: the distribution of g|[s,t]
depends on g[0,1]\[s,t] only via g(s), g(t).
And the Dirichlet process Q
0 on G0 has a remarkable self-similarity property: if we restrict the
functions g onto a given interval [s, t] and then linearly rescale their domain and range in order
to make them again elements of G0 then this new process is distributed according to Qβ
0 with
β′ = β · |t− s|.
Proposition 3.12. For each β > 0, and each s, t ∈ [0, 1], s < t
g|[s,t] ∈ .
∣ g[0,1]\[s,t]
g|[s,t] ∈ .
∣ g(s), g(t)
(3.13)
(Ξs,t)∗Q
0 = Q
β·|t−s|
0 (3.14)
where Ξs,t : G0 → G0 with Ξs,t(g)(r) = g((1−r)s+rt)−g(s)g(t)−g(s) for r ∈ [0, 1].
Proof. Both properties follow immediately from the representation in Proposition 3.10.
Corollary 3.13. The probability measures Q
0 , β > 0 on G0 are uniquely characterized by the
self-similarity property (3.14) and the distributions of g1/2:
0 (g1/2 ∈ dx) =
Γ(β/2)2
· [x(1− x)]β/2−1dx.
Proposition 3.14. (i) For β → 0 the measures Qβ0 weakly converge to a measure Q00 defined
as the uniform distribution on the set {1[t,1] : t ∈ ]0, 1]} ⊂ G0.
Analogously, the measures Qβ weakly converge for β → 0 to a measure Q0 defined as the uniform
distribution on the set of constant maps {t : t ∈ S1} ⊂ G.
(ii) For β → ∞ the measures Qβ0 (or Qβ) weakly converge to the Dirac mass δe on the identity
map e of [0, 1] (or S1, resp.).
Proof. (i) Since the space G0 (equipped with the L2-topology) is compact, so is P(G0) (equipped
with the weak topology). Hence the family Q
0 , β > 0 is pre-compact. Let Q
0 denote the limit of
any converging subsequence of Q
0 for β → 0. According to the formula for the one-dimensional
distributions, for each t ∈ ]0, 1[
0 (gt ∈ dx) =
Γ(βt)Γ(β(1 − t)) · x
βt−1(1− x)β(1−t)−1dx
−→ (1− t)δ{0}(dx) + tδ{1}(dx)
as β → 0. Hence, Q00 is the uniform distribution on the set {1[t,1] : t ∈ ]0, 1]} ⊂ G0.
(ii) Similarly, Q
0 (gt ∈ dx) → δt(dx) as β → ∞. Hence, δe with e : t 7→ t will be the unique
accumulation point of Q
0 for β → ∞.
Restating the previous results in terms of the entropic measures, yields that the entropic mea-
sures P
0 converge weakly to the uniform distribution P
0 on the set {(1 − t)δ{0} + tδ{1} : t ∈
[0, 1]} ⊂ P0; and the measures Pβ converge weakly to the uniform distribution P0 on the set
{δ{t} : t ∈ S1} ⊂ P whereas for β → ∞ both, Pβ0 and Pβ, will converge to δLeb, the Dirac mass
on the uniform distribution of [0, 1] or S1, resp.
The assertions of Proposition 3.12 imply the following Markov property and self-similarity prop-
erty of the entropic measure.
Proposition 3.15. For each each x, y ∈ [0, 1], x < y
µ|[x,y] ∈ .
∣µ|[0,1]\[x,y]
µ|[x,y] ∈ .
∣µ([x, y]
µ|[x,y] ∈ .
∣µ([x, y]) = α
0 (µx,y ∈ . )
with µx,y ∈ P0 (’rescaling of µ|[x,y]’) defined by µx,y(A) = 1µ([x,y])µ(x+(y−x) ·A) for A ⊂ [0, 1].
3.7 Dirichlet Processes on General Measurable Spaces
Recall Ferguson’s notion of a Dirichlet process on a general measurable spaceM with parameter
measure m on M . This is a probability measure Qm
on P(M), uniquely defined by the fact
that for any finite measurable partition M = ˙
i=1 Mi and σi := m(Mi).
P(M) (µ : µ(M1) ∈ dx1, . . . , µ(MN ) ∈ dxN )
Γ(m(M))
i=1 Γ(σi)
1 · · · x
)σN+1−1dx1 · · · dxN ,
If a map h :M →M leaves the parameter measure m invariant then obviously the induced map
ĥ : P(M) → P(M), µ 7→ h∗µ leaves the Dirichlet process QmP(M) invariant.
In the particular case M = [0, 1] and m = β · Leb, the Dirichlet process Qm
can be obtained
as push forward of the measure Q
0 (introduced before) under the isomorphism ζ : G0 → P([0, 1])
which assigns to each g the induced Lebesgue-Stieltjes measure dg (the inverse ζ−1 assigns to
each probability measure its distribution function):
P([0,1]) = ζ∗Q
0 . (3.15)
Note that the support properties of the measure Qm
P([0,1])
are completely different from those of
the measure P
0 . In particular, Q
P([0,1])
-almost every µ ∈ P([0, 1]) is discrete and has full topo-
logical support, cf. Corollary 3.11. The invariance properties of Qm
P([0,1])
under push forwards by
means of measure preserving transformations of [0, 1] seems to have no intrinsic interpretation
in terms of Q
4 The Change of Variable Formula for the Dirichlet Process and
for the Entropic Measure
Our main result in this chapter will be a change of variable formula for the Dirichlet process.
To motivate this formula, let us first present an heuristic derivation based on the formal repre-
sentation (3.1).
4.1 Heuristic Approaches to Change of Variable Formulae
Let us have a look on the change of variable formula for the Wiener measure. On a formal level,
it easily follows from Feynman’s heuristic interpretation
dPβ(g) =
g′(t)2dt dP(g)
with the (non-existing) ’uniform distribution’ P. Assuming that the latter is ’translation invari-
ant’ (i.e. invariant under additive changes of variables, – at least in ’smooth’ directions h) we
immediately obtain
dPβ(h+ g) =
(h+g)′(t)2dt dP(h + g)
h′(t)2dt−β
h′(t)g′(t)dt · e−
g′(t)2dt dP(g)
h′(t)2dt−β
h′(t)dg(t) dPβ(g).
If we interpret
h′(t)dg(t) as the Ito integral of h′ with respect to the Brownian path g then
this is indeed the famous Cameron-Martin-Girsanov-Maruyama theorem.
In the case of the entropic measure, the starting point for a similar argumentation is the heuristic
interpretation
0 (g) =
log g′(t)dt dQ0(g),
again with a (non-existing) ’uniform distribution’ Q0 on G0. The natural concept of ’change
of variables’, of course, will be based on the semigroup structure of the underlying space G0;
that is, we will study transformations of G0 of the form g 7→ h ◦ g for some (smooth) element
h ∈ G0. It turns out that Q0 should not be assumed to be invariant under translations but
merely quasi-invariant:
dQ0(h ◦ g) = Y 0h (g) dQ0(g)
with some density Yh. This immediately implies the following change of variable formula for Q
0 (h ◦ g) =
log(h◦g)′(t)dt dQ0(h ◦ g)
log h′(g(t))dt · eβ
log g′(t)dt · Y 0h (g) dQ0(g)
log g′(t)dt · Y 0h (g) dQ
0 (g).
This is the heuristic derivation of the change of variables formula. Its rigorous derivation (and
the identification of the density Yh) is the main result of this chapter.
4.2 The Change of Variables Formula on the Sphere
For g, h ∈ G with h ∈ C2 we put
Y 0h (g) :=
h′ (g(a−)) · h′ (g(a+))
δ(h◦g)
, (4.1)
where Jg ⊂ S1 denotes the set of jump locations of g and
δ(h ◦ g)
(a) :=
h (g(a+)) − h (g(a−))
g(a+)− g(a−) .
To simplify notation, here and in the sequel (if no ambiguity seems possible), we write y − x
instead of |[x, y]| to denote the length of the positively oriented segment from x to y in S1. We
will see below that the infinite product in the definition of Y 0h (g) converges for Q
β-a.e. g ∈ G.
Moreover, for β > 0 we put
h (g) := exp
log h′ (g(s)) ds
h (g) := X
h (g) · Y
h (g). (4.2)
Theorem 4.1. Each C2-diffeomorphism h ∈ G induces a bijective map τh : G → G, g 7→ h ◦ g
which leaves the measure Qβ quasi-invariant:
dQβ(h ◦ g) = Y βh (g) dQ
β(g).
In other words, the push forward of Qβ under the map τ−1h = τh−1 is absolutely continuous w.r.t.
Qβ with density Y
d(τh−1)∗Q
dQβ(g)
h (g).
The function Y
h is bounded from above and below (away from 0) on G.
By means of the canonical isometry χ : G → P, g 7→ g∗Leb, Theorem 4.1 immediately implies
Corollary 4.2. For each C2-diffeomorphism h ∈ G the entropic measure Pβ is quasi-invariant
under the transformation µ 7→ h∗µ of the space P:
dPβ(h∗µ) = Y
−1(µ)) dPβ(µ).
The density Y
−1(µ)) introduced in (4.2) can be expressed as follows
−1(µ)) = exp
log h′(s)µ(ds)
I∈gaps(µ)
h′(I−) · h′(I+)
|h(I)|/|I|
where gaps(µ) denotes the set of segments I = ]I−, I+[⊂ S1 of maximal length with µ(I) = 0
and |I| denotes the length of such a segment.
4.3 The Change of Variables Formula on the Interval
From the representation of Qβ as a product of Q
0 and Leb (see Remark 3.7) and the change of
variable formulae for Qβ and Leb, one can deduce a change of variable formula for Q
0 similar to
that of Theorem 4.1 but containing an additional factor 1
h′(0)
. In this case, one has to restrict
to translations by means of C2-diffeomorphisms h ∈ G with h(0) = 0.
More generally, one might be interested in translations of G0 by means of C2-diffeomorphisms
h ∈ G0. In contrast to the previous situation, it now may happen that h′(0) 6= h′(1).
For g ∈ G0 and C2-ismorphism h : [0, 1] → [0, 1] we put
h,0(g) := X
h (g) · Yh,0(g) (4.3)
Yh,0(g) =
h′(0) · h′(1)
· Y 0h (g)
and X
(g) and Y 0h (g) defined as before in (4.1), (4.2). Note that here and in the sequel by a
C2-isomorphism h ∈ G0 we understand an increasing homeomorphism h : [0, 1] → [0, 1] such that
h and h−1 are bounded in C2([0, 1]), which in particular implies h′ > 0.
Theorem 4.3. Each translation τh : G0 → G0, g 7→ h ◦ g by means of a C2-isomorphism h ∈ G0
leaves the measure Q
0 quasi-invariant:
0 (h ◦ g) = Y
(g) dQ
0 (g)
or, in other words,
d(τh−1)∗Q
0 (g)
0 (g)
The function Y
h,0 is bounded from above and below (away from 0) on G0.
Corollary 4.4. For each C2-isomorphism h ∈ G0 the entropic measure Pβ0 is quasi-invariant
under the transformation µ 7→ h∗µ of the space P0:
0 (h∗µ)
0 (µ)
= exp
log h′(s)µ(ds)
h′(0) · h′(1)
I∈gaps(µ)
h′(I−) · h′(I+)
|h(I)|/|I|
where gaps(µ) denotes the set of intervals I = ]I−, I+[⊂ [0, 1] of maximal length with µ(I) = 0
and |I| denotes the length of such an interval.
Remark 4.5. Theorem 4.3 seems to be unrelated to the quasi-invariance of the measure Qm
P([0,1])
under the transformation dg → h · dg/〈h, dg〉 shown in [Han02]. Nor is it anyhow implied by
the quasi-ivarariance formula for the general measure valued gamma process as in [TVY01] with
respect to a similar transformation. In our present case the latter would correspond to the
mapping dγ → h · dγ of the (measure valued) Gamma process dγ.
4.4 Proofs for the Sphere Case
Lemma 4.6. For each C2-diffeomorphism h ∈ G
h (g) = lim
h (g(ti+1))− h (g(ti))
g(ti+1)− g(ti)
]β(ti+1−ti)
(4.4)
Here ti =
for i = 0, 1, . . . , k − 1 and tk = 0. Thus ti+1 − ti := |[ti, ti+1]| = 1k for all i.
Proof. Without restriction, we may assume β = 1. According to Taylor’s formula
h (g(ti+1)) = h (g(ti)) + h
′ (g(ti)) · (g(ti+1)− g(ti)) + 12h
′′(γi) · (g(ti+1)− g(ti))2
for some γi ∈ [g(ti), g(ti+1)]. Hence,
h (g(ti+1))− h (g(ti))
g(ti+1)− g(ti)
]ti+1−ti
= lim
h′ (g(ti)) +
h′′(γi) · (g(ti+1)− g(ti))
]ti+1−ti
= lim
log h′ (g(ti)) + log
1 + 1
h′′(γi)
h′ (g(ti))
(g(ti+1)− g(ti))
· (ti+1 − ti)
= exp
log h′ (g(ti)) · (ti+1 − ti)
= exp
log h′ (g(t)) dt
= X1h(g).
Here (⋆) follows from the fact that
1 + 1
h′′(γi)
h′ (g(ti))
· (g(ti+1)− g(ti)) =
h (g(ti+1))− h (g(ti))
g(ti+1)− g(ti)
h′ (g(ti))
= h′(ηi) ·
h′ (g(ti))
≥ ε > 0
for some ηi ∈ [g(ti), g(ti+1)] and some ε > 0, independent of i and k. Thus
1 + 1
h′′(γi)
h′ (g(ti))
· (g(ti+1)− g(ti))
· (ti+1 − ti)
≤ C1 ·
h′′(γi)
h′ (g(ti))
· (g(ti+1)− g(ti)) · (ti+1 − ti)
≤ C2 ·
(g(ti+1)− g(ti)) · (ti+1 − ti)
≤ C3 · 1k .
Lemma 4.7. For each C3-diffeomorphism h ∈ G
Y 0h (g) := lim
h′ (g(ti)) ·
g(ti+1)− g(ti)
h (g(ti+1))− h (g(ti))
(4.5)
where ti =
for i = 0, 1, . . . , k − 1 and tk = 0.
Proof. Let h and g be given. Depending on some ε > 0 let us choose l ∈ N large enough (to be
specified in the sequel) and let a1, . . . , al denote the l largest jumps of g. Put J
g = Jg\{a1, . . . , al}
and for simplicity al+1 := a1. For k very large (compared with l) and j = 1, . . . , l let kj denote
the index i ∈ {0, 1, . . . , k − 1}, for which aj ∈ [ti, ti+1[. Then again by Taylor’s formula
kj+1−1
i=kj+1
h′ (g(ti)) ·
g(ti+1)− g(ti)
h (g(ti+1))− h (g(ti))
kj+1−1
i=kj+1
1 + 1
h′′ (g(ti))
h′ (g(ti))
· (g(ti+1)− g(ti)) + 16
h′′′(ηi)
h′ (g(ti))
· (g(ti+1)− g(ti))2
≤ exp
kj+1−1
i=kj+1
log h′
(g(ti)) +
· (g(ti+1)− g(ti))
≤ eε/l · exp
kj+1−1
i=kj+1
log h′
(g(ti)) · (g(ti+1)− g(ti))
provided l and k are chosen so large that
|g(ti+1)− g(ti)| ≤
C1 · l
for all i ∈ {0, . . . , k − 1} \ {k1, . . . , kl}, where C1 = sup
|h′′′(x)|
6·h′(y)
On the other hand,
g(tkj+1)
g(tkj+1)
) = exp
∫ g(tkj+1 )
g(tkj+1)
log h′
(s) ds
= exp
kj+1−1
i=kj+1
log h′
(g(ti)) · (g(ti+1)− g(ti)) +
log h′
(γi) · 12 (g(ti+1)− g(ti))
≥ e−ε/l · exp
kj+1−1
i=kj+1
log h′
(g(ti)) · (g(ti+1)− g(ti))
provided l and k are chosen so large that
|g(ti+1)− g(ti)| ≤
C2 · l
for all i ∈ {0, 1, . . . , k − 1} \ {k1, . . . , kl}, where C2 = sup
log h′
Therefore,
i∈{0,1,...,k−1}\{k1,...,kl}
h′ (g(ti)) ·
g(ti+1)− g(ti)
h (g(ti+1))− h (g(ti))
≤ e2ε ·
g(tkj+1)
g(tkj+1)
) = (I).
In order to derive the corresponding lower estimate, we can proceed as before in (1a) and (2)
(replacing ε by −ε and ≤ by ≥ and vice versa). To proceed as in (1b) we have to argue as
follows
kj+1−1
i=kj+1
log h′
(g(ti))− εl
· (g(ti+1)− g(ti))
≥ e−ε/l · exp
kj+1−1
i=kj+1
(1− ε) ·
log h′
(g(ti)) · (g(ti+1)− g(ti))
provided l and k are chosen so large that
log (1 + C3 · (g(ti+1)− g(ti))) ≥ (1− ε) · C3 · (g(ti+1)− g(ti))
for all i ∈ {0, 1, . . . , k − 1} \ {k1, . . . , kl}, where C3 = sup
log h′
Thus we obtain the following lower estimate
i∈{0,1,...,k−1}\{k1,...,kl}
h′ (g(ti)) ·
g(ti+1)− g(ti)
h (g(ti+1))− h (g(ti))
≥ e−2ε ·
g(tkj+1)
g(tkj+1)
≥ e−2ε · C−ε/23 ·
g(tkj+1)
g(tkj+1)
) = (II),
since
g(tkj+1)
g(tkj+1)
= exp
log h′
g(tkj+1)
− log h′
g(tkj+1)
≤ exp
g(tkj+1)− g(tkj+1)
≤ exp
where C3 = sup
∣(log h′)
Now for fixed l as k → ∞ the bound (I) converges to
(I′) = e2ε ·
h′ (g(aj+1−))
h′ (g(aj+))
and the bound (II) to
(II′) = e−2ε · C−ε/23 ·
h′ (g(aj+1−))
h′ (g(aj+))
Finally, it remains to consider
i∈{k1,...,kl}
h′ (g(ti)) ·
g(ti+1)− g(ti)
h (g(ti+1))− h (g(ti))
= (III).
Again for fixed l and k → ∞ this obviously converges to
(III′) =
h′ (g(aj−))
· δ(h ◦ g)
Putting together these estimates and letting l → ∞, we obtain the claim.
Lemma 4.8. (i) For all g, h ∈ G with h ∈ C2 strictly increasing, the infinite product in the
definition of Y 0h (g) converges. There exists a constant C = C(β, h) such that ∀g ∈ G
≤ Y β
(g) ≤ C.
(ii) If hn → h in C2 then Y 0hn(g) → Y
h (g).
(iii) Let Y 0h,k,X
denote the sequences used in Lemma 4.6 and 4.7 to approximate Y 0h ,X
Then there exists a constant C = C(β, h) such that ∀g ∈ G, ∀k ∈ N
≤ Y βh,k(g) ≤ C.
Proof. (i) Put C = sup |(log h′)′|. Given g ∈ G and ǫ > 0, we choose k large enough such that
a∈Jg(k)
|g(a+)−g(a−)| ≤ ǫ where Jg(k) = Jg \{a1, a2, . . . , ak} denotes the ’set of small jumps’
of g. Here we enumerate the jump locations a1, a2, . . . ∈ Jg according to the size of the respective
jumps. Then with suitable ξa ∈ [g(a−), g(a+)]
a∈Jg(k)
h′(g(a−))
h′(g(a+))
δ(h◦g)
a∈Jg(k)
log h′(g(a−)) + 1
log h′(g(a−)) − log h′(ξ(a))
a∈Jg(k)
|C · (g(a+)− g(a−))| = C · ǫ.
Hence, the infinite sum
h′(g(a−))
h′(g(a+))
δ(h◦g)
= lim
a∈Jg(k)
h′(g(a−))
h′(g(a+))
δ(h◦g)
is absolutely convergent and thus also infinite product in the definition of Y 0h (g) converges. The
same arguments immediately yield
∣log Y 0h (g)
log h′(g(a−)) + 1
log h′(g(a−)) − log h′(ξ(a))
≤ C. (4.6)
(ii) In order to prove the convergence Y 0hn(g) → Y
h (g), for given g ∈ G we split the product over
all jumps into a finite product over the big jumps and an infinite product over all small jumps.
Obviously, the finite products will converge (for any choice of k)
a∈{a1,...,ak}
h′n(g(a−))
h′n(g(a+))
δ(hn◦g)
a∈{a1,...,ak}
h′(g(a−))
h′(g(a+))
δ(h◦g)
as n → ∞ provided hn → h in C2. Now let C = supn supx |(log h′n)′(x)| and choose k as before.
Then uniformly in n
a∈Jg\{a1,...,ak}
h′n(g(a−))
h′n(g(a+))
δ(hn◦g)
≤ C · ǫ.
(iii) Let C1 = sup
|h′(x)| and C2 = sup
∣(log h′)
∣. Then for all g and k:
Xh,k(g) =
h′(ηi)
ti+1−ti ≤ C1
Y 0h,k(g) =
h′ (g(ti))
h′(γi)
= exp
log h′
(ζi) · (g(ti)− γi)
≤ exp
|g(ti)− γi|
≤ exp(C2)
(with suitable γi, ηi ∈ [g(ti), g(ti+1)] and ζi ∈ [g(ti), γi]). Analogously, the lower estimates
follow.
Proof of Theorem 4.1. In order to prove the equality of the two measures under consideration, it
suffices to prove that all of their finite dimensional distributions coincide. That is, for each m ∈
N, each ordered family t1, . . . , tm of points in S
1 and each bounded continuous u : (S1)m −→ R
one has to verify that
h−1 (g(t1)) , h
−1 (g(t2)) , . . . , h
−1 (g(tm))
dQβ(g)
u (g(t1), g(t2), . . . , g(tm)) · Y βh (g) dQ
β(g).
Without restriction, we may restrict ourselves to equidistant partitions, i.e. ti =
for i =
1, . . . ,m. Let us fix m ∈ N, u and h. For simplicity, we first assume that h is C3. Then by
Lemmas 4.6 - 4.8 and Lebesgue’s theorem
, . . . , g (1)
· Y βh (g) dQ
, . . . , g (1)
· lim
(g) dQβ(g)
= lim
, . . . , g (1)
dQβ(g)
= lim
[Γ(β/km)]km
u(xk, x2k, . . . , xmk)
h′(xi) ·
[h(xi+1)− h(xi)]
dx1 . . . dxmk
= lim
[Γ(β/km)]km
u(xk, x2k, . . . , xmk) ·
[h(xi+1)− h(xi)]
dh(x1) . . . dh(xmk)
= lim
[Γ(β/km)]km
h−1(yk), h
−1(y2k), . . . , h
−1(ymk)
[yi+1 − yi]
dy1 . . . dymk
, h−1
, . . . , h−1 (g (1))
dQβ(g).
Now we treat the general case h ∈ C2. We choose a sequence of C3-functions hn ∈ G with hn → h
in C2. Then
h−1 (g(t1)) , h
−1 (g(t2)) , . . . , h
−1 (g(tm))
dQβ(g)
= lim
h−1n (g(t1)) , h
n (g(t2)) , . . . , h
n (g(tm))
dQβ(g)
= lim
u (g(t1), g(t2), . . . , g(tm)) · Y βhn(g) dQ
u (g(t1), g(t2), . . . , g(tm)) · Y βh (g) dQ
β(g).
For the last equality, we have used the dominated convergence Y
(g) → Y βh (g) (due to Lemma
4.8).
4.5 Proof for the Interval Case
The proof of Theorem 4.3 uses completely analogous arguments as in the previous section. To
simplify notation, for h ∈ C1([0, 1]), k ∈ N let Xh,k, Y 0h,k : G0 → R be defined by
Xh,k(g) :=
h (g(ti+1))− h (g(ti))
g(ti+1)− g(ti)
]ti+1−ti
Y 0h,k(g) :=
g(t1)− g(t0)
h (g(t1))− h (g(t0))
] k−1
h′ (g(ti)) ·
g(ti+1)− g(ti)
h (g(ti+1))− h (g(ti))
where ti =
with i = 0, 1, . . . , k. Similar to the proof of theorem 4.1 the measure Q
0 satisfies
the following finite dimensional quasi-invariance formula.
For any u : [0, 1]m−1 → R, m, l ∈ N and C1-isomorphism h : [0, 1] → [0, 1]
h−1 (g(t1)) , h
−1 (g(t2)) , . . . , h
−1 (g(tm−1))
0 (g)
u (g(t1), g(t2), . . . , g(tm−1)) ·Xβh,l·m(g) · Y
h,l·m(g) dQ
0 (g),
where ti =
, i = 1, · · · ,m−1. The passage to the limit for letting first l and then m to infinity
is based on the following assertions.
Lemma 4.9. (i) For each C2-isomorphism h ∈ G0 and g ∈ G0
Xh(g) = lim
Xh,k(g).
(ii) For each C3-isomorphism h ∈ G0 and g ∈ G0
Y 0h,k(g) =
h′(g(a+)) · h′(g(a−))
δ(h◦g)
h′(g(0)) · h′(g(1−))
1 if g(1−) = g(1)
h′(g(1−))
δ(h◦g)
else,
where Jg ⊂]0, 1[ is the set of jump locations of g on ]0, 1[. In particular,
Y 0h,k(g) = Yh,0(g) for Q
0 -a.e.g.
(iii) For all g ∈ G0 and C2-isomorphism h ∈ G0, the infinite product in the definition of Yh,0(g)
converges. There exists a constant C = C(β, h) such that ∀g ∈ G0
≤ Y β
(g) ≤ C.
(iv) If hn → h in C2([0, 1], [0, 1]) with h as above, then Y 0,hn(g) → Y0,h(g).
(v) For each C3-isomorphism h ∈ G0 there exists a constant C = C(β, h) such that ∀g ∈ G,
∀k ∈ N
≤ Xβh,k(g) · Y
h,k(g) ≤ C.
Proof. The proofs of (i) and (iii)-(iv) carry over from their respective counterparts on the sphere,
lemmas 4.6 and 4.8 above. We sketch the proof of statement (ii) which needs most modification.
For ε > 0 choose l ∈ N large enough and let a2, . . . , al−1 denote the l − 2 largest jumps of g
on ]0, 1[. For k very large (compared with l) we may assume that a2, . . . , al−2 ∈] 2k , 1 −
[. Put
a1 :=
, al := 1 − 1k . For j = 1, . . . , l let kj denote the index i ∈ {1, . . . , k − 1}, for which
aj ∈ [ti, ti+1[. In particular, k1 = 1 and kl = k−1. Then using the same arguments as in lemma
4.7 one obtains, for k and l sufficiently large, the two sided bounds
(I) = e2ε ·
g(tkj+1)
g(tkj+1)
i∈{1,...,k−1}\{k1,...,kl}
h′ (g(ti)) ·
g(ti+1)− g(ti)
h (g(ti+1))− h (g(ti))
≥ e−2ε · C−ε/23 ·
g(tkj+1)
g(tkj+1)
) = (II)
For fixed l and k → ∞ the bounds (I) and (II) converge to
(I′) = e2ε
h′(g(a2−))
h′(g(0))
h′ (g(aj+1−))
h′ (g(aj+))
h′(g(1−))
h′(g(al−1+))
(II′) = e−2ε · C−ε/23 ·
h′(g(a2−))
h′(g(0))
h′ (g(aj+1−))
h′ (g(aj+))
h′(g(1−))
h′(g(al−1+))
It remains to consider the three remaining terms
(III) =
i∈{k2,...,kl−1}
h′ (g(ti)) ·
g(ti+1)− g(ti)
h (g(ti+1))− h (g(ti))
which for fixed l and k → ∞ converges to
(III′) =
h′ (g(aj−))
· δ(h ◦ g)
(IV) =
)− g(0)
− h (g(0))
)− g( 1
converging by right continuity of g to
(IV′) = h′(g(0))
(V) =
k − 1
g(1) − g(k−1
h (g(1)) − h
g(k−1
which tends, also for k → ∞, to
(V′) =
1 if g continuous in 1
δ(h◦g)
(1) 1
h′(g(1−))
else.
Combining these estimates and letting l → ∞, we obtain the first claim. The second claim in
statement (ii) follows from the fact that g is continuous in t = 1 Q
0 -almost surely.
5 The Integration by Parts Formula
In order to construct Dirichlet forms and Markov processes on G, we will consider it as an infinite
dimensional manifold. For each g ∈ G, the tangent space TgG will be an appropriate completion
of the space C∞(S1,R). The whole construction will strongly depend on the choice of the norm
on the tangent spaces TgG. Basically, we will encounter two important cases:
• in Chapter 6 we will study the case TgG = Hs(S1,Leb) for some s > 1/2, independent
of g; this approach is closely related to the construction of stochastic processes on the
diffeomorphism group of S1 and Malliavin’s Brownian motion on the homeomorphism
group on S1, cf. [Mal99].
• in Chapters 7-9 we will assume TgG = L2(S1, g∗Leb); in terms of the dynamics on the
space P(S1) of probability measures, this will lead to a Dirichlet form and a stochastic
process associated with the Wasserstein gradient and with intrinsic metric given by the
Wasserstein distance.
In this chapter, we develop the basic tools for the differential calculus on G. The main result
will be an integration by parts formula. These results will be independent of the choice of the
norm on the tangent space.
5.1 The Drift Term
For each ϕ ∈ C∞(S1,R), the flow generated by ϕ is the map eϕ : R × S1 → S1 where for each
x ∈ S1 the function eϕ(., x) : R → S1, t 7→ eϕ(t, x) denotes the unique solution to the ODE
= ϕ(xt) (5.1)
with initial condition x0 = x. Since eϕ(t, x) = etϕ(1, x) for all ϕ, t, x under consideration, we
may simplify notation and write etϕ(x) instead of eϕ(t, x).
Obviously, for each ϕ ∈ C∞(S1,R) the family etϕ, t ∈ R is a group of orientation preserving,
C∞-diffeomorphism of S1. (In particular, e0 is the identity map e on S1, etϕ ◦ esϕ = e(t+s)ϕ for
all s, t ∈ R and (eϕ)−1 = e−ϕ.)
Since ∂
etϕ(x)|t=0 = ϕ(x) we obtain as a linearization for small t
etϕ(x) ≈ x+ tϕ(x). (5.2)
More precisely,
|etϕ(x)− (x+ tϕ(x))| ≤ C · t2
as well as
etϕ(x)− (1 + t
ϕ(x))| ≤ C · t2
uniformly in x and |t| ≤ 1.
For ϕ ∈ C∞(S1,R) and β > 0 we define functions V βϕ : G → R by
V βϕ (g) := V
ϕ (g) + β
ϕ′(g(x))dx
where
V 0ϕ (g) :=
ϕ′(g(a+)) + ϕ′(g(a−))
− ϕ(g(a+)) − ϕ(g(a−))
g(a+)− g(a−)
. (5.3)
Lemma 5.1. (i) The sum in (5.3) is absolutely convergent. More precisely,
|V 0ϕ (g)| ≤
ϕ′(g(a+)) + ϕ′(g(a−))
− ϕ(g(a+)) − ϕ(g(a−))
g(a+)− g(a−)
|ϕ′′(x)|dx
|V βϕ (g)| ≤ (1/2 + β) ·
|ϕ′′(x)|dx.
(ii) For each β ≥ 0
V βϕ (g) =
Y βetϕ(g)
e+tϕ(g)
. (5.4)
Proof. (i) According to Taylor’s formula, for each a ∈ Jg
ϕ′(g(a+)) + ϕ′(g(a−))
− δ(ϕ ◦ g)
(a) =
2(g(a+) − g(a−))
∫ g(a+)
g(a−)
∫ g(a+)
g(a−)
sgn(y−x) ·ϕ′′(y)dydx.
Hence,
ϕ′(g(a+)) + ϕ′(g(a−))
− δ(ϕ ◦ g)
(g(a+)− g(a−))
∫ g(a+)
g(a−)
∫ g(a+)
g(a−)
sgn(y − x) · ϕ′′(y)dydx
∫ g(a+)
g(a−)
|ϕ′′(y)|dy = 1
|ϕ′′(y)|dy.
Finally,
ϕ′(g(x))dx| ≤ sup
|ϕ′(y)| ≤
|ϕ′′(y)|dy.
(ii) Let us first consider the case β = 0.
log Y 0etϕ(g)
etϕ)(g(a+)) +
etϕ)(g(a−)) − log
δ(etϕ ◦ g)
etϕ)(g(a+)) +
etϕ)(g(a−)) − log
δ(etϕ ◦ g)
In order to justify that we may interchange differentiation and summation, we decompose (as
we did several times before) the infinite sum over all jumps in Jg into a finite sum over big
jumps a1, . . . , ak and an infinite sum over small jumps in Jg(k) = Jg \ {a1, . . . , ak}. Of course,
the finite sum will make no problem. We are going to prove that the contribution of the small
jumps is arbitrarily small. Recall from Lemma 4.8 that
a∈Jg(k)
etϕ)(g(a+)) +
etϕ)(g(a−)) − log
δ(etϕ ◦ g)
≤ Ct·
a∈Jg(k)
[g(a+)− g(a−)]
where Ct := supx
log( ∂
etϕ)(x)
∣. Now Ct ≤ C · |t| for all |t| ≤ 1 and an appropriate constant
C. Thus for any given ǫ > 0
a∈Jg(k)
etϕ)(g(a+)) +
etϕ)(g(a−)) − log
δ(etϕ ◦ g)
provided k is chosen large enough (i.e. such that C ·
a∈Jg(k)
|g(a+)−g(a−)| ≤ ǫ). This justifies
the above interchange of differentiation and summation.
Now for each x ∈ S1
etϕ(x)
= ϕ′(x)
since the linearization of etϕ for small t yields
etϕ(x) ≈ x+ tϕ(x),
etϕ(x) ≈ 1 + tϕ′(x).
Similarly, for small t we obtain
δ(etϕ ◦ g)
(a) ≈ 1 + t · δ(ϕ ◦ g)
and thus
δ(etϕ ◦ g)
δ(ϕ ◦ g)
Therefore,
log Y 0etϕ(g)
= V 0ϕ (g).
On the other hand, obviously
log Y 0etϕ(g)
Y 0etϕ(g)
since Y 0e0(g) = 1.
Finally, we have to consider the derivative of Xetϕ . Based on the previous arguments and using
the fact that ∂
(x) is uniformly bounded in t ∈ [−1, 1] and x ∈ S1 we immediately
logXetϕ(g)
(g(y))dy
(g(y)) dy =
ϕ′(g(y))dy.
Again Xe0(g) = 1. Therefore,
= β ·
ϕ′(g(y))dy
and thus
Y βetϕ(g)
= V βϕ (g).
this proves the first identity in (5.4). The proof of the second one V
ϕ (g) =
e+tϕ(g)
similar (even slightly easier).
5.2 Directional Derivatives
For functions u : G → R we will define the directional derivative along ϕ ∈ C∞(S1,R) by
Dϕu(g) := lim
[u(etϕ ◦ g)− u(g)] (5.5)
provided this limit exists. In particular, this will be the case for the following ’cylinder functions’.
Definition 5.2. We say that u : G → R belongs to the class Sk(G) if it can be written as
u(g) = U(g(x1), . . . , g(xm)) (5.6)
for some m ∈ N, some x1, . . . , xm ∈ S1 and some Ck-function U : (S1)m → R.
It should be mentioned that functions u ∈ Sk(G) are in general not continuous on G.
Lemma 5.3. The directional derivative exists for all u ∈ S1(G). In particular, for u as above
Dϕu(g) = lim
[u(g + t · ϕ ◦ g) − u(g)]
∂iU(g(x1), . . . , g(xm)) · ϕ(g(xi))
with ∂iU :=
U . Moreover, Dϕ : S
k(G) → Sk−1(G) for all k ∈ N ∪ {∞} and
‖Dϕu‖L2(Qβ) ≤
m · ‖∇U‖∞ · ‖ϕ‖L2(S1).
Proof. The first claim follows from
Dϕu(g) =
U(etϕ(g(x1)), . . . , etϕ(g(xm)))
∂iU(etϕ(g(x1)), . . . , etϕ(g(xm))) ·
etϕ(g(xi))
∂iU(g(x1), . . . , g(xm)) · ϕ(g(xi))
U(g(x1) + tϕ(g(x1)), . . . , g(xm) + tϕ(g(xm)))
= lim
[u(g + t · ϕ ◦ g)− u(g)] .
For the second claim,
‖Dϕu‖2L2(Qβ) =
∂iU(g(x1), . . . , g(xm)) · ϕ(g(xi))
dQβ(g)
(∂iU)
2(g(x1), . . . , g(xm)) ·
ϕ2(g(xi))
dQβ(g)
≤ ‖∇U‖2∞ ·
ϕ2(g(xi)) dQ
= m · ‖∇U‖2∞ ·
ϕ2(y) dy.
5.3 Integration by Parts Formula on P(S1)
For ϕ ∈ C∞(S1,R) let D∗ϕ denote the operator in L2(G,Qβ) adjoint to Dϕ with domain S1(G).
Proposition 5.4. Dom(D∗ϕ) ⊃ S1(G) and for all u ∈ S1(G)
D∗ϕu = −Dϕu− V βϕ · u. (5.7)
Proof. Let u, v ∈ S1(G). Then
Dϕu · v dQβ = lim
[u(etϕ ◦ g) − u(g)] · v(g) dQβ(g)
= lim
u(g) · v(e−tϕ ◦ g) · Y βe−tϕ − u(g) · v(g)
dQβ(g)
= lim
u(g) · [v(e−tϕ ◦ g)− v(g)] dQβ(g)
+ lim
u(g) · v(g) ·
Y βe−tϕ − 1
dQβ(g)
+ lim
u(g) · [v(e−tϕ ◦ g) − v(g)] ·
Y βe−tϕ − 1
dQβ(g)
u ·Dϕv dQβ(g)−
u · v · V βϕ dQβ(g) + 0.
To justify the last equality, note that according to Lemma 4.8 | log Y βetϕ | ≤ C · |t| for |t| ≤ 1.
Hence, the claim follows with dominated convergence and Lemma 5.4.
Corollary 5.5. The operator (Dϕ,S
1(G)) is closable in L2(Qβ). Its closure will be denoted by
(Dϕ,Dom(Dϕ)).
In other words, Dom(Dϕ) is the closure (or completion) of S
1(G) with respect to the norm
[u2 + (Dϕu)
2] dQβ
Of course, the space Dom(Dϕ) will depend on β but we assume β > 0 to be fixed for the sequel.
Remark 5.6. The bilinear form
Eϕ(u, v) :=
Dϕu ·Dϕv dQβ , Dom(Eϕ) := Dom(Dϕ) (5.8)
is a Dirichlet form on L2(G,Qβ) with form core S∞(G). Its generator (Lϕ,Dom(Lϕ)) is the
Friedrichs extension of the symmetric operator
(−D∗ϕ ◦Dϕ, S2(G)).
5.4 Derivatives and Integration by Parts Formula on P([0, 1])
Now let us have a look on flows on [0, 1]. To do so, let a function ϕ ∈ C∞([0, 1],R) with
ϕ(0) = ϕ(1) = 0 be given. (Note that each such function can be regarded as ϕ ∈ C∞(S1,R)
with ϕ(0) = 0.) The flow equation (5.1) now defines a flow etϕ, t ∈ R, of order preserving C∞
diffeomorphisms of [0, 1]. In particular, etϕ(0) = 0 and etϕ(1) = 1 for all t ∈ R.
Lemma 5.1 together with Theorem 4.3 immediately yields
Lemma 5.7. For ϕ ∈ C∞([0, 1],R) with ϕ(0) = ϕ(1) = 0 and each β ≥ 0
etϕ,0
= V βϕ (g)−
ϕ′(0) + ϕ′(1)
ϕ,0(g). (5.9)
For functions u : G0 → R we will define the directional derivative along ϕ ∈ C∞([0, 1],R) with
ϕ(0) = ϕ(1) = 0 as before by
Dϕu(g) := lim
[u(etϕ ◦ g)− u(g)] (5.10)
provided this limit exists. We will consider three classes of ’cylinder functions’ for which the
existence of this limit is guaranteed.
Definition 5.8. (i) We say that a function u : G0 → R belongs to the class Ck(G0) (for k ∈
N ∪ {0,∞}) if it can be written as
u(g) = U
~f(t)g(t)dt
(5.11)
for some m ∈ N, some ~f = (f1, . . . , fm) with fi ∈ L2([0, 1],Leb) and some Ck-function U : Rm →
R. Here and in the sequel, we write
~f(t)g(t)dt =
f1(t)g(t)dt, . . . ,
fm(t)g(t)dt
(ii) We say that u : G0 → R belongs to the class Sk(G0) if it can be written as
u(g) = U (g(x1), . . . , g(xm)) (5.12)
for some m ∈ N, some x1, . . . , xm ∈ [0, 1] and some Ck-function U : Rm → R.
(iii) We say that u : G0 → R belongs to the class Zk(G0) if it can be written as
u(g) = U
~α(gs)ds
(5.13)
with U as above, ~α = (α1, . . . , αm) ∈ Ck([0, 1],Rm) and
~α(gs)ds =
α1(gs)ds, . . . ,
αm(gs)ds
Remark 5.9. For each ϕ ∈ C∞(S1,R) with ϕ(0) = 0 (which can be regarded as ϕ ∈ C∞([0, 1],R)
with ϕ(0) = ϕ(1) = 0), the definitions of Dϕ in (5.5) and (5.10) are consistent in the following
sense. Each cylinder function u ∈ S1(G0) defines by v(g) := u(g − g0) (∀g ∈ G) a cylinder
function v ∈ S1(G) with Dϕv = Dϕu on G0. Conversely, each cylinder function v ∈ S1(G)
defines by u(g) := v(g) (∀g ∈ G0) a cylinder function u ∈ S1(G0) with Dϕv = Dϕu on G0.
Lemma 5.10. (i) The directional derivative Dϕu(g) exists for all u ∈ C1(G0)∪S1(G0)∪Z1(G0)
(in each point g ∈ G0 and in each direction ϕ ∈ C∞([0, 1],R) with ϕ(0) = ϕ(1) = 0) and
Dϕu(g) = limt→0
[u(g + t · ϕ ◦ g)− u(g)]. Moreover,
Dϕu(g) =
~f(t)g(t)dt
fi(t)ϕ(g(t))dt
for each u ∈ C1(G0) as in (5.11),
Dϕu(g) =
∂iU(g(x1), . . . , g(xm)) · ϕ(g(xi))
for each u ∈ S1(G0) as in (5.12), and
Dϕu(g) =
~α(gs)ds
α′i(gs)ϕ(gs)ds
for each u ∈ Z1(G0) as in (5.13).
(ii) For ϕ ∈ C∞([0, 1],R) with ϕ(0) = ϕ(1) = 0 let D∗ϕ,0 denote the operator in L2(G0,Q
adjoint to Dϕ. Then for all u ∈ C1(G0) ∪S1(G0) ∪ Z1(G0)
D∗ϕ,0u = −Dϕu− V
ϕ,0 · u. (5.14)
Proof. See the proof of the analogous results in Lemma 5.3 and Proposition 5.4.
Remark 5.11. The operators (Dϕ,C
1(G0)), (Dϕ,S1(G0)), and (Dϕ,Z1(G0)) are closable in
0 ). The closures of (Dϕ,C
1(G0)), (Dϕ,Z1(G0)) and (Dϕ,S1(G0)) coincide. They will be
denoted by (Dϕ,Dom(Dϕ)). See (proof of) Corollary 6.11.
6 Dirichlet Form and Stochastic Dynamics on on G
At each point g ∈ G, the directional derivative Dϕu(g) of any ’nice’ function u on G defines
a linear form ϕ 7→ Dϕu(g) on C∞(S1). If we specify a pre-Hilbert norm ‖.‖g on C∞(S1) for
which this linear form is continuous then there exists a unique element Du(g) ∈ TgG with
Dϕu(g) = 〈Du(g), ϕ〉g for all ϕ ∈ C∞(S1). Here TgG denotes the completion of C∞(S1) w.r.t.
the norm ‖.‖g.
The canonical choice of a Dirichlet form on G will then be (the closure of)
E(u, v) =
〈Du(g),Dv(g)〉g dQβ(g), u, v ∈ S1(G). (6.1)
Given such a Dirichlet form, there is a straightforward procedure to construct an operator (’gen-
eralized Laplacian’) and a Markov process (’generalized Brownian motion’). Different choices of
‖.‖g in general will lead to completely different Dirichlet forms, operators and Markov processes.
We will discuss in detail two choices: in this chapter we will choose ‖.‖g (independent of g) to
be the Sobolev norm ‖.‖Hs for some s > 1/2; in the remaining chapters, ‖.‖g will always be the
L2-norm ϕ 7→ (
ϕ(gt)
2dt)1/2 of L2(S1, g∗Leb).
For the sequel, fix – once for ever – the number β > 0 and drop it from the notations, i.e.
Q := Qβ, Vϕ := V
ϕ etc.
6.1 The Dirichlet Form on G
Let (ψk)k∈N denote the standard Fourier basis of L
2(S1). That is,
ψ2k(x) =
2 · sin(2πkx), ψ2k+1(x) =
2 · cos(2πkx)
for k = 1, 2, . . . and ψ1(x) = 1. It constitutes a complete orthonormal system in L
2(S1): each
ϕ ∈ L2(S1) can uniquely be written as ϕ(x) =
k=1 ck · ψk(x) with Fourier coefficients of ϕ
given by ck :=
ϕ(y)ψk(y)dy. In terms of these Fourier coefficients we define for each s ≥ 0
the norm
‖ϕ‖Hs :=
c21 +
k2s · (c22k + c22k+1)
(6.2)
on C∞(S1). The Sobolev space Hs(S1) is the completion of C∞(S1) with respect to the norm
‖.‖Hs . It has a complete orthonormal system consisting of smooth functions (ϕk)k∈N. For
instance, one may choose
ϕ2k(x) =
2 · k−s · sin(2πkx), ϕ2k+1(x) =
2 · k−s · cos(2πkx) (6.3)
for k = 1, 2, . . . and ϕ1(x) = 1.
A linear form A : C∞(S1) → R is continuous w.r.t. ‖.‖Hs — and thus can be represented as
A(ϕ) = 〈ψ,ϕ〉Hs for some ψ ∈ Hs(S1) with ‖ψ‖Hs = ‖A‖Hs — if and only if
‖A‖Hs :=
|A(ψ1)|2 +
k2s · (|A(ψ2k)|2 + |A(ψ2k+1)|2)
<∞. (6.4)
Proposition 6.1. Fix a number s > 1/2. Then for each cylinder function u ∈ S(G) and each
g ∈ G, the directional derivative defines a continuous linear form ϕ 7→ Dϕu(g) on C∞(S1) ⊂
Hs(S1). There exists a unique tangent vector Du(g) ∈ Hs(S1) such that Dϕu(g) = 〈Du(g), ϕ〉Hs
for all ϕ ∈ C∞(S1).
In terms of the family Φ = (ϕk)k∈N from (6.3)
Du(g) =
Dϕku(g) · ϕk(.)
‖Du(g)‖2Hs =
|Dϕku(g)|2. (6.5)
Proof. It remains to prove that the RHS of (6.5) is finite for each u and g under consideration.
According to Lemma 5.3, for any u ∈ S(G) represented as in (5.12)
|Dϕku(g)|2 =
∂iU(g(x1), . . . , g(xm)) · ϕk(g(xi))
≤ m · ‖∇U‖2∞ · ‖
ϕ2k‖∞ = m · ‖∇U‖2∞ · (1 + 4
k−2s).
And, indeed, the latter is finite for each s > 1/2.
For the sequel, let us now fix a number s > 1/2 and define
E(u, v) =
〈Du(g),Dv(g)〉Hs dQ(g) (6.6)
for u, v ∈ S1(G). Equivalently, in terms of the family Φ = (ϕk)k∈N from (6.3)
E(u, v) =
Dϕku(g) ·Dϕkv(g) dQ(g). (6.7)
Theorem 6.2. (i) (E ,S1(G)) is closable. Its closure (E ,Dom(E)) is a regular Dirichlet form
on L2(G,Q) which is strongly local and recurrent (hence, in particular, conservative).
(ii) For u ∈ S1(G) with representation (5.6)
E(u, u) =
∂iU(g(x1), . . . , g(xm)) · ϕk(g(xi))
dQ(g).
The generator of the Dirichlet form is the Friedrichs extension of the operator L given on S2(G)
Lu(g) =
i,j=1
∂i∂jU (g(x1), . . . , g(xm))ϕk(g(xi))ϕk(g(xj))
∂iU (g(x1), . . . , g(xm)) [ϕ
k(g(xi)) + Vϕk(g)]ϕk(g(xi)).
(iii) Z1(G) is a core for Dom(E) (i.e. it is contained in the latter as a dense subset). For
u ∈ Z1(G) with representation (5.13)
E(u, u) =
~α(gt)dt) ·
α′i(gt)ϕk(gt)dt
dQ(g).
The generator of the Dirichlet form is the Friedrichs extension of the operator L given on Z2(G)
Lu(g) =
i,j=1
∂i∂jU
~α(gt)dt
α′i(gt)ϕk(gt)dt ·
α′j(gt)ϕk(gt)dt
~α(gt)dt
{Vϕk(g) +
[α′′i (gt)ϕ
k(gt) + α
i(gt)ϕ
k(gt)ϕk(gt)]dt}.
(iv) The intrinsic metric ρ can be estimated from below in terms of the L2-metric:
ρ(g, h) ≥ 1√
‖g − h‖L2 .
Remark 6.3. All assertions of the above Theorem remain valid for any E defined as in (6.7)
with any choice of a sequence Φ = (ϕk)k∈N of smooth functions on S
1 with
C := ‖
ϕ2k‖∞ <∞. (6.8)
(This condition is satisfied for the sequence from (6.3) if and only if s > 1/2.)
The proof of the Theorem will make use of the following
Lemma 6.4. (i) Dom(E) contains all functions u which can be represented as
u(g) = U(‖g − f1‖L2 , . . . , ‖g − fm‖L2) (6.9)
with some m ∈ N, some f1, . . . , fm ∈ G and some U ∈ C1(Rm,R).
For each u as above, each ϕ ∈ C∞(S1) and Q-a.e. g ∈ G
Dϕu(g) =
∂iU(‖g − f1‖L2 , . . . , ‖g − fm‖L2) ·
sign(g(t)− fi(t))
|g(t) − fi(t)|
‖g − fi‖L2
ϕ(g(t))dt
where sign(z) := +1 for z ∈ S1 with |[0, z]| ≤ 1/2 and sign(z) := −1 for z ∈ S1 with |[z, 0]| < 1/2.
(ii) Moreover, Dom(E) contains all functions u which can be represented as
u(g) = U(gǫ1(x1), . . . , gǫm(xm)) (6.10)
with some m ∈ N, some x1, . . . , xm ∈ S1, some ǫ1, . . . , ǫm ∈ ]0, 1[ and some U ∈ C1((S1)m,R).
Here gǫ(x) :=
∫ x+ǫ
g(t)dt ∈ S1 for x ∈ S1 and 0 < ǫ < 1. More precisely,
gǫ(x) := π(
∫ x+ǫ
π−1g(t)dt)
where π : G(R) → G (cf. section 2.2) denotes the projection and π−1 : G → G(R) the canonical
lift with π−1(g)(t) ∈ [g(x), g(x) + 1] ⊂ R for t ∈ [x, x+ 1] ⊂ R.
For each u as above, each ϕ ∈ C∞(S1) and each g ∈ G
Dϕu(g) =
∂iU(gǫ1(x1), . . . , gǫm(xm)) ·
∫ xi+ǫi
ϕ(g(t))dt.
(iii) The set of all u of the form (6.10) is dense in Dom(E).
Proof. (i) Let us first prove that for each f ∈ G, the map u(g) = ‖g − f‖L2 lies in Dom(E). For
n ∈ N, let πn : G → G be the map which replaces each g by the piecewise constant map:
πn(g)(t) := g(
) for t ∈ [ i
Then by right continuity πn(g) → g as n→ ∞ and thus
|g( i
)− f( i
)|2 −→
|g(t) − f(t)|2dt.
Therefore, for each g ∈ G as n→ ∞
un(g) := Un(g(0), g(
), . . . , g(
n − 1
)) −→ u(g) (6.11)
where Un(x1, . . . , xn) :=
i=0 dn(xi+1 − f( in))
and dn is a smooth approximation of
the distance function x 7→ |x| on S1 (which itself is non-differentiable at x = 0 and x = 1
) with
|d′n| ≤ 1 and dn(x) → |x| as n→ ∞. Obviously, un ∈ S1(G).
By dominated convergence, (6.11) also implies that un → u in L2(G,Q). Hence, u ∈ Dom(E) if
(and only if) we can prove that
E(un) <∞.
E(un) =
∂iUn(g(0), g(
), . . . , g(
n − 1
)) · ϕ(g( i − 1
dQ(g)
ϕ2k(g(
i − 1
)) dQ(g) =
‖ϕk‖2L2 <∞,
uniformly in n ∈ N. This proves the claim for the function u(g) = ‖g − f‖L2 .
From this, the general claim follows immediately: if vn, n ∈ N, is a sequence of S1(G) approx-
imations of g 7→ ‖g − 0‖L2 then un(g) := U(vn(g − f1), . . . , vn(g − fm)) defines a sequence of
S1(G) approximations of u(g) = U(‖g − f1‖L2 , . . . , ‖g − fm‖L2).
(ii) Again it suffices to treat the particular case m = 1 and U = id, that is, u(g) = gǫ(x)
for some x ∈ S1 and some 0 < ǫ < 1. Let g̃ ∈ G(R) be the lifting of g and recall that
u(g) = π(1
∫ x+ǫ
g̃(t)dt). Define un ∈ S1(G) for n ∈ N by un(g) = π( 1n
i=0 g̃(x +
ǫ)). Right
continuity of g̃ implies un → u as n → ∞ pointwise on G and thus also in L2(G,Q). To see the
boundedness of E(un) note that Dϕun(g) = 1n
i=0 ϕ(g(x +
ǫ)). Thus
E(un) ≤
ϕ2k(g(x +
ǫ))dQ(g) =
‖ϕk‖2L2 <∞.
(iii) We have to prove that each u ∈ S1(G) can be approximated in the norm (‖.‖2 + E(.))1/2 by
functions un of type (6.10). Again it suffices to treat the particular case u(g) = g(x) for some
x ∈ S1. Choose un(g) = g1/n(x). Then by right continuity of g, un → u pointwise on G and
thus also in L2(G,Q). Moreover, Dϕun(g) = n
∫ x+1/n
ϕ(g(t))dt (for all ϕ and g) and therefore
E(un) ≤
∫ x+1/n
ϕ2k(g(t))dtdQ(g) =
‖ϕk‖2L2 <∞.
Proof of the Theorem. (a) The sum E of closable bilinear forms with common domain S1(G)
is closable, provided it is still finite on this domain. The latter will follow by means of Lemma
5.3 which implies for all u ∈ S1(G) with representation (5.11)
E(u, u) =
∂iU(g(x1), . . . , g(xm)) · ϕk(g(xi))
dQ(g)
≤ m · ‖∇U‖2∞ ·
‖ϕk‖2L2(S1) <∞.
Hence, indeed E is finite on S1(G).
(b) The Markov property for E follows from that of the Eϕk(u, v) =
Dϕku ·Dϕkv dQ.
(c) According to the previous Lemma, the class of continuous functions of type (6.10) is dense
in Dom(E). Moreover, the class of finite energy functions of type (6.9) is dense in C(G) (with
the L2 topology of G ⊂ L2(S1), cf. Proposition 2.1). Therefore, the Dirichlet form E is regular.
(e) The estimate for the intrinsic metric is an immediate consequence of the following estimate
for the norm of the gradient of the function u(g) = ‖g − f‖L2 (which holds for each f ∈ G
uniformly in g ∈ G):
‖Du(g)‖2 =
sign(g(t)− fi(t))
|g(t) − fi(t)|
‖g − fi‖L2
ϕk(g(t))dt
ϕ2k(g(t))dt ≤ ‖
ϕ2k‖∞ =: C.
(f) The locality is an immediate consequence of the previous estimate: Given functions u, v ∈
Dom(E) with disjoint supports, one has to prove that E(u, v) = 0. Without restriction, one may
assume that supp[u] ⊂ Br(g) and supp[v] ⊂ Br(h) with ‖g − h‖L2 > 2r + 2δ. (The general
case will follow by a simple covering argument.) Without restriction, u, v can be assumed to be
bounded. Then |u| ≤ Cwδ,g and |v| ≤ Cwδ,h for some constant C where
wδ,g(f) =
(r + δ − ‖f − g‖L2) ∧ 1
Given un ∈ S1(G) with un → u in Dom(E) put
un = (un ∧ wδ,g) ∨ (−wδ,g).
Then un → u in Dom(E). Analogously, vn → v in Dom(E) for vn = (vn ∧ wδ,h) ∨ (−wδ,h). But
obviously, E(un, vn) = 0 since un · vn = 0. Hence, E(u, v) = 0.
(g) In order to prove that Z1(G) is contained in Dom(E) it suffices to prove that each u ∈ Z1(G)
of the form u(g) =
α(gt)dt can be approximated in Dom(E) by un ∈ S1(G). Given u as above
with α ∈ C1(S1,R) put un(g) = 1n
i=1 α(gi/n). Then un ∈ S1(G), un → u on G and
Dϕun(g) =
α′(gi/n)ϕ(gi/n) →
α′(gt)ϕ(gt)dt = Dϕu(g).
Moreover,
E(un, un) =
α′(gi/n)ϕ(gi/n)
dQ(g)
≤ C ·
α′(gi/n)
2 dQ(g) = C ·
α′(t)2dt
uniformly in n ∈ N. Hence, u ∈ Dom(E) and
E(u, u) = lim
E(un, un) =
α′(gt)ϕk(gt)dt
dQ(g).
(h) The set Z1(G) is dense in Dom(E) since according to assertion (ii) of the previous Lemma
already the subset of all u of the form (6.10) is dense in Dom(E).
Finally, one easily verifies that Z2(G) is dense in Z1(G) and (using the integration by parts
formula) that L is a symmetric operator on Z2(G) with the given representation.
Corollary 6.5. There exists a strong Markov process (gt)t≥0 on G, associated with the Dirichlet
form E. It has continuous trajectories and it is reversible w.r.t. the measure Q. Its generator
has the form
DϕkDϕk +
Vϕk ·Dϕk
with {ϕk}k∈N being the Fourier basis of Hs(S1).
Remark 6.6. This process (gt)t≥0 is closely related to the stochastic processes on the diffeo-
morphism group of S1 and to the ’Brownian motion’ on the homeomorphism group of S1,
studied by Airault, Fang, Malliavin, Ren, Thalmaier and others [AMT04, AM06, AR02, Fan02,
Fan04, Mal99]. These are processes with generator 1
kDϕkDϕk . For instance, in the
case s = 3/2 our process from the previous Corollary may be regarded as ’Brownian motion
plus drift’. All the previous approaches are restricted to s ≥ 3/2. The main improvements of
our approach are:
• identification of a probability measure Q such that these processes — after adding a
suitable drift — are reversible;
• construction of such processes in all cases s > 1/2.
6.2 Finite Dimensional Noise Approximations
In the previous section, we have seen the construction of the diffusion process on G under minimal
assumptions. However, the construction of the process is rather abstract. In this section, we try
to construct explicitly a diffusion process associated with the generator of the Dirichlet form E
from Theorem 6.2. Here we do not aim for greatest generality.
Let a finite family Φ = (ϕk)k=1,...,n of smooth functions on S
1 be given and let (Wt)t≥0 with
Wt = (W
t , . . . ,W
t ) be a n-dimensional Brownian motion, defined on some probability space
(Ω,F ,P). For each x ∈ S1 we define a stochastic processes (ηt(x))t≥0 with values in S1 as the
strong solution of the Ito differential equation
dηt(x) =
ϕk(ηt(x))dW
ϕ′k(ηt(x))ϕk(ηt(x))dt (6.12)
with initial condition η0(x) = x. Equation (6.12) can be rewritten in Stratonovich form as
follows
dηt(x) =
ϕk(ηt(x)) ⋄ dW kt . (6.13)
Obviously, for every t and for P-a.e. ω ∈ Ω, the function x 7→ ηt(x, ω) is an element of the
semigroup G. (Indeed, it is a C∞-diffeomorphism.) Thus (6.13) may also be interpreted as a
Stratonovich SDE on the semigroup G:
dηt =
ϕk(ηt) ⋄ dW kt , η0 = e. (6.14)
This process on G is right invariant: if gt denotes the solution to (6.14) with initial condition
g0 = g for some initial condition g ∈ G then gt = ηt ◦ g. One easily verifies that the generator
of this process (gt)t≥0 is given on S
2(G) by 1
k=1DϕkDϕk . What we aim for, however, is a
process with generator
D∗ϕkDϕk =
DϕkDϕk +
Vϕk ·Dϕk .
Define a new probability measure Pg on (Ω,F), given on Ft by
dPg = exp
Vϕk(ηs ◦ g)dW ks −
|Vϕk(ηs ◦ g)|2ds
dP (6.15)
and a semigroup (Pt)t≥0 acting on bounded measurable functions u on G as follows
Ptu(g) =
u(ηt(g(.), ω)) dP
g(ω).
Proposition 6.7. (Pt)t≥0 is a strongly continuous Markov semigroup on G. Its generator is an
extension of the operator 1
L = −1
Dϕk with domain S
2(G). That is, for all u ∈ S2(G)
and all g ∈ G
(Ptu(g)− u(g)) =
Lu(g). (6.16)
Proof. The strong continuity follows easily from the fact that ηt(x, .) → x a.s. as t → 0 which
implies by dominated convergence
Ptu(g) =
u(ηt ◦ g) dPg → u(g)
for each continuous u : G → R.
Now we aim for identifying the generator. According to Girsanov’s theorem, under the measure
Pg the processes
W̃ kt =W
Vϕk(ηs ◦ g)ds
for k = 1, . . . , n will define n independent Brownian motions. In terms of these driving processes,
(6.12) can be reformulated as
dgt(x) =
ϕk(gt(x))dW̃
[ϕ′k(gt(x)) + Vϕk(gt)]ϕk(gt(x))dt (6.17)
(recall that gs = ηs ◦ g). The chain rule applied to a smooth function U on (S1)m, therefore,
yields
dU (gt(y1), . . . , gt(ym))
U (gt(y1), . . . , gt(ym)) dgt(yi)
i,j=1
∂xi∂xj
U (gt(y1), . . . , gt(ym)) d〈g.(yi), g.(yj)〉t
U (gt(y1), . . . , gt(ym))ϕk(gt(yi))dW̃
U (gt(y1), . . . , gt(ym)) [ϕ
k(gt(yi)) + Vϕk(gt)]ϕk(gt(yi))dt
i,j=1
∂xi∂xj
U (gt(y1), . . . , gt(ym))ϕk(gt(yi))ϕk(gt(yj))dt.
Hence, for a cylinder function of the form u(g) = U(g(y1), . . . , g(ym)) we obtain
(Ptu(g) − u(g))
= lim
[U (gt(y1), . . . , gt(ym))− U (g0(y1), . . . , g0(ym))] dPg
= lim
U (gs(y1), . . . , gs(ym)) [ϕ
k(gs(yi)) + Vϕk(gs)]ϕk(gs(yi))
i,j=1
∂xi∂xj
U (gs(y1), . . . , gs(ym))ϕk(gs(yi))ϕk(gs(yj))
 ds dPg
U (g(y1), . . . , g(ym)) [ϕ
k(g(yi)) + Vϕk(g)]ϕk(g(yi))
i,j=1
∂xi∂xj
U (g(y1), . . . , g(ym))ϕk(g(yi))ϕk(g(yj))
[DϕkDϕku(g) + Vϕk(g) ·Dϕku(g)] = −
D∗ϕkDϕku(g).
In order to justify (∗), we have to verify continuity in s in all the expressions preceding (∗). The
only term for which this is not obvious is Vϕk(gs). But gs = ηs ◦g with a function ηs(x, ω) which
is continuous in x and in s. Thus Vϕk(ηs(., ω) ◦ g) is continuous in s.
Remark 6.8. All the previous argumentations in principle also apply to infinite families of
(ϕk)k=1,2,..., provided they have sufficiently good integrability properties. For instance, the
family (6.3) with s > 5
will do the job. There are three key steps which require a careful
verification:
• the solvability of the Ito equation (6.12) and the fact that the solutions are homeomor-
phisms of S1; here s ≥ 3
suffices, cf. [Mal99];
• the boundedness of the quadratic variation of the drift to justify Girsanov’s transformation
in (6.15); for s > 5
this will be satisfied since Lemma 5.1 implies (uniformly in g)
|Vϕk(g)|
2 ≤ (β + 1)2
|ϕ′′k(x)|2dx ≤ 4(β + 1)2
k4−2s;
• the finiteness of the generator and Ito’s chain rule for C2-cylinder functions; here s > 3
will be sufficient.
Remark 6.9. Another completely different approximation of the process (gt)t≥0 in terms of
finite dimensional SDEs is obtained as follows. For N ∈ N, let S1N denote the set of cylinder
functions u : G → R which can be represented as u(g) = U(g(1/N), g(2/N), . . . , g(1)) for some
U ∈ C1((S1)N ). Denote the closure of (E ,S1N ) by (EN ,Dom(EN )). It is the image of the
Dirichlet form (EN ,Dom(EN )) on ΣN ⊂ (S1)N given by
EN (U) =
i,j=1
∂iU(x)∂jU(x) aij(x)ρ(x) dx (6.18)
aij(x) =
ϕk(xi)ϕk(xj), ρ(x) =
Γ(β/N)N
(xi+1 − xi)β/N−1dx.
and (as before) ΣN =
(x1, . . . , xN ) ∈ (S1)N :
i=1 |[xi, xi+1]| = 1
. That is,
EN (u) = EN (U)
for cylinder functions u ∈ S1N as above. Let (Xt,Px)t≥0,x∈ΣN be the Markov process on ΣN
associated with EN . Then the semigroup associated with EN is given by
TNt u(g) = Eg(1/N),...,g(1) [U(Xt)] .
Now let (gt,Pg)t≥0,g∈G and (Tt)t≥0 denote the Markov process and the L
2-semigroup associated
with E . Then as N → ∞
t → Tt strongly in L2
since
E2N ց E
in the sense of quadratic forms, [RS80], Theorem S.16. (Note that ∪N∈NS12N is dense inDom(E).)
6.3 Dirichlet Form and Stochastic Dynamics on G1 and P
In order to define the derivative of a function u : G1 → R we regard it as a function ũ on G with
the property ũ(g) = ũ(g ◦θz) for all z ∈ S1. This implies that Dϕũ(g) = (Dϕũ)(g ◦θz) whenever
one of these expressions is well-defined. In other words, Dϕũ defines a function on G1 which will
be denoted by Dϕu and called the directional derivative of u along ϕ.
Corollary 6.10. (i) Under assumption (6.8), with the notations from above,
E(u, u) =
|Dϕku|
2 dQ.
defines a regular, strongly local, recurrent Dirichlet form on L2(G1,Q).
(ii) The Markov process on G analyzed in the previous section extends to a (continuous, re-
versible) Markov process on G1.
In order to see the second claim, let g, g̃ ∈ G with g̃ = g ◦ θz for some z ∈ S1. Then obviously,
g̃t(., ω) = ηt(g̃(.), ω) = ηt(g(. + z), ω) = gt(., ω) ◦ θz.
Moreover,
Pg̃ = Pg
since Vϕ(g ◦ θz) = Vϕ(g) for all ϕ under consideration and all z ∈ S1.
The objects considered previously – derivative, Dirichlet form and Markov process on G1 – have
canonical counterparts on P. The key to these new objects is the bijective map χ : G1 → P.
The flow generated by a smooth ’tangent vector’ ϕ : S1 → R through the point µ ∈ P will be
given by ((etϕ)∗µ)t∈R. In these terms, the directional derivative of a function u : P → R at the
point µ ∈ P in direction ϕ ∈ C∞(S1,R) can be expressed as
Dϕu(µ) = lim
[u((etϕ)∗µ)− u(µ)] ,
provided this limit exists. The adjoint operator to Dϕ in L
2(P,P) is given (on a suitable dense
subspace) by
D∗ϕu(µ) = −Dϕ(µ)− Vϕ(χ−1(µ)) · u(µ).
The drift term can be represented as
−1(µ)) = β
ϕ′(s)µ(ds) +
I∈gaps(µ)
ϕ′(I−) + ϕ
′(I+)
− ϕ(I+)− ϕ(I−)|I|
Given a sequence Φ = (ϕk)k∈N of smooth functions on S
1 satisfying (6.8), we obtain a (regular,
strongly local, recurrent) Dirichlet form E on L2(P,P) by
E(u, u) =
|Dϕku(µ)|2dP(µ). (6.19)
It is the image of the Dirichlet form defined in (6.7) under the map χ. The generator of E is
given on an appropriate dense subspace of L2(P,P) by
L = −
D∗ϕkDϕk . (6.20)
For P-a.e. µ0 ∈ P, the associated Markov process (µt)t≥0 on P starting in µ0 is given as
µt(ω) = gt(ω)∗Leb
where (gt)t≥0 is the process on G, starting in g0 := χ−1(µ0). (As mentioned before, (gt)t≥0 admits
a more direct construction provided we restrict ourselves to a finite sequence Φ = (ϕk)k=1,...,n.)
6.4 Dirichlet Form and Stochastic Dynamics on G0 and P0
For s > 0 and ϕ : [0, 1] → R let the Sobolev norm ‖ϕ‖Hs be defined as in (6.2) and let Hs0([0, 1])
denote the closure of C∞c (]0, 1[), the space of smooth ϕ : [0, 1] → R with compact support
in ]0, 1[. If s ≥ 1/2 (which is the only case we are interested in) Hs0([0, 1]) can be identified
with {ϕ ∈ Hs([0, 1]) : ϕ(0) = ϕ(1) = 0} or equivalently with {ϕ ∈ Hs(S1) : ϕ(0) = 0}.
For the sequel, fix s > 1/2 and a complete orthonormal basis Φ = {ϕk}k∈N of Hs0([0, 1]) with
C := ‖
k‖∞ <∞, and define
E0(u, u) =
|Dϕk ,0u(g)|2 dQ0(g).
Corollary 6.11. (E0,S1(G0)), (E0,Z1(G0)) and (E0,C1(G0)) are closable. Their closures coincide
and define a regular, strongly local, recurrent Dirichlet form (E0,Dom(E0)) on L2(G0,Q0).
Proof. For the closability (and the equivalence of the respective closures) of (E0,S1(G0)) and
(E0,Z1(G0)), see the proof of Theorem 6.2. Also all the assertions on the closure are deduced in
the same manner. For the closability of (E0,C1(G0)) (and the equivalence of its closure with the
previously defined closures), see the proof of Theorem 7.8 below.
As explained in the previous subsection, these objects (invariant measure, derivative, Dirichlet
form and Markov process) on G0 have canonical counterparts on P0 defined by means of the
bijective map χ : G0 → P0.
7 The Canonical Dirichlet Form on the Wasserstein Space
7.1 Tangent Spaces and Gradients
The aim of this chapter is to construct a canonical Dirichlet form on the L2-Wasserstein space
P0. Due to the isometry χ : G0 → P0 this is equivalent to construct a canonical Dirichlet form
on the metric space (G0, ‖.‖L2). This can be realized in two geometric settings which seem to be
completely different:
• Like in the preceding two chapters, G0 can be considered as a group, with composition of
functions as group operation. The tangent space TgG0 is the closure (w.r.t. some norm)
of the space of smooth functions ϕ : [0, 1] → R with ϕ(0) = ϕ(1) = 0. Such a function ϕ
induces a flow on G0 by (g, t) 7→ etϕ ◦ g ≈ g + t ϕ ◦ g and it defines a directional derivative
by Dϕu(g) = limt→0
[u(etϕ ◦ g)−u(g)] for u : G0 → R. The norm on TgG0 we now choose
to be ‖ϕ‖Tg := (
ϕ(gs)
2ds)1/2. That is,
TgG0 := L2([0, 1], g∗Leb).
For given u and g as above, a gradient Du(g) ∈ TgG0 exists with
Dϕu(g) = 〈Du(g), ϕ〉Tg (∀ϕ ∈ Tg)
if and only if supϕ
Dϕu(g)
‖ϕ◦g‖
• Alternatively, we can regard G0 as a closed subset of the space L2([0, 1],Leb). The lin-
ear structure of the latter (with the pointwise addition of functions as group operation)
suggests to choose as tangent space
TgG0 := L2([0, 1],Leb).
An element f ∈ TgG0 induces a flow by (g, t) 7→ g+tf and it defines a directional derivative
(’Frechet derivative’) by Dfu(g) = limt→0
[u(g + tf) − u(g)] for u : G0 → R, provided u
extends to a neighborhood of G0 in L2([0, 1],Leb) or the flow (induced by f) stays within
G0. A gradient Du(g) ∈ TgG0 exists with
Dfu(g) = 〈Du(g), f〉L2 (∀ϕ ∈ L2)
if and only if supf
Dfu(g)
<∞. In this case, Du(g) is the usual L2-gradient.
Fortunately, both geometric settings lead to the same result.
Lemma 7.1. (i) For each g ∈ G0, the map ιg : ϕ 7→ ϕ ◦ g defines an isometric embedding
of TgG0 = L2([0, 1], g∗Leb) into TgG0 = L2([0, 1],Leb). For each (smooth) cylinder function
u : G0 → R
Dϕu(g) = Dϕ◦gu(g).
If Du ∈ L2(Leb) exists then Du ∈ L2(g∗Leb) also exists.
(ii) For Q0-a.e. g ∈ G0, the above map ιg : TgG0 → TgG0 is even bijective. For each u as above
Du(g) = Du(g) ◦ g−1 and
‖Du(g)‖Tg = ‖Du(g)‖Tg .
Proof. (i) is obvious, (ii) follows from the fact that for Q0-a.e. g ∈ G0 the generalized inverse
g−1 is continuous and thus g−1(gt) = t for all t (see sections 3.5 and 2.1). Hence, the map
ιg : TgG0 → TgG0 is surjective: for each f ∈ TgG0
ιg(f ◦ g−1) = f ◦ g−1 ◦ g = f.
Example 7.2. (i) For each u ∈ Z1(G0) of the form u(g) = U(
~α(gt)dt) with U ∈ C1(Rm,R)
and ~α = (α1, . . . , αm) ∈ C1([0, 1],Rm), the gradients Du(g) ∈ TgG0 = L2([0, 1], g∗Leb) and
Du(g) ∈ TgG0 = L2([0, 1],Leb) exist:
Du(g) =
~α(gt)dt) · α′i(g(.)), Du(g) =
~α(gt)dt) · α′i(.)
and their norms coincide:
‖Du(g)‖2Tg = ‖Du(g)‖
~α(gt)dt) · α′i(g(s))
(ii) For each u ∈ C1(G0) of the form u(g) = U(
~f(t)g(t)dt) with U ∈ C1(Rm,R) and ~f =
(f1, . . . , fm) ∈ L2([0, 1],Rm), the gradient
Du(g) =
~f(t)g(t)dt) · αi(.) ∈ L2([0, 1],Leb)
exists and
‖Du(g)‖2Tg =
~f(t)g(t)dt) · fi(s)
For u ∈ C1(G0) ∪ Z1(G0), the gradient Du can be regarded as a map G0 × [0, 1] → R, (g, t) 7→
Du(g)(t). More precisely,
D : C1(G0) ∪ Z1(G0) → L2(G0 × [0, 1],Q0 ⊗ Leb).
Proposition 7.3. The operator D : Z1(G0) → L2(G0×[0, 1],Q0⊗Leb) is closable in L2(G0,Q0).
Proof. LetW ∈ L2(G0×[0, 1],Q0⊗Leb) be of the formW (g) = w(g)·ϕ(gt) with some w ∈ Z1(G0)
and some ϕ ∈ C∞([0, 1]) satisfying ϕ(0) = ϕ(1) = 0. Then according to the integration by parts
formula for each u ∈ Z1(G0) with u(g) = U(
~α(gs)ds)
G0×[0,1]
Du ·W d(Q0 ⊗ Leb) =
~α(gs)ds)α
i(gt)w(g)ϕ(gt)dtdQ0(g)
Dϕu(g)w(g) dQ0(g) =
u(g)D∗ϕw(g) dQ0(g).
To prove the closability of D, consider a sequence (un)n in Z
1(G0) with un → 0 in L2(Q0) and
Dun → V in L2(Q0 ⊗ Leb). Then
V ·W d(Q0 ⊗ Leb) = lim
Dun ·W d(Q0 ⊗ Leb) = lim
ϕw dQ0 = 0 (7.1)
for all W as above. The linear hull of the latter is dense in L2(Q0 ⊗ Leb). Hence, (7.1) implies
V = 0 which proves the closability of D.
The closure of (D,Z1(G0)) will be denoted by (D,Dom(D). Note that a priori it is not clear
whether D coincides with D on C1(G0). (See, however, Theorem 7.8 below.)
7.2 The Dirichlet Form
Definition 7.4. For u, v ∈ Z1(G0) ∪ C1(G0) we define the ’Wasserstein Dirichlet integral’
E(u, v) =
〈Du(g),Dv(g)〉L2 dQ0(g). (7.2)
Theorem 7.5. (i) (E,Z1(G0)) is closable. Its closure (E,Dom(E)) is a regular, recurrent
Dirichlet form on L2(G0,Q0).
Dom(E) = Dom(D) and for all u, v ∈ Dom(D)
E(u, v) =
G0×[0,1]
Du · Dv d(Q0 ⊗ Leb).
(ii) The set Z∞0 (G0) of all cylinder functions u ∈ Z∞(G0) of the form u(g) = U(
~α(gs)ds) with
U ∈ C∞(Rm,R) and ~α = (α1, . . . , αm) ∈ C∞([0, 1],Rm) satisfying α′i(0) = α′i(1) = 0 is a core
for (E,Dom(E)).
(iii) The generator (L,Dom(L) of (E,Dom(E)) is the Friedrichs extension of the operator
(L,Z∞0 (G0) given by
Lu(g) = −
D∗αiui(g)
i,j=1
∂i∂jU
~α(gs)ds
α′i(gs)α
j(gs)ds +
~α(gs)ds
· V β
where ui(g) := ∂iU(
~α(gs)ds) and V
(g) denotes the drift term defined in section 5.1 with
ϕ = α′i; β > 0 is the parameter of the entropic measure fixed throughout the whole chapter.
(iv) The Dirichlet form (E,Dom(E)) has a square field operator given by
Γ(u, v) := 〈Du,Dv〉L2(Leb) ∈ L1(G0,Q0)
with Dom(Γ) = Dom(E) ∩ L∞(G0,Q0). That is, for all u, v, w ∈ Dom(E) ∩ L∞(G0,Q0)
w · Γ(u, v) dQ0 = E(u, vw) + E(uw, v) − E(uv,w). (7.3)
Proof. (a) The closability of the form (E,Z1(G0)) follows immediately from the previous Propo-
sition 7.3. Alternatively, we can deduce it from assertion (iii) which we are going to prove first.
(b) Our first claim is that E(u,w) = −
u · Lw dQ0 for all u,w ∈ Z∞0 (G0). Let u(g) =
~α(gs)ds) and w(g) = W (
~γ(gs)ds) with U,W ∈ C∞(Rm,R) and ~α = (α1, . . . , αm), ~γ =
(γ1, . . . , γm) ∈ C∞([0, 1],Rm) satisfying α′i(0) = α′i(1) = γ′i(0) = γ′i(1) = 0. Observe that
〈Du(g),Dw(g)〉L2 =
i,j=1
~α(gs)ds) · ∂jW (
~γ(gs)ds) ·
α′i(gs)γ
j(gs)ds
ui(g) ·Dα′
w(g).
Hence, according to the integration by parts formula from Proposition 5.10
E(u,w) =
〈Du(g),Dw(g)〉L2 dQ0(g)
ui(g) ·Dα′
w(g) dQ0(g)
ui(g) · w(g) dQ0(g)
Lu(g) · w(g) dQ0(g).
This proves our first claim. In particular, (L,Z∞0 (G0)) is a symmetric operator. Therefore, the
form (E,Z∞0 (G0)) is closable and its generator coincides with the Friedrichs extension of L.
(c) Now let us prove that Z∞0 (G0) is dense in Z1(G0). That is, let us prove that each function
u ∈ Z1(G0) can be approximated by functions uǫ ∈ Z∞0 (G0). For simplicity, assume that u is of the
form u(g) = U(
α(gs)ds) with U ∈ C1(R) and α ∈ C1([0, 1]). (That is, for simplicity, m = 1.)
Let Uǫ ∈ C∞(R) for ǫ > 0 be smooth approximations of U with ‖U − Uǫ‖∞ + ‖U ′ − U ′ǫ‖∞ → 0
as ǫ → 0 and let αǫ ∈ C∞(R) with α′ǫ(0) = α′ǫ(1) = 0 be smooth approximations of α with
‖α−αǫ‖∞ → 0 and α′ǫ(t) → α′(t) for all t ∈]0, 1[ as ǫ → 0. Moreover, assume that supǫ ‖α′‖∞ <
Define uǫ ∈ Z∞0 (G0) as uǫ(g) = Uǫ(
αǫ(gs)ds). Then uǫ → u in L2(G0,Q0) by dominated
convergence relative Q0.
Since
U ′ǫ(
αǫ(g(s))ds)
[0,1]
α′ǫ(gs)
2ds ≤ C,
(α′ǫ)
2(g(s))
ǫ→0−→ α′(gs)2 ∀s ∈ [0, 1] \
{g = 0} ∩ {g = 1}
[0, 1] \
{g = 0} ∩ {g = 1}
=]0, 1[ for Q0-almost all g ∈ G0
one finds by dominated convergence in L2([0, 1],Leb), for Q0-almost all g ∈ G0
U ′ǫ(
αǫ(gs)ds)
[0,1]
α′ǫ(gs)
ǫ→0−→
α(gs)ds)
[0,1]
α′(gs)
Hence also with
E(uǫ, uǫ) =
U ′ǫ(
αǫ(gs)ds)
α′ǫ(gs)
2dsQ0(dg)
ǫ→0−→
α(gs)ds)
α′(gs)
2dsQ0(dg)
by dominated convergence in L2(G0,Q0). In particular, {uǫ}ǫ constitutes a Cauchy sequence
relative to the norm ‖v‖2
E,1 := ‖v‖2L2(G,Q) + E(v, v). In fact, since the sequence uǫ is uniformly
bounded w.r.t. to ‖.‖E,1, by weak compactness there is a weakly converging subsequence in
(Dom(E), ‖.‖E,1). Since the associated norms converge, the convergence is actually strong in
(Dom(E), ‖.‖E,1). Moreover, since uǫ → u in L2(G0,Q0), this limit is unique. Hence the entire
sequence converges to u ∈ (Dom(E), ‖.‖E,1), such that in particular E(u, u) = limǫ→0E(uǫ, uǫ).
This proves our second claim. In particular, it implies that also (E,Z1(G0)) is closable and that
the closures of Z∞0 (G0) and Z1(G0) coincide.
(d) Obviously, (E,Dom(E)) has the Markovian property. Hence, it is a Dirichlet form. Since
the constant functions belong to Dom(E), the form is recurrent. Finally, the set Z1(G0) is dense
in (C(G0), ‖.‖∞) according to the theorem of Stone-Weierstrass since it separates the points in
the compact metric space G0. Hence, (E,Dom(E)) is regular.
(e) According to Leibniz’ rule, (7.3) holds true for all u, v, w ∈ Z1(G0). Arbitrary u, v, w ∈
Dom(E)∩L∞(G0,Q0) can be approximated in (E(.) + ‖.‖2)1/2 by un, vn, wn ∈ Z1(G0) which are
uniformly bounded on G0. Then unvn → uv, unwn → uw and vnwn → vw in (E(.) + ‖.‖2)1/2.
Moreover, we may assume that wn → w Q0-a.e. on G0 and thus
|wΓ(u, v) − wnΓ(un, vn)| dQ0 ≤
|w−wn|Γ(u, v)dQ0 +
|wn| · |Γ(u, v)−Γ(un, vn)|dQ0 → 0
by dominated convergence. Hence, (7.3) carries over from Z1(G0) to Dom(E) ∩L∞(G0,Q0).
Lemma 7.6. For each f ∈ G0 the function u : g 7→ 〈f, g〉L2 belongs to Dom(E).
Proof. (a) For f, g ∈ G0 put µf = f∗Leb and µg = g∗Leb. Recall that by Kantorovich duality
‖f − g‖2L2 =
d2W (µf , µg)
= sup
ϕdµf +
= sup
ϕ(ft)dt+
ψ(gt)dt
where the supϕ,ψ is taken over all (smooth, bounded) ϕ ∈ L1([0, 1], µf ), ψ ∈ L1([0, 1], µg)
satisfying ϕ(x) + ψ(y) ≤ 1
|x − y|2 for µf -a.e. x and µg-a.e. y in [0, 1]. Replacing ϕ(x) by
|x|2/2 − ϕ(x) (and ψ(y) by . . .) this can be restated as
〈f, g〉L2 = inf
ϕ(ft)dt+
ψ(gt)dt
(7.4)
where the infϕ,ψ now is taken over all (smooth, bounded) ϕ ∈ L1([0, 1], µf ), ψ ∈ L1([0, 1], µg)
satisfying ϕ(x) + ψ(y) ≥ 〈x, y〉 for µf -a.e. x and µg-a.e. y in [0, 1]. If g is strictly increasing
then ψ can be chosen as
ψ′ = f ◦ g−1,
cf. [Vil03], sect. 2.1 and 2.2.
(b) Now fix a countable dense set {gn}n∈N of strictly increasing functions in G0 and an arbitrary
function f ∈ G0. Let (ϕn, ψn) denote a minimizing pair for (f, gn) in (7.4) and define un : G0 → R
un(g) := min
i=1,...,n
ϕ(fi(t))dt+
ψi(g(t))dt
Note that ψ′i = f ◦ g
i and thus un(gi) = 〈f, gi〉 for all i = 1, . . . , n. Therefore,
|un(g)− un(g̃)| ≤ max
|ψi(g(t))dt − ψi(g̃(t))|dt ≤ max
‖ψ′i‖∞ ·
|g(t)− g̃(t)|dt ≤ ‖g − g̃‖L1
for all g, g̃ ∈ G0. Hence, un → u pointwise on G0 and in L2(G0,Q0) where u(g) := 〈f, g〉.
(c) The function un is in the class Z
0(G0):
un(g) = Un
~α(gt)dt
with Un(x1, . . . , xn) = min{c1 + x1, . . . , cn + xn}, ci =
ϕi(f(t))dt and αi = ψi. The function
Un can be easily approximated by C1 functions in order to verify that un ∈ Dom(E) and
Dun(g) =
1Ai(g) · ψ′i(g(.))
with a suitable disjoint decomposition G0 = ∪iAi. (More precisely, Ai denotes the set of all
g ∈ G0 satisfying
ϕ(fi(t))dt +
ψi(g(t))dt <
ϕ(fj(t))dt +
ψj(g(t))dt for all j < i and
ϕ(fi(t))dt+
ψi(g(t))dt ≤
ϕ(fi(t))dt+
ψi(g(t))dt for all j > i.) Thus
‖Dun(g)‖2 =
1Ai(g) ·
ψ′i(g(t))
E(un) ≤ max
‖ψ′i ◦ g‖2L2dQ0(g).
In particular, since |ψ′i| ≤ 1,
E(un) ≤ 1
and thus u ∈ Dom(E).
Lemma 7.7. For all u ∈ Z1(G0) and all w ∈ C1(G0) ∩Dom(E)
E(u,w) =
〈Du(g),Dw(g)〉L2dQ0(g) (7.5)
(with Du(g) and Dw(g) given explicitly as in Example 7.2).
Proof. Recall that for u ∈ Z∞0 (G0) of the form u(g) = U(
~α(gt)dt)
Lu(g) =
ui(g)
with ui(g) = ∂iU(
~α(gt)dt). Hence, for w ∈ C1(G0) of the form w(g) =W (〈~h, g〉)
E(u,w) = −
Lu(g)w(g) dQ0(g)
ui(g)w(g) dQ0(g) =
ui(g)Dα′
ui(g)w(g) dQ0(g)
i,j=1
~α(gt)dt) · ∂jW (
~h(t)g(t)dt) ·
α′i(g(t))hj(t)dt dQ0(g)
〈Du(g),Dw(g)〉dQ0(g).
This proves the claim provided u ∈ Z∞0 (G0). By density this extends to all u ∈ Z1(G0).
Theorem 7.8. (i) (E,C1(G0)) is closable and its closure coincides with (E,Dom(E)). Similarly,
(D,C1(G0)) is closable and its closure coincides with (D,Dom(D)).
(ii) For all u,w ∈ Z1(G0) ∪ C1(G0)
Γ(u,w)(g) = 〈Du(g),Dw(g)〉L2 , (7.6)
in particular, E(u,w) =
〈Du(g),Dw(g)〉L2dQ0(g) (with Du(g) and Dw(g) given explicitly as
in Example 7.2).
(iii) For each f ∈ G0 the function uf : g 7→ ‖f − g‖L2 belongs to Dom(E) and Γ(uf , uf ) ≤ 1
Q0-a.e. on G0.
(iv) (E,Dom(E)) is strongly local.
Proof. (a) Claim: For each f ∈ L2([0, 1],Leb) the function uf : g 7→ 〈f, g〉L2 belongs to Dom(E)
and E(uf , uf ) = ‖f‖2L2 .
Indeed, if f ∈ L2 ∩ C1 then f = c0 + c1f1 + c2f2 with f1, f2 ∈ G0 and c0, c1, c2 ∈ R. Hence,
uf ∈ Dom(E) according to Lemma 7.6 and E(uf , uf ) =
‖Duf‖2dQ0 = ‖f‖2 according to
Lemma 7.7. Finally, each f ∈ L2 can be approximated by fn ∈ L2 ∩ C1 with ‖f − fn‖ → 0.
Hence, uf ∈ Dom(E) and E(uf , uf ) = ‖f‖2.
(b) Claim: C1(G0) ⊂ Dom(E).
Let u ∈ C1(G0) be given with u(g) = U(〈~f , g〉), U ∈ C1(Rm,R), ~f = (f1, . . . , fm) ∈ L2([0, 1],Rm).
For each i = 1, . . . ,m let (wi,n)n∈N be an approximating sequence in (Z
1(G0), (E + ‖.‖2)1/2) for
wi : g 7→ 〈fi, g〉. Put un(g) = U(w1,n(g), . . . , wm,n(g)). Then un ∈ Z1(G0), un → u pointwise on
G0 and in L2(G0,Q0). Moreover,
E(un, un) =
∂iU(w1,n(g), . . . , wm,n(g))Dwi,n(g)‖2L2 dQ0(g)
∂iU(〈f1, g〉, . . . , 〈fm, g〉)Dwi(g)‖2L2 dQ0(g)
‖Du(g)‖2dQ0(g).
Hence, u ∈ Dom(E) and E(u, u) =
‖Du(g)‖2dQ0(g).
(c) Assertion (ii) then follows via polarization and bi-linearity. Assertion (iii) is an immediate
consequence of assertion (ii). Assertion (iii) allows to prove the locality of the Dirichlet form
(E,Dom(E)) in the same manner as in the proof of Theorem 6.2.
(d) Claim: C1(G0) is dense in Dom(E).
We have to prove that each u ∈ Z1(G0) can be approximated by un ∈ C1(G0). As usual, it suffices
to treat the particular case u(g) =
α(gt)dt for some α ∈ C1([0, 1]). Put Un(x1, . . . , xn) =
i=1 α(xi) and fn,i(t) = n · 1[ i−1
(t). Then
un(g) := Un(〈fn,1, g〉, . . . 〈fn,n, g〉) =
defines a sequence in C1(G0) with un(g) → u(g) pointwise on G0 and in L2(G0,Q0).
Moreover,
Dun(g) =
· 1[ i−1
[(.) (7.7)
and therefore
E(un) =
dQ0(g) −→
α′(gt)
2dtdQ0(g) = E(u). (7.8)
Thus (un)n is Cauchy in Dom(E) and un → u in Dom(E).
7.3 Rademacher Property and Intrinsic Metric
We say that a function u : G0 → R is 1-Lipschitz if
|u(g)− u(h)| ≤ ‖g − h‖L2 (∀g, h ∈ G0).
Theorem 7.9. Every 1-Lipschitz function u on G0 belongs to Dom(E) and Γ(u, u) ≤ 1 Q0-a.e.
on G0.
Before proving the theorem in full generality, let us first consider the following particular case.
Lemma 7.10. Given n ∈ N, let {h1, . . . , hn} be a orthonormal system in L2([0, 1],Leb) and let
U be a 1-Lipschitz function on Rn. Then the function u(g) = U(〈h1, g〉, . . . , 〈hn, g〉) belongs to
Dom(E) and Γ(u, u) ≤ 1 Q0-a.e. on G0.
Proof. Let us first assume that in addition U is C1. Then according to Theorem 7.8, u is in
Dom(E) and Du(g) =
i=1 ∂iU(〈~h, g〉) · hi. Thus
Γ(u, u)(g) = ‖Du(g)‖L2 =
|∂iU(〈~h, g〉)|2 ≤ 1.
In the case of a general 1-Lipschitz continuous U on Rn we choose an approximating sequence
of 1-Lipschitz functions Uk, k ∈ N, in C1(Rn) with Uk → U uniformly on Rn and put uk(g) =
Uk((〈~h, g〉) for g ∈ G0. Then uk → u pointwise and in L2(G0,Q0). Hence, u ∈ Dom(E) and
Γ(u, u) ≤ 1 Q0-a.e. on G0.
Proof of Theorem 7.9. Every 1-Lipschitz function u on G0 can be extended to a 1-Lipschitz
function ũ on L2([0, 1],Leb) (’Kirszbraun extension’). Hence, without restriction, assume that
u is a 1-Lipschitz function on L2([0, 1],Leb). Choose a complete orthonormal system {hi}i∈N of
the separable Hilbert space L2([0, 1],Leb) and define for each n ∈ N the function Un : Rn → R
Un(x1, . . . , xn) = u
for x = (x1, . . . , xn) ∈ Rn. This function Un is 1-Lipschitz on Rn:
|Un(x)− Un(y)| ≤
xihi −
≤ |x− y|.
Hence, according to the previous Lemma the function
un(g) = Un(〈h1, g〉, . . . , 〈hn, g〉)
belongs belongs to Dom(E) and Γ(un, un) ≤ 1 Q0-a.e. on G0.
Note that
un(g) = u
〈hi, g〉hi
for each g ∈ L2([0, 1],Leb). Therefore, un → u on L2([0, 1],Leb) since
i=1〈hi, g〉hi → g
on L2([0, 1],Leb) and since u is continuous on L2([0, 1],Leb). Thus, finally, u ∈ Dom(E) and
Γ(u, u) ≤ 1 Q0-a.e. on G0.
Our next goal is the converse to the previous Theorem.
Theorem 7.11. Every continuous function u ∈ Dom(E) with Γ(u, u) ≤ 1 Q0-a.e. on G0 is
1-Lipschitz on G0.
Lemma 7.12. For each u ∈ C1(G0) ∪ Z1(G0) and all g0, g1 ∈ G0
u(g1)− u(g0) =
〈Du ((1− t)g0 + tg1) , g1 − g0〉L2dt. (7.9)
Proof. Put gt = (1− t)g0 + tg1 and consider the C1 function η : [0, 1] → R defined by ηt = u(gt).
η̇t = Dg1−g0u(gt) = 〈Du(gt), g1 − g0〉
and thus
η1 − η0 =
η̇tdt =
〈Du(gt), g1 − g0〉dt.
Lemma 7.13. Let g0, g1 ∈ G0 ∩C3 and put gt = (1− t)g0 + tg1. Then for each u ∈ Dom(E) and
each bounded measurable Ψ : G0 → R
[u(g1 ◦ h)− u(g0 ◦ h)]Ψ(h) dQ0(h) =
〈Du(gt ◦ h, (g1 − g0) ◦ h〉Ψ(h)Q0(h)dt. (7.10)
Proof. Given g0, g1, Ψ and u ∈ Dom(E) as above, choose an approximating sequence in Z1(G0)∪
C1(G0) with un → u in Dom(E) as n→ ∞. According to the previous Lemma for each n
[un(g1◦h)−un(g0◦h)]Ψ(h) dQ0(h) =
〈Dun (gt ◦ h) , (g1−g0)◦h〉Ψ(h) dQ0(h)dt. (7.11)
By assumption un → u in L2(G0,Q0) and Dun → Du in L2(G0 × [0, 1],Q0 ⊗ Leb) as n → ∞.
Using the quasi-invariance of Q0 (Theorem 4.3) this implies
|u(gt ◦ h)− un(gt ◦ h)|Ψ(h) dQ0(h) =
[u(h)− un(h)|Ψ(g−1t ◦ h) · Y
(h) dQ0(h) → 0
as n→ ∞ as well as
‖Du(gt ◦ h)− Dun(gt ◦ h)|2L2Ψ(h)Q0(h)
‖Du(h) − Dun(h)|2L2Ψ(g
t ◦ h) · Y
(h)Q0(h) → 0
Hence, we may pass to the limit n→ ∞ in (7.11) which yields the claim.
Proof of Theorem 7.11. Let a continuous u ∈ Dom(E) be given with Γ(u, u) ≤ 1 Q0-a.e. on G0.
We want to prove that u(g1)− u(g0) ≤ ‖g1 − g0‖L2 for all g0, g1 ∈ G0. By density of G0 ∩ C3 in
G0 and by continuity of u it suffices to prove the claim for g0, g1 ∈ G0 ∩ C3.
Choose a sequence of bounded measurable Ψk : G0 → R+ such that the probability measures
ΨkdQ0 on G0 converge weakly to δe, the Dirac mass in the identity map e ∈ G0. Then according
to the previous Lemma and the assumption ‖Du‖ ≤ 1
[u(g1 ◦ h)− u(g0 ◦ h)]Ψk((h)dQ0(h)
〈Du(gt ◦ h, (g1 − g0) ◦ h〉Ψk(h) dQ0(h)dt
‖Du(gt ◦ h)‖L2 · ‖(g1 − g0) ◦ h‖L2 ·Ψk(h) dQ0(h)dt
‖(g1 − g0) ◦ h‖L2 ·Ψk(h) dQ0(h).
Now the integrands on both sides, h 7→ u(g1 ◦h)−u(g0 ◦h) as well as h 7→ ‖(g1 − g0) ◦h‖L2 , are
continuous in h ∈ G0. Hence, as k → ∞ by weak convergence ΨkdQ0 → δe we obtain
u(g1)− u(g0) ≤ ‖g1 − g0‖L2 .
Corollary 7.14. The intrinsic metric for the Dirichlet form (E,Dom(E)) is the L2-metric:
‖g1 − g0‖L2 = sup {u(g1)− u(g0) : u ∈ C(G0) ∩Dom(E), Γ(u, u) ≤ 1Q0-a.e. on G0}
for all g0, g1 ∈ G0.
7.4 Finite Dimensional Noise Approximations
The goal of this section is to present representations – and finite dimensional approximations –
of the Dirichlet form
E(u, v) =
〈Du(g),Dv(g)〉L2 dQ0(g)
in terms of globally defined vector fields.
If (ϕi)i∈N is a complete orthonormal system in Tg = L
2([0, 1], g∗Leb) for a given g ∈ G0 then
obviously
〈Du(g),Dv(g)〉L2 =
Dϕiu(g)Dϕiv(g). (7.12)
Unfortunately, however, there exists no family (ϕi)i∈N which is simultaneously orthonormal in
all Tg = L
2([0, 1], g∗Leb), g ∈ G0. For a general family, the representation (7.12) should be
replaced by
〈Du(g),Dv(g)〉L2 =
i,j=1
Dϕiu(g) · aij(g) ·Dϕjv(g) (7.13)
where a(g) = (aij(g))i,j∈N is the ’generalized inverse’ to Φ(g) = (Φij(g))i,j∈N with
Φij(g) := 〈ϕi, ϕj〉Tg =
ϕi(gt)ϕj(gt)dt.
In order to make these concepts rigorous, we have to introduce some notations.
For fixed n ∈ N let S+(n) ⊂ Rn×n denote the set of symmetric nonnegative definite real (n×n)-
matrices. For each A ∈ S+(n) a unique element A−1 ∈ S+(n), called generalized inverse to A,
is defined by
A−1x :=
0 if x ∈ Ker(A),
y if x ∈ Ran(A) with x = Ay
This definition makes sense since (by the symmetry of A) we have an orthogonal decomposition
Rn = Ker(A)⊕ Ran(A). Obviously,
A−1 ·A = A · A−1 = πA
where πA denotes the projection onto Ran(A).
Moreover, for each A ∈ S+(n) there exists a unique element A1/2 ∈ S+(n), called nonnegative
square root of A, satisfying
A1/2 · A1/2 = A.
Let Ψ(n) denote the map A 7→ A−1, regarded as a map from S+(n) ⊂ Rn×n to Rn×n, with
ij (A) = (A
−1)ij for i, j = 1, . . . , n. Similarly, put
Ξ(n) : S+(n) → S+(n), A 7→ (A1/2)−1 = (A−1)1/2.
Note that Ψ(n)(A) = Ξ(n)(A) · Ξ(n)(A) for all A ∈ S+(n).
The maps Ψ(n) and Ξ(n) are smooth on the subset of positive definite matrices A ∈ S+(n) but
unfortunately not on the whole set S+(n). However, they can be approximated from below
(in the sense of quadratic forms) by smooth maps: there exists a sequence of C∞ maps Ξ(n,l) :
Rn×n → Rn×n with
ξ · Ξ(n,k)(A) · ξ ≤ ξ · Ξ(n,l)(A) · ξ
for all A ∈ S+(n), ξ ∈ Rn and all k, l ∈ N with k ≤ l and
(n,l)
ij (A) → Ξ
ij (A) = (A
−1/2)ij
for all A ∈ S+(n), i, j ∈ {1, . . . , n} as l → ∞. Put Ψ(n,l)(A) = Ξ(n,l)(A) ·Ξ(n,l)(A) for A ∈ Rn×n.
Then the sequence (Ψ(n,l))l∈N approximates Ψ
(n) from below in the sense of quadratic forms.
Now let us choose a family {ϕi}i∈N of smooth functions ϕi : [0, 1] → R which is total in C0([0, 1])
w.r.t. uniform convergence (i.e. its linear hull is dense). Put
Φij(g) := 〈ϕi, ϕj〉Tg =
ϕi(gx)ϕj(gx)dx
(n,l)
ij (g) = Ψ
(n,l)
ij (Φ(g)) , σ
(n,l)
ij (g) = Ξ
(n,l)
ij (Φ(g)).
Note that the maps g 7→ a(n,l)ij (g) and g 7→ σ
(n,l)
ij (g) (for each choice of n, l, i, j) belong to the
class Z∞(G0). Moreover, put
ij (g) = Ψ
ij (Φ(g)) .
Then obviously the orthogonal projection πn onto the linear span of {ϕ1, . . . , ϕn} ⊂ Tg =
L2([0, 1], g∗Leb) is given by
πnu =
i,j=1
ij (g) · 〈u, ϕi〉Tg · ϕj
〈πnu, πnv〉Tg =
i,j=1
〈u, ϕi〉Tg · a
ij (g) · 〈v, ϕj〉Tg
for all u, v ∈ Tg.
Theorem 7.15. (i) For each n, l ∈ N the form (E(n,l),Z1(G0)) with
(n,l)(u, v) =
i,j=1
Dϕiu(g) · a
(n,l)
ij (g) ·Dϕjv(g) dQ0(g)
is closable. Its closure is a Dirichlet form with generator being the Friedrichs extension of the
symmetric operator (L(n,l),Z2(G0)) given by
(n,l) =
i,j=1
(n,l)
ij ·DϕiDϕj +
i,j=1
(n,l)
ij + a
(n,l)
ij · V
Dϕj . (7.14)
(ii) As l → ∞
(n,l) ր E(n)
where
(n)(u, v) =
i,j=1
Dϕiu(g) · a
ij (g) ·Dϕjv(g) dQ0(g).
for u, v ∈ Z1(G0). Hence, in particular, E(n) is a Dirichlet form.
(iii) As n→ ∞
(n) ր E
(which provides an alternative proof for the closability of the form (E,Z1(G0))).
Proof. (i) The function a
(n,l)
i,j on G0 is a cylinder function in the class Z1(G0). The integration
by parts formula for the Dϕi , therefore, implies that for all u, v ∈ Z2(G0)
(n,l)(u, v) =
Dϕiu(g)Dϕjv(g)a
(n,l)
ij (g)dQ0(g)
u(g) ·D∗ϕi
(n,l)
ij Dϕjv
(g) dQ0(g) = −
u(g) · L(n,l)v(g) dQ0(g).
(n,l) = −
i,j=1
(n,l)
ij Dϕj
Hence, (E(n,l),Z2(G0)) is closable and the generator of its closure is the Friedrichs extension of
(L(n,l),Z2(G0)).
(ii) The monotone convergence E(n,l) ր E(n) of the quadratic forms is an immediate consequence
of the fact that a(n,l)(g) ր a(n)(g) (in the sense of symmetric matrices) for each g ∈ G0 which in
turn follows from the defining properties of the approximations Ψ(n,l) of the generalized inverse
Ψ(n).
The limit of an increasing sequence of Dirichlet forms is itself again a Dirichlet form provided it
is densely defined which in our case is guaranteed since it is finite on Z2(G0).
(iii) Obviously, the En, n ∈ N constitute an increasing sequence of Dirichlet forms with En ≤ E
for all n. Moreover, Z1(G0) is a core for all the forms under consideration. Hence, it suffices to
prove that for each u ∈ Z1(G0) and each ǫ > 0 there exists an n ∈ N such that
(n)(u, u)− E(u, u)
To simplify notation, assume that u is of the form u(g) = U(
α(gt)dt) for some U ∈ C1c (R)
and some α ∈ C1([0, 1]). By assumption, the set {ϕi, i ∈ N} is total in C0([0, 1]) w.r.t. uniform
convergence. Hence, for each δ > 0 there exist n ∈ N and ϕ ∈ span(ϕ1, . . . , ϕn) with ‖α′−ϕ‖sup ≤
δ which implies
〈α′, ϕ〉Tg
‖ϕ‖Tg
≥ ‖ϕ‖Tg − δ ≥ ‖α′‖Tg − 2δ.
E(u, u) ≥ E(n)(u, u) ≥
α(gt)dt)
2 · 〈α′, ϕ〉2Tg ·
‖ϕ‖2Tg
dQ0(g)
α(gt)dt)
‖α′‖Tg − 2δ
dQ0(g)
α(gt)dt)
1 + δ
‖α′‖2Tg − 4δ
dQ0(g)
1 + δ
E(u, u)− 4δ‖U ′‖2sup.
Hence, for δ sufficiently small, E(u, u) and E(n)(u, u) are arbitrarily close to each other.
Remark 7.16. For any given g0 ∈ G0, let (gt)t≥0 with gt : (x, ω) 7→ gxt (ω) be the solution to the
dgxt =
i,j=1
(n,l)
ij (gt) · ϕj(g
t ) dW
i,j=1
(n,l)
ij (gt) · ϕj(g
t ) ·
ϕ′i(g
t ) + V
i,j=1
k,m=1
(n,l)
ij (Φ(gt)) · 〈(ϕkϕm)
′, ϕi〉Tg · ϕj(gxt )dt
where ∂kmΨ
(n,l)
ij for (k,m) ∈ {1, . . . , n}2 denotes the 1st order partial derivative of the function
(n,l)
ij : R
n×n → R with respect to the coordinate xkm. Then the generator of the process coincides
on Z2(G0) with the operator 12L
(n,l) from (7.14), the generator of the Dirichlet form E(n,l).
Let us briefly comment on the various terms in the SDE from above:
• The first one,
i,j=1 σ
(n,l)
ij (gt) · ϕj(gxt ) dW it is the diffusion term, written in Ito form;
• the second one, 1
i,j=1 a
(n,l)
ij (gt) ·ϕj(gxt ) ·ϕ′i(gxt )dt is a drift which comes from the trans-
formation between Stratonovich and Ito form (it would disappear if we wrote the diffusion
term in Stratonovich form).
• The next one, 1
i,j=1 a
(n,l)
ij (gt) ·ϕj(gxt ) · V
ϕi(gt)dt is a drift which arises from our change
of variable formula. Actually, since
V βϕi(g) = β
ϕ′i(g(y))dy +
ϕ′i(g(a+)) + ϕ
i(g(a−))
− ϕi(g(a+)) − ϕi(g(a−))
g(a+)− g(a−)
it consists of two parts, one originates in the logarithmic derivative of the entropy of the
g’s (which finally will force the process to evolve as a stochastic perturbation of the heat
equation), the other one is created by the jumps of the g’s.
• The last term, 1
i,j=1
k,m=1 ∂kmΨ
(n,l)
ij (Φ(gt)) · 〈(ϕkϕm)′, ϕi〉Tg · ϕj(gxt )dt involves the
derivative of the diffusion matrix. It arises from the fact that the generator is originally
given in divergence form.
7.5 The Wasserstein Diffusion (µt) on P0
The objects considered previously – derivative, Dirichlet form and Markov process on G0 – have
canonical counterparts on P0. The key to these objects is the bijective map χ : G0 → P0,
g 7→ g∗Leb.
We denote by Zk(P0) the set of all (’cylinder’) functions u : P0 → R which can be written as
u(µ) = U
α1dµ, . . . ,
(7.15)
with some m ∈ N, some U ∈ Ck(Rm) and some ~α = (α1, . . . , αm) ∈ Ck([0, 1],Rm) . The subset
of u ∈ Zk(P0) with α′i(0) = α′i(1) = 0 for all i = 1, . . . ,m will be denoted by Zk0(P0). For
u ∈ Z1(P0) represented as above we define its gradient Du(µ) ∈ L2([0, 1], µ) by
Du(µ) =
~αdµ) · α′i(.)
with norm
‖Du(µ)‖L2(µ) =
~αdµ) · α′i
The tangent space at a given point µ ∈ P0 can be identified with L2([0, 1], µ). The action of a
tangent vector ϕ ∈ L2([0, 1], µ) on µ (’exponential map’) is given by the push forward ϕ∗µ.
Theorem 7.17. (i) The image of the Dirichlet form defined in (7.2) under the map χ is the
regular, strongly local, recurrent Wasserstein Dirichlet form E on L2(P0,P0) defined on its core
Z1(P0) by
E(u, v) =
〈Du(µ),Dv(µ)〉2L2(µ)dP0(µ). (7.16)
The Dirichlet form has a square field operator, defined on Dom(E) ∩ L∞, and given on Z1(P0)
Γ(u, v)(µ) = 〈Du(µ),Dv(µ)〉2L2(µ).
The intrinsic metric for the Dirichlet form is the L2-Wasserstein distance dW . More precisely,
a continuous function u : P0 → R is 1-Lipschitz w.r.t. the L2-Wasserstein distance if and only
if it belongs to Dom(E) and Γ(u, u)(µ) ≤ 1 for P0-a.e. µ ∈ P0.
(ii) The generator of the Dirichlet form is the Friedrichs extension of the symmetric operator
(L,Z20(P0) on L2(P0,P0) given as L = L1 + L2 + β · L3 with
L1u(µ) =
i,j=1
∂i∂jU(
~αdµ) ·
L2u(µ) =
~αdµ) ·
I∈gaps(µ)
α′′i (I−) + α
i (I+)
i(I+)− α′i(I−)
i (0) + α
i (1)
L3u(µ) =
~αdµ) ·
α′′i dµ.
Recall that gaps(µ) denotes the set of intervals I = ]I−, I+[⊂ [0, 1] of maximal length with
µ(I) = 0 and |I| denotes the length of such an interval.
(iii) For P0-a.e. µ0 ∈ P0, the associated Markov process (µt)t≥0 on P0 starting in µ0, called
Wasserstein diffusion, with generator 1
L is given as
µt(ω) = gt(ω)∗Leb
where (gt)t≥0 is the Markov process on G0 associated with the Dirichlet form of Theorem 7.5,
starting in g0 := χ
−1(µ0).
For each u ∈ Z20(P0) the process
u(µt)− u(µ0)−
Lu(µs)ds
is a martingale whenever the distribution of µ0 is chosen to be absolutely continuous w.r.t. the
entropic measure P0. Its quadratic variation process is
Γ(u, u)(µs)ds.
Remark 7.18. L1 is the second order part (’diffusion part’) of the generator L, L2 and L3
are first order operators (’drift parts’). The operator L1 describes the diffusion on P0 in all
directions of the respective tangent spaces. This means that the process (µt) at each time t ≥ 0
experiences the full ’tangential’ L2([0, 1], µt)-noise.
L3 is the generator of the deterministic semigroup (’Neumann heat flow’) (Ht)t≥0 on L
2(P0,P0)
given by
Htu(µ) = u(htµ).
Here ht is the heat kernel on [0, 1] with reflecting (’Neumann’) boundary conditions and htµ(.) =
ht(., y)µ(dy). Indeed, for each u ∈ Z10(P0) given as u(g) = U(
~αdµ) we obtain Htu(µ) =
~α(x)ht(x, y)µ(dy)dx
and thus
∂tHtu(µ) =
∂iU(htµ) · ∂t
αi(x)ht(x, y)µ(dy)dx
∂iU(htµ) ·
αi(x)h
t (x, y)µ(dy)dx
∂iU(htµ) ·
α′′i (x)ht(x, y)µ(dy)dx = L3Htu(µ).
Note that L depends on β only via the drift term L3 and
L → L3 as β → ∞.
The following statement, which in the finite dimensional case is known as Varadhan’s formula,
exhibits another close relationship between (µt) and the geometry of (P([0, 1]), dW ). The Gaus-
sian short time asymptotics of the process (µt)t≥0 are governed by the L
2-Wasserstein distance.
Corollary 7.19. For measurable sets A,B ∈ P0 with positive P0-measure, let dW (A,B) =
inf{dW (ν, ν̃) | ν ∈ A, ν̃ ∈ B} and pt(A,B) =
pt(ν, dν̃)P0(dν) where pt(ν, dν̃) denotes the
transition semigroup for the process (µt)t≥0.
t log pt(A,B) = −
dW (A,B)
. (7.17)
Proof. This type of result is known as Varadhan’s formula. Its respective form for (E,Dom(E) on
L2(P0,P0) holds true by the very general results of [HR03] for conservative symmetric diffusions,
and the identification of the intrinsic metric as dW in our previous Theorem.
Due to the sample path continuity of (µt) the Wasserstein diffusion is equivalently characterized
by the following martingale problem. Here we use the notation 〈α, µt〉 =
α(x)µt(dx).
Corollary 7.20. For each α ∈ C2([0, 1]) with α′(0) = α′(1) = 0 the process
Mt = 〈α, µt〉 −
〈α′′, µs〉ds
I∈gaps(µs)
α′′(I−) + α
′′(I+)
′(I+)− α′(I−)
′′(0) + α′′(1)
is a continuous martingale with quadratic variation process
[M ]t =
〈(α′)2, µs〉ds.
Remark 7.21. For illustration one may compare corollary 7.20 for (µt) in the case β = 1 to
the respective martingale problems for four other well-known measure valued process, say on
the real line, namely the so-called super-Brownian motion or Dawson-Watanabe process (µDWt ),
the Fleming-Viot process (µFW ), both of which we can consider with the Laplacian as drift, the
Dobrushin-Doob process (µDDt ) which is the empirical measure of independent Brownian motions
with locally finite Poissonian starting distribution, cf. [AKR98], and finally simply the empirical
measure process of a single Brownian motion (µBMt = δXt). For each i ∈ {DW,FV,DD,BM}
and sufficiently regular α : R → R the process M it := 〈α, µit〉 − 12
〈α′′, µis〉ds is a continuous
martingale with quadratic variation process
[MDW ]t =
〈α2, µDWs 〉ds,
[MFV ]t =
[〈α2, µFVs 〉 − (〈α, µFVs 〉)2]ds,
[MDD]t =
〈(α′)2, µDDs 〉ds,
[MBM ]t =
〈(α′)2, µBMs 〉ds.
In view of corollary 7.19 the apparent similarity of µDD and µBM to the Wasserstein diffusion
µ is no suprise. However, the effective state spaces of µDD, µBM and µt are as much different
as their invariant measures.
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	Introduction
	Spaces of Probability Measures and Monotone Maps
	The Spaces P0=P([0,1]) and G0
	The Spaces G, G1 and P=P(S1)
	Dirichlet Process and Entropic Measure
	Gibbsean Interpretation and Heuristic Derivation of the Entropic Measure
	The Measures Q and P 
	The Measures Q0 and P0 
	The Dirichlet Process as Normalized Gamma Process
	Support Properties
	Scaling and Invariance Properties
	Dirichlet Processes on General Measurable Spaces
	The Change of Variable Formula for the Dirichlet Process and for the Entropic Measure
	Heuristic Approaches to Change of Variable Formulae
	The Change of Variables Formula on the Sphere
	The Change of Variables Formula on the Interval
	Proofs for the Sphere Case
	Proof for the Interval Case
	The Integration by Parts Formula
	The Drift Term
	Directional Derivatives
	Integration by Parts Formula on P(S1)
	Derivatives and Integration by Parts Formula on P([0,1])
	Dirichlet Form and Stochastic Dynamics on on G
	The Dirichlet Form on G
	Finite Dimensional Noise Approximations
	Dirichlet Form and Stochastic Dynamics on G1 and P
	Dirichlet Form and Stochastic Dynamics on G0 and P0
	The Canonical Dirichlet Form on the Wasserstein Space
	Tangent Spaces and Gradients
	The Dirichlet Form
	Rademacher Property and Intrinsic Metric
	Finite Dimensional Noise Approximations
	The Wasserstein Diffusion (t) on P0
