Heat ﬂow on Alexandrov spaces
Nicola Gigli∗†, Kazumasa Kuwada‡§, Shin-ichi Ohta¶kJanuary 19, 2012

Abstract
We prove that on compact Alexandrov spaces with curvature bounded
below the gradient ﬂow of the Dirichlet energy in the L2 -space produces
the same evolution as the gradient ﬂow of the relative entropy in the L2 Wasserstein space. This means that the heat ﬂow is well deﬁned by either
one of the two gradient ﬂows. Combining properties of these ﬂows, we are able
to deduce the Lipschitz continuity of the heat kernel as well as Bakry-Émery
gradient estimates and the Γ2 -condition. Our identiﬁcation is established bypurely metric means, unlike preceding results relying on PDE techniques.
Our approach generalizes to the case of heat ﬂow with drift.

1

Introduction

The heat equation is one of the most important evolutionary PDEs. It is a wellknown fact in modern analysis that such an equation, say in Rn , can be seen as
the gradient ﬂow of the Dirichlet energy
Z1|∇f |2 dx
(1.1)
2 Rn
∗

Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115 Bonn,
Germany / current aﬃliation: Department of Mathematics, University of Nice, Valrose 06108
Nice Cedex 02, France (gigli@unice.fr)†
Partially supported by SFB 611‡
Graduate School of Humanities and Sciences, Ochanomizu University, Ohtsuka 2-1-1,
Bunkyo-ku, Tokyo 112-8610, Japan & Institut für Angewandte Mathematik, Universität Bonn,
Endenicher Allee 60, 53115 Bonn, Germany (kuwada.kazumasa@ocha.ac.jp)§
Partially supported by the JSPS fellowship for research abroad & the Grant in Aid for Young
Scientists (B) 22740083¶
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan & Max-PlanckInstitut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany (sohta@math.kyoto-u.ac.jp)kPartially supported by the Grant-in-Aid for Young Scientists (B) 20740036

1

in the space L2 (Rn , Ln ). This viewpoint has been extended to the concept ofDirichlet form (see e.g. [6, 12]) and it has grown up into a huge research ﬁeld
in potential analysis and probability theory. More recently, Jordan, Kinderlehrer
and Otto [15] understood that the same equation can be seen as the gradient ﬂow
of the relative entropyZρ log ρ dx

(1.2)

Rn

in the L2 -Wasserstein space (P2 (Rn ), W2 ), where P2 (Rn ) stands for the spaceof Borel probability measures on Rn with ﬁnite second moment. This intuition,with the further studies of Otto [31], has been one of the fundamental ingredientsthat drove the research in the ﬁeld of gradient ﬂows in relation with optimaltransportation problems in the past decade (see, e.g., [1] and [42]).
The aim of this paper is to carry on the study of the heat ﬂow as gradient
ﬂow of the two very diﬀerent functionals, (1.1) and (1.2) in the two metric spaces
L2 (Rn , Ln ) and (P2 (Rn ), W2 ), in a non-smooth setting. The point is the following. On the one hand, it is known that these two gradient ﬂows produce the same
evolution ‘in all the smooth settings’, i.e., it has been proved that in Rn , in Riemannian manifolds with Ricci curvature bounded below and in compact Finsler
manifolds (for the latter see the work of Sturm and the third author [28]), the gradient ﬂow of the Dirichlet energy with respect to the L2 -distance coincides withthe gradient ﬂow of the relative entropy with respect to the Wasserstein distanceW2 (see also [10], [16] for related work on diﬀerent kinds of spaces). On the other
hand, these ﬂows are studied well also in some non-smooth settings. These raise
the natural question: do these two notions always coincide?
We remark that a natural abstract setting where one could try to give ananswer to this question is the one of metric measure spaces with Ricci curvaturebounded below (deﬁned by Sturm [40] and Lott and Villani [26], see Deﬁnition 2.4).
Indeed, in order to deﬁne either one of the two gradient ﬂows, one usually needs
both a metric and a measure on the space considered. Furthermore, it is known
that in the Riemannian setting the fact that the heat ﬂow does not lose mass isstrictly related to the bound from below on the Ricci curvature, so that one has toassume some kind of lower Ricci curvature bound also in the non-smooth setting.
In this paper, we restrict our attention to the case of ﬁnite dimensional, compact Alexandrov spaces (X, d, Hn ) of curvature bounded below without boundary, equipped with the Hausdorﬀ measure Hn . An Alexandrov space of curvaturebounded from below by k with k ∈ R is a metric space of sectional curvaturebounded from below by k in the sense of the triangle comparison property (seeDeﬁnition 2.1). As naturally expected and recently shown in [34] and [43], such
spaces actually have Ricci curvature bounded below in the sense of Lott-Sturm-

2Villani (see Remark 2.5). Our following main result is the ﬁrst one establishing
the equivalence of the two gradient ﬂows in a genuinely non-smooth setting (seeTheorem 3.1 for the slightly more general statement):
Theorem 1.1 Let (X, d, Hn ) be a ﬁnite-dimensional compact Alexandrov space
without boundary. Then for any f0 ∈ L2 (X, Hn ) with f0 Hn ∈ P(X), the gradientﬂow (ft )t∈[0,∞) of the Dirichlet energy with respect to the L2 -distance gives the gradient ﬂow (ft Hn )t∈[0,∞) of the relative entropy with respect to the L2 -Wasserstein
distance W2 , and vice versa.
On Alexandrov spaces, the structure of the heat ﬂow as gradient ﬂow has alreadybeen studied in [20] for the Dirichlet energy approach and in [27] by the third
author (see also [14] and [38]) for the relative entropy approach. However, up to
now it was not clear that these two notions coincide.
We observe that the idea behind our proof of the theorem is completely different from the ones used in the aforementioned smooth settings. Indeed, all the
proofs available in the smooth setting had the following structure:• one studies the gradient ﬂow of the Dirichlet energy with respect to L2 andwrites down the equation that it satisﬁes;• one studies the gradient ﬂow of the relative entropy with respect to W2 andwrites down the equation that it satisﬁes;• one realizes that the two equations are actually the same and calls intoplay the uniqueness of the solution of the equation to conclude that the two
gradient ﬂows coincide.
Our approach, instead, is completely diﬀerent and in particular does not pass fromthe study of the heat equation in the Alexandrov setting. Read back in Rn our
proof gives a new and purely metric way to prove the coincidence of these two
gradient ﬂows.
After having proved the identiﬁcation, we can combine them: their interaction
gives fruitful applications. Among them, the Lipschitz continuity (Theorem 4.4) of
the heat kernel is derived as a corollary of a Bakry-Émery type gradient estimate(Theorem 4.3), which follows with the aid of a result of the second author [19].
The Lipschitz regularity improves the Hölder regularity established in [20]. We
believe that it is curious and worth to be underlined that such Lipschitz property
immediately follows when one knows the equivalence of the two gradient ﬂows,
but is not at all trivial if one sticks to either one of the two approaches alone.
Our Lipschitz regularity enables us to deduce the Γ2 -condition (Theorem 4.6)
from a Lott-Sturm-Villani Ricci curvature bound. The Γ2 -condition as well as
3the Wasserstein contraction of heat ﬂow and the Bakry-Émery gradient estimate
for heat semigroup are known to be analytic characterizations of a lower Ricci
curvature bound (see [37, 3, 23]). We show the equivalence of those three analytic
conditions even on Alexandrov spaces (with sharp constants, Theorem 4.8).
It should be remarked that our approach easily generalizes to the case of heat
ﬂow with drift, where the drift is given by the gradient of a semiconvex potential
V . In other words, the current approach can be used to study the heat ﬂow onweighted Alexandrov spaces of the kind (X, d, e−V Hn ) (see the end of Section 4).
After having completed the work on this paper, we got aware of a paper [44]by Zhang and Zhu where the Lipschitz continuity of the heat kernel has beenstudied, via a completely diﬀerent argument. In particular, their proof relies on
the Lipschitz continuity of harmonic functions due to Petrunin [33]. They evenproceeded their study to [45] and [35], where they showed a Bochner type formulaand Li-Yau estimates under their own notion of lower Ricci curvature bound whichis stronger than the one of Lott-Sturm-Villani. It is not discussed in [44, 45, 35]
whether their approach generalizes to the case of weighted Alexandrov spaces.
The organization of the article is as follows. Section 2 is devoted to preliminaries for Alexandrov geometry and known results on the gradient ﬂow of the
Dirichlet energy as well as the gradient ﬂow of the relative entropy on compact
Alexandrov spaces. We prove our main theorem in Section 3, and discuss its
applications in Section 4.

2

Preliminaries and notations

2.1

Alexandrov spaces

We ﬁrst review the basics of Alexandrov geometry, see [8], [30] and [7] for details.
Let (X, d) be a metric space. A rectiﬁable curve γ : [0, l] → X is called ageodesic if it is locally length minimizing and parametrized with constant speed.
(Precisely, for any t0 ∈ [0, l], there is ε > 0 such that d(γ(s), γ(t))/|s−t| is constantfor all s, t ∈ [t0 − ε, t0 + ε] ∩ [0, l].) If γ is minimizing between endpoints, then wecall it a minimal geodesic. We say that (X, d) is a geodesic space if any pair ofpoints in X are connected by a minimal geodesic.
For k ∈ R, we denote by M2 (k) the simply-connected, two-dimensional space
form of constant sectional curvaturek. Given three points x, y, z ∈ X, with√d(x, y) + d(y, z) + d(z, x) < 2π/ k if k > 0, we can take a comparison triangle
4x̃ỹz̃ ⊂ M2 (k) such that d(x̃, ỹ) = d(x, y), d(ỹ, z̃) = d(y, z) and d(z̃, x̃) = d(z, x).
Such a triangle is unique up to a diﬀerence of isometry.

4Deﬁnition 2.1 (Alexandrov spaces) For k ∈ R, a complete geodesic space(X, d) is called an Alexandrov space of curvature bounded from below
by k if, for√
any three points x, y, z ∈ X (with d(x, y) + d(y, z) + d(z, x) < 2π/ k if k > 0) andany minimal geodesic γ : [0, 1] → X from y to z, we have d(x, γ(t)) ≥ d(x̃, γ̃(t))
for all t ∈ [0, 1], where 4x̃ỹz̃ ⊂ M2 (k) is a comparison triangle of 4xyz andγ̃ : [0, 1] → M2 (k) is the unique minimal geodesic from ỹ to z̃.
Example 2.2 (a) A complete Riemannian manifold is an Alexandrov space of
curvature bounded from below by k if and only if its sectional curvature is greaterthan or equal to k everywhere.
(b) If (X, d) is an Alexandrov space of curvature bounded from below by k,then the scaled metric space (X, c · d) with c > 0 is an Alexandrov space of
curvature bounded from below by k/c2 .
(c) For a convex domain D in the Euclidean space Rn , the boundary ∂Dequipped with the length distance is an Alexandrov space of nonnegative curvature.
(d) Let (M, g) be a Riemannian manifold of nonnegative sectional curvatureand G be a compact group acting on M by isometries. Then the quotient space
M/G equipped with the quotient metric is an Alexandrov space of nonnegative
curvature.
(e) If a sequence of Alexandrov spaces of curvature bounded from below by kis convergent with respect to the Gromov-Hausdorﬀ distance, then its limit space
is again an Alexandrov space of curvature bounded from below by k.
Fix x ∈ X and let Σ̂x be the set of all unit speed geodesics γ : [0, l] → X with
γ(0) = x. For γ, η ∈ Σ̂x , thanks to the curvature bound, the joint limit¶µs2 + t2 − d(γ(s), η(t))2∠x (γ, η) := arccos lim
s,t→0
2st
exists and is a distance on Σ̂x /∼ where γ ∼ η holds if ∠x (γ, η) = 0. We deﬁne thespace of directions (Σx , ∠x ) at x as the completion of Σ̂x /∼ with respect to ∠x .
The tangent cone (Kx , d) is the Euclidean cone over (Σx , ∠x ), i.e.,
Kx := Σx × [0, ∞)/Σx × {0},p¢d (γ, s), (η, t) := s2 + t2 − 2st cos ∠x (γ, η).
¡

The inner product on Kx is deﬁned by h(γ, s), (η, t)ix := st cos ∠x (γ, η). In Riemannian manifolds, spaces of directions and tangent cones correspond to unit
tangent spheres and tangent spaces, respectively.

5The Hausdorﬀ dimension of X is an integer or inﬁnity. From here on, we
consider a compact n-dimensional Alexandrov space of curvature bounded frombelow by k without boundary equipped with the n-dimensional Hausdorﬀ measure
Hn (see [8] for the deﬁnition of the boundary of Alexandrov spaces). We saythat x ∈ X is a singular point if Kx is not isometric to the Euclidean space Rn ,and denote the set of singular points by SX . We remark that Hn (SX ) = 0 holds
whereas SX can be dense in X (see [8], [30]).

2.2

Dirichlet energy and the associated gradient ﬂow

We introduce the Sobolev space and the Dirichlet energy following [20], and will
see that it coincides with other notions of Sobolev spaces. We begin by discussing a C 1 -diﬀerentiable structure of the set of regular points X \ SX established in [8] and [30]. We remark that Perelman extends this to DC 1 -structure(via ‘diﬀerence of concave functions’, see [32]), but the C 1 -structure is enough forconsidering the Sobolev space. There is a weak C 1 -atlas {(Uφ , Vφ , φ)}φ∈Φ in thensense that Uφ ⊂S X is an open set, φ : Uφ → R is a bi-Lipschitz embedding,Vφ ⊂ Uφ with φ∈Φ Vφ ⊃ X \ SX , and that the coordinate change φ2 ◦ φ−1
1 is1
C on φ1 (Vφ1 ∩ Vφ2 ∩ (X \ SX )) if Vφ1 ∩ Vφ2 6= ∅ ([30, Theorem 4.2(1)]). Such
charts are constructed through the distance function. Precisely, φ is introduced as
φ(x) := (d(x, p1 ), d(x, p2 ), . . . , d(x, pn )) for suitable p1 , p2 , . . . , pn ∈ X, and thenVφ is chosen as the set of regular points x such that a minimal geodesic between
x and pi is unique for all i ([30, Section 3]). It is worth mentioning that, for any
ε > 0, φ can be (1+ε)-bi-Lipschitz by taking smaller Uφ (cf. [7, Theorem 10.9.16]).
We also remark that it is possible to modify φ by taking an average so as to satisfy
Vφ = Uφ ([30, Section 5]), but it is unnecessary for our discussion (just like [5,Remark 2.9]).
We say that a function f on X is diﬀerentiable at a regular point x ∈ X \ SX if
x ∈ Vφ and f ◦φ−1 is diﬀerentiable at φ(x) for some φ ∈ Φ. Then we can deﬁne thegradient vector ∇f (x) ∈ Kx by identifying Kx and Rn through φ. (To be precise,by virtue of the ﬁrst variation formula ([30, Theorem 3.5]), each di := d(·, pi ) is
diﬀerentiable at x with ∇di (x) = −vi , where vi ∈ Kx is the tangent vector of theunique, minimal,
unit speed geodesic from x to pi . Then Kx is linearly identiﬁedPnwith R as i ai vi = (−ai ).) Moreover, again due to the ﬁrst variation formula,
we obtain the Taylor expansion¡¢
f γ(t) = f (x) + th∇f (x), γ̇(0)ix + ox (t),(2.1)
where γ : [0, δ] → X is a minimal geodesic emanating from x and ox (t) is independent of the choice of γ (see [5, Lemma 3.4], and note that the remainder
term in the ﬁrst variation formula for d(·, pi ) at x indeed depends only on x and6pi ). Another important fact we will use is the Rademacher theorem, namely a
Lipschitz function f on X is diﬀerentiable Hn -a.e.. This easily follows from theusual Rademacher
theorem for f ◦ φ−1 (see [5, Corollary 2.14]). It follows fromp(2.1) that h∇f (x), ∇f (x)i coincides with the local Lipschitz constant |∇L f |(x)given by|f (x) − f (y)||∇L f |(x) := lim.
y→xd(x, y)Based on the notion of gradient vector, we deﬁne the Sobolev space and theDirichlet energy as follows (see [20] and [22] for details). For a function f : X → R
such that f ◦ φ−1 ∈ W 1,2 (φ(Uφ )) for all φ ∈ Φ, we introduce the weak gradientvector ∇f (x) ∈ Kx for a.e. x ∈ Vφ as the element corresponding to the weakgradient vector ∇(f ◦ φ−1 )(φ(x)). We deﬁne the Sobolev space W 1,2 (X) and theDirichlet energy E by
¯Z
½
¾
¯
1,2
2
n ¯
n
W (X) := f ∈ L (X, H ) ¯ h∇f, ∇f i dH < ∞ ,XZn
E(f, g) :=h∇f, ∇gi dH
for f, g ∈ W 1,2 (X).
X

We do not divide E by 2 for notational simplicity. Note that E coincides withthe energy functional introduced by Korevaar and Schoen [18] (we can reduce
the argument to the Euclidean case by using a (1 + ε)-bi-Lipschitz chart; see [22,Theorem 6.2]). We also remark that the set of Lipschitz functions C Lip (X) isdense in W 1,2 (X) with respect to the Sobolev norm kf k2W 1,2 = kf k2L2 + E(f, f )([20, Theorem 1.1]).
Furthermore, if f is a Lipschitz function, then the weak gradientvector ∇f (x)pcoincides with the gradient vector as in (2.1) a.e. x, and hence h∇f (x), ∇f (x)i =|∇L f |(x) holds a.e. x. Therefore E also coincides with Cheeger’s energy functional ([9]), because the local Lipschitz constant is the minimal generalized upper
gradient ([9, Theorem 6.1]) and Lipschitz functions are dense in both Sobolev
spaces (thanks to the weak Poincaré inequality for upper gradients and the volume doubling condition, [9, Theorem 4.24]). Indeed, in our framework, the volume
doubling condition directly follows from the Bishop-Gromov volume comparison
theorem and the Poincaré inequality is a consequence of [21] and [36], for instance.
By following the general theory of bilinear forms, there exists a nonpositive
selfadjoint operator (∆, D(∆)) on L2 (X, Hn ) associated with (E, W 1,2 (X)). It ischaracterized by the following identity:
Z
E(g, f ) = −g∆f dHn , f ∈ D(∆), g ∈ W 1,2 (X).
(2.2)
X

7

We call ∆ the Laplacian as in the classical case. Based on a general theory offunctional analysis, the one-parameter semigroup of contractive symmetric linear
operators Tt = et∆ on L2 (X, Hn ) is deﬁned associated with ∆. For any f ∈L2 (X, Hn ), Tt f solves the (linear) heat equation ∂t u = ∆u with u(0, ·) = f in thesense that Tt f ∈ D(∆) for t > 0 andlimt↓0Tt f − f= ∆f in L2 (X, Hn ) for f ∈ D(∆),
tlim Tt f = f in L2 (X, Hn ) for f ∈ L2 (X, Hn ).
t↓0Note that Tt is Markovian in the sense that 0 ≤ Tt f ≤ 1 holds whenever 0 ≤ f ≤ 1.
As shown in [20, Theorem 1.5], there exists a continuous function (t, x, y) 7→
pt (x, y) on (0, ∞) × X × X satisfying the following properties:(i) For any f ∈ L2 (X, Hn ), t > 0 and Hn -a.e. x ∈ X,
Z
Tt f (x) =pt (x, y)f (y) Hn (dy).

(2.3)

X

(ii) For any s, t > 0 and x, y ∈ X,Z

pt (x, y) = pt (y, x),Z
ps+t (x, y) =ps (x, z)pt (z, y) Hn (dz),

(2.4)
(2.5)

X

pt (x, z) Hn (dz) = 1,

(2.6)

X

pt (x, y) > 0.
See [20, Theorems 1.4, 1.5(3)] for the continuity of pt (x, y). The equality (2.6)
follows from the fact that 1 ∈ W 1,2 (X) and E(1, 1) = 0, because X is assumed tobe compact. As in the classical case, we call pt (x, y) the heat kernel. The existence
and these properties of pt (x, y) are deduced from the Poincaré inequality for E andthe volume doubling condition, together with results in [39].
Remark 2.3 The bilinear form (E, W 1,2 (X)) is a symmetric strongly local regular Dirichlet form (see [20] for it and further details; see [6, 12] for basics on
Dirichlet forms). Moreover, Tt enjoys the strong Feller property. As a result,
there exists a diﬀusion process ((Xt )t≥0 , (Px )x∈X ) on the whole space X associated with (E, W 1,2 (X)) in the sense that Ex [f (Xt )] = Tt f (x) for f ∈ C(X), x ∈ X
and t > 0.
8

Denote by P(X) the set of all Borel probability measures on X. Let us deﬁne
a positive Borel measure Tt ν for ν ∈ P(X) and t ≥ 0 by¶µZ
pt (x, y) ν(dx) Hn (dy) t > 0,Tt ν(dy) :=
X

ν(dy)
t = 0.
Thanks to (2.6), Tt ν ∈ P(X) holds. By deﬁnition, Tt ν is absolutely continuouswith respect to Hn for t > 0. When dν = f dHn , it holds dTt ν = Tt f dHn . In
this paper, we call the evolution (t, ν) 7→ Tt ν the gradient ﬂow of the Dirichlet
energy (since the Dirichlet energy is a functional on the L2 -space of functions,this terminology should be interpreted in an extended sense). Indeed, it is easy
to see fromZ
1
1
E(f + εg, f + εg) = E(f, f ) − εg∆f dHn + O(ε2 )22Xthat the Radon-Nikodym derivative dTt ν/dHn is the gradient ﬂow of E/2 withrespect to the L2 -norm.
Before closing this subsection, we review the derivation property of W 1,2 (X). It
is formulated as follows: For f1 , . . . , fk , g ∈ W 1,2 (X)∩L∞ (X, Hn ) and Φ : Rk → R
which is C 1 on the range of (f1 , . . . , fk ), Φ(f1 , . . . , fk ) belongs to W 1,2 (X) andh∇Φ(f1 , . . . , fk ), ∇gi =

k
X
∂Φ
j=1

∂xj

(f1 , . . . , fk ) h∇fj , ∇gi

Hn -a.e..

(2.7)

This identity directly follows from the deﬁnition of W 1,2 (X) or from the strong locality of the Dirichlet form (E, W 1,2 (X)) (see [6, Corollary I.6.1.3] and [12, Section
3.2] for the latter).

2.3

Gradient ﬂows in the Wasserstein space

We next introduce the Wasserstein space and a purely metric notion of gradientﬂows in it. We refer to [1] and [42] for the basic theory as well as the recent diverse
developments.
Given µ, ν ∈ P(X), a probability measure π ∈ P(X × X) is called a couplingof µ and ν if π(A×X) = µ(A) and π(X ×A) = ν(A) hold for all Borel sets A ⊂ X.
Then, for 1 ≤ p < ∞, we deﬁne the Lp -Wasserstein distance asµZ¶1/pd(x, y)p π(dxdy)

Wp (µ, ν) := inf
π

X×X

9

,

where π runs over all couplings of µ and ν. In most parts, we work in the quadraticcase p = 2. The L2 -Wasserstein space (P(X), W2 ) becomes a metric space andinherits the compactness from (X, d). Moreover, (P(X), W2 ) is a geodesic space.
If µ is absolutely continuous with respect to Hn , then a minimal geodesic (µt )t∈[0,1]from µ to any ν is unique and µt is also absolutely continuous for all t ∈ (0, 1) (see[5] for a more detailed characterization of µt , and [11] for the absolute continuity).
For µ ∈ P(X), we deﬁne the relative entropy byZ
Ent(µ) :=ρ log ρ dHn
X

when µ = ρHn with ρ ∈ L1 (X, Hn ), and Ent(µ) := ∞ otherwise. Set P ∗ (X) :={µ ∈ P(X) | Ent(µ) < ∞}. Note that Ent is lower semi-continuous with respect
to W2 and satisﬁes Ent(µ) ≥ − log Hn (X) by Jensen’s inequality.
Deﬁnition 2.4 (The curvature-dimension condition) For K ∈ R, we saythat (X, d, Hn ) satisﬁes the curvature-dimension condition CD(K, ∞) if Ent is
K-geodesically convex in the sense that any pair µ, ν ∈ P(X) admits a minimal
geodesic (µt )t∈[0,1] from µ to ν such that
Ent(µt ) ≤ (1 − t) Ent(µ) + t Ent(ν) −K(1 − t)tW2 (µ, ν)22

holds for all t ∈ [0, 1].
We remark that the above inequality is obvious if µ 6∈ P ∗ (X) or ν 6∈ P ∗ (X).
Therefore it is suﬃcient to consider µ, ν ∈ P ∗ (X), and then a minimal geodesic
between them is unique.
Remark 2.5 (i) The curvature-dimension condition CD(K, ∞) for general metricmeasure spaces is introduced and studied independently in [40] and [26], andknown to be equivalent to the lower Ricci curvature bound Ric ≥ K for complete
Riemannian manifolds equipped with the Riemannian distance and the volume
measure ([37]).
(ii) The condition CD(K, N ) for N ∈ (1, ∞) is also introduced in [41] and [24].
In general, CD(K, N ) implies CD(K, ∞). In the Riemannian case, CD(K, N ) is
equivalent to Ric ≥ K and dim ≤ N .
(iii) It is recently demonstrated in [34] and [43] that n-dimensional Alexandrov
spaces of curvature bounded from below by k satisfy CD((n − 1)k, n) (and hence
CD((n − 1)k, ∞)), as is naturally expected from the relation between the sectional
and the Ricci curvatures.

10There is a well established theory on the gradient ﬂow of geodesically convex
functions as comprehensively discussed in [1]. For later convenience, we recall acouple of notions in a general form. We say that a curve (µt )t∈I ⊂ P(X) on aninterval I ⊂ R is absolutely continuous if there is f ∈ L1 (I) such thatZ s
W2 (µt , µs ) ≤f (r) dr
(2.8)
t

for all t, s ∈ I with t ≤ s. Note that absolutely continuous curves are continuous.
For an absolutely continuous curve (µt )t∈I , the metric derivative|µ̇t | := lim

h→0W2 (µt , µt+h )|h|is well-deﬁned for a.e. t ∈ I ([1, Theorem 1.1.2]). Moreover, |µ̇t | belongs to
L1 (I) and is the minimal function for which (2.8) holds. Given a functional E :P(X) → R ∪ {+∞}, we consider a gradient ﬂow of E solving “µ̇t = −∇E(µt )” in
the following sense. For µ ∈ P(X) with E(µ) < ∞, we deﬁne the local slope asmax{E(µ) − E(ν), 0}.
ν→µW2 (µ, ν)|∇− E|(µ) := lim

If E is K-geodesically convex, then we have
Z s|µ̇r | · |∇− E|(µr ) dr
|E(µt ) − E(µs )| ≤(2.9)

(2.10)tfor all t, s ∈ I with t < s along any absolutely continuous curve (µt )t∈I with valuesin P(X) with E(µt ) < ∞. As a consequence, it holds
Z
Z
1 s
1 s
2
E(µt ) ≤ E(µs ) +|µ̇r | dr +|∇− E|2 (µr ) dr
(2.11)
2 t
2 t
for all t < s. From (2.10) and (2.11), it is natural to give the following deﬁnition.
Deﬁnition 2.6 (Gradient ﬂows) Let E : P(X) → R∪{+∞} be a K-geodesicallyconvex functional. We say that an absolutely continuous curve (µt )t∈[0,∞) in P(X)
is a gradient ﬂow of E provided E(µt ) < ∞ for t ≥ 0 and
Z
Z
1 s
1 s
2
E(µt ) = E(µs ) +|µ̇r | dr +|∇− E|2 (µr ) dr
(2.12)
2 t
2 t
for all 0 ≤ t < s. The equation (2.12) is called the energy dissipation identity.
11

The existence of such a gradient ﬂow comes from the general theory presented
in [1, Corollary 2.4.11]. Furthermore, as shown in [27, Theorem 6.2] (see also [14,Theorem 4.2]), the gradient ﬂow produces a contraction semigroup in the sensethat for any µ, ν ∈ P(X) with E(µ) < ∞ and E(ν) < ∞, the gradient ﬂows(µt )t∈[0,∞) , (νt )t∈[0,∞) starting from µ, ν satisfy
W2 (µt , νt ) ≤ e−Kt W2 (µ, ν)

for all t ≥ 0,

(2.13)

where K is the modulus of convexity of E. In particular, the uniqueness follows
from considering µ = ν. (Though the strategy of the construction in [27] is diﬀerent from [1], the resulting ﬂow is the same by uniqueness, see [14, Remark 2.7].)
Remark 2.7 (i) In [38, Theorem 7] the contractivity (from the geodesical convexity) is shown on spaces satisfying the local angle condition. Alexandrov spaces
satisfy this condition.
(ii) The ﬁrst author [13, Theorem 15] proved the uniqueness of the gradient
ﬂow of the relative entropy on general metric measure spaces satisfying CD(K, ∞),without relying on the contractivity. In this generality, the contractivity fails.
More precisely, the heat ﬂow on a ﬁnite-dimensional (Minkowski) normed space is
not contractive (except for inner product spaces; see [29]).
In the sequel, we mainly study the gradient ﬂow of the relative entropy. Then
(2.13) allows us to continuously and uniquely extend the gradient ﬂow semigroupto the full P(X) (since the closure of P ∗ (X) is P(X)). Such an extension alsosatisﬁes (2.13).

3

Identiﬁcation of the two gradient ﬂows

This section contains our main result:
Theorem 3.1 (Identiﬁcation of the two gradient ﬂows) Let (X, d) be a compact n-dimensional Alexandrov space without boundary. For any ν ∈ P(X), thegradient ﬂow of the Dirichlet energy starting from ν is the gradient ﬂow of therelative entropy, and vice versa.
Recall that CD(K, ∞) holds with K = (n − 1)k, and we use this, e.g., in the
proof of Proposition 3.2 below. The main technical obstacle in the proof of thistheorem is to let the L2 and the W2 structures ‘interact’. Our strategy consistsin picking a gradient ﬂow of the Dirichlet energy and in proving that it obeys theenergy dissipation identity (2.12) for the relative entropy. We start with a boundon the local slope (2.9) by the Fisher information.
12

Proposition 3.2 (Bound on the local slope) Let µ = f Hn ∈ P ∗ (X) withf ∈ W 1,2 (X). Then it holds
Zh∇f, ∇f i
2|∇− Ent | (µ) ≤
dHn .
f
X
Proof. We ﬁrst assume that f is Lipschitz and bounded away from 0. Then weknow that h∇f, ∇f i = |∇L f |2 Hn -a.e., so that the conclusion follows from [42,Theorem 20.1] together with CD(K, ∞). Thus all we need to do is to proceed byapproximation. Suppose< c ≤ f Hn -a.e. for some c ∈ R. Since C Lip (X) is√ that 01,2
1,2
dense in W (X) and f ∈ W (X),√we can ﬁnd a sequence {gi }i∈N of Lipschitz
functions such that gi converges to f as i → ∞ with respect to the Sobolev√
norm. Substituting Ci max{gi , c} for some Ci > 0 if necessary, we can assume√
that ci ≤ gi for some ci > 0 Hn -a.e. as well as kgi kL2 = 1 for all i. Set fi := gi2 .
As fi is Lipschitz and bounded away from 0, we haveZh∇fi , ∇fi i
2
n
dHn .
|∇− Ent | (fi H ) ≤fiX
On √the √one hand,
the right-hand side is equal to 4E(gi , gi ) and converges to
R
4E( f , f ) = X h∇f, ∇f i/f dHn by construction. On the other hand, we deducefrom [1, Corollary 2.4.10] that|∇− Ent |(f Hn ) ≤ lim |∇− Ent |(fi Hn ).

(3.1)

i→∞

Combining these, we prove the claim for f . It remains to remove the assumptionthat f is bounded away from 0. To do this we just consider fi := (1 + i−1 )−1 (f +i−1 ) and apply (3.1) again.
¤
Let us turn to considering the gradient ﬂow of the Dirichlet energy (Tt ν)t∈[0,∞) .
For simplicity of notations, we write ft = dTt ν/dHn in the remainder of thesection. By (2.3), ft ∈ C(X) ⊂ L2 (X, Hn ) holds for t > 0. Then (2.5), (2.4) and(2.3) imply ft ∈ D(∆). For any t > 0, there is εt > 0 such that ft ≥ εt holds sincept (x, y) is positive and continuous. As a well-known fact, the following bound also
holds.
Lemma 3.3 (Maximum principle) Let f0 ∈ L2 (X, Hn ). If f0 ≥ c a.e., thenTt f0 ≥ c a.e. for every t ≥ 0. The same holds for bounds from above.
Proof. Take A ⊂ X measurable. Since Tt is Markovian, Tt 1A ≥ 0 Hn -a.e.. Thus
Z
Z
Z
n
n
(Tt f0 − c) dH =Tt (f0 − c) dH =Tt 1A · (f0 − c) dHn ≥ 0.
A

AX

Since A is arbitrary, the assertion holds. Bounds from above follow by applying
the same argument to −f0 .
¤13

By virtue of Lemma 3.3 with the remark before it, for any ε > 0, there arec, C > 0 such thatc ≤ ft ≤ C for all t ≥ ε.
(3.2)
Proposition 3.4 (Entropy dissipation) Let ν ∈ P(X) and ft = dTt ν/dHn .
Then the function t 7→ Ent(ft Hn ) is locally Lipschitz in (0, ∞) and, moreover, it
holds
Zh∇ft , ∇ft idEnt(ft Hn ) = −dHna.e. t.
dtftX
Proof. As the function e(s) := s log s is C 1 in [c, C] and t 7→ ft is locally Lipschitz
in (0, ∞) with values in L2 (X, Hn ), we deduce from (3.2) that t 7→ Ent(ft Hn )isin (0, ∞). Applying formulas (2.2) and (2.7) and recalling
R locally Lipschitzn
X ∆ft dH = 0, we obtain
Z
Z
¡
¢
d
n
0
n
Ent(ft H ) =e (ft )∆ft dH =log(ft ) + 1 ∆ft dHn
dt
X
Z
Z X
h∇ft , ∇ft i=−h∇ log(ft ), ∇ft i dHn = −
dHn .
f
t
X
X
¤
For the next argument, we brieﬂy recall some properties of the Hamilton-Jacobi
semigroup in our context. For f ∈ C Lip (X) and t > 0, we deﬁne Qt f : X → R by·¸
d2 (x, y)
Qt f (x) := inf f (y) +
.
(3.3)
y∈X
2t
Also, set Q0 f := f . The following is shown in [4, Theorem 2.5(iv)] and [25,
Theorem 2.5(viii)] in the framework of general metric measure spaces supporting
the volume doubling condition and the Poincaré inequality for upper gradients.
Proposition 3.5 (Hamilton-Jacobi semigroup) It holds Qt f ∈ C Lip (X) for
every t ≥ 0, the map [0, ∞) 3 t 7→ Qt f ∈ C(X) is Lipschitz in the uniform norm,andd|∇L Qt f |2 (x)Qt f (x) +
= 0,
(3.4)
dt
2
for almost every t, x.
Remark 3.6 The equation (3.3) has been called the Hopf-Lax formula or the
Moreau-Yosida approximation also in the literature. The former name is mainly
used in the PDE context to a special solution to the Hamilton-Jacobi equation(3.4). The latter one is typically used for an approximation of an (unbounded)
operator in functional analysis.
14

Proposition 3.7 (Absolute continuity with respect to W2 ) For ν ∈ P(X),the curve t 7→ Tt ν is absolutely continuous in the space (P(X), W2 ) and its metricspeed |Tt˙ ν| satisﬁes
Zh∇ft , ∇ft i2
˙|Tt ν| ≤dHn ,for a.e. t.
(3.5)
ft
X
Proof. Fix t, s > 0. By the Kantorovich duality (cf., e.g., [1, Theorem 6.1.1] and[42, Theorem 5.10]) together with (3.3), we obtain
·Z
¸
Z
¢
1 2¡
n
n
W Tt ν, Tt+s ν = sup(Q1 ϕ)ft+s dH −ϕft dH .
(3.6)
2 2
X
X
ϕ∈C Lip (X)
By Proposition 3.5, the map r 7→ Qr ϕ from [0, 1] to L2 (X, Hn ) is Lipschitz.
Moreover, it is diﬀerentiable in L2 (X, Hn ) and the derivative is determined by(3.4). The curve [0, 1] 3 r 7→ ft+rs ∈ L2 (X, Hn ) is Lipschitz as well. Thus [0, 1] 3
r 7→ (Qr ϕ)ft+rs ∈ L1 (X, Hn ) is Lipschitz and its derivative can be calculated with
the Leibniz rule. Thus we haveZ
Zn
(Q1 ϕ)ft+s dH −ϕft dHnXX¸
Z 1 ·Zdn(Qr ϕ)ft+rs dH dr
=
0 dr
X
¶
Z 1Z µ|∇L Qr ϕ|2−
ft+rs + s(Qr ϕ)∆ft+rs dHn dr.
(3.7)
=
2
0
X
Using formulas (2.2), (3.2) and the trivial inequality− h∇g, ∇g̃i ≤1sh∇g, ∇gi + h∇g̃, ∇g̃i ,2s2

Hn -a.e. for s > 0,

we haveZ(Qr ϕ)∆ft+rs dHn
XZ=−h∇Qr ϕ, ∇ft+rs i dHnÀ
ZX ¿∇ft+rsft+rs dHn=−
∇Qr ϕ,
f
t+rs
ZX
Z
1
s
h∇ft+rs , ∇ft+rs in≤h∇Qr ϕ, ∇Qr ϕi ft+rs dH +
dHn .
2s X
2 X
ft+rs

15

Plugging this inequality in (3.7) and recalling that h∇Qr ϕ, ∇Qr ϕi = |∇L Qr ϕ|2
Hn -a.e. (since Qr ϕ is Lipschitz), we obtain
Z
Z
Z Z
s2 1
h∇ft+rs , ∇ft+rs inn
ϕft dH ≤dHn dr.
(Q1 ϕ)ft+s dH −
2
f
t+rs
X
X
0
X
This bound does not depend on ϕ, thus from (3.6) we deduce
Z 1Z¡¢
h∇ft+rs , ∇ft+rs i22dHn dr.
W2 Tt ν, Tt+s ν ≤ s
ft+rs
0
X

(3.8)

Since we have (3.2) and the Dirichlet energy decreases along the ﬂow t 7→ ft , weobtainZ Z
Z¡¢ s2 1s2
W22 Tt ν, Tt+s ν ≤h∇ft+rs , ∇ft+rs i dHn dr ≤h∇ft , ∇ft i dHn ,
c 0 X
c X
which gives that the map t 7→ Tt ν ∈ P(X) is locally Lipschitz. The bound (3.5)
follows directly from (3.8) (by recalling the deﬁnition of the absolutely continuous
curves (2.8)).
¤
Now we are ready to prove Theorem 3.1.
Proof of Theorem 3.1. By Propositions 3.7, 3.2 and 3.4, we haveZh∇ft , ∇ft id
1 ˙ n2 1
2n|ft H | + |∇− Ent | (ft H ) ≤dHn = − Ent(ft Hn )
2
2
f
dt
t
X
a.e. t. As the reverse inequality (2.11) is always true, equality (2.12) holds for all
0 < t ≤ s. Since (ft Hn )t∈[0,∞) is continuous also at t = 0, it is the gradient ﬂow
of the relative entropy. The converse immediately follows from the uniqueness ofthe gradient ﬂow of the relative entropy.
¤
Remark 3.8 (i) When ν = f0 Hn with f0 ∈ L2 (X, Hn ), we can give a proof of
Theorem 3.1 without relying on the positivity improving property (3.2). Indeed,by virtue of the contraction property of both ﬂows, it is possible to use Propositions 3.4, 3.7 only for 0 < c ≤ f0 ≤ C and prove the theorem via approximation.
It suggests that our argument possibly works in a more general framework where
the existence of the density pt (x, y) does not follow from the theory of Dirichletforms. In such a case, we could ‘construct’ pt (x, ·) as the gradient ﬂow startingfrom the Dirac measure δx ∈ P(X) (via the contraction property).
(ii) In the proof of Theorem 3.1, we heavily rely on the fact h∇f, ∇f i = |∇L f |2
for f ∈ C Lip (X) for which we used the local structure of Alexandrov spaces.
Actually, we use |∇L f | in Propositions 3.2, 3.7 and |∇f | in Propositions 3.4, 3.7.
16

4

Applications

In this section, we assume that (X, d) is a compact Alexandrov space withoutboundary satisfying CD(K, ∞), and prove some applications of Theorem 3.1. Itshould be stressed that the ‘sectional curvature bound’ k in the sense of Alexandrov appears nowhere in the sequel, and the ‘Ricci curvature bound’ K is essential
instead (recall Remark 2.5). Indeed, those results involving K are natural extensions of the corresponding ones on a Riemannian manifold with Ric ≥ K. In thissense, the emergence of K instead of k is natural and gives sharper estimates.
Since the gradient ﬂow of the Dirichlet energy Tt ν is obviously linear andsymmetric, we immediately obtain the following:
Theorem 4.1 (Linearity and symmetry) For ν ∈ P(X), let (µνt )t≥0 be thegradient ﬂow of the relative entropy on P(X) with µν0 = ν. Then the following
hold true.
(i) For ν0 , ν1 ∈ P(X), λ ∈ [0, 1] and t ≥ 0,(1−λ)ν0 +λν1

µt

= (1 − λ)µνt 0 + λµνt 1 .

(ii) For f, g ∈ L1 (X, Hn ) with f, g ≥ 0 and kf kL1 = kgkL1 = 1,ZZnnf dµgH=g dµtf H .
tXXWe remark that the linearity, but not symmetry, also follows from the gradient
ﬂow approach under the local angle condition ([38, Theorem 8]). In general, theseproperties are completely nontrivial, and the linearity indeed fails in the Finsler
setting ([28]).
A new property for (Tt ν)t≥0,ν∈P(X) coming from our identiﬁcation with thegradient ﬂow of the relative entropy is the L2 -Wasserstein contraction (2.13). Together with [19, Corollary 3.4], we obtain the following:
Theorem 4.2 (Contraction for the heat ﬂow) For 1 ≤ p ≤ 2,
Wp (Tt ν0 , Tt ν1 ) ≤ e−Kt Wp (ν0 , ν1 )(4.1)

holds for every ν0 , ν1 ∈ P(X).
Furthermore, by the duality result [19, Theorem 2.2], Theorem 4.2 yields thefollowing Bakry-Émery type L2 -gradient estimate|∇L Tt f |(x) ≤ e−Kt Tt (|∇L f |2 )(x)1/2
17

(4.2)

for any f ∈ C Lip (X) and x ∈ X. By combining (4.2) with the regularity of theheat kernel, we can extend it to f ∈ W 1,2 (X) as follows:
Theorem 4.3 (Gradient estimate) Let f ∈ W 1,2 (X) and t > 0. Then Tt f ∈C Lip (X) and|∇L Tt f |(x) ≤ e−Kt Tt (|∇f |2 )(x)1/2(4.3)
holds for all x ∈ X. In particular,|∇Tt f |(x) ≤ e−Kt Tt (|∇f |2 )(x)1/2
pand |∇L Tt f | ≤ e−Kt kTt kL1 →L∞ E(f, f ) hold.

for a.e. x(4.4)

Proof.
Take {fi }i∈N ⊂ C Lip (X) such that fi → f in W 1,2 (X) as i → ∞.
Then (4.2) yields Tt fi ∈ C Lip (X). Let y ∈ X and γ : [0, l] → X a unit speed
minimal geodesic from x to y. Since |∇L Tt fi | is an upper gradient of Tt fi (see [9,Proposition 1.11] for instance), we have
Z l
|Tt fi (y) − Tt fi (x)| ≤|∇L Tt fi |(γ(s)) ds
0
Z l
−Kt
≤e
Tt (|∇L fi |2 )(γ(s))1/2 ds,(4.5)
0

where the second inequality follows from (4.2). Thanks to (2.3) and the boundedness of pt , Tt fi converges pointwisely to Tt f as i → ∞. Since Tt (|∇L fi |2 )(z) =Tt (|∇fi |2 )(z) for z ∈ X, Tt (|∇L fi |2 ) converges pointwisely to Tt (|∇f |2 ) in a similarmanner. Thus, by letting i → ∞ in (4.5), we obtainZ l−Kt|Tt f (x) − Tt f (y)| ≤ e
Tt (|∇f |2 )(γ(s))1/2 ds.
(4.6)
0

The boundedness of pt yields that there is C > 0 satisfying
Tt g(z) ≤ CkgkL1 (X,Hn )(4.7)

for all g ∈ L1 (X, Hn ) and z ∈ X. Since l = d(x, y), the estimate (4.7) forg = |∇f |2 together with (4.6) implies Tt f ∈ C Lip (X). In order to show (4.3),
choose a sequence {yi }i∈N in X so that yi → x as i → ∞ and|Tt f (x) − Tt f (yi )|= |∇L Tt f |(x).
i→∞d(x, yi )lim

By the continuity of pt , Tt (|∇f |2 ) ∈ C(X) holds. Thus, applying (4.6) for y = yi ,dividing both sides of it by d(x, yi ) and letting i → ∞ yield (4.3).
¤
18

As an easy but important consequence of Theorem 4.3, we obtain the Lipschitz
continuity of the heat kernel pt (x, y) as well as that of eigenfunctions. Recall
that −∆ has discrete spectrum consisting of nonnegative eigenvalues with ﬁnite
multiplicity ([20, Corollary 1.1]).
Theorem 4.4 (Lipschitz continuity) (i) For ν ∈ P(X) and t > 0, let ft =dTt ν/dHn . Then ft ∈ C Lip (X) for t > 0. In particular, we have pt (x, ·) ∈C Lip (X) and Tt f ∈ C Lip (X) for all x ∈ X and f ∈ L1 (X, Hn ).
(ii) Let f be an L2 -eigenfunction of ∆ correspondingpto an eigenvalue −λ < 0.
Then f ∈ C Lip (X). Moreover, |∇L f | ≤ e(λ−K)t λkTt kL1 →L∞ kf kL2 (X,Hn )holds for each t > 0.
Proof. Since ft ∈ W 1,2 (X), Theorem 4.3 yields that ft = Tt/2 ft/2 ∈ C Lip (X).
For the second assertion, note that f = eλt Tt f . Then the ﬁrst assertion andE(f, f ) = λkf k2L2 (X,Hn ) yield the conclusion.
¤
Remark 4.5 (i) It has been known that the heat kernel pt (x, y) is (locally) Hölder
continuous of some fractional exponent, that follows from the parabolic Harnack
inequality shown in [20].
(ii) The existence and the continuity of pt are used in the proof of Theorems 4.3,
4.4 in an essential way (cf. Remark 3.8).
(iii) To obtain a useful estimate of |∇L Tt f | along our argument, we need anice bound for kTt kL1 →L∞ . For instance, the parabolic Harnack inequality, or the
Nash inequality, impliesC(4.8)
pt (x, y) ≤ n √H (B t (x))
with some constant C > 0 being independent of t, x, y, for small t. It gives a boundfor kTt kL1 →L∞ . By a general argument, (4.8) follows from the local Poincaréinequality and the volume doubling condition, both of which depend only on thedimension n of X and a lower curvature bound (see [20, 21, 36, 39] for instance).
However, we should be careful if we want to know whether C in (4.8) depends onthe diameter and/or the volume of X. Indeed, estimates of type (4.8) are mainly
studied on non-compact state spaces and hence they did not seem to pay so muchattentions on such a dependency in the literature.
In what follows, we consider two additional applications of the Bakry-Émery gradient estimate (4.4) by employing the Lipschitz continuity of Tt f . The ﬁrst one is
the following inequality:

19

Theorem 4.6 (Γ2 -condition) Let f ∈ D(∆) with ∆f ∈ W 1,2 (X). Then, forg ∈ D(∆) ∩ L∞ (X, Hn ) with g ≥ 0 and ∆g ∈ L∞ (X, Hn ), we haveZZZ
1∆g h∇f, ∇f i dHn −g h∇∆f, ∇f i dHn ≥ K
g h∇f, ∇f i dHn . (4.9)
2 X
X
X
Remark 4.7 (i) By virtue of the analyticity of Tt and Theorem 4.4, we have
Tt f ∈ D(∆m+1 ) and ∆m Tt f ∈ C Lip (X) for any f ∈ L2 (X, Hn ), t > 0 and m ≥ 0.
Thus there are fairly many f and g’s satisfying the condition in Theorem 4.6.
(ii) The inequality (4.9) is nothing but a weak form of the Γ2 -condition
Γ2 (f, f ) :=ª1©
∆(h∇f, ∇f i) − 2 h∇f, ∇∆f i ≥ K h∇f, ∇f i .
2(4.10)

This inequality is known to be equivalent to (4.4) in an abstract framework (see[3, 23] and references therein, for instance). However, the assumption involves
the existence of a nice core A ⊂ D(∆) and it seems hopeless to verify it on
Alexandrov spaces. When X is a complete Riemannian manifold, the inequality(4.10) is equivalent to Ric ≥ K. Indeed,Γ2 (f, f ) = Ric(∇f, ∇f ) + | Hess f |2
holds by the Bochner identity.
Proof. We ﬁrst show the claim for f ∈ D(∆) ∩ C Lip (X) with ∆f ∈ D(∆) ∩L∞ (X, Hn ). By (4.4), we obtain
Z
Z
n
−2Kt
g h∇Tt f, ∇Tt f i dH ≤ egTt (h∇f, ∇f i) dHn .
(4.11)X

X

The derivation property (2.7) yields that, for t ≥ 0,
Z
Z
Z
n
n
g h∇Tt f, ∇Tt f i dH =h∇(gTt f ), ∇Tt f i dH −Tt f h∇g, ∇Tt f i dHn
X
X
X
Z
Z
­
®
1
n
∇g, ∇(Tt f )2 dHn=−
gTt f ∆Tt f dH −
2 X
ZX
Z
1
=−
gTt f Tt ∆f dHn +∆g(Tt f )2 dHn .
(4.12)
2
X
X

20

Hence we obtain
¯
Z
¯
d
n¯
g h∇Tt f, ∇Tt f i dH ¯
dt X
Z
Z t=0
Z
2
n
2
n
=−
g(∆f ) dH −gf ∆ f dH +∆gf ∆f dHn
X
X
X
Z
Z
Z
n
n
=
h∇(g∆f ), ∇f i dH +h∇(gf ), ∇∆f i dH −h∇g, ∇(f ∆f )i dHn
X
X
X
Z
=2
g h∇∆f, ∇f i dHn(4.13)
X

by using the derivation property again. Since f ∈ C Lip (X), h∇f, ∇f i = |∇L f |2 ∈L∞ (X, Hn ) holds. Hence we have
µ
¶¯
Z
¯
d
−2Kt
n ¯
e
gTt (h∇f, ∇f i) dH ¯
dt
t=0
µ X Z
¶¯
¯
d
−2Kt
n ¯
e
(Tt g) h∇f, ∇f i dH ¯
=
dt
X
t=0
Z
Z
=
∆g h∇f, ∇f i dHn − 2K
g h∇f, ∇f i dHn .
(4.14)
X

X

Since (4.11) implies
¯
µ
¶¯
Z
Z
¯
¯
d
d
−2Kt
n¯
n ¯
e
g h∇Tt f, ∇Tt f i dH ¯
gTt (h∇f, ∇f i) dH ¯ , (4.15)
≤
dt X
dt
X
t=0
t=0
we obtain (4.9) by combining (4.15) with (4.13) and (4.14).
Next we consider the case that f ∈ D(∆) with ∆f ∈ W 1,2 (X). Then, by
the above discussion, Tδ f and g satisfy (4.9) for δ > 0 (cf. Remark 4.7(i)).
Since g, ∆g ∈ L∞ (X, Hn ) and ∆Tδ f = Tδ ∆f , it suﬃces to show the claim thatlimδ→0 h∇Tδ h, ∇Tδ h0 i = h∇h, ∇h0 i weakly in L1 (X, Hn ) for h, h0 ∈ W 1,2 (X). By
polarization, we may assume h = h0 . The spectral decomposition yields
lim E(Tδ h − h, Tδ h − h) = 0

δ→0(4.16)(see [12, Lemma 1.3.3], for instance). Let ψ ∈ L∞ (X, Hn ). Then the Schwarz
inequality yields
¯Z
¯
Z
¯
¯
¯
n
n¯
ψ h∇Tδ h, ∇Tδ hi dH −ψ h∇h, ∇hi dH ¯
¯
¯ X
¯
X
Ãµ Z
¶1/2 µZ¶1/2 !
≤
ψ 2 |∇Tδ h|2 dHn+
ψ 2 |∇h|2 dHn
E(Tδ h − h, Tδ h − h)1/2
XX≤ 2kψkL∞ E(h, h)1/2 E(Tδ h − h, Tδ h − h)1/2 .
21(4.17)

Hence the desired claim follows from (4.16) and (4.17).

¤

While we proved the implication from (4.1) with p = 2 to (4.4) and (4.9), these
conditions are equivalent to each other on complete Riemannian manifolds (see[37]). Such an equivalence still holds in our framework with a sharp constant,which can be diﬀerent from K in our hypothesis CD(K, ∞):Theorem 4.8 (Equivalence of “Ricci curvature bound” inequalities) Given
K0 ∈ R, the following are equivalent.
(i) (4.1) holds for ν0 , ν1 ∈ P(X) and t ≥ 0, with p = 2 and K = K0 .
(ii) (4.4) holds for f ∈ W 1,2 (X) and t ≥ 0 with K = K0 .
(iii) (4.9) holds with K = K0 for f ∈ D(∆) with ∆f ∈ W 1,2 (X) and g ∈D(∆) ∩ L∞ (X, Hn ) with g ≥ 0 and ∆g ∈ L∞ (X, Hn ).
Proof. “(i) ⇒ (ii) ⇒ (iii)” is already shown in Theorem 4.3 and Theorem 4.6.
For “(iii) ⇒ (ii)”, it follows from a standard argument of the so-called Γ2 calculus (see [3, 23] for instance). For completeness, we give a sketch of theproof. Take g0 ∈ C(X) with g0 ≥ 0 arbitrary and let g = Tδ g0 . Then g ∈D(∆) ∩ L∞ (X, Hn ) with g ≥ 0 and ∆g ∈ L∞ (X, Hn ). Let us deﬁne Ψ : [0, t] → R
byZ
gTs (|∇Tt−s f |2 ) dHn .

Ψ(s) :=
X

By a similar calculation as in (4.12), we can easily prove that Ψ is continuous on[0, t) and C 1 on (0, t). A similar argument as in (4.17) yields that Ψ is continuous
at t. Here we use the ultracontractivity kTt kL1 →L∞ < ∞. A similar calculation
as in the proof of Theorem 4.3 together with the assumption in (iii) leads to theinequality Ψ0 (s) ≥ 2K0 Ψ(s). Hence (4.4) follows by integrating it.
For “(ii) ⇒ (i)”, we claim that (4.4) implies (4.3) for every x ∈ X. Indeed, byusing a bi-Lipschitz chart, we can bring the problem locally on an open set in a
Euclidean space. Then, by applying [2, Lemma 3.2.1] and by using the continuityof Tt (|∇f |2 ), the claim follows. Then we can apply [19, Theorem 2.2] to conclude(i) from (4.3).
¤
Remark 4.9 By the same argument as in “(ii) ⇒ (i)” of the last proof, we cangive a proof of Tt f ∈ C Lip (X) for f ∈ W 1,2 (X) under the condition (4.4). In other
word, a priori regularity Tt f ∈ C Lip (X) by Theorem 4.3 is not used in the last
proof.

22

As the second application of (4.4), we mention that (4.4) together with Theorem 4.4 implies some functional inequalities by means of [17, Theorem 1.3]. Since
Tt is Markovian, we can restrict Tt to a contraction on L∞ (X, Hn ). Then we canfurther extend Tt to a contraction on Lp (X, Hn ) for any 1 ≤ p ≤ ∞ by interpolation and the symmetry of Tt . Let us denote the inﬁnitesimal generator of Tt in(q)
Lp (X, Hn ) by ∆p . Let us deﬁne Rα f := |∇((α − ∆p )−q/2 f )|.
Corollary 4.10 Let 2 ≤ p < ∞, q > 1 and α > max{(−K), 0}. Then we havethe following:(i) There exists a constant CR > 0 which depends only on p, q and max{(α +K), α} such thatkRα(q) f kLp ≤ CR kf kLp
for f ∈ Lp (X, Hn ).
(ii) Suppose q < 2. Then there exists Cp,q > 0 such that³´k|∇Tt f |kLp ≤ Cp,q kRα(q) kLp →Lp αq/2 + t−q/2 kf kLp
for t > 0 and f ∈ Lp (X, Hn ).
Proof.
It is suﬃcient to verify the assumption of [17, Theorem 1.3]. Set
Lip
A := C (X) ∩ D(∆2 ). Since we already know (4.4), we only need to show
the following claim: A is dense in W 1,2 (X) and f 2 ∈ D(∆1 ) holds for any
f ∈ A. By Theorem 4.4, Tt f ∈ A holds. Thus A is dense in W 1,2 (X) sinceE(Tt f − f, Tt f − f ) → 0 as t → 0 (cf. (4.16)). By [6, Proposition I.2.4.3], it isenough to prove f 2 ∈ D(∆2 ) for f ∈ A. Take g ∈ W 1,2 (X) arbitrary. Recall that|∇f |2 = |∇L f |2 ∈ L∞ (X, Hn ). The derivation property yields
¯Z
¯
¯
¯
2
n
|E(f , g)| = 2 ¯¯ f h∇f, ∇gi dH ¯¯
¯ZX
¯
¯
¯
n¯
¯
= 2 ¯ (h∇f, ∇(gf )i − g h∇f, ∇f i)dH ¯
¯
¯ZX
¯
¯
n
= 2 ¯¯ (f ∆f − h∇f, ∇f i)g dH ¯¯¡ X¢≤ 2 kf kL∞ k∆f kL2 + k|∇f |2 kL2 kgkL2 .
This estimate means f 2 ∈ D(∆2 ) and hence the proof is completed.

23¤

Finally, we observe that all of our results are generalized to the heat equation
with drift, in other words, the Fokker-Planck equation. Given a potential function
V ∈ C Lip (X), we modify the Dirichlet energy and the relative entropy intoZV
E (f, g) =h∇f, ∇gi e−V dHn
f, g ∈ W 1,2 (X),X
Z
V
Ent (µ) = Ent(µ) +V dµµ ∈ P(X).
X

We regard E V as a bilinear form on L2 (X, e−V Hn ). Observe that EntV is nothing
but the relative entropy with respect to e−V Hn . Note that the semigroup TtV on
L2 (X, e−V Hn ) associated with E V solves the following diﬀusion equationdu = ∆u − h∇V, ∇ui .
dt(4.18)

Since e−V is bounded and away from 0, e−V Hn is equivalent to Hn . Hence(X, d, e−V Hn ) satisﬁes the volume doubling condition as well as the Poincaré inequality for upper gradients. Moreover, as E and E V are equivalent, the Poincaréinequality for E V is also valid. Therefore a continuous density pVt for TtV exists.
Under the assumption that EntV is K-geodesically convex, we can apply the general theory of the gradient ﬂow on (P(X), W2 ) to obtain the gradient ﬂow µt of
EntV . Furthermore, every argument in Sections 3, 4 works verbatim and gives
similar results for the equation (4.18). Note that, under CD(K, ∞) for (X, d, Hn ),the K 0 -convexity of V implies the (K + K 0 )-convexity of EntV .

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