First variation formula in Wasserstein spaces
over compact Alexandrov spaces
Nicola Gigli∗and Shin-ichi Ohta†
July 29, 2010
Abstract
We extend results proven by the second author (Amer. J. Math., 2009) for
nonnegatively curved Alexandrov spaces to general compact Alexandrov spaces X
with curvature bounded below: the gradient ﬂow of a geodesically convex functionalon the quadratic Wasserstein space (P(X), W2 ) satisﬁes the evolution variational
inequality. Moreover, the gradient ﬂow enjoys uniqueness and contractivity. These
results are obtained by proving a ﬁrst variation formula for the Wasserstein distance.

1

Introduction

This paper should be considered as an addendum to [Oh] of the second author. In [Oh], itis studied the quadratic Wasserstein space (P(X), W2 ) built over a compact Alexandrov
space X with curvature bounded below, and proven the existence of Euclidean tangent
cones (see also [Gi]). This result is particularly interesting for Alexandrov spaces with anegative curvature bound, as it is known that Wasserstein spaces built over them do not
admit lower curvature bounds in the sense of Alexandrov.
The existence of such tangent cones has then been used in [Oh] to perform studies ofgradient ﬂows of geodesically convex functionals on (P(X), W2 ). In particular, existenceof gradient ﬂows of such functionals has been proven via an approach inspired by [Ly](which diﬀers from the ‘purely metric’ approach in [AGS] without using tangent cones).
For technical reasons, however, uniqueness and contraction of such gradient ﬂows havebeen obtained only in the case where X has the nonnegative curvature.
In this paper we extend these latter results to general Alexandrov spaces with curvaturebounded below possibly by a negative value (Theorem 4.2). A key tool in our approach is aﬁrst variation formula for the Wasserstein distance (Theorem 3.4) from which it also easily
follows that gradient ﬂows satisfy the evolution variational inequality (Proposition 4.1).
See Remarks 3.7, 4.5 for the diﬀerence from the argument in [Oh].
Section 2 is devoted to recalling known results on the geometric structure of andgradient ﬂows in (P(X), W2 ). We show the ﬁrst variation formula in Section 3, and use
it to study gradient ﬂows in Section 4.
∗

Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany(nicolagigli@googlemail.com)†
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan (sohta@math.kyoto-u.ac.jp);
Supported in part by the Grant-in-Aid for Young Scientists (B) 20740036.

1

2

Preliminaries

2.1

Wasserstein spaces over compact Alexandrov spaces

Let (X, d) be a metric space. A rectiﬁable curve γ : [0, l] −→ X is called a geodesic if it
is locally minimizing and has a constant speed. We say that γ is minimal if it is globally
minimizing (i.e., d(γ(s), γ(t)) = (|s − t|/l) · d(γ(0), γ(l)) for all s, t ∈ [0, l]). If any two
points in X are joined by some minimal geodesic, then X is called a geodesic space.
Throughout the article, (X, d) will be a compact Alexandrov space of curvature ≥ −1.
By this we mean that (X, d) is a geodesic space such that every triangle in X is thicker
than a geodesic triangle (with the same side lengths) in the hyperbolic plane H2 (−1) ofconstant sectional curvature −1 (see [Oh] for the detailed deﬁnition, [BGP], [OtSh] and[BBI] for the basic theory). We remark that (X, d) can be inﬁnite-dimensional, so thatits local structure may be very wild.
Denote by P(X) the set of all Borel probability measures on X. Given µ, ν ∈ P(X),
we consider the L2 -Wasserstein distance
{∫
}1/2
2
W2 (µ, ν) := inf
d(x, y) dπ(x, y)
,
π

X×X

where π ∈ P(X × X) runs over all couplings of µ and ν. Note that W2 (µ, ν) is ﬁnite and(P(X), W2 ) is compact as X is assumed to be compact. We refer to [AGS] and [Vi] formore on Wasserstein geometry and optimal transport theory.
It is known that, if (X, d) has nonnegative curvature, then so does (P(X), W2 ) ([St2,
Proposition 2.10], [LV, Theorem A.8]). Although the analogues implication is false fornegative curvature bounds (essentially because it is not a scaling invariant condition, [St2,
Proposition 2.10]), we obtain the following generalization of the 2-uniform smoothness in
Banach space theory.
Proposition 2.1 ([Oh, Proposition 3.1, Lemma 3.3], [Sa]) For all µ, ν, ω ∈ P(X), allminimal geodesics α : [0, 1] −→ P(X) from ν to ω and for all τ ∈ [0, 1], we have()2
W2 µ, α(τ ) ≥ (1 − τ )W2 (µ, ν)2 + τ W2 (µ, ω)2 − S 2 (1 − τ )τ W2 (ν, ω)2 ,where S =(1)√
1 + (diam X)2 .

In fact, the 2-uniform smoothness (1) holds in (X, d) and descends to (P(X), W2 ) withthe same constant S. Although (P(X), W2 ) is not an Alexandrov space, it is possible toshow the following (see also [Oh, Theorem 3.6]):
Theorem 2.2 ([Gi, Theorem 3.4, Remark 3.5]) Given µ ∈ P(X) and unit speed geodesics
α, β : [0, δ) −→ P(X) with α(0) = β(0) = µ, the joint limit
s2 + t2 − W2 (α(s), β(t))2
s,t→0
2st
lim

exists.

2

Theorem 2.2 means that (P(X), W2 ) looks like a Riemannian space (rather than a
Finsler space), and we can investigate its inﬁnitesimal structure according to the theoryof Alexandrov spaces. For µ ∈ P(X), denote by Σ0µ [P(X)] the set of all (nontrivial) unit
speed minimal geodesics emanating from µ. Given α, β ∈ Σ0µ [P(X)], Theorem 2.2 veriﬁes
that the angle()
s2 + t2 − W2 (α(s), β(t))2∠µ (α, β) := arccos lim∈ [0, π]s,t→0
2stis well-deﬁned and provides an appropriate (pseudo-)distance structure of Σ0µ [P(X)]. We
deﬁne the space of directions (Σµ [P(X)], ∠µ ) as the completion of (Σ0µ [P(X)]/ ∼, ∠µ ),where α ∼ β holds if ∠µ (α, β) = 0. Then the tangent cone (Cµ [P(X)], σµ ) is deﬁned as
the Euclidean cone over (Σµ [P(X)], ∠µ ):()/()
Cµ [P(X)] := Σµ [P(X)] × [0, ∞)Σµ [P(X)] × {0} ,√()σµ (α, s), (β, t) := s2 + t2 − 2st cos ∠µ (α, β).
Note that σµ is a distance on Cµ [P(X)] and that()()σµ (α, s), (β, t) = lim W2 α(sε), β(tε) /ε
ε→0

(2)

holds by the deﬁnition of ∠µ . We denote the origin of Cµ [P(X)] by oµ and deﬁne()
h(α, s), (β, t)iµ := 2st cos ∠µ (α, β),|(α, s)|µ := s = σµ oµ , (α, s)for (α, s), (β, t) ∈ Cµ [P(X)]. The subscript µ will be omitted if the space under consideration is clearly understood. We sometimes abbreviate like t · (α, s) := (α, st) and identify
α ∈ Σµ [P(X)] with (α, 1) ∈ Cµ [P(X)].
Using this inﬁnitesimal structure, we introduce a class of ‘diﬀerentiable curves’.
Deﬁnition 2.3 (Right diﬀerentiability) We say that a curve ξ : [0, l) −→ P(X) is rightdiﬀerentiable at t ∈ [0, l) if there is v ∈ Cξ(t) [P(X)] such that, for any sequences {εi }i∈Nof positive numbers tending to zero and {αi }i∈N of unit speed minimal geodesics from ξ(t)to ξ(t + εi ), the sequence {(αi , W2 (ξ(t), ξ(t + εi ))/εi )}i∈N ⊂ Cξ(t) [P(X)] converges to v.
˙ = v.
Such v is clearly unique if it exists, and then we write ξ(t)˙ ξ(t) . We also remark that
In particular, we have limε→0 W2 (ξ(t), ξ(t + ε))/ε = |ξ(t)|every minimal geodesic α : [0, l] −→ P(X) is right diﬀerentiable at all t ∈ [0, l). Thisis because Alexandrov spaces are known to satisfy the non-branching property, which isinherited by (P(X), W2 ) (cf. [Vi, Corollary 7.32]). Therefore α|[0,t] is a unique minimal
geodesic between α(0) and α(t) for all t ∈ (0, l), and α̇(0) = (α, W2 (α(0), α(t))/t).

2.2

Gradient ﬂows in Wasserstein spaces

The contents of this subsection will come into play in Section 4. The readers interested
only in the ﬁrst variation formula can skip to Section 3.

3

Consider a lower semi-continuous function f : P(X) −→ (−∞, +∞] which is K-convex
for some K ∈ R in the sense that
(
)
(
)
(
) K
(
)2
f α(τ ) ≤ (1 − τ )f α(0) + τ f α(1) − (1 − τ )τ W2 α(0), α(1)2(3)

holds along all minimal geodesics α : [0, 1] −→ P(X) and all τ ∈ [0, 1]. We also supposethat f is not identically +∞, and deﬁneP ∗ (X) := {µ ∈ P(X) | f (µ) < ∞}.
The K-convexity guarantees that minimal geodesics between points in P ∗ (X) are again
included in P ∗ (X), and hence it makes sense to consider Σµ [P ∗ (X)] as well as Cµ [P ∗ (X)]for µ ∈ P ∗ (X).
Given µ ∈ P ∗ (X) and α ∈ Σµ [P ∗ (X)], we setf (β(t)) − f (µ),
t→0tDµ f (α) := lim inf lim
β→αwhere β ∈ Σ0µ [P ∗ (X)] is a unit speed geodesic and the convergence β → α is with respectto ∠µ . Clearly Dµ f is well-deﬁned on Σµ [P ∗ (X)]. Deﬁne the absolute gradient (called the
local slope in [AGS]) of f at µ ∈ P ∗ (X) by}{f (µ) − f (ν).
|∇− f |(µ) := max 0, lim sup
W2 (µ, ν)ν→µNote that −Dµ f (α) ≤ |∇− f |(µ) holds for any α ∈ Σµ [P ∗ (X)]. According to the argument
in [PP] and [Ly], we ﬁnd a negative gradient vector of f at each point in P ∗ (X) withﬁnite absolute gradient.
Lemma 2.4 ([Oh, Lemma 4.2]) For each µ ∈ P ∗ (X) with 0 < |∇− f |(µ) < ∞, there
exists unique α ∈ Σµ [P ∗ (X)] satisfying Dµ f (α) = −|∇− f |(µ). Moreover, for any β ∈Σµ [P ∗ (X)], it holds that Dµ f (β) ≥ −|∇− f |(µ)hα, βiµ .
The second assertion is regarded as a ﬁrst variation formula for f . Using α in theabove lemma, we deﬁne the negative gradient vector of f at µ as()∇− f (µ) := α, |∇− f |(µ) ∈ Cµ [P ∗ (X)].
In the case of |∇− f |(µ) = 0, we simply set ∇− f (µ) := oµ .
Deﬁnition 2.5 (Gradient curves) A continuous curve ξ : [0, l) −→ P ∗ (X) which is locally
Lipschitz continuous on (0, l) is called a gradient curve of f if |∇− f |(ξ(t)) < ∞ for all˙t ∈ (0, ∞) and if it is right diﬀerentiable with ξ(t)= ∇− f (ξ(t)) at all t ∈ [0, l) with|∇− f |(ξ(t)) < ∞. We say that a gradient curve ξ is complete if it is deﬁned on [0, ∞).
Again along the discussion in [PP] and [Ly], despite some technical diﬃculties as(P(X), W2 ) is not an Alexandrov space, we can show the existence of complete gradientcurves.

4

Theorem 2.6 ([Oh, Theorem 5.11]) For any µ ∈ P ∗ (X), there exists a complete gradientcurve ξ : [0, ∞) −→ P ∗ (X) of f with ξ(0) = µ.
Remark 2.7 Let us make a more detailed comment on the construction of gradientcurves. The strategy in [Oh] (following [PP] and [Ly]) is that we ﬁrst construct a unit
speed curve η with (f ◦η)0 (t) = −|∇− f |(η(t)) a.e. t, then an appropriate reparametrizationof η provides a gradient curve. Another way is the direct construction comprehensively
discussed in [AGS]. In fact, a generalized minimizing movement u : [0, ∞) −→ P(X) is
locally Lipschitz continuous on (0, ∞) and satisﬁes
}2
{
(
)2
f (u(t + ε)) − f (u(t))
W2 (u(t), u(t + ε))lim= −|∇− f | u(t) = − lim
ε↓0
ε↓0
ε
ε
at every t ∈ (0, ∞) ([AGS, Theorem 2.4.15]), and therefore the discussion as in [Oh,Lemma 5.5] shows that u is a gradient curve in the sense of Deﬁnition 2.5. Moreover, theuniqueness of gradient curves proved below (Theorem 4.2) ensures that both constructions
give rise to the same curve.

3

A ﬁrst variation formulaThis section contains our main results. These are shown after a series of lemmas.
Lemma 3.1 For any minimal geodesic α : [0, t] −→ P(X) and ν ∈ P(X), we have
W2 (α(t), ν)2 − W2 (α(0), ν)2t)2W2 (α(s), ν)2 − W2 (α(0), ν)2 S 2 (+ W2 α(0), α(t) ,≤ lim infs→0stwhere S =√1 + (diam X)2 .

Proof. For s ∈ (0, t), the 2-uniform smoothness (1) shows()2
W2 α(s), ν(()))2 s (
)2
)2
(
s
s s (
2
≥ 1−
W2 α(0), ν + W2 α(t), ν − S 1 −W2 α(0), α(t) .
ttt tDividing both sides by s yields
W2 (α(t), ν)2 − W2 (α(0), ν)2
t
(
)
(
)2
W2 (α(s), ν)2 − W2 (α(0), ν)2 S 2s+≤
1−
W2 α(0), α(t) ,stt
and letting s tend to zero completes the proof.

52

Lemma 3.2 For any pair of minimal geodesics α : [0, δ) −→ P(X), β : [0, 1] −→ P(X)with α(0) = β(0) =: µ, we have
lim sups→0W2 (α(s), β(1))2 − W2 (µ, β(1))2≤ −2hα̇(0), β̇(0)iµ .
s(4)Proof. For any s ∈ (0, δ) and t ∈ (0, s−1 ), the triangle inequality gives
W2 (α(s), β(1)) − W2 (µ, β(1))
W2 (α(s), β(st)) + W2 (β(st), β(1)) − W2 (µ, β(1))≤
ssW2 (α(s), β(st)) − W2 (µ, β(st))=.
sThus we deduce from (2) that, for any t > 0,lim sups→0()
W2 (α(s), β(1)) − W2 (µ, β(1))≤ σ α̇(0), tβ̇(0) − t|β̇(0)|s|α̇|2 − 2t|α̇||β̇| cos ∠(α, β)(0).
=σ(α̇, tβ̇) + t|β̇|

Letting t go to inﬁnity implies
W2 (α(s), β(1))2 − W2 (µ, β(1))2
s
s→0
(
)
W2 (α(s), β(1)) − W2 (µ, β(1))= 2W2 µ, β(1) lim supss→0−2|α̇||β̇| cos ∠(α, β)≤ 2|β̇|(0) = −2hα̇(0), β̇(0)i.
2|β̇|lim sup2
We remark that equality holds in (4) if X is nonnegatively curved (cf. [BBI, Theorem 4.5.6]).
Lemma 3.3 For any triplet v1 , v2 , w ∈ Cµ [P(X)], we have
hv1 , wiµ ≤ hv2 , wiµ + |w|µ · σµ (v1 , v2 ).
Proof. We just calculate, putting vi = (αi , si ) and w = (β, t),{}2|hv1 , wi − hv2 , wi|2 = t2 s1 cos ∠(α1 , β) − s2 cos ∠(α2 , β)[2 2
= t s1 {1 − sin2 ∠(α1 , β)} + s22 {1 − sin2 ∠(α2 , β)}]− 2s1 s2 cos ∠(α1 , β) cos ∠(α2 , β)[{}]
≤ t2 s21 + s22 − 2s1 s2 cos ∠(α1 , β) cos ∠(α2 , β) + sin ∠(α1 , β) sin ∠(α2 , β){()}
= t2 s21 + s22 − 2s1 s2 cos ∠(α1 , β) − ∠(α2 , β){}
≤ t2 s21 + s22 − 2s1 s2 cos ∠(α1 , α2 )= t2 σ(v1 , v2 )2 .
2
6

Now we are ready to prove our main theorem.
Theorem 3.4 (First variation formula) Let ξ : [0, δ) −→ P(X) be a curve right diﬀerentiable at 0, and let β : [0, 1] −→ P(X) be a minimal geodesic from µ := ξ(0) to ν. Then
we have
W2 (ξ(t), ν)2 − W2 (µ, ν)2˙
lim sup≤ −2hξ(0),β̇(0)iµ .
tt→0Proof. For each small t > 0, let αt : [0, t] −→ P(X) be a minimal geodesic from µ to˙
ξ(t). Then it follows from the right diﬀerentiability of ξ that α̇t (0) converges to ξ(0)in
Cµ [P(X)]. We deduce from Lemmas 3.1, 3.2, 3.3 that
W2 (ξ(t), ν)2 − W2 (µ, ν)2W2 (αt (t), ν)2 − W2 (µ, ν)2
=
t
t
)2
S2 (
≤ −2hα̇t (0), β̇(0)i + W2 µ, αt (t)t() S2 ()2˙˙≤ −2hξ(0),β̇(0)i + 2|β̇(0)| · σ ξ(0),α̇t (0) + W2 µ, ξ(t) .
tLetting t tend to zero, we complete the proof.

2

The following simple lemma (valid for general metric spaces) is useful.
Lemma 3.5 Let ξ, ζ : [0, δ) −→ Y be curves in a metric space (Y, d), and z ∈ Y be a
midpoint of x := ξ(0) and y := ζ(0) (i.e., d(z, x) = d(z, y) = d(x, y)/2). Then we haved(ξ(t), ζ(t))2 − d(x, y)2t
t→0d(ξ(t), z)2 − d(x, z)2d(ζ(t), z)2 − d(y, z)2≤ 2 lim sup+ 2 lim sup
.
t
t
t→0
t→0

lim sup

Proof. The triangle inequality immediately impliesd(ξ(t), ζ(t))2 − d(x, y)2d(ξ(t), ζ(t)) − d(x, y)= 2d(x, y) lim sup
t
t
t→0
t→0
d(ξ(t), z) + d(ζ(t), z) − d(x, y)≤ 2d(x, y) lim sup
t
{ t→0
}
d(ξ(t), z) − d(x, z)d(ζ(t), z) − d(y, z)≤ 2d(x, y) lim sup+ lim sup
t
t
t→0
t→0
2
2
2
d(ξ(t), z) − d(x, z)d(ζ(t), z) − d(y, z)2= 2 lim sup+ 2 lim sup
.
t
t
t→0
t→0

lim sup2
Combining this with Theorem 3.4 yields the following ﬁrst variation formula for the
distance between two right diﬀerentiable curves.

7

Corollary 3.6 Let ξ, ζ : [0, δ) −→ P(X) be two curves right diﬀerentiable at 0. Putµ := ξ(0), ν := ζ(0), let α : [0, 1] −→ P(X) be a minimal geodesic from µ to ν and letβ(τ ) := α(1 − τ ) be its converse curve. Then we have
lim supt→0

W2 (ξ(t), ζ(t))2 − W2 (µ, ν)2˙≤ −2hξ(0),α̇(0)iµ − 2hζ̇(0), β̇(0)iν .
t

Proof. Apply Lemma 3.5 to ξ and ζ with z = α(1/2) = β(1/2). Then Theorem 3.4 yields
the desired estimate.
2
Remark 3.7 If X is nonnegatively curved, then (P(X), W2 ) is an Alexandrov space andhence the right diﬀerentiability gives a better control at the level of P(X) (not only inCµ [P(X)]). Such strong right diﬀerentiability ([Oh, (6.1)]) and the ﬁrst variation formula
along geodesics (Lemma 3.2) immediately lead us to the formula along right diﬀerentiable
curves (see [Oh, Lemma 6.1]).

Applications for gradient ﬂows in P(X)

4

In this section, we use the ﬁrst variation formula in the previous section to extend resultsin [Oh, Section 6] where we assumed that (X, d) is nonnegatively curved.

4.1

Uniqueness and contraction

As in Subsection 2.2, let f : P(X) −→ (−∞, +∞] be a lower semi-continuous, K-convex
function for some K ∈ R such that P ∗ (X) = f −1 ((−∞, +∞)) is nonempty. We ﬁrst
verify the evolution variational inequality (see [AGS, (4.0.13)]) as a consequence of ﬁrst
variation formulas Lemma 2.4 and Theorem 3.4.
Proposition 4.1 (Evolution variational inequality) Let ξ : [0, ∞) −→ P ∗ (X) be a gradient curve of f . Then we have, for any t ∈ (0, ∞) and ν ∈ P(X),lim supε↓0)2()
W2 (ξ(t + ε), ν)2 − W2 (ξ(t), ν)2 K (+ W2 ξ(t), ν + f ξ(t) ≤ f (ν).
2ε2Proof. The assertion is clear if f (ν) = ∞, so that we assume ν ∈ P ∗ (X). We observefrom Theorem 3.4 thatlim supε↓0W2 (ξ(t + ε), ν)2 − W2 (ξ(t), ν)2˙≤ −hξ(t),β̇(0)iξ(t) ,2ε˙where β : [0, 1] −→ P ∗ (X) is a minimal geodesic from ξ(t) to ν. As ξ(t)= ∇− f (ξ(t)),
Lemma 2.4 and the K-convexity (3) of f together imply
)2
(
) K (
f (β(τ )) − f (ξ(t))≤ f (ν) − f ξ(t) − W2 ξ(t), ν .
τ →0
τ
2

˙
−hξ(t),
β̇(0)iξ(t) ≤ lim

2This completes the proof.
A similar argument shows the contraction property of gradient curves.
8

Theorem 4.2 (Contraction and uniqueness) Given any pair of gradient curves ξ, ζ :[0, ∞) −→ P ∗ (X) of f ,()()
W2 ξ(t), ζ(t) ≤ e−tK W2 ξ(0), ζ(0)holds for all t ∈ [0, ∞). In particular, each µ ∈ P ∗ (X) admits a unique complete gradientcurve starting from µ.
Proof. Put h(t) := W2 (ξ(t), ζ(t))2 , ﬁx t ∈ (0, ∞), let α : [0, 1] −→ P(X) be a minimal
geodesic from ξ(t) to ζ(t) and put β(τ ) := α(1 − τ ). Then Corollary 3.6 and Lemma 2.4
show thath(t + ε) − h(t)˙lim sup≤ −2hξ(t),α̇(0)iξ(t) − 2hζ̇(t), β̇(0)iζ(t)
ε
ε↓0
}
{
f (β(τ )) − f (β(0))
f (α(τ )) − f (α(0))≤ 2 lim+ lim≤ −2Kh(t).
τ →0
τ →0
τ
τ
We used the K-convexity (3) along α in the last inequality. Therefore we obtain h0 ≤−2Kh for a.e. t, and hence h(t) ≤ e−2tK h(0) by Gronwall’s theorem.
2
We deﬁne the gradient ﬂow G : P ∗ (X) × [0, ∞) −→ P ∗ (X) as ξ(t) := G(µ, t) is theunique gradient curve starting from ξ(0) = µ. Note that G is continuous by virtue of thecontraction property.
Corollary 4.3 The gradient ﬂow G : P ∗ (X) × [0, ∞) −→ P ∗ (X) extends uniquely andcontinuously to G : P ∗ (X) × [0, ∞) −→ P ∗ (X). Moreover, G satisﬁes the contraction
property:()()
W2 G(µ, t), G(ν, t) ≤ e−tK W2 G(µ, 0), G(ν, 0)for µ, ν ∈ P ∗ (X) and t ∈ (0, ∞); as well as the semigroup property:()
G(µ, s + t) = G G(µ, s), t
for µ ∈ P ∗ (X) and s, t ∈ [0, ∞).

4.2

Heat ﬂow as gradient ﬂow on Riemannian manifolds

Until here, we only deal with the triangle comparison property of the Wasserstein space.
In this last subsection, in order to see that gradient ﬂow of the free energy produces asolution to the Fokker-Planck equation, we use the structure of the underlying space (thatwas implicitly avoided in [Oh]). This kind of interpretation of evolution equations goesback to celebrated work of Jordan et al. [JKO]. It is recently demonstrated that there isalso a remarkable connection with the Ricci ﬂow (see [MT]).
Although some parts also work in Alexandrov spaces, we consider only compact Riemannian manifolds for brevity (the compactness guarantees that the sectional curvature
is bounded below). Since Theorem 4.2 ensures that our gradient curve coincides withthe one constructed as in [AGS] (see Remark 2.7), the realization of solutions to the
Fokker-Planck equation as gradient ﬂows of the free energy is a well established fact in9the Riemannian setting. Here, however, we present a way of completing the self-contained
proof in [Oh, Subsection 6.2] along our notion of gradient curves for thoroughness.
Let (M, g) be a compact Riemannian manifold equipped with the associated Riemannian distance d and the volume measure volg . Thanks to McCann’s theorem [Mc], we canrepresent each v ∈ Cµ [P(M )] as a (measurable) vector ﬁeld on M which will be again
denoted by v∫ by a slight abuse of notation. Moreover, for v, w ∈ Cµ [P(M )], we haveσµ (v, w)2 = M |v(x) − w(x)|2 dµ(x). In other words, Otto’s [Ot] Riemannian structure
coincides with ours induced from Theorem 2.2.
Due to the compactness of M , the Taylor expansion immediately gives the following:
Lemma 4.4 For any h ∈ C ∞ (M ) and any geodesic α : [0, l) −→ P(M ), we have
∫
∫
∫
(
)
h dµt =
h dµ + t
hv, ∇hi dµ + Oh W2 (µ, µt )2 ,M

M

M

where we set µ := α(0), µt := α(t) and v := α̇(0) ∈ Cµ [P(M )].
Let f : P(M ) −→ (−∞, +∞] be as in Subsections 2.2, 4.1. Take µ ∈ P ∗ (M ) and letξ : [0, ∞) −→ P ∗ (M ) be the unique complete gradient curve with ξ(0) = µ. Fix t > 0
˙ = ∇− f (ξ(t)). For any h ∈ C ∞ (M ), sinceand recall that ξ is right diﬀerentiable with ξ(t)the remainder term Oh in Lemma 4.4 is uniform in the choice of geodesics α, we obtain
{∫
} ∫
∫
1
h dµt+δ −h dµt =h∇− f (µt ), ∇hi dµt ,
lim
δ↓0 δ
M
M
M
where we set µt := ξ(t). For each δ > 0, choose some νδ ∈ P ∗ (M ) attaining the inﬁmumof the function
W2 (µt , ν)2∗
P (M ) 3 ν 7−→ f (ν) +.
2δSuch νδ indeed exists since P(M ) is compact and f is lower semi-continuous. We alsochoose a minimal geodesic βδ : [0, lδ ] −→ P(M ) from µt to νδ where lδ := W2 (µt , νδ ).
Then we know that (βδ , lδ /δ) ∈ Cµt [P(M )] converges to ∇− f (µt ) as δ tends to zero ([Oh,Lemma 6.4]). Thus Lemma 4.4 also shows that, for any h ∈ C ∞ (M ),
{∫
} ∫
∫
1
lim
h dνδ −
h dµt =h∇− f (µt ), ∇hi dµt .
δ↓0 δ
M
M
M
Therefore we conclude that
1
lim
δ↓0 δ

{∫

}

∫
h dµt+δ −

M

h dνδ

=0

(5)

M

holds for all h ∈ C ∞ (M ).
Remark 4.5 If X has the nonnegative curvature, then the convergence of (βδ , lδ /δ) to∇− f (µt ) implies limδ↓0 W2 (νδ , µt+δ )/δ = 0 (see [Oh, Lemma 6.4]). Thus the KantorovichRubinstein theorem yields (5) without using Lemma 4.4, moreover, the convergence (5)is uniform for all 1-Lipschitz functions h.

10For µ ∈ P(M ), we deﬁne the relative entropy as{ ∫ρ log ρ dvolg if µ = ρ · volg ,M
Ent(µ) :=+∞otherwise.
Given V ∈ C ∞ (M ), let fV : P(M ) −→ (−∞, ∞] be the associated free energy:∫
f (µ) := Ent(µ) +V dµ.
MNote that fV is lower semi-continuous and the corresponding subset P ∗ (M ) ⊂ P(M )satisﬁes P ∗ (M ) = P(M ). Furthermore, the K-convexity of fV is known to be equivalent
to the lower bound of the Bakry-Émery tensor: Ric + Hess V ≥ K ([St1]) (in particular,
the K-convexity of Ent is equivalent to Ric ≥ K, [vRS]). Hence fV is K-convex for someK ∈ R by virtue of the compactness of M .
The estimate (5) is enough to follow the proof of [Oh, Theorem 6.6] and yields thefollowing:
Theorem 4.6 Given V ∈ C ∞ (M ), a gradient curve ξ = ρ · volg : [0, ∞) −→ P ∗ (M ) offV produces a weak solution ρ to the associated Fokker-Planck equation:∂ρ= ∆ρ + div(ρ · ∇V ).
∂t
In particular, the gradient ﬂow of Ent coincides with the heat ﬂow.
To be precise, for any h ∈ C ∞ (R × M ) and 0 ≤ t1 < t2 < ∞, we have}∫∫
∫ t2 ∫ {∂ht
ht2 dµt2 −+ ∆ht − h∇ht , ∇V i dµt dt,
ht1 dµt1 =
∂t
M
M
t1
M
where we set µt := ξ(t) and ht := h(t, ·).
Remark 4.7 At this point, it should be recalled that in the Riemannian setting there are
two ways to see the heat ﬂow as gradient ﬂow: as gradient ﬂow of the relative entropy withrespect to the distance W2 as we just did, or as gradient ﬂow of the Dirichlet energy withrespect to the L2 -distance (associated with the volume measure). Recently the second
author and Sturm [OhSt] proved that these two approaches coincide also in the Finsler
setting.
For ﬁnite-dimensional Alexandrov spaces, the construction of the heat kernel via the
Dirichlet energy has been performed in [KMS]. It is yet to be proven that such heat
kernel coincides with the gradient ﬂow of the entropy with respect to W2 in the genuine
Alexandrov setting.

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