On the Equivalence of the Entropic Curvature-Dimension Condition and Bochner's Inequality on Metric Measure Spaces

Introduction

Bochner’s inequality is one of the most fundamental estimates in geometric analysis. It states that

The curvature-dimension condition ${\operatorname{CD}(K,N)}$ was introduced by Sturm in . It was later adopted and slightly modified by Lott & Villani, see also the elaborate presentation in the monograph . The ${\operatorname{CD}(K,N)}$-condition for finite $N$ is a sophisticated tightening up of the much simpler $\operatorname{CD}(K,\infty)$-condition introduced as a synthetic Ricci bound for metric measure spaces independently by Sturm and Lott & Villani . From the very beginning, a disadvantage of the ${\operatorname{CD}(K,N)}$-condition for finite $N$ was the lack of a local-to-global result. To overcome this drawback, Bacher & Sturm introduced the reduced curvature-dimension condition ${\operatorname{CD}^{*}(K,N)}$ which has a local-to-global property and which is equivalent to the local version of ${\operatorname{CD}(K,N)}$. The curvature-dimension condition ${\operatorname{CD}(K,N)}$ has been verified for Riemannian manifolds , Finsler spaces , Alexandrov spaces , , cones and warped products of Riemannian manifolds . Actually, in all these cases the conditions ${\operatorname{CD}(K,N)}$ and ${\operatorname{CD}^{*}(K,N)}$ turned out to be equivalent.

A completely different approach to generalized curvature-dimension bounds was set forth in the pioneering work of Bakry and Émery . It applies to the general setting of Dirichlet forms and the associated Markov semigroups and is formulated using the (iterated) carré du champ operators built from the generator of the semigroup. This energetic curvature-dimension condition $\operatorname{BE}(K,N)$ has proven a powerful tool in particular in infinite dimensional situations. It yields hypercontractivity of the semigroup and has successfully been used to derive functional inequalities like the logarithmic Sobolev inequalities in a variety of examples. Among the remarkable analytic consequences of the Bakry–Émery condition $\operatorname{BE}(K,\infty)$ we single out the point-wise gradient estimates for the semigroup $H_{t}$. It implies that for any $f$ in a large class of functions

where $\Gamma$ is the carré du champ operator.

The relation between the two notions of curvature bounds based on optimal transport and Dirichlet forms has been studied in large generality by Ambrosio, Gigli and Savaré in a series of recent works , see also . The key tool of their analysis is a powerful calculus on metric measure spaces which allows them to match the two settings. Starting from a metric measure structure they introduce the so called Cheeger energy which takes over the role of the ’standard’ Dirichlet energy and is obtained by relaxing the $L^{2}$-norm of the slope of Lipschitz functions. A key result is the identification of the $L^{2}$-gradient flow of the Cheeger energy with the Wasserstein gradient flow of the entropy. This is the mms equivalent of the famous result by Jordan–Kinderlehrer–Otto and allows one to define unambiguously a heat flow in metric measure spaces.

We say that a metric measure space is infinitesimally Hilbertian if the heat flow is linear. This is equivalent to the Cheeger energy being the associated Dirichlet form. We denote its domain by $W^{1,2}$. Under the assumption of linearity of the heat flow, Ambrosio–Gigli–Savaré prove that $\operatorname{CD}(K,\infty)$ implies $\operatorname{BE}(K,\infty)$ and the converse also holds under an additional regularity assumption. Combining linearity of the heat flow with the $\operatorname{CD}(K,\infty)$ condition leads to the Riemannian curvature condition $\operatorname{RCD}(K,\infty)$ introduced in . This concept again turns out to be stable under Gromov–Hausdorff convergence and tensorization.

Recently, also Bochner’s inequality has been extended to singular spaces. Ohta & Sturm proved it for Finsler spaces and Gigli, Kuwada & Ohta and Zhang & Zhu for Alexandrov spaces. Finally, Ambrosio, Gigli & Savaré established the Bochner inequality without the dimension term (i.e. with $N=\infty$) in $\operatorname{RCD}(K,\infty)$ spaces. However, in the classical setting, the full strength of Bochner’s inequality only comes to play if also the dimension effect is taken into account, i.e. with finite $N$. This can be seen for example from the famous results of Li–Yau who derive from it a differential Harnack inequality, eigenvalue estimates for the Laplacian and Gaussian heat kernel bounds.

We prove the equivalence of curvature-dimension bounds via optimal transport and via the Bakry–Émery approach in full generality for infinitesimally Hilbertian metric measure spaces. In particular, we establish the full Bochner inequality on such metric measure spaces.

holds in distribution sense. A function $S$ on a geodesic space is called $(K,N)$-convex if it is $(K,N)$-convex along each unit speed geodesic – or at least along each curve within a class of unit speed geodesics which connect each pair of points in $X$. This way, $(K,N)$-convexity is a weak formulation of

For a essentially non-branching mms (see Definition 3.10) the entropic curvature-dimension condition ${\operatorname{CD}^{e}(K,N)}$ is equivalent to the reduced curvature-dimension condition ${\operatorname{CD}^{*}(K,N)}$.

We say that a metric measure space satisfies the Riemannian curvature-dimension condition ${\operatorname{RCD}^{*}(K,N)}$ if it is infinitesimally Hilbertian and satisfies ${\operatorname{CD}^{e}(K,N)}$ or ${\operatorname{CD}^{*}(K,N)}$. This notion turns out to have the natural stability properties. Namely, we prove (see Theorems 3.22, 3.23, 3.25) that the ${\operatorname{RCD}^{*}(K,N)}$ condition is preserved under measured Gromov–Hausdorff convergence as well as under tensorization of metric measure spaces and that it has a local–to–global property.

The geometric intuition coming from the analysis of $(K,N)$-convex functions and their gradient flows leads to a new form of the Evolution Variation Inequality ${\operatorname{EVI}_{K,N}}$ on the Wasserstein space taking into account also the effect of the dimension bound. Until now, the notion of ${\operatorname{EVI}_{K,N}}$ gradient flow was known only without dimension term (i.e. with $N=\infty$). These Evolution Variational Inequalities first appeared in the setting of Hilbert spaces where they characterize uniquely the gradient flows of $K$-convex functionals. In a general metric setting and in connection with optimal transport these inequalities have been extensively studied in . In particular, it turned out that $\operatorname{RCD}(K,\infty)$ spaces can be characterized by the fact that the heat flow is an $\operatorname{EVI}_{K,\infty}$ gradient flow of the entropy. Here we obtain a reinforcement of this result. Namely, the new Riemannian curvature-dimension condition ${\operatorname{RCD}^{*}(K,N)}$ is equivalent to the existence of an ${\operatorname{EVI}_{K,N}}$ gradient flow of the entropy in the following sense.

A mms $(X,d,m)$ satisfies ${\operatorname{RCD}^{*}(K,N)}$ if and only if $(X,d)$ is a length space, $m$ satisfies an integrability condition (3.6) and every $\mu_{0}\in\mathscr{P}_{2}(X,d)$ is the starting point of a curve $(\mu_{t})_{t\geq 0}$ in $\mathscr{P}_{2}(X,d)$ such that for any other $\nu\in\mathscr{P}_{2}(X,d)$ and a.e. $t>0$:

Here U_{N}(\mu)=\exp\Big{(}-\frac{1}{N}\operatorname{Ent}(\mu)\Big{)} and {\mathfrak{s}}_{\kappa}(r)=\sqrt{1/\kappa}\sin\big{(}\sqrt{\kappa}r\big{)} provided $\kappa>0$ and {\mathfrak{s}}_{\kappa}(r)=\sqrt{1/(-\kappa)}\sinh\big{(}\sqrt{-\kappa}r\big{)},\ {\mathfrak{s}}_{0}(r)=r for $\kappa<0$ resp. $\kappa=0$.

This curve is unique, in fact, it is the heat flow which we denote in the following by $\mu_{t}=H_{t}\mu_{0}$.

The Evolution Variation Inequality ${\operatorname{EVI}_{K,N}}$ as stated above immediately implies new, sharp contraction estimates (or, more precisely, expansion bounds) in Wasserstein metric for the heat flow.

Let $(X,d,m)$ be a ${\operatorname{RCD}^{*}(K,N)}$ space. Then for any $\mu,\nu\in\mathscr{P}_{2}(X,d)$ and $s,t>0$:

The latter implies the slightly weaker bound

Due to the work of Kuwada , it is well known that $W_{2}$-expansion bounds are intimately related to pointwise gradient estimates. The next result is a particular case of a more general equivalence that will be the subject of a forthcoming publication .

Assume that the mms $(X,d,m)$ is infinitesimally Hilbertian and satisfies a regularity assumption (Assumption 4.2). If the $W_{2}$-expansion bound (1.5) holds then for any $f$ of finite Cheeger energy:

Assume that the mms $(X,d,m)$ is infinitesimally Hilbertian and satisfies the gradient estimate (1.6). Then for all $f\in D(\Delta)$ with $\Delta f\in W^{1,2}(X,d,m)$ and all $g\in D(\Delta)$ bounded and non-negative with $\Delta g\in L^{\infty}(X,m)$ we have

Assume that the mms $(X,d,m)$ is infinitesimally Hilbertian and satisfies Assumption 4.2. Then the Bochner inequality $\operatorname{BE}(K,N)$ (1.7) implies the entropic curvature-dimension condition ${\operatorname{CD}^{e}(K,N)}$.

Thus we have closed the circle. All the previous key properties are equivalent to each other, at least if we require the heat flow to be linear.

Let $(X,d,m)$ be an infinitesimally Hilbertian metric measure space. Then the following properties are equivalent:

$(X,d)$ is a length space, (3.6) and the existence of the ${\operatorname{EVI}_{K,N}}$ gradient flow of the entropy starting from every $\mu\in\mathscr{P}_{2}(X,d)$.

If one of them is satisfied, we obtain the following:

The Bakry–Ledoux pointwise gradient estimate $\operatorname{BL}(K,N)$ (1.6),

The Bochner inequality $\operatorname{BE}(K,N)$ (1.7).

Moreover, under Assumption 4.2, all of properties (i)–(vi) are equivalent.

Finally, let us point out – on a more heuristic level – two remarkable links between $(K,N)$-convexity and the Bakry-Émery condition $\operatorname{BE}(K,N)$:

The $(K,N)$-convexity of a function $V$ on a Riemannian manifold $(M,g)$ can be interpreted as the $BE(K,N)$-condition for the re-scaled drift diffusion

The $\operatorname{BE}(K,N)$-condition for the Brownian motion or heat flow on $M$ is equivalent to the $(K,N)$-convexity of the function $S=\operatorname{Ent}(.)$ on the Wasserstein space $\mathscr{P}_{2}(M)$.

Both links are related to each other since the heat flow is the solution to the ODE (”without diffusion”)

on $\mathscr{P}_{2}(M)$ (regarded as infinite dimensional Riemannian manifold). The link (II) is the main result of this paper.

[Prop. 4.21]. In the Wasserstein picture, the $\operatorname{BE}(\frac{1}{\alpha}K,\frac{1}{\alpha}N)$-condition for $\widetilde{L}$ translates into the $(\frac{1}{\alpha}K,\frac{1}{\alpha}N)$-convexity of the functional $\widetilde{S}(\mu)=\operatorname{Ent}(\mu)+\frac{1}{\alpha}\int V\,d\mu$ [Thm. 7]. The latter in turn is equivalent to the $(K,N)$-convexity of $S(\mu)=\alpha\operatorname{Ent}(\mu)+\int V\,d\mu$ on $\mathscr{P}_{2}(M)$ [Lemma 2.9].

Note that this also makes perfectly sense for $\alpha=0$ in which case the associated gradient flow equation on the Wasserstein space $\mathscr{P}_{2}(M)$ reads

This is the $(K,N)$-convexity of $V$ on $M$.

Organization of the article. First we illustrate the new concept of $(K,N)$-convexity in a smooth and finite dimensional setting. Since many of the arguments which relate geodesic convexity, the Evolution Variational Inequality and space-time expansion bounds for the gradient flow are of a purely metric nature we study $(K,N)$-convexity, $\operatorname{EVI}_{K,N}$ and its consequences in the general setting of metric spaces in Section 2. In Section 3 we turn to the study of $(K,N)$-convexity of the entropy on the Wasserstein space. The entropic curvature-dimension condition is introduced in Section 3.1 and its basic properties are established. In particular we prove equivalence with the reduced curvature-dimension condition for essentially non-branching spaces. In Section 3.3 we prove that the entropic curvature-dimension condition plus linearity of the heat flow is equivalent to the existence of an $\operatorname{EVI}_{K,N}$ gradient flow of the entropy which leads to the Riemannian curvature-dimension condition. Here we also prove the stability results for ${\operatorname{RCD}^{*}(K,N)}$. Finally, in Section 4 we prove the equivalence of the entropic curvature-dimension condition, space-time Wasserstein expansion bounds, pointwise gradient estimates and the Bochner inequality for infinitesimally Hilbertian metric measure spaces. As applications, new functional inequalities deduced from ${\operatorname{CD}^{e}(K,N)}$ are studied in Section 3.4 and the sharp Lichnerowicz bound for ${\operatorname{RCD}^{*}(K,N)}$ spaces is established in Section 4.3.

(K,N)𝐾𝑁(K,N)-convex functions and their EVI gradient flows

In order to illustrate the concept of $(K,N)$-convexity of the entropy and the consequences for its gradient flow, we consider in this section a smooth and finite-dimensional setting.

in the sense that for all $x\in M$ and $v\in T_{x}M$ we have

A direct calculation shows that (2.1) can equivalently be written as:

This condition can be thought of as a “concavity” property of $U_{N}$. As with concavity, it can be expressed in an integrated form. To this end we introduce the following functions.

For each constant speed geodesic $(\gamma_{t})_{t\in}$ in $M$ and all $t\in$ we have with $d:=d(\gamma_{0},\gamma_{1})$:

For each constant speed geodesic $(\gamma_{t})_{t\in}$ in $M$ we have that

(i)$\Rightarrow$(ii): Let $(\gamma_{t})_{t\in}$ be a constant speed geodesic. Then in particular $|{\dot{\gamma_{t}}}|_{\gamma_{t}}=d$ and (2.2) immediately yields that the function $u:t\mapsto U_{N}(\gamma_{t})$ satisfies

(ii)$\Rightarrow$(iii): This follows immediately by subtracting $U_{N}(\gamma_{0})$ on both sides of (2.3), dividing by $t$ and letting $t\searrow 0$.

(iii)$\Rightarrow$(i): Let $\gamma:\to M$ be a constant speed geodesic with $\gamma_{0}=x$ and $\dot{\gamma}_{0}=v$, i.e. $d=d(\gamma_{0},\gamma_{1})=|{v}|$. Using (2.4) for the rescaled geodesics $\gamma^{\prime}:\to M,\;t\mapsto\gamma_{\varepsilon t}$ and $\gamma^{\prime\prime}:\to M,\;t\mapsto\gamma_{-\varepsilon t}$ and adding up we obtain

Dividing by $\varepsilon^{2}$ and using the fact that {\mathfrak{c}}_{K/N}\big{(}\varepsilon d\big{)}=1-\frac{K}{N}\varepsilon^{2}d^{2}+o(\varepsilon^{2}) finally yields

Assume that $S$ is $(K,N)$-convex and differentiable. A smooth curve $x:[0,\infty)\to M$ is a solution to the gradient flow equation

if and only if the following Evolution Variation Inequality ($EVI_{K,N}$) holds: for all $z\in M$ and all $t>0$:

To prove the only if part, fix $t\geq 0$, $z\in M$ and a constant speed geodesic $\gamma:\to M$ connecting $x_{t}$ to $z$. Observe that by (2.6) and the first variation formula we have

Combining this with the $(K,N)$-convexity condition in the form (2.4) we obtain with $d=d(x_{t},z)$:

it is immediate to see that the last inequality is equivalent to (2.7).

For the if part, fix $t\geq 0$ and a constant speed geodesic $\gamma:\to M$ with $\gamma_{0}=x_{t}$. Using the Evolution Variational inequality in the form (2.8) with $z=\gamma_{\varepsilon}$ for some $\varepsilon>0$ we obtain

where $v=\dot{\gamma}_{0}$. Dividing by $\varepsilon$ and letting $\varepsilon\searrow 0$, taking into account that {\mathfrak{c}}_{K/N}\big{(}\varepsilon d\big{)}=1+o(\varepsilon) and ${\mathfrak{s}}_{K/N}\left(\varepsilon d\right)=\varepsilon d+o(\varepsilon^{2})$, we obtain

Since the direction of $v\in T_{x_{t}}M$ was arbitrary we obtain (2.6). ∎

We conclude this section by exhibiting some 1-dimensional models of $(K,N)$-convex functions.

Each of the following are $(K,N)$-convex functions. Note that the domain of definition is maximal in each case.

The cases (i) and (iv) of the previous example canonically extend to multidimensional spaces.

Let $(M,g)$ be a $n$-dimensional Riemannian manifold, $z\in M$ be any point and $N>0$ be any real number.

Indeed, analogous statements hold true on geodesic spaces with generalized bounds for the sectional curvature in the sense of Alexandrov .

2. (K,N)𝐾𝑁(K,N)-convexity in metric spaces

We proceed our study of $(K,N)$-convexity in a purely metric setting. Let $(X,d)$ be a complete and separable metric space and let $S:X\to[-\infty,\infty]$ be a functional on $X$. We denote by $D(S):=\{x\in X~{}:~{}S(x)<\infty\}$ the proper domain of $S$. Given a number $N\in(0,\infty)$ we define the functional $U_{N}:X\to[0,\infty)$ by setting

If (2.11) holds for every geodesic $\gamma:\to D(S)$ we say that $S$ is strongly $(K,N)$-convex.

For investigating $(K,N)$-convexity (especially for the strong form), the following equivalent conditions will be helpful in the sequel.

For each constant speed geodesic $\gamma:\to X$ and $t\in$, $\displaystyle u^{\prime\prime}(\gamma_{t})~{}\leq~{}-\kappa d(\gamma_{0},\gamma_{1})^{2}u(\gamma_{t})$ in the distributional sense, i.e.

for any $\varphi\in C_{0}^{\infty}((0,1))$ with $\varphi\geq 0$.

For each constant speed geodesic $\gamma$ on $X$ and $t\in$,

For each constant speed geodesic $\gamma$ on $X$, there is $\delta=\delta_{\gamma}>0$ such that for all $0\leq s\leq t\leq 1$ with $t-s\leq\delta$ and $\alpha\in$,

For each constant speed geodesic $\gamma$ on $X$ and $t\in$,

with $g(t,r)=\min\{(1-t)r,(1-r)t\}$ being the Green function on the interval $$.

In particular, when $-\infty\notin S(X)$ and $S$ is lower semi-continuous, $S$ is strongly $(K,N)$-convex if and only if $u=U_{N}$ and $\kappa=K/N$ satisfies one of these conditions.

For simplicity of presentation, we denote $\theta=\theta_{\gamma}=d(\gamma_{0},\gamma_{1})$ in this proof whenever a fixed geodesic is under consideration. we also denote the restriction of $\gamma$ on $[s,t]$ for $0\leq s<t\leq 1$ by $\gamma^{[s,t]}:\to X$, that is, $\gamma^{[s,t]}_{r}:=\gamma_{(1-r)s+rt}$.

for any $\varphi\in C^{\infty}_{0}((s,t))$ with $\varphi\geq 0$, (i) implies $(u(\gamma_{\cdot})-\kappa\theta^{2}u_{*})^{\prime\prime}\leq 0$ on $$in the distributional sense. Thus the distributional characterization of convex functions (see [38, Theorem 1.29], for instance) yields that$u(\gamma_{\cdot})-\kappa\theta^{2}u_{*}$coincides with a concave function a.e. and hence concave because$u$is upper semi-continuous. It immediately implies (iv) since$u_{*}(0)=u_{*}(1)=1$.

(iv) $\Rightarrow$ (i): Note first that $u(\gamma_{t})$ is continuous. Indeed, the condition (iv) together with the upper semi-continuity of $u$ implies that $u(\gamma_{t})$ is continuous at $t=0,1$. Thus the continuity follows by applying the same for $\gamma^{[0,s]}$ and $\gamma^{[s,1]}$. For $s\in(0,1)$ and $h>0$ with $s+h,s-h\in$, we apply (iv) to $\gamma^{[s-h,s+h]}$ and $t=1/2$ to obtain

Then (i) follows by multiplying $\varphi\in C_{0}^{\infty}((0,1))$, integrating w.r.t. $t$ (for sufficiently small $h$), dividing by $h^{2}$ and $h\to 0$ with a change of variable.

Then (i) implies $\widetilde{u}_{\varepsilon}^{\prime\prime}(t)\leq-\kappa\theta^{2}\widetilde{u}_{\varepsilon}(t)$ for each $t\in[a_{\varepsilon},1]$ for some $a_{\varepsilon}>0$. Note that $a_{\varepsilon}$ can be chosen so that $\lim_{\varepsilon\to 0}a_{\varepsilon}=0$. Thus, in the same way as in Lemma 2.2, we obtain

By virtue of the equivalence (i) $\Leftrightarrow$ (iv), $u\circ\gamma$ is continuous and hence $\widetilde{u}_{\varepsilon}\to u\circ\gamma$ as $\varepsilon\to 0$ uniformly on $$. Thus the conclusion follows by letting$\varepsilon\to 0$.

(ii) $\Rightarrow$ (iii): It follows by considering (ii) for $\gamma^{[s,t]}$.

(iii) $\Rightarrow$ (i): We imitate the proof of the implication (iv) $\Rightarrow$ (i) by using the following:

We conclude this section with some remarks about $(K,N)$-convexity. The first property is immediate from the definition.

If $S$ is $(K,N)$-convex, then for $\lambda>0$ the functional $\lambda\cdot S$ is $(\lambda K,\lambda N)$-convex.

Let $S^{1}:X\to(-\infty,\infty]$ be a $(K_{1},N_{1})$-convex functional and $S^{2}:X\to(-\infty,\infty]$ a strongly $(K_{2},N_{2})$-convex functional. Then the functional $S:=S^{1}+S^{2}$ is $(K_{1}+K_{2},N_{1}+N_{2})$-convex. In particular, $S$ is strongly $(K_{1}+K_{2},N_{1}+N_{2})$-convex if $S^{1}$ is strongly $(K_{1},N_{1})$-convex.

Let us set $K=K_{1}+K_{2}$ and $N=N_{1}+N_{2}$ and given $x_{0},x_{1}\in D(S)=D(S^{1})\cap D(S^{2})$ take a constant speed geodesic $\gamma:\to X$ from $x_{0}$ to $x_{1}$ according to the convexity assumption of $S^{1}$. By the convexity assumption on $S^{1}$ and $S^{2}$ we have

where the function $G_{t}$ is given by (2.14). By Lemma 2.11 below, $G_{t}$ is convex. Hence we obtain

Taking the exponential on both sides yields the claim. The last assertion is obvious from the proof. ∎

Note that we have $\sigma^{(s)}_{\kappa}(\theta)=\sigma^{(s)}_{\kappa\theta^{2}}(1)$ for $s\in$, $\theta\geq 0$ and $\kappa\in(-\infty,\pi^{2}/\theta^{2})$. It is useful to apply this lemma.

We define the function $g^{(t)}:\kappa\mapsto\log\sigma^{(t)}_{\kappa}(1)$ on $(-\infty,\pi^{2})$ and write

where F(u,v)=\log\big{(}e^{u}+e^{v}\big{)}. The claim then follows by noting that the function $F$ is convex, $a\mapsto F(u+a,v+a)$ is increasing and that the functions $g^{(t)}$ are convex. ∎

Finally we remark that the notion of $(K,N)$-convexity is consistent in the parameters $K$ and $N$.

If $S$ is $(K,N)$-convex then it is also $(K^{\prime},N^{\prime})$-convex for all $K^{\prime}\leq K$ and $N^{\prime}\geq N$. Moreover, it is $K$-convex in the sense that for each pair $x_{0},x_{1}\in D(S)$ there exist a constant speed geodesic $\gamma:\to X$ connecting $x_{0}$ to $x_{1}$ such that for all $t\in$:

Consistency in $K$ is immediate from the fact that for any fixed $t$ and $\theta$ the coefficient \sigma^{(t)}_{K/N}\big{(}\theta\big{)} is increasing in $K$. Consistency in $N$ is a consequence e.g. of Lemma 2.10 and the trivial observation that for any $N^{\prime}>N$ the constant functional $S^{0}\equiv 0$ is $(0,N^{\prime}-N)$-convex.

Using the consistency in $N$ we can derive (2.15) by subtracting $1$ on both sides of (2.11), multiplying with $N$ and passing to the limit $N\nearrow\infty$. Here we use the fact that \sigma^{(t)}_{K/N}\big{(}\theta\big{)}=t+-K(t^{3}-t)\theta^{2}/(6N)+o(1/N) and $U_{N}(x)=1-S(x)/N+o(1/N)$. ∎

3. Evolution Variational Inequalities in metric spaces

In this section we study the Evolution Variational Inequality with parameters $K$ and $N$ and the associated notion of gradient flow in a purely metric setting. In particular, we investigate the relation with geodesic convexity. Our approach extends the results obtained in where the case $N=\infty$ has been considered.

Let $(X,d)$ be a complete separable geodesic metric space and $S:X\to(-\infty,\infty]$ a lower semi-continuous functional. Note that our framework is slightly more restrictive than that in the last section. We define the descending slope of $S$ at $x\in D(S)$ as

for some $g\in L^{1}(I)$. For an absolutely continuous curve $\gamma$ the metric speed, defined by

exists for a.e. $t\in I$ and is the minimal $g$ in (2.16) (see e.g. [3, Thm. 1.1.2]). The following is a classical notion of gradient flow in a metric space, see e.g. .

We say that a locally absolutely continuous curve $x:[0,\infty)\to X$ with $x_{0}\in D(S)$ is a (downward) gradient flow of $S$ starting in $x_{0}$ if the Energy Dissipation Equality holds:

We introduce here a more restrictive notion of gradient flow based on the Evolution Variational Inequality.

If $(x_{t})_{t}$ is an ${\operatorname{EVI}_{K,N}}$ flow for $S$, then it is also an $\operatorname{EVI}_{K^{\prime},N^{\prime}}$ flow for $S$ for any $K^{\prime}\leq K$ and $N^{\prime}\geq N$. Moreover, $(x_{t})$ is an $\operatorname{EVI}_{K}$ flow for $S$, i.e. for all $z\in D(S)$ and a.e. $t>0$:

Using the (2.9) one checks that (2.18) is equivalent to either of the following inequalities:

are increasing in $N$. (2.19) follows immediately from (2.21) by passing to the limit as $N\to\infty$. For this we note that

This shows consistency with the theory of $\operatorname{EVI}_{K}$ gradient flows of geodesically $K$-convex functions. It can be thought of as the limiting case $N=\infty$. By taking the limit $N\to\infty$ in the estimates obtained in this section we recover the corresponding results for $\operatorname{EVI}_{K}$ flows established in .

Let $(x_{t})$ be an ${\operatorname{EVI}_{K,N}}$ gradient flow of $S$ starting in $x_{0}$. Then the following statements hold:

If $x_{0}\in D(S)$ then $(x_{t})$ is also a metric gradient flow in the sense of Definition 2.13. In particular, the map $t\mapsto S(x_{t})$ is non-increasing.

If $S$ is bounded below we have the uniform continuity estimate

By Lemma 2.15 $(x_{t})$ is an $\operatorname{EVI}_{K}$ flow of $S$ and hence a metric gradient flow by [1, Prop. 3.9]. (2.22) follows immediately from (2.24) in Proposition 2.18 below by taking $t_{0}=0$. The uniform continuity estimate (2.23) is obtained similarly by taking $z=x_{t_{0}}$. ∎

Let $D\subset D(S)$ be dense in energy and let $x:(0,\infty)\to D(S)$ be a locally absolutely continuous curve with $\lim_{t\to 0}x_{t}=x_{0}$. Then $(x_{t})$ is an ${\operatorname{EVI}_{K,N}}$ gradient flow of $S$ if and only if one of the following statements holds:

The differential inequality (2.18) holds for all $z\in D$ and a.e. $t>0$.

For all $z\in D$ and all $0\leq t_{0}\leq t_{1}$:

We prove the equivalence of Definition 2.14 and (ii). Assume that $(x_{t})$ is an ${\operatorname{EVI}_{K,N}}$ flow and note that the right hand side of (2.18) can be rewritten as

Integrating from $t_{0}$ to $t_{1}$ and using that the map $t\mapsto U_{N}(x_{t})$ is non-decreasing by (i) of Proposition 2.17 then yields (2.24) for all $z\in D(S)$. Conversely, differentiating (2.24) yields (2.18). The fact that (2.24) holds for all $z\in D(S)$ if and only if it holds for all $z\in D$ is obvious. Similar arguments show the equivalence of Definition 2.14 with (i) and (iii). ∎

An important property of ${\operatorname{EVI}_{K,N}}$ flows is the following expansion bound.

Let $(x_{t}),(y_{t})$ be two ${\operatorname{EVI}_{K,N}}$ gradient flows of $S$ starting from $x_{0}$ resp. $y_{0}$. Then for all $s,t\geq 0$:

Let us fix $s,t>0$. Choose $\lambda,r>0$ such that $\lambda r=t$ and $\lambda^{-1}r=s$, i.e. $\lambda=\sqrt{\frac{t}{s}}$ and $r=\sqrt{ts}$. From (2.24) applied to $(x_{t})$ with $z=y_{\lambda^{-1}r}$ and $t_{0}=\lambda r,t_{1}=\lambda(r+\varepsilon)$ for some $\varepsilon>0$ we obtain

Similarly, choosing $z=x_{\lambda(r+\varepsilon)}$ and $t_{0}=\lambda^{-1}r,t_{1}=\lambda^{-1}(r+\varepsilon)$ and applying (2.24) to $(y_{s})$ we obtain

Multiplying (2.27) and (2.28) after taking square roots and using Young’s inequality, $2\sqrt{ab}\leq\lambda a+\lambda^{-1}b$, we deduce the estimate

and take the limit as $\varepsilon\searrow 0$ in (2.29) we obtain

By an application of Gronwall’s lemma we deduce that

Rewriting $r,\lambda$ in terms of $s,t$ finally yields (2.26). ∎

In the limit $d(x_{0},y_{0})\to 0$ and $s\to t$ the contraction estimate (2.26) reads asymptotically as follows:

For each $x_{0}\in\overline{D(S)}$ there exist at most one ${\operatorname{EVI}_{K,N}}$ gradient flow of $S$ starting from $x_{0}$. The maps $P_{t}:x_{0}\mapsto x_{t}$, where $(x_{t})$ is the unique gradient flow starting from $x_{0}$ constitute a continuous semigroup defined on a closed (possibly empty) subset of $\overline{D(S)}$.

The previous expansion estimate in Theorem 2.26 implies a slightly weaker estimate directly for the distance $d$ not involving the functions $\mathfrak{s}_{K/N}$. More precisely, we have the following:

The expansion bound (2.26) implies the following bound: For each $x_{0},x_{1}\in X$ and $s,t\geq 0$, $x_{t}:=P_{t}x_{0}$ and $y_{s}:=P_{s}y_{0}$ satisfies

where $\tau(s,t)=2(t+\sqrt{ts}+s)/3$. In particular, setting $t=s$ yields the following estimate:

For $0<s^{\prime}<t^{\prime}$, let $\Phi:\to[s^{\prime},t^{\prime}]$ be given by $\Phi(r):=(\sqrt{s^{\prime}}+(\sqrt{t^{\prime}}-\sqrt{s^{\prime}})r)^{2}$. Let $(\gamma_{u})_{u\in}$ be a constant speed geodesic. By (2.26), there exists $C_{1}>0$ such that

Let $\lambda\geq 1$, $\tau,h>0$, $s^{\prime}=\lambda^{-1}(\tau+h)$, $t^{\prime}=\lambda(\tau+h)$, $\gamma_{0}:=P_{\lambda^{-1}r}y_{0}$ and $\gamma_{1}:=P_{\lambda r}x_{0}$. Then the last inequality implies

Thus the conclusion follows from this estimate as in the proof of Theorem 2.19. ∎

We now investigate the relation between the Evolution Variational Inequality and geodesic convexity of the functional $S$.

Assume that for every starting point $x_{0}\in\overline{D(S)}$ the ${\operatorname{EVI}_{K,N}}$ flow for $S$ exists. Then $S$ is strongly $(K,N)$-convex.

Let $P$ denote the ${\operatorname{EVI}_{K,N}}$ gradient flow semigroup of $S$. We treat the case $K\neq 0$ first. So let $(\gamma_{s})_{s\in}$ be a constant speed geodesic. Let us fix $s\in,t>0$ and set $\gamma_{s}^{t}:=P_{t}\gamma_{s}$. We can assume that $d:=d(\gamma_{0},\gamma_{1})\neq 0$. Using the identity (2.9) we see that (2.24) can be rewritten as

Using (2.33) with $t_{0}=0,t_{1}=t,x=\gamma_{s}$ and $z=\gamma_{0}$ respectively $z=\gamma_{1}$ we immediately obtain

To conclude, we apply this with $\alpha=(1-s)d$, $\alpha^{\prime}=sd$ and $\varepsilon=d(\gamma_{s}^{t},\gamma_{1})-(1-s)d$, $\varepsilon^{\prime}=d(\gamma_{s}^{t},\gamma_{0})-sd$ and note that $\varepsilon+\varepsilon^{\prime}\geq 0$ by the triangle inequality.

Finally, we treat the case $K=0$. By Lemma 2.15 $P$ is a $\operatorname{EVI}_{K^{\prime},N}$ flow for every $K^{\prime}<0$. Thus by the previous argument (2.11) holds with $K^{\prime}$ instead of $K$ and we can pass to the limit as $K^{\prime}\nearrow 0$. ∎

Entropic and Riemannian curvature-dimension conditions

In this section we introduce a new curvature-dimension condition for metric measure spaces based on $(K,N)$-convexity of the entropy on the Wasserstein space.

Let $(X,d,m)$ be a metric measure space, i.e. $(X,d)$ is a complete and separable metric space and $m$ is a locally finite, $\sigma$-finite Borel measure on $X$. We denote by $\mathscr{P}_{2}(M,d)$ the $L^{2}$-Wasserstein space over $(X,d)$, i.e. the set of all Borel probability measures $\mu$ satisfying

for some, hence any, $x_{0}\in X$. The subspace of all measures absolutely continuous w.r.t. $m$ is denoted by $\mathscr{P}_{2}(X,d,m)$. The $L^{2}$-Wasserstein distance between $\mu_{0},\mu_{1}\in\mathscr{P}_{2}(X,d)$ is defined by

Given a measure $\mu\in\mathscr{P}_{2}(X,d)$ we define its relative entropy by

if $\mu=\rho m$ is absolutely continuous w.r.t. $m$ and $(\rho\log\rho)_{+}$ is integrable. Otherwise we set $\operatorname{Ent}(\mu)=+\infty$. The subset of probability measures with finite entropy will be denoted by $\mathscr{P}_{2}^{*}(X,d,m)$. Moreover, for a number $N\in(0,\infty)$ we introduce the functional $U_{N}:\mathscr{P}_{2}(X,d)\to[0,\infty]$ by

If (3.1) holds for any constant speed geodesic $(\mu_{t})_{t\in}$ in $\mathscr{P}^{*}_{2}(X,d,m)$ we say that $(X,d,m)$ is a strong ${\operatorname{CD}^{e}(K,N)}$ space.

In other words, the ${\operatorname{CD}^{e}(K,N)}$-condition means that the entropy is $(K,N)$-convex along Wasserstein geodesic. As an immediate consequence of Lemma 2.12 we obtain the following consistency result.

If $(X,d,m)$ satisfies the ${\operatorname{CD}^{e}(K,N)}$ condition, then it also satisfies $\operatorname{CD}^{e}(K^{\prime},N^{\prime})$ for any $K^{\prime}\leq K$ and $N^{\prime}\geq N$. Moreover, it satisfies the $\operatorname{CD}(K,\infty)$ condition.

As an application of the additivity of $(K,N)$-convexity we note the following

Take the logarithm on both sides of (3.2). By virtue of Lemma 2.11, we can use Jensen’s inequality when integrating it w.r.t. $\pi$ to obtain

and Lemma 2.10. The latter assertion is obvious from the proof. ∎

We will now derive some first geometric consequences of the entropic curvature-dimension condition.

Assume that $(X,d,m)$ satisfies the condition ${\operatorname{CD}^{e}(K,N)}$ with $N\geq 1$. Then for all measurable sets $A_{0},A_{1}\subset X$ with $m(A_{0}),m(A_{1})>0$ and all $t\in$ we have

where $\bar{m}$ is the completion of $m$, $A_{t}$ denotes the set of $t$-midpoints and $\Theta$ the minimal/maximal distance between points in $A_{0}$ and $A_{1}$, i.e.

We first prove the assertion under the assumption that $m(A_{0}),m(A_{1})<\infty$, the general case then follows by approximating the sets $A_{0},A_{1}$ by sets of finite volume. Applying the condition $\operatorname{CD}^{e}(K,N)$ to $\mu_{i}=m(A_{i})^{-1}{{\bf 1}}_{A_{i}}m$ for $i=0,1$ yields

where $\mu_{t}=\rho_{t}m$ is the $t$-midpoint of a geodesic connecting $\mu_{0}$ and $\mu_{1}$. Since $\mu_{t}$ is concentrated on $A_{t}$, which is a Souslin set, a double application of Jensen’s inequality gives that

Hence (3.3) follows by noting that \theta\mapsto\sigma^{(t)}_{K/N}\big{(}\theta\big{)} is increasing if $K\geq 0$ and decreasing if $K<0$ and that $W_{2}(\mu_{0},\mu_{1})\geq\Theta$ (resp. $\leq\Theta$). ∎

The Brunn–Minkowski inequality entails further geometric consequences like a Bishop–Gromov type volume growth estimate and a generalized Bonnet–Myers theorem. The following results can be deduced from Proposition 3.4 using similar arguments as in and replacing the coefficients $\tau^{(t)}_{K/N}(\cdot)$ by \sigma^{(t)}_{K/N}\big{(}\cdot\big{)}.

The estimates presented below are not sharp, yet they provide necessary local compactness results for example. We will see below that under the assumption that $(X,d,m)$ is non-branching the ${\operatorname{CD}^{e}(K,N)}$ condition is equivalent to the ${\operatorname{CD}^{*}(K,N)}$ condition. It has been proven by Cavaletti & Sturm that under the same assumption ${\operatorname{CD}^{*}(K,N)}$ implies the measure contraction property $\text{MCP}(K,N)$ from which a sharp Bishop–Gromov and Lichnerowicz inequality can be derived, see .

To state the volume growth estimate we introduce the following notation. Given a metric measure space $(X,d,m)$ and a point $x_{0}\in\operatorname{supp}[m]$ we denote by

the volume of the closed ball of radius $r$ around $x_{0}$. Moreover, we set

for the volume of the corresponding sphere.

Assume that $(X,d,m)$ satisfies the condition ${\operatorname{CD}^{e}(K,N)}$ with $N\geq 1$. Then each bounded closed set $M\subset\operatorname{supp}[m]$ is compact and has finite volume. More precisely, for each $x_{0}\in\operatorname{supp}[m]$ and $0<r<R\leq\pi\sqrt{N/(K\vee 0)}$,

If $(X,d,m)$ satisfies the condition ${\operatorname{CD}^{e}(K,N)}$ with $K>0$ and $N\geq 1$, then the support of $m$ is compact and its diameter $L$ can be bounded as $L\leq\pi\sqrt{N/K}$.

${\operatorname{CD}^{e}(K,N)}$ or $\operatorname{CD}(K,\infty)$ yields that $\mathscr{P}(X,d)$ is a length space and hence so is $(\operatorname{supp}m,d)$ [39, Rem. I.4.6(iii), Prop. 2.11(iii)]. Thus, by the local compactness ensured in Proposition 3.6, if $(X,d,m)$ is a ${\operatorname{CD}^{e}(K,N)}$ space then $(\operatorname{supp}m,d)$ and hence $\mathscr{P}_{2}(\operatorname{supp}m,d)$ is a geodesic space (see e.g. [14, Thm. 2.5.23]). In addition, the volume growth estimate (3.5) implies in particular that for any $x_{0}\in X$ and $c>0$:

It is well known that the latter implies that $\operatorname{Ent}$ does not take the value $-\infty$ on $\mathscr{P}_{2}(X,d)$ and is lower semi-continuous w.r.t. $W_{2}$ (see e.g. [6, Sec. 7]). Thus, when $\operatorname{supp}m=X$, Definition 3.1 fits well into the setting of Section 2.3, where we assumed these additional regularity properties.

It turns out that under mild assumptions the modified curvature-dimension condition ${\operatorname{CD}^{e}(K,N)}$ is equivalent to the reduced curvature-dimension condition ${\operatorname{CD}^{*}(K,N)}$ introduced in . We recall here the definition. Denote by $\mathscr{P}_{\infty}(X,d,m)$ the set of measures in $\mathscr{P}_{2}(X,d,m)$ with bounded support.

We say that a metric measure space $(X,d,m)$ satisfies the reduced curvature-dimension condition ${\operatorname{CD}^{*}(K,N)}$ if and only if for each pair $\mu_{0}=\rho_{0}m,\mu_{1}=\rho_{1}m\in\mathscr{P}_{\infty}(X,d,m)$ there exist an optimal coupling $q$ of them and a geodesic $(\mu_{t})_{t\in}$ in $\mathscr{P}_{\infty}(X,d,m)$ connecting them such that for all $t\in$ and $N^{\prime}\geq N$:

If (3.7) holds for any geodesic $(\mu_{t})_{t\in}$ in $\mathscr{P}_{\infty}(X,d,m)$ we say that $(X,d,m)$ is a strong ${\operatorname{CD}^{*}(K,N)}$ space.

The assumption we need to prove equivalence of the different curvature-dimension conditions is the following weak form of non-branching.

We say that a metric measure space $(X,d,m)$ is essentially non-branching if any dynamic optimal coupling $\pi\in\mathscr{P}(\operatorname{Geo}(X))$ between two absolutely continuous measures is supported in a set of non-branching geodesics, i.e. there exists $A\subset\operatorname{Geo}(X)$ such that $\pi(A)=1$ and for all $\gamma,\widetilde{\gamma}\in A$:

This condition has been introduced in and it has been shown that strong $\operatorname{CD}(K,\infty)$ spaces are essentially non-branching. It has also been noted there that the essential non-branching condition is equivalent to the following apparently stronger condition: Every dynamic optimal coupling $\pi$ between absolutely continuous measures is concentrated on a set of geodesics that do not meet at intermediate times, i.e. there is $A^{\prime}\subset\operatorname{Geo}(X)$ such that $\pi(A^{\prime})=1$ and for all $\gamma,\widetilde{\gamma}\in A^{\prime}$:

Indeed, assuming the existence of a dynamic optimal coupling where such crossings happen with positive probability, one can reshuffle the geodesics before and after the crossing to produce a dynamic optimal coupling of the same marginals where branching happens with positive probability, contradicting the essentially non-branching assumption.

An immediate consequence of this observation is the following adaption of [9, Lem. 2.8].

Let $(X,d,m)$ be an essentially non-branching metric measure space and let $\pi$ be a dynamic optimal coupling. Assume that $\pi=\sum_{k=1}^{n}\alpha_{k}\pi^{k}$ for suitable $\alpha_{k}>0$ and dynamic optimal couplings $\pi^{k}$. For given $t\in(0,1)$ and $i\in\{0,t\}$ we set $\mu_{i}^{k}=(e_{i})_{\#}\pi^{k}$. If the family $\{\mu^{k}_{0}\}_{k}$ is mutually singular, then also the family $\{\mu_{t}^{k}\}_{k}$ is mutually singular.

Let $(X,d,m)$ be an essentially non-branching metric measure space. Then the following assertions are equivalent:

$(X,d,m)$ satisfies ${\operatorname{CD}^{*}(K,N)}$,

For each pair $\mu_{0},\mu_{1}\in\mathscr{P}_{\infty}(X,d,m)$ there is a dynamic optimal coupling $\pi$ of them such that we have $(e_{t})_{\#}\pi\ll m$ and

for $\pi$-a.e. $\gamma\in\operatorname{Geo}(X)$, where $\rho_{t}$ denotes the density of $(e_{t})_{\#}\pi$ w.r.t. $m$.