Super-Ricci flows and improved gradient and transport estimates

TL;DR

This paper extends heat flow and transport estimates to super-Ricci flows, enabling the construction of Brownian motions on evolving metric measure spaces and improving gradient inequalities.

Contribution

It introduces new transport estimates for heat flow on super-Ricci flows and constructs Brownian motions in time-dependent metric spaces.

Findings

Transport estimates hold for all L^p-Kantorovich distances.

Brownian motions can be constructed on time-dependent metric measure spaces.

Refined Bochner's inequality is established under stronger metric assumptions.

Abstract

We show that the heat flow on super-Ricci flows in the sense of Sturm satisfies transport estimates with respect to every $L^p$-Kantorovich distance, $p\in[1,\infty]$. As an application we construct Brownian motions on time-dependent metric measure spaces and present transport estimates for their trajectories. The proof is inspired by the approach from Savar\'e and Bakry respectively and takes advantage of the self-improvement of the gradient estimates. For this we prove a refined version of Bochner's inequality under strengthened assumptions on the metric.

Equations

$$W_2(P_t\mu ,P_t\nu )\le e^{-Kt}{W}_2(\mu ,\nu ),\Gamma (P_{t}u)\le e^{-2Kt}{P}_t\Gamma (u),$$

$$(\Gamma (P_{t}u){)}^{\alpha }\le e^{-2\alpha Kt}{P}_t(\Gamma (u{)}^{\alpha }),$$

$$W_p(P_t\mu ,P_t\nu )\le e^{-Kt}{W}_p(\mu ,\nu )p\in [1,\infty ].$$

$$\partial_{a}S_{t}(\mu^{a})\big{|}_{a=1}-\partial_{a}S_{t}(\mu^{a})\big{|}_{a=0}\geq-\frac{1}{2}\partial_{t}^{-}W_{t}^{2}(\mu^{0},\mu^{1}),$$

$${\partial }_t{u}_t={\Delta }_t{u}_t,\text{ for }t>s\text{ and }{u}_s=u,$$

$${\Gamma }_t(P_{t,s}u)\le P_{t,s}({\Gamma }_s(u)),$$

$$W_{2,s}({\hat{P}}_{t,s}\mu ,{\hat{P}}_{t,s}\nu )\le W_{2,t}(\mu ,\nu ),$$

$$W_{p,s}({\hat{P}}_{t,s}\mu ,{\hat{P}}_{t,s}\nu )\le W_{p,t}(\mu ,\nu ),$$

$$d_s(B_s^1,B_s^2)\le d_t(B_t^1,B_t^2)\text{ almost surely}.$$

$$({\Gamma }_t(P_{t,s}u){)}^{\alpha }\le P_{t,s}({\Gamma }_s(u{)}^{\alpha })m\text{-a.e.},$$

$$W_{p,s}({\hat{P}}_{t,s}\mu ,{\hat{P}}_{t,s}\nu )\le W_{p,t}(\mu ,\nu ),$$

$$d_s(B_s^1,B_s^2)\le d_t(B_t^1,B_t^2)\text{ almost surely}.$$

$$\frac{1}{2}{\Delta }_t({\Gamma }_t(u))-{\Gamma }_t(u,{\Delta }_{t}u)\ge \frac{1}{2}({\partial }_t{\Gamma }_t)(u).$$

$$({\Gamma }_t(P_{t,s}u){)}^{\alpha }\le P_{t,s}({\Gamma }_s(u{)}^{\alpha })m\text{-a.e..}$$

$${\mathrm{Ric}}_t\ge -\frac{1}{2}{\partial }_t{g}_t$$

$${\mathrm{Ric}}_t+{\nabla }_t^2{f}_t\ge -\frac{1}{2}{\partial }_t{g}_t,$$

$$f_t(x)=⟨x,{\alpha }_t{⟩}^2+⟨x,{\beta }_t⟩+{\gamma }_t,$$

$$d_t^2=d^2(1-2Kt),$$

$$\cos (d_{\Sigma }((x,s),(x^{\prime },s^{\prime }))):=\cos s\cos s^{\prime }+\sin s\sin s^{\prime }\cos (d(x,x^{\prime })\wedge \pi )$$

$$g_t=(1-8t)g_0.$$

$$|f_t(x)|\le C,|f_t(x)-f_t(y)|\le C{d}_t(x,y),|f_t(x)-f_s(x)|\le L|t-s|,$$

$$|\log (d_t(x,y)/d_s(x,y))|\le L|t-s|$$

$$\frac{1}{2}\int {\Delta }_{t}g{\Gamma }_t(f)dm-\int g{\Gamma }_t({\Delta }_{t}f,f)dm\ge K\int g{\Gamma }_t(f)dm+\frac{1}{N}\int g({\Delta }_{t}f{)}^{2}dm$$

$$\displaystyle\mathrm{Ch}_{t}(u):=\inf\Big{\{}\liminf_{n\to\infty}\frac{1}{2}\int_{X}\mathrm{lip}_{t}(u_{n})^{2}\,dm_{t}\Big{\}}$$

$${\mathrm{lip}}_{t}u(x):=\underset{y\to x}{\mathrm{limsup}}\frac{|u(y)-u(x)|}{d_t(x,y)},$$

$${\mathrm{Ch}}_t(u)=\frac{1}{2}{\int }_X|{\nabla }_{t}u{|}_{\ast }^{2}d{m}_t,$$

$${\Gamma }_t(u)=|{\nabla }_{t}u{|}_{\ast }^2$$

$${\mathcal{E}}_t(u)={\int }_X{\Gamma }_t(u)d{m}_t.$$

$${\Gamma }_t(u,v):=\frac{1}{4}({\Gamma }_t(u+v)-{\Gamma }_t(u-v)).$$

$${\Gamma }_t(\theta (u),v)={\theta }^{\prime }(u){\Gamma }_t(u,v),{\Gamma }_t(uv,w)=u{\Gamma }_t(v,w)+v{\Gamma }_t(u,w),$$

Full text

1 Introduction and statement of the main results

The heat flow on a metric measure space $(X,d,m)$ can be understood either as the gradient flow of the Cheeger energy $\mathrm{Ch}$ on the Hilbert space $L^{2}(X,m)$ or as the gradient flow of the relative entropy $S$ on the space of probability measures $\mathcal{P}_{2}(X)$ endowed with the $L^{2}$-Kantorovich distance $W_{2}$. It has been shown in [1] that these two notions coincide under the assumption that $(X,d,m)$ satisfies a lower Ricci curvature bound in the sense of Lott-Sturm-Villani.

A metric measure space is said to have a lower Ricci curvature bound $K$, in short CD$(K,\infty)$, if the relative entropy is $K$-convex along $L^{2}$-Kantorovich geodesics. This definition is consistent with the Riemannian case, i.e. a Riemannian manifold has Ricci curvature bounded from below by $K$ if and only if its relative entropy is $K$-convex. But then even more holds true. Each of the following properties characterize a lower curvature bounds of the manifold: $L^{p}$-transport and gradient estimates of the heat flow for $p\in[1,\infty]$, and pathwise contraction for Brownian trajectories, see e.g. [34].

On general CD$(K,\infty)$-spaces these properties fail, and the heat flow does not even have to be linear. In order to obtain a more Riemannian-like behavior, Ambrosio, Gigli and Savaré introduced in [2] the notion of RCD$(K,\infty)$-spaces, i.e. CD$(K,\infty)$-spaces whose heat flow is linear. This notion is characterized by one single formula, namely the Evolution variational inequality of the heat flow with respect to the $L^{2}$-Kantorovich distance. Moreover one immediately recovers the $L^{2}$-transport and gradient estimates

[TABLE]

where $\Gamma$ denotes the Carré du champ operator. On a Riemannian manifold $(M,g)$, $\Gamma(u)$ corresponds to $|\nabla u|_{g}^{2}$.

Savaré proved in [28] that the gradient estimate implies the stronger estimate

[TABLE]

where $\alpha\in[1/2,2]$. This has been first noticed by Bakry in [6] in the framework of Dirichlet forms. Crucial for this estimate is the self-improvement of the Bochner inequality. By Kuwada’s duality [19] the stronger gradient estimate further implies stronger $L^{p}$-transport estimates

[TABLE]

In this paper we aim to show similar estimates for time-dependent metric measures spaces $(X,d_{t},m_{t})_{t\in I}$, $I=(0,T)$, which evolve under a super-Ricci flow. Moreover we introduce Brownian motions, proving their existence and uniqueness. We parametrize them backward in time, which seems to be unconventional at first glance, but is the right thing to do when considering super-Ricci flows, which we parametrize forward in time. We refer to this parametrization of Brownian motion as backward Brownian motion and prove transport estimates of their trajectories.

$(X,d_{t},m_{t})_{t\in I}$ is said to be a super-Ricci flow if for a.e. $t\in I$ and every $W_{t}$-geodesic $(\mu^{a})_{a\in[0,1]}$ in $\mathcal{P}(X)$

[TABLE]

where $S_{t}$ denotes the Boltzmann entropy at time $t$ and $W_{t}=W_{2,t}$ the $L^{2}$-Kantorovich distance with respect to $d_{t}$. This notion goes back to Sturm in [31] and is a generalization to super-Ricci flows for manifolds (“$\mathrm{Ric}_{t}\geq-\frac{1}{2}\partial_{t}g_{t}$”) on one hand and to CD$(K,\infty)$-spaces on the other.

In [18] the authors show existence and uniqueness of the heat flow $(P_{t,s})_{t\geq s}$ on $(X,d_{t},m_{t})_{t\in I}$, i.e. $P_{t,s}u$ solves

[TABLE]

and moreover, that super-Ricci flows are characterized by the time-dependent gradient estimate

[TABLE]

or equivalently, by the $L^{2}$-Kantorovich transport estimate

[TABLE]

where $\hat{P}_{t,s}$ denotes the dual heat flow on probability measures. See also Definition 2.5 below. The spaces $(X,d_{t},m_{t})_{t\in I}$ are assumed to satisfy RCD$(K,N)$ for some $K,N\in\mathbb{N}$, i.e. they satisfy RCD$(K,\infty)$ with dimensions bounded from above by $N$. For more details we refer to Section 2.

In the same setting as in [18] we will introduce backward Brownian motions (see Definition 3.1) and prove their existence and uniqueness (Proposition 3.2). Moreover we construct couplings of backward Brownian motions satisfying a pathwise contraction estimate assuming that stronger $L^{p}$-transport estimates hold (Theorem 3.8).

Theorem 1.1**.**

Let $(X,d_{t},m_{t})_{t\in I}$ be a family of RCD$(K,N)$-spaces.

Then there exists a unique backward Brownian motion $(B_{s})_{s\leq t}$ with terminal distribution $\mu$, i.e. a sample-continuous Markov process with transition probabilities $\hat{P}_{s^{\prime},s}(\delta_{x})$.

Moreover, assuming that

[TABLE]

holds for all $s\leq t$, $p\in[0,\infty]$ and $\mu,\nu\in\mathcal{P}(X)$, there exists a coupling $(B_{s}^{1},B_{s}^{2})_{s\leq t}$ of backward Brownian motions satisfying

[TABLE]

The question is under what circumstances the estimate in (2) is satisfied in the setting of time-dependent metric measure spaces. The answer is it holds if and only if $(X,d_{t},m_{t})$ is a super-Ricci flow. Moreover the estimate in (3) characterize super-Ricci flows as well. This is summarized in the following theorem. For the precise statement see Theorem 4.3.

Theorem 1.2**.**

Let $(X,d_{t},m_{t})_{t\in I}$ be a family of RCD$(K,N)$-spaces such that $t\mapsto\Gamma_{t}(u)$ is a $\mathcal{C}^{1}$-function for all $u\in\mathrm{Lip}(X)$. Then, $(X,d_{t},m_{t})_{t\in I}$ is a super-Ricci flow if and only if one of the following equivalent properties holds

1. i)

for all $s\leq t$, $\alpha\in[1/2,1]$ and $u\in\mathrm{Lip}(X)$

[TABLE] 2. ii)

for all $s\leq t$, $p\in[1,\infty]$ and $\mu,\nu\in\mathcal{P}(X)$

[TABLE] 3. iii)

there exists a coupling of backward Brownian motions $(B_{s}^{1},B_{s}^{2})_{s\leq t}$ such that for all $s\leq t$

[TABLE]

The proof of this is subdivided into developing two results, Theorem 1.3 and Theorem 1.4, which we will state separately, since they are interesting in itself. In Theorem 1.3 we derive a pointwise dynamic version of Bochner’s inequality, which is crucial to deduce the improved gradient estimate in Theorem 1.4. Having established the improved gradient estimate, Kuwada’s duality [19, Theorem 2.2] and Theorem 1.1 immediately imply Theorem 1.2.

Theorem 1.3**.**

Let $(X,d_{t},m_{t})_{t\in I}$ be a family of RCD$(K,N)$-spaces such that $t\mapsto\Gamma_{t}(u)$ is a $\mathcal{C}^{1}$-function for all $u\in\mathrm{Lip}(X)$. Then, if $(X,d_{t},m_{t})_{t\in I}$ is a super-Ricci flow the pointwise dynamic Bochner inequality

[TABLE]

holds for all $t\in I$.

For the precise statement see Theorem 4.12. Note that the authors in [18] derive a dynamic version of Bochner’s inequality, where the test functions $u$ appear as heat flows $P_{t,s}u_{s}$ and the estimate holds in an almost everywhere sense. The pointwise dynamic Bochner inequality is an essential modification, since from this we obtain the improved gradient estimates $i)$ in Theorem 1.2. We summarize this in the following theorem, see also Theorem 4.5.

Theorem 1.4**.**

Let $(X,d_{t},m_{t})_{t\in I}$ be a family of RCD$(K,N)$-spaces such that $t\mapsto\Gamma_{t}(u)$ is a $\mathcal{C}^{1}$-function for all $u\in\mathrm{Lip}(X)$. Then, if $(X,d_{t},m_{t})_{t\in I}$ is a super-Ricci flow for every $\alpha\in[1/2,1]$ we have for $s\leq t$ and every $u\in\mathrm{Lip}(X)$

[TABLE]

Let us remark that several difficulties arise due to the time-dependence, especially regarding the regularity of the heat flow, e.g. the Laplacian and the semigroup do not commute and the domain of the Laplacian is time-dependent.

In the following we visualize super-Ricci flows satisfying the properties in Theorem 1.2 by giving a few examples:

Example 1**.**

Static spaces: Let $(X,d,m)$ be an RCD$(K,N)$-space. Then the constant space $(X,d_{t},m_{t})_{t\in I}$ is a super-Ricci flow if and only if $K=0$.

Example 2**.**

Smooth spaces: Let $(M,g_{t},\textrm{vol}_{t})_{t\in I}$ be a family of smooth manifolds, where $(g_{t})_{t\in I}$ is a family of smooth Riemannian metrics with associated volume measures $(\textrm{vol}_{t})_{t\in I}$. The metrics satisfy the super-Ricci flow estimate

[TABLE]

if and only if $i),ii)$ or $iii)$ of Theorem 1.2 is satisfied. See also Corollary 1 in [24], Theorem 4.1 in [4], and Theorem 1.5 in [17].

E.g. the Ricci flow $\mathrm{Ric}_{t}=-\frac{1}{2}\partial_{t}g_{t}$ introduced by Hamilton in [16] defines a super-Ricci flow. But this “minimal” Ricci flow is not induced by $i),ii)$ or $iii)$ of Theorem 1.2. Haslhofer and Naber characterize Ricci flows via functional inequalities on the path space, see Theorem 1.22 in [17].

Example 3**.**

Smooth weighted spaces: Let $(M,g_{t},m_{t})_{t\in I}$ be a family of smooth manifolds, where $(g_{t})_{t\in I}$ are smooth Riemannian metrics and the volume measures $(m_{t})_{t\in I}$ are given by $dm_{t}=e^{-f_{t}}\,d\textrm{vol}_{t}$. Then, $g_{t}$ and $f_{t}$ satisfy the estimate

[TABLE]

if and only if $i),ii)$ or $iii)$ of Theorem 1.2 is satisfied, cf. Theorem 0.2 in [31].

Example 4**.**

Wandering Gaussian: Let $(X,d_{t},m_{t})_{t\in I}$ such that $X=\mathbb{R}^{n}$, $d_{t}(x,y)=||x-y||$ and $dm_{t}=e^{-f_{t}}dx$ with

[TABLE]

where $\alpha,\beta\colon I\to\mathbb{R}^{n}$ and $\gamma\colon I\to\mathbb{R}$ are arbitrary functions. Then this is a super-Ricci flow if and only if $i),ii)$ or $iii)$ of Theorem 1.2 is satisfied.

Example 5**.**

Homothetic flows: Let $(X,d_{t},m_{t})_{t\in I}$, where $m_{t}=m$ and

[TABLE]

such that $(X,d,m)$ satisfies RCD$(K,N)$. Then this is a super-Ricci flow if and only if $i),ii)$ or $iii)$ of Theorem 1.2 is satisfied, cf. Proposition 2.7 in [31] and Example 6 below.

Example 6**.**

A non-smooth example for an homothetic flow can be constructed on the spherical cone $\Sigma$ over $M=S^{2}(1/\sqrt{3})\times S^{2}(1/\sqrt{3})$ with product metric $d$ as in [18]: The spherical cone is defined as the quotient of $M\times[0,\pi]$ by contracting $M\times\{0\}$ to the south pole $S$ and $M\times\{\pi\}$ to the north pole $N$ with metric

[TABLE]

and measure $dm_{\Sigma}(x,s):=d\textrm{vol}(x)\otimes(\sin^{4}s\,ds)$.

This space is a RCD$(4,5)$-space (see [5, Thorem 3.5]), and the punctured cone $\Sigma_{0}:=\Sigma\setminus\{S,N\}$ is a 5-dimensional (non-complete) Riemannian manifold with metric $g_{0}$ and constant curvature $\mathrm{Ric}_{g_{0}}=4$ (see [5, Lemma 3.6]). A possible Ricci flow on the punctured cone is then given by

[TABLE]

The associated metric measure spaces of $(\Sigma_{0},g_{t},\textrm{vol}_{t})_{t\in I}$ build a super-Ricci flow for all $\bar{I}\subset[0,1/8)$, since the geodesics over the poles play no role in optimal transport, cf. [5, Theorem 3.2]. Let us remark that the sectional curvature of the punctured spherical cone is neither bounded from below nor from above.

Let us briefly make some remarks about related literature.

Ricci flows have been introduced by Hamilton in [16]. They gained a lot of attention when Perelman utilized them for proving the Poincaré conjecture for 3-manifolds with positive Ricci curvature [25, 27, 26].

In respect of optimal transport, McCann and Topping [24] showed that the $L^{2}$-transport estimate holds if and only if the underlying compact manifolds evolve as a super-Ricci flow, indicating how super-Ricci flows should be defined in a weaker context.

Coming from a probabilistic viewpoint, Arnaudon, Coulibaly and Thalmaier [4] refined McCann’s and Topping’s result by showing a pathwise estimate: They prove the existence of coupled Brownian motions with pathwise contraction. Their result implies the stronger $L^{1}$-gradient estimate.

Kuwada and Philipowski [20] construct couplings of Brownian motions such that the normalized Perelman’s $\mathcal{L}$-distance of the coupling is a supermartingale, which implies transport estimates for more general costs, see also the results in [32]. This construction is obtained on smooth Riemannian manifolds evolving as a backward super-Ricci flow.

Haslhofer and Naber characterize in [17] Ricci flows in terms of functional inequalities on the path space of smooth Riemannian manifolds. Inserting one-point cylinder functions implies the characterization of super-Ricci flows in terms of a gradient estimate with $\alpha=1$ and $\alpha=1/2$ and in terms of a Bochner formula.

The plan of the paper goes as follows. First we will recall the notion of heat flows on time-dependent metric measure spaces introduced in [18] and state the characterization of super-Ricci flows in terms of the heat flow. In Section 3 we introduce backward Brownian motions on time-dependent metric measure spaces and prove their existence and uniqueness. We anticipate the $p$-transport estimates (2) and prove that they imply contraction estimates for backward Brownian motions. The proof of Theorem 1.2 is arranged in Section 4. For this we first introduce a new dynamic Bochner inequality from which we deduce the gradient estimate with exponent $\alpha\in[1/2,1]$. In Section 4.2 we show that this dynamic Bochner inequality indeed holds under super-Ricci flow. Finally we gather in Section 4.3 the last steps to prove Theorem 1.2.

2 Preliminaries

In the sequel let $(X,d_{t},m_{t})_{t\in I}$, where $I=(0,T)$, be a one-parameter family of geodesic Polish metric measure spaces such that the following holds:

There exists a finite reference measure $m$ with full topological support such that $m_{t}=e^{-f_{t}}m$ with Borel functions $(f_{t})$ satisfying

[TABLE]

with constants $C,L>0$ independent of $x,y\in X$ and $s,t\in I$. 2. 2.

the distance is “log-Lipschitz” continuous, i.e.

[TABLE]

for all $x,y\in X$ and all $s,t\in I$, 3. 3.

there exist constants $K,N\in\mathbb{R}$ such that for each $t\in I$ the space $(X,d_{t},m_{t})$ satisfies the Riemannian curvature-dimension bound RCD$(K,N)$.

RCD$(K,N)$-spaces are metric measure spaces which are infinitesimally Hilbertian and satisfy the curvature dimension condition CD$(K,N)$. CD$(K,N)$-spaces have been introduced independently by Sturm [29] and Lott-Villani [22]. In order to rule out Finsler spaces Ambrosio, Gigli and Savaré introduced in [2] the notion infinitesimally Hilbertian, also called Riemannian. CD$(K,N)$-spaces are stable under measured Gromov-Hausdorff convergence, infinitesimally Hilbertian spaces not. As observed in [15] RCD$(K,N)$-spaces are stable, which has been finally proven in [13].

Under mild regularity assumptions “$(X,d_{t},m_{t})$ satisfies RCD$(K,N)$” means that $(X,d_{t},m_{t})$ is infinitesimally Hilbertian, i.e. $\Delta_{t}$ is a linear operator, and

[TABLE]

holds for all $g\in\mathit{Dom}(\Delta_{t})$ bounded and nonnegative with $\Delta_{t}g$ bounded and all $f\in\mathit{Dom}(\Delta_{t})$ with $\Delta_{t}f\in W^{1,2}(X,m)$, see [13].

In the following we will explain the metric setup in more detail, i.e. briefly recalling the main ideas from [1, 2, 3, 15].

The Cheeger energy $\mathrm{Ch}_{t}$ at time $t\in I$ is defined as the convex and lower-semicontinuous functional in $L^{2}(X,m_{t})$

[TABLE]

where the infimum is taken over all bounded Lipschitz functions $u_{n}\in\mathrm{Lip}_{b}(X)$ such that $u_{n}\to u$ in $L^{2}(X,m_{t})$ (cf. [1, 31]). Here, $\mathrm{lip}_{t}u$ denotes the local Lipschitz constant w.r.t. the metric $d_{t}$

[TABLE]

and $\mathrm{Ch}_{t}$ admits the local representation formula

[TABLE]

where $|\nabla_{t}u|_{*}$ is the minimal relaxed gradient [1]. Since $(X,d_{t},m_{t})$ satisfies a Riemannian curvature bound, (in particular $\mathrm{Ch}_{t}$ is quadratic) $\mathcal{E}_{t}:=2\mathrm{Ch}_{t}$ is a strongly local Dirichlet form with Carré du Champ

[TABLE]

cf. [28, 3, 2], i.e.

[TABLE]

Thanks to (8), $\mathcal{E}(u,v)=\int_{X}\Gamma_{t}(u,v)\,dm_{t}$ where

[TABLE]

$\Gamma(\cdot,\cdot)$ satisfies the chain rule and the Leibniz rule

[TABLE]

where $u,v,w\in\mathit{Dom}(\mathcal{E}_{t})$ and $\theta\in\mathrm{Lip}(\mathbb{R})$, $\theta(0)=0$. We call the linear generator $\Delta_{t}$ the Laplacian and

[TABLE]

with domain $\mathit{Dom}(\Delta_{t})\subset\mathit{Dom}(\mathcal{E}_{t})$.

Due to our assumptions (6) and (7), the sets $L^{2}(X,m_{t})$ and $W^{1,2}(X,d_{t},m_{t}):=\mathcal{D}(\mathcal{E}_{t})$ do not depend on $t$ and the respective norms for varying $t$ are equivalent to each other. We put $\mathcal{H}=L^{2}(X,m)$ and $\mathcal{F}=\mathit{Dom}(\mathcal{E}_{t_{0}})$ for some fixed $t_{0}$ as well as

[TABLE]

for each $0\leq s<\tau\leq T$.

The heat equations

A function $u$ is called solution to the heat equation

[TABLE]

if $u\in\mathcal{F}_{(s,\tau)}$ and if for all $w\in\mathcal{F}_{(s,\tau)}$

[TABLE]

where $\langle\cdot,\cdot\rangle_{\mathcal{F}^{*},\mathcal{F}}=\langle\cdot,\cdot\rangle$ denotes the dual pairing. Note that thanks to (6), w\in L^{2}\big{(}(s,\tau)\to\mathcal{F}\big{)} if and only if we^{-f}\in L^{2}\big{(}(s,\tau)\to\mathcal{F}\big{)}.

Further a function $v$ is called solution to the adjoint heat equation

[TABLE]

if $v\in\mathcal{F}_{(\sigma,t)}$ and if for all $w\in\mathcal{F}_{(\sigma,t)}$

[TABLE]

We recall the following results from [18].

Theorem 2.1**.**

(i) For each $0\leq s<\tau\leq T$ and each $h\in\mathcal{H}$ there exists a unique solution $u\in\mathcal{F}_{(s,\tau)}$ to the heat equation $\partial_{t}u_{t}=\Delta_{t}u_{t}$ on $(s,\tau)\times X$ with $u_{s}=h$.

(ii) The heat propagator $P_{t,s}:h\mapsto u_{t}$ admits a kernel $p_{t,s}(x,y)$ w.r.t. $m_{s}$, i.e.

[TABLE]

If $X$ is bounded, for each $(s^{\prime},y)\in(s,T)\times X$ the function $(t,x)\mapsto p_{t,s}(x,y)$ is a solution to the heat equation on $(s^{\prime},T)\times X$.

(iii) All solutions $u:(t,x)\mapsto u_{t}(x)$ to the heat equation on $(s,\tau)\times X$ are Hölder continuous in $t$ and $x$. All nonnegative solutions satisfy a scale invariant parabolic Harnack inequality of Moser type.

(iv) The heat kernel $p_{t,s}(x,y)$ is Hölder continuous in all variables, it is Markovian

[TABLE]

and has the propagator property

[TABLE]

Theorem 2.2**.**

(i) For each $0\leq\sigma<t\leq T$ and each $g\in\mathcal{H}$ there exists a unique solution $v\in\mathcal{F}_{(0,t)}$ to the adjoint heat equation $\partial_{s}v_{s}=-\Delta_{s}v_{s}+(\partial_{s}f_{s})v_{s}$ on $(\sigma,t)\times X$ with $v_{t}=g$.

(ii) This solution is given as $v_{s}(y)=P^{*}_{t,s}g(y)$ in term of the adjoint heat propagator

[TABLE]

If $X$ is bounded, for each $(t^{\prime},x)\in(0,t)\times X$ the function $(s,y)\mapsto p_{t,s}(x,y)$ is a solution to the adjoint heat equation on $(0,t^{\prime})\times X$.

(iii) All solutions $v:(s,y)\mapsto v_{s}(y)$ to the adjoint heat equation on $(\sigma,t)\times X$ are Hölder continuous in $s$ and $y$. All nonnegative solutions satisfy a scale invariant parabolic Harnack inequality of Moser type.

By duality, the propagator $(P_{t,s})_{s\leq t}$ acting on bounded continuous functions induces a dual propagator $(\hat{P}_{t,s})_{s\leq t}$ acting on probability measures as follows

[TABLE]

where $\mathcal{P}(X)$ denotes the space of all Borel probability measures.

The time-dependent function $v_{t}(x)=P_{t,s}u(x)$ is a solution to the heat equation, whereas the time-dependent measure $\nu_{s}(dy)=\hat{P}_{t,s}\mu(dy)$ is a solution to the dual heat equation

[TABLE]

Again $\hat{\Delta}_{s}$ is defined by duality: $\int u\,d(\hat{\Delta}_{s}\mu)=\int\Delta_{s}u\,d\mu\quad\forall u,\forall\mu.$

Lemma 2.3** (Lemma 3.8 in [18]).**

Let $u,g\in\mathcal{F}$ and $t\in I$ with $g\in L^{1}(X,m_{t})$. Then,

[TABLE]

Theorem 2.4** (Theorem 2.12 in [18]).**

For all $0<s<\tau<T$ and for all solutions $u\in\mathcal{F}_{(s,T)}$ to the heat equation

• (i)

$u_{t}\in\mathit{Dom}(\Delta_{t})$* for a.e. $t\in(s,\tau)$.*

• (ii)

If the initial condition $u_{s}\in\mathcal{F}$ then

[TABLE]

More precisely,

[TABLE]

• (iii)

For all solutions $v$ to the adjoint heat equation on $(\sigma,t)\times X$ and all $s\in(\sigma,t)$

[TABLE]

Moreover, $v_{s}\in\mathit{Dom}(\Delta_{s})$ for a.e. $s\in(\sigma,t)$.

Super-Ricci flows

Let us recall the notion of super-Ricci flows as defined in [31]. We denote the relative entropy by $S\colon I\times\mathcal{P}(X)\to(-\infty,\infty]$, i.e.

[TABLE]

whenever $\mu=\rho m_{t}$, and $S_{t}(\mu)=\infty$ otherwise.

We set for each $p\in[1,\infty)$

[TABLE]

where $\Pi(\mu_{1},\mu_{2})$ is the space of all measures in $\mathcal{P}(X\times X)$ whose marginals $(e_{i})_{\#}\mu$ coincide with $\mu_{i}$. We also set

[TABLE]

with essential supremum $||d||_{L^{\infty}(\gamma)}=\inf\{C\geq 0|d(x,y)\leq C\,\gamma\text{-a.e. }x,y\}$. For the second equality see e.g. Lemma 3.2 in [19].

Contraction estimates for the dual heat flow under $W_{2,t}=W_{t}$ characterize, among others, super-Ricci flows in the sense of [31]. This is the main theorem in [18] and is repeated in the following definition.

Definition 2.5** (Theorem 1.7 in [18]).**

We say that $(X,d_{t},m_{t})_{t\in(0,T)}$ is a super-Ricci flow if one of the following equivalent assertions holds

1. i)

For a.e. $t\in(0,T)$ and every $W_{t}$-geodesic $(\mu^{a})_{a\in[0,1]}$ in $\mathcal{P}(X)$ with $\mu^{0},\mu^{1}\in\mathit{Dom}(S)$

[TABLE]

(‘dynamic convexity’). 2. ii)

For all $0\leq s<t\leq T$ and $\mu,\nu\in\mathcal{P}(X)$

[TABLE]

(‘transport estimate’). 3. iii)

For all $u\in\mathit{Dom}(\mathcal{E})$ and all $0<s<t<T$

[TABLE]

(‘gradient estimate’). 4. iv)

For all $0<s<t<T$ and for all $u_{s},g_{t}\in\mathcal{F}$ with $g_{t}\geq 0$, $g_{t}\in L^{\infty}$, $u_{s}\in\mathrm{Lip}(X)$ and for a.e. $r\in(s,t)$

[TABLE]

(‘dynamic Bochner inequality’ or ‘dynamic Bakry-Emery condition’) where $u_{r}=P_{r,s}u_{s}$ and $g_{r}=P^{*}_{t,r}g_{t}$.

Here

[TABLE]

denotes the distribution valued $\Gamma_{2}$-operator (at time $r$) applied to $u_{r}$ and tested against $g_{r}$ and

[TABLE]

denotes any subsequential weak limit of \frac{1}{2\delta}\big{(}\Gamma_{r+\delta}-\Gamma_{r-\delta}\big{)}(u_{r}) in $L^{2}((s,t)\times X)$.

3 Brownian motions and transport estimates

In the following we introduce backward Brownian motions on time-dependent metric measure spaces $(X,d_{t},m_{t})_{t\in I}$ from Section 2, and prove their existence.

Recall that by Theorem 2.1 $(p_{t,s})_{s<t}$ are Markov kernels which satisfy the propagator property

[TABLE]

Definition 3.1**.**

Let $\mu\in\mathcal{P}(X)$ and $t\in I$. We call a stochastic process $(B_{s})_{s\leq t}$ on a probability space $(\Omega,\Sigma,\mathbb{P})$ with values in $X$ a backward Brownian motion on $X$ with terminal distribution $\mu$ if it is a sample-continuous Markov process with transition probabilities

[TABLE]

for all $s<s^{\prime}<t$ and every Borel set $A$ such that $\mathbb{P}\circ B_{t}^{-1}=\mu$.

Remark**.**

We define backward Brownian motions with terminal data, since we later want to use them to characterize super-Ricci flows (Theorem 4.3). Parametrizing them forward in time, we would have to consider backward super-Ricci flows in order to get the equivalent result.

Remark**.**

We consider the time-dependent generator $\Delta_{s}$ instead of $\frac{1}{2}\Delta_{s}$. This is only for convenience and the stochastic process with generators $(\frac{1}{2}\Delta_{s})_{s\leq 2t}$ is given by $(\tilde{B}_{s})_{s\leq 2t}$, where $\tilde{B}_{s}:=B_{\frac{s}{2}}$.

In order to prove existence of backward Brownian motions we consider for fixed $t\in I$ the finite subset $J=\{t_{1},\cdots,t_{r}\}$ of $(0,t]$, where $0<t_{1}<t_{2}<\ldots<t_{r}<t$ and the finite dimensional distribution $P_{J}^{\mu}$, where $\mu\in\mathcal{P}(X)$, defined by

[TABLE]

The family of probability measures $\{P_{J}^{\mu}|J\text{ finite }\subset(0,t]\}$ defines a projective family, hence the Kolmogorov extension theorem [8, Theorem 35.5] implies that there exists a unique probability measure $P_{(0,t]}^{\mu}$ on $(X^{(0,t]},\mathcal{B}(X)^{(0,t]})$ such that $(\pi_{J})_{\#}P_{(0,t]}^{\mu}=P_{J}^{\mu}$. Here, $\pi_{J}$ denotes the projection $\omega\mapsto(\omega(t_{1}),\ldots,\omega(t_{r}))$ from $X^{(0,t]}$ to $X^{r}$.

For every $s\in(0,t]$ the map $\pi_{s}\colon\omega\mapsto\omega(s)$ from $X^{(0,t]}$ to $X$ is a stochastic process with finite-dimensional distributions $(P_{J}^{\mu})_{J}$. The following Proposition yields existence of a continuous modification $(B_{s})_{s\leq t}$, and hence a backward Brownian motion.

Proposition 3.2**.**

For each $t\in I$ and each $\mu\in\mathcal{P}(X)$ there exists a backward Brownian motion on $X$ with terminal distribution $\mu$, which is unique in law.

Proof.

We need to show that there exists positive constants $\alpha,\beta,c>0$ such that the above mentioned process $\pi_{s}$ satisfies

[TABLE]

for all $s^{\prime},s\in(0,t]$. Then the Kolmogorov continuity theorem [8, Theorem 39.3] implies that there exists a modification $(B_{s})_{s\leq t}$ such that the map $s\mapsto B_{s}(\omega)$ is continuous for $P_{(0,t]}^{\mu}$-a.e $\omega$. Hence the process $(B_{s})_{s\leq t}$ on the probability space $(X^{(0,t]},\mathcal{B}(X)^{(0,t]},P_{(0,t]}^{\mu})$ yields the desired properties. For $\alpha>2$ (17) follows from (LABEL:continuitylpproof) in the proof of Lemma 3.3 below.

Since all finite-dimensional distributions are uniquely determined this process is unique in law. ∎

Lemma 3.3**.**

Let $\mu_{s}=\hat{P}_{t,s}\mu$. Then there exist constants $c,c^{\prime}>0$ depending only on $K,N$ and $L$ such that

[TABLE]

for all $0\leq s,s^{\prime}\leq t$.

Proof.

Assume $0<s<s^{\prime}<t$. Then by $\mu_{s}=\hat{P}_{s^{\prime},s}\mu_{s^{\prime}}$ we estimate

[TABLE]

By virtue of the Gaussian upper bounds ([21, Section 4]) and the Bishop Gromov volume comparison in RCD$(K,N)$ spaces ([29, Theorem 2.3]) we obtain for $\sigma=s^{\prime}-s$ and $B_{t}(r,x)$ denoting the ball of radius $r$ around $x$ in the metric space $(X,d_{t})$

[TABLE]

for $R\geq r$ where $A(r,x)=\partial_{r+}m_{t}(B_{t}(r,x))$ and thus (by integrating from 0 to $\sqrt{\sigma}$)

[TABLE]

for $R\geq\sqrt{\sigma}$. Then estimating further yields (with varying constants)

[TABLE]

∎

Couplings of Brownian motions

In the remainder of this section we will study couplings of backward Brownian motions under the additional assumption that the following transport estimate holds: For every $\mu,\nu\in\mathcal{P}(X)$ and every $p\in[1,\infty]$ we have

[TABLE]

In Section 4 we show that, under additional regularity assumptions, $(X,d_{t},m_{t})_{t\in I}$ is a super-Ricci flow if and only if (19) holds.

Using (19) we construct couplings of two backward Brownian motions $(B_{s}^{1})_{s\leq t},\\ (B_{s}^{2})_{s\leq t}$ on $X$ such that the distance $d_{s}$ between $B_{s}^{1}$ and $B_{s}^{2}$ does not exceed the distance $d_{t}$ between $B_{t}^{1}$ and $B_{t}^{2}$.

We adapt the strategy in [30] and introduce the $\sigma$-field

[TABLE]

of universally measurable subsets of $X^{2}$, i.e. the intersection of all $\mathcal{B}^{\nu}(X^{2})$, where $\nu$ runs through the set $\mathcal{P}(X^{2})$ and where $\mathcal{B}^{\nu}(X^{2})$ denotes the completion of the Borel $\sigma$-field on $X^{2}$ w.r.t. $\nu\in\mathcal{P}(X^{2})$. Let $\mathrm{D}:=\{k2^{-n}|k,n\in\mathbb{N}\}\cap(0,t]$ denote the set of nonnegative dyadic number s in $(0,t]$ and $\mathrm{D}_{n}:=\{k2^{-n}|k\in\mathbb{N}\}\cap(0,t]$ for fixed $n\in\mathbb{N}$.

Lemma 3.4**.**

Assume that (19) holds. Then for each $s\leq t$ there exists a Markov kernel $q_{t,s}^{*}$ on $(X^{2},\mathcal{B}^{u}(X^{2}))$ with the following properties:

1. i)

For each $(x,y)\in X^{2}$ the probability measure $q_{t,s}^{*}((x,y),\cdot)$ is a coupling of the probability measures $p_{t,s}(x,\cdot)$ and $p_{t,s}(y,\cdot)$. 2. ii)

For each $(x,y)\in X^{2}$ and $q_{t,s}^{*}((x,y),\cdot)$-a.e. $(x^{\prime},y^{\prime})\in X^{2}$

[TABLE]

Proof.

By virtue of the transport estimate (19) there exists at least one probability measures with properties i) and ii). Indeed, define $\mu_{s}=\hat{P}_{t,s}\delta_{x}$, $\nu_{s}=\hat{P}_{t,s}\delta_{y}$ and let $\gamma_{p}\in\Pi(\mu_{s},\nu_{s})$ such that $W_{p,s}(\mu_{s},\nu_{s})=||d_{s}||_{L^{p}(\gamma_{p})}$. Since $\gamma_{p}\in\Pi(\mu_{s},\nu_{s})$, $(\gamma_{p})_{p\in\mathbb{N}}$ is tight ([33, Lemma 4.4]) and hence there exists a subsequence $p_{k}$ and a probability measure $\gamma$ such that $\gamma_{p_{k}}$ weakly converges to $\gamma$. Since $\Pi(\mu_{s},\nu_{s})$ is closed we obtain that $\gamma\in\Pi(\mu_{s},\nu_{s})$. Moreover, since $d_{s}\wedge R\in\mathcal{C}_{b}(X\times X)$

[TABLE]

where the second inequality follows from the Hölder inequality and the last from Corollary 2.15. Letting $R\to\infty$ and $p\to\infty$, we obtain

[TABLE]

Hence the set of all these couplings $\gamma$ is non-empty and satisfies i) and ii). Moreover, for given $x,y\in X$ this set is closed w.r.t. weak convergence in $\mathcal{P}(X^{2})$. According to the measurable selection theorem [9, Theorem 6.9.2] we may choose a coupling $q_{t,s}^{*}((x,y),\cdot)$ such that the map

[TABLE]

is measurable. ∎

Lemma 3.5**.**

Assuming (19), for each $n\in\mathbb{N}$ and $s,s^{\prime}\in\mathrm{D}_{n}$ there exists a Markov kernel $q_{s,s^{\prime}}^{(n)}$ on $(X^{2},\mathcal{B}^{u}(X^{2}))$ with the following properties:

1. i)

For each $(x,y)\in X^{2}$ the probability measure $q_{s,s^{\prime}}^{(n)}((x,y),\cdot)$ is a coupling of $p_{s,s^{\prime}}(x,\cdot)$ and $p_{s,s^{\prime}}(y,\cdot)$. 2. ii)

For each $(x,y)\in X^{2}$

[TABLE]

for $q_{s,s^{\prime}}^{(n)}((x,y),\cdot)$-a.e. $(x^{\prime},y^{\prime})$.

Proof.

For $s=l2^{-n}$ and $s^{\prime}=k2^{-n}$ with $l\geq k$ we put

[TABLE]

Obviously we have for $r\in\mathrm{D}_{n}$ such that $s^{\prime}\leq r\leq s$,

[TABLE]

and the properties i) and ii) hold by iteration, cf. Lemma 2.3 in [30]. ∎

We fix a distribution $\nu\in\mathcal{P}(X^{2})$ with marginals $\nu_{1}$ and $\nu_{2}$. Similarly as before for any finite subset $J=\{t_{1},\ldots,t_{r}\}$ of $\mathrm{D}_{n}$ we consider the finite-dimensional distribution $Q_{J}^{(n)}$ on $(X^{2})^{|J|}$

[TABLE]

where $q_{t,t_{r}}^{*}=q_{l2^{-n},t_{r}}^{(n)}\circ q^{*}_{t,l2^{-n}}$ whenever $l2^{-n}<t<(l+1)2^{-n}$.

Lemma 3.6**.**

For fixed finite $J\subset\mathrm{D}_{m}$ the family $\{Q_{J}^{(n)}|n\in\mathbb{R},n\geq m\}$ is a tight family of probability measures on $(X^{2})^{|J|}$.

Proof.

Let $J=\{t_{1},\ldots,t_{r}\}$ with each $t_{i}\in\mathrm{D}_{m}$. The families $\{\hat{P}_{t,t_{i}}(\nu_{1})|i=1,\ldots,r\}$ and $\{\hat{P}_{t,t_{i}}(\nu_{2})|i=1,\ldots,r\}$ are tight by virtue of Prokhorov’s theorem, see e.g. [9]. This means that given $\varepsilon>0$ there exist compact sets $B_{1},B_{2}\subset X$ such that for all $i=1,\ldots,r$

[TABLE]

Applying $A_{1}\times A_{2}\subset X\times A_{2}\cup A_{1}\times X$ and (20) yields for the compact set $\vec{B}=(B_{1}\times B_{2})^{r}$ and $n\in\mathbb{N}$

[TABLE]

where the last two inequalities follow from i) of Lemma 3.5 and the tightness of $\{\hat{P}_{t,t_{i}}(\nu_{j})\}_{i}$ respectively. Hence the family $\{Q_{J}^{(n)}|n\in\mathbb{R},n\geq m\}$ is tight. ∎

For $J=\{t_{1},\ldots,t_{r}\}$ as above we set

[TABLE]

and similarly for $\vec{e_{2}}$.

Proposition 3.7**.**

There exists a projective family $\{Q_{J}^{\nu}|J\text{ finite }\subset\mathrm{D}\}$ of probability measures and a subsequence $(n_{l})_{l\in\mathbb{N}}$ such that for each finite $J\subset\mathrm{D}$

1. i)

$Q_{J}^{(n_{l})}\to Q_{J}^{\nu}$* weakly in $\mathcal{P}((X^{2})^{|J|})$ as $l\to\infty$,* 2. ii)

and $(\vec{e_{1}})_{\#}Q_{J}^{\nu}=P_{J}^{\nu_{1}}$, $(\vec{e_{2}})_{\#}Q_{J}^{\nu}=P_{J}^{\nu_{2}}$.

In particular there exists a probability measure $Q_{\mathrm{D}}^{\nu}\in\mathcal{P}((X^{2})^{\mathrm{D}})$ such that for all finite $J\subset\mathrm{D}$

[TABLE]

and

[TABLE]

Proof.

Lemma 3.6 yields for each fixed $J$ the existence of a weakly converging subsequence $Q_{J}^{(n_{l})}$ by virtue of Prokhorov’s theorem. By a diagonal argument we may choose a subsequence such that $Q_{J}^{(n_{l})}$ weakly converges for all finite $J\subset\mathrm{D}$. Note that

[TABLE]

and hence the same holds true for the limit. We obtain the last assertion by applying Kolmogorov’s extension theorem. ∎

The next theorem is in particular true for super-Ricci flows satisfying additionally (21) and (LABEL:assumption4). This is summarized in Theorem 4.3 which we prove in Section 4.3.

Theorem 3.8**.**

Let $(X,d_{t},m_{t})_{t\in I}$ be a family of RCD$(K,N)$ spaces such that (6) and (7) hold. Moreover we assume that the transport estimate (19) holds for every $p\in[1,\infty]$. Then, for each $x,y\in X$ there exists a continuous stochastic process $(B_{s})_{s\leq t}$ such that $(B_{s})_{s\leq t}$ is a coupling of the backward Brownian motions $(B_{s}^{1})_{s\leq t}$ and $(B_{s}^{2})_{s\leq t}$ with values in $X$ and terminal distributions $\delta_{x}$ and $\delta_{y}$ respectively and it satisfies for $Q_{\mathrm{D}}^{(\delta_{x},\delta_{y})}$-a.e. path

[TABLE]

for each $s\leq t$.

Proof.

Set $\nu=(\nu_{1},\nu_{2})=(\delta_{x},\delta_{y})$. Consider the coordinate process $\pi_{s}=(\pi_{s}^{1},\pi_{s}^{2})\colon(X^{2})^{\mathrm{D}}\to X^{2}$. Under $Q_{\mathrm{D}}^{\nu}$ the process $(\pi_{s}^{1})_{s\in\mathrm{D}}$ has distribution $P_{\mathrm{D}}^{\nu_{1}}$ and satisfies the continuity property (17). The corresponding statement holds true for the process $(\pi_{s}^{2})_{s\in\mathrm{D}}$. Hence, the process $\pi_{t}=(\pi_{s}^{1},\pi_{s}^{2})$ satisfies the Kolmogorov continuity theorem for $\alpha>2$ since

[TABLE]

with product metric $\hat{d}^{2}((x^{1},y^{1}),(x^{2},y^{2}))=d^{2}(x^{1},x^{2})+d^{2}(y^{1},y^{2})$. Consequently there exists a continuous modification $(B_{s})_{s\leq t}=(B_{s}^{1},B_{s}^{2})_{s\leq t}$ defined by $B_{s}=\lim_{s^{\prime}\to s,s\in\mathrm{D}}\pi_{s^{\prime}}$ for $Q_{\mathrm{D}}^{\nu}$-a.e. $\omega$ and all $s\leq t$, cf. Lemma 63.5 in [7]. The process $(B_{s}^{i})_{s\leq t}$, $i=1,2$ is a backward Brownian motion by continuity of $s\mapsto p_{t,s}(x,dy)$.

We need to justify that for $Q_{\mathrm{D}}^{\nu}$-a.e. path

[TABLE]

For each $n\in\mathbb{N}$ let $Q_{\mathrm{D}_{n}}^{(n)}$ be the projective limit of the family $(Q_{J}^{(n)})_{J\subset\mathrm{D}_{n}}$, which exists thanks to the Kolmogorov extension theorem. Consider the coordinate process $(\pi_{s}^{(n)})_{s\in\mathrm{D}_{n}}=(\pi_{s}^{1,(n)},\pi_{s}^{2,(n)})_{s\in\mathrm{D}_{n}}$ from $(X^{2})^{\mathrm{D}_{n}}\to X^{2}$. Then $Q_{\mathrm{D}_{n}}^{(n)}$-a.e. we have $d_{s}(\pi_{s}^{1,(n)},\pi_{s}^{2,(n)})\leq d(x,y)$ by virtue of Lemma 3.5. Applying Proposition 3.7 and ii) of Lemma 3.5 we obtain for a subsequence

[TABLE]

for each $s\in\mathrm{D}$. Letting $R$ and $p$ tend to $\infty$ we find for each $s\in\mathrm{D}$

[TABLE]

Since the process $(B_{s})_{s\in\mathrm{D}}$ is a modification we get for each $s\in\mathrm{D}$ and $Q_{\mathrm{D}}^{\nu}$-a.e. $d_{s}(B_{s}^{1},B_{s}^{2})\leq d_{t}(x,y)$. Since $\mathrm{D}\subset(0,t]$ is a dense and countable subset we obtain the result by continuity of $s\mapsto B_{s}(\omega)$.

∎

4 New characterization of super-Ricci flows

Let $(X,d_{t},m_{t})_{t\in I}$ be as in Section 2. The main task of this section is to show that (19) holds if and only if $(X,d_{t},m_{t})_{t\in I}$ is a super-Ricci flow. From Definition 2.5 we know already that the latter is equivalent to (19) for $p=2$. So we still need to show that (19) holds for $p>2$. Crucial for this is to establish a stronger dynamic Bochner inequality.

For this we afford more regularity of the map $r\mapsto\log d_{r}(x,y)$. We assume that there exists a $\mathcal{C}^{0}$ map $r\mapsto h_{r}(x,y)$, uniformly bounded $|h_{r}(x,y)|\leq C$ such that for each $s,t\in I$ and $x,y\in X$

[TABLE]

Consequently, for each $x,y\in X$, $r\mapsto\log d_{r}(x,y)$ is continuously differentiable with derivative $h_{r}(x,y)=\frac{d}{dr}\log d_{r}(x,y)$.

Moreover we assume that

[TABLE]

We obtain the following lemma.

Lemma 4.1**.**

Let $u\in\mathrm{Lip}(X)$. Then, assuming (21) and (LABEL:assumption4), for all $s,t\in I$ and $x\in X$

[TABLE]

Proof.

For $s<t$, we obtain from the very definition of the local slope

[TABLE]

where we applied dominated convergence. Changing the roles of $s$ and $t$ yields

[TABLE]

which proves the assertion. ∎

We apply our observation to the minimal relaxed gradient. We say that $G\in L^{2}(X,m_{t})$ is a $t$-relaxed gradient of $u\in L^{2}(X,m_{t})$ if there exists Lipschitz functions $u_{n}\in L^{2}(X,m_{t})$ such that

[TABLE]

$G$ is the minimal $t$-relaxed gradient $|\nabla_{t}u|_{*}$ if its $L^{2}(X,m_{t})$ norm is minimal among all relaxed gradients, see [1, Definition 4.2]. The collection of all $t$-relaxed gradients is convex and closed in $L^{2}(X,m_{t})$ [1, Lemma 4.3].

Proposition 4.2**.**

Assume (21) and (LABEL:assumption4) hold. Then, for $m$-a.e. $x\in X$, we have

[TABLE]

for each $u\in\mathcal{F}$ and for all $s,t\in I$. In particular for $m$-a.e. $x\in X$, $t\mapsto|\nabla_{t}u|_{*}(x)$ is continuously differentiable.

Proof.

Assume $s\leq t$. Let $u_{n}\in L^{2}(X,m_{s})$ be a sequence of Borel Lipschitz functions such that $u_{n}\to u$ and $\mathrm{lip}_{s}u_{n}\to|\nabla_{s}u|_{*}$ in $L^{2}(X,m_{s})$, see Lemma 4.3 in [1]. Then since $H$ is uniformly bounded

[TABLE]

This implies that $|\nabla_{s}u|_{*}(\cdot)e^{-\int_{s}^{t}H_{r}(\cdot)\,dr}$ is a relaxed gradient of $u$ with respect to the $d_{t}$ norm, and hence from Lemma 4.4 in [1]

[TABLE]

Changing the roles of $s$ and $t$ yields that

[TABLE]

Choosing $s$ and $t$ from a dense and countable set $D$ in $I$ the argument from above implies that $m$-a.e. in X

[TABLE]

for each $s$ and $t$ in $D$. Since the dependence of the left and the right side of the equality is continuous with respect to $s$ and $t$, we conclude that for $m$-a.e. $x\in X$, $|\nabla_{t}u|_{*}(\cdot)=|\nabla_{s}u|_{*}(\cdot)e^{-\int_{s}^{t}H_{r}(\cdot)\,dr}$ holds for every $s$ and $t$ in $I$.

Similarly, we choose $u$ in a dense and countable set $C$ in $\mathcal{F}$ ([2, Proposition 4.10]) and obtain that $m$-a.e. equation (23) holds for every $s,t\in I$ and every $u\in C$. Given $u\in\mathcal{F}$ we approximate $u$ by a sequence $u_{n}\in C$, i.e. $|\nabla_{t}u_{n}|\to|\nabla_{t}u|$ in $L^{2}(X,m_{t})$. Then there exists a subsequence $u_{n_{k}}$ such that for $m$-a.e. $x\in X$, $|\nabla_{t}u_{n_{k}}|(x)\to|\nabla_{t}u|(x)$. Equality (23) implies that for the same subsequence $|\nabla_{s}u_{n_{k}}|(x)\to|\nabla_{s}u|(x)$ for $m$-a.e. $x$. Hence we showed that for $m$-a.e. $x\in X$, (23) holds for every $u\in\mathcal{F}$ and every $s,t\in I$.

The last assertion follows directly from the fact that $r\mapsto H_{r}(x)$ is supposed to be continuous for all $x\in X$.

∎

We will now state the main theorem of this section.

Theorem 4.3**.**

Let $(X,d_{t},m_{t})_{t\in I}$ be a one-parameter family of geodesic Polish metric measure spaces satisfying (6), (7), (21) and (LABEL:assumption4) such that each $(X,d_{t},m_{t})$ is a RCD$(K,N)$ space.. Then, $(X,d_{t},m_{t})_{t\in I}$ is a super-Ricci flow if and only if one of the following equivalent properties holds

1. i)

for all $s\leq t$, $\alpha\in[1/2,1]$ and $u\in\mathit{Dom}(\mathrm{Ch})$

[TABLE] 2. ii)

for all $s\leq t$, $p\in[1,\infty]$ and $\mu,\nu\in\mathcal{P}(X)$

[TABLE] 3. iii)

there exists a coupling of backward Brownian motions $(B_{s}^{1},B_{s}^{2})_{s\leq t}$ terminating in $x$ and $y$ respectively such that for all $s\leq t$

[TABLE]

The proof of this is given in Section 4.3. In the following two subsections we prepare the necessary ingredients for the proof.

4.1 Dynamic Bochner inequality and gradient estimates

As stated in Definition 2.5, the dynamic Bochner inequality (16) is equivalent to the $L^{2}$-gradient estimate (15). In the static case (“$\partial_{t}\Gamma_{t}=0$”) it is well-known [28, 6] that it is also equivalent to the stronger $L^{1}$-gradient estimate, which yields stronger transport estimates and contraction estimates of couplings of Brownian motions [34, 30].

The aim in this section is to prove a time-dependent version of the $L^{1}$-gradient estimate

[TABLE]

which in return will imply the stronger transport estimates (19) by Kuwada’s duality. For this we give a new, more appealing definition of a time-dependent Bochner inequality.

Definition 4.4**.**

We say that the pointwise dynamic Bochner inequality holds at time $t$ if for all $u\in\mathit{Dom}(\Delta_{t})\cap L^{\infty}(X,m_{t})$ such that $\Gamma_{t}(u)\in L^{\infty}(X,m_{t})$, and all $g\in\mathit{Dom}(\Delta_{t})\cap L^{\infty}(X,m_{t})$ with $g\geq 0$

[TABLE]

This is a “real” Bochner inequality in the sense that on the one hand $u$ and $g$ do not have to arise as a heat flow (see [18, Definition 5.5]), and on the other we employ the time-derivative $\partial_{t}\Gamma_{t}(u)$ in contrast to [18, Definition 5.5, Definition 5.6]). We will show in Section 4.2 that (25) holds if and only if $(X,d_{t},m_{t})_{t\in I}$ is a super-Ricci flow, provided (21) and (LABEL:assumption4) hold. Under sufficient regularity of $u$ ($\Delta_{t}u\in\mathit{Dom}(\mathcal{E})$, $\Gamma_{t}(u)\in\mathit{Dom}(\Delta_{t})$), the left hand side can be expressed with the usual $\Gamma_{2,t}(u)=\frac{1}{2}\Delta_{t}\Gamma_{t}(u)-\Gamma_{t}(u,\Delta_{t}u)$, but for the purpose of Section 4.2, we deal with less regularity.

In the following theorem we assume that

[TABLE]

where $u_{r}=P_{r,s}u$ and $u$ is some Lipschitz function. This is not a restriction since the $L^{2}$-gradient estimate (15) in Definition 2.5 will imply this.

Theorem 4.5**.**

Let $(X,d_{t},m_{t})_{t\in I}$ be a one-parameter family of geodesic Polish metric measure spaces satisfying (6), (7), (21) and (LABEL:assumption4) such that each $(X,d_{t},m_{t})$ is a RCD$(K,N)$ space. Then, if the pointwise dynamic Bochner inequality (25) for each $s\leq r\leq t$ and the regularity assumption (26) is satisfied, for every $\alpha\in[1/2,1]$ we have for a.e. $\tau\leq t$ and $\sigma\geq s$ and every $u\in\mathit{Dom}(\mathrm{Ch})$

[TABLE]

In order to prove the theorem we adapt the strategy in [28], but we have to take care of the new term on the right hand side of (25) and of the different domains in (25) compared to [28].

Quasi-regular Dirichlet forms

We briefly recall the notion of quasi-regular Dirichlet forms developed in [23] and [12]. We denote by $\mathcal{F}=\{u\in L^{2}(X,m)|\mathcal{E}(u)<\infty\}$ the domain of a Dirichlet form $\mathcal{E}\colon L^{2}(X,m)\to[0,\infty]$, where $X$ is a Polish space and $m$ is a $\sigma$-finite Borel measure. $\mathcal{F}$ is a Hilbert space with norm $||u||_{\mathcal{F}}^{2}=||u||^{2}_{L^{2}(X,m)}+\mathcal{E}(u)$. If $F$ is a closed set in $X$ we denote

[TABLE]

Definition 4.6** ([23],[12]).**

*Given a Dirichlet form $\mathcal{E}$ on a Polish space $X$, an $\mathcal{E}$-nest is an increasing sequence of closed subsets $(F_{k})_{k\in\mathbb{N}}\subset X$ such that $\cup_{k\in\mathbb{N}}\mathcal{F}_{F_{k}}$ is dense in $\mathcal{F}$.

A set $N\subset X$ is $\mathcal{E}$-polar if there is an $\mathcal{E}$-nest $(F_{k})_{k\in\mathbb{N}}$ such that $N\subset X\setminus\cup_{k\in\mathbb{N}}F_{k}$. If a property holds in a complement of an $\mathcal{E}$-polar set we say that it holds $\mathcal{E}$-quasi-everywhere ($\mathcal{E}$-q.e.).

A function $u\colon X\to\mathbb{R}$ is said to be $\mathcal{E}$-quasi-continuous if there exists an $\mathcal{E}$-nest $(F_{k})_{k\in\mathbb{N}}$ such that every restriction $f_{|F_{k}}$ is continuous on $F_{k}$.

The Dirichlet form $\mathcal{E}$ is said to be quasi-regular if the following three properties hold.

1. i)

There exists an $\mathcal{E}$-nest $(F_{k})_{k\in\mathbb{N}}$ consisting of compact sets. 2. ii)

There exists a dense subset of $\mathcal{F}$ whose elements have $\mathcal{E}$-quasi-continuous representatives. 3. iii)

There exists an $\mathcal{E}$-polar set $N\subset X$ and a countable collection of $\mathcal{E}$-quasi-continuous functions $(f_{k})_{k\in\mathbb{N}}\subset\mathcal{F}$ separating the points of $X\setminus N$.

For every $u\in\mathcal{F}$ the quasi-regularity implies that $u$ admits an $\mathcal{E}$-quasi-continuous representative $\tilde{u}$. The representative is unique $q.e.$ and

[TABLE]

The following Lemma is taken from [28].

Lemma 4.7** ([28, Lemma 2.6]).**

Let $\mathcal{E}$ be a strongly local, quasi-regular Dirichlet form with linear generator $\Delta$. Let $\psi\in L^{1}(X,m)\cap L^{\infty}(X,m)$ nonnegative and $\varphi\in L^{1}(X,m)\cap L^{2}(X,m)$ such that

[TABLE]

for any nonnegative $g\in\mathcal{F}\cap L^{\infty}(X,m)$ with $\Delta g\in L^{\infty}(X,m)$. Then $\psi\in\mathcal{F}$ with

[TABLE]

and there exists a unique finite Borel measure $\mu:=\mu_{+}-\varphi m$ with $\mu_{+}\geq 0$, $\mu_{+}(X)\leq\int\varphi\,dm$ such that every $\mathcal{E}$-polar set is $|\mu|$-negligible, the q.c. representative of any function in $\mathcal{F}$ belongs to $L^{1}(X,|\mu|)$ and

[TABLE]

We denote by $\Delta^{*}u$ the measured valued Laplacian, i.e. the signed measure $\mu=\mu_{+}-\mu_{-}$ such that

[TABLE]

Contraction estimates for the heat flows $P_{t,s}$ and $\hat{P}_{t,s}$

Recall that on a family of closed Riemannian manifolds $(M,g_{t})$ we obtain the equality

[TABLE]

Further note that $|\langle\nabla_{t}^{2}u\nabla_{t}g,\nabla_{t}h\rangle_{g_{t}}|\leq|\nabla^{2}_{t}u|_{\textrm{HS}}|\nabla_{t}g||\nabla_{t}h|$, where $|\cdot|_{\textrm{HS}}$ denotes the Hilbert-Schmidt norm. If the manifold has Ricci curvature bounded from below by some $K\in\mathbb{R}$ then with $||\cdot||_{2}=||\cdot||_{L^{2}}$ and $K_{-}=\max\{-K,0\}$

[TABLE]

where we used the static Bochner inequality and integration by parts.

For each $t\in I$ we define the “Hessian”

[TABLE]

Moreover, we define the distribution valued $\Gamma_{2}$-operator

[TABLE]

as in [18].

Definition 4.8**.**

For each $u\in\mathit{Dom}(\Delta_{t})$ such that $u,\Gamma_{t}(u)\in L^{\infty}(X,m_{t})$ we define

[TABLE]

where $g\in\mathcal{F}$ such that $g\in L^{1}(X,m_{t})\cap L^{\infty}(X,m_{t})$.

Note that thanks to the static RCD$(K,\infty)$-condition the domain of the Laplacian is contained in the domain of the Hessian, i.e. $\mathit{Dom}(\Delta_{t})\subset W^{2,2}(X,d_{t},m_{t})$, and thus $\Gamma_{2,t}(u)(g)\in\mathbb{R}$ for $u,g$ as above. Indeed, using [14, Corollary 3.3.9], we get

[TABLE]

and thus the following estimate holds

[TABLE]

cf. Section 5 in [18]. Moreover, each $\mathcal{E}_{t}=2\mathrm{Ch}_{t}$ defines a quasi-regular Dirichlet form ([28, Theorem 4.1]).

Proposition 4.9**.**

Let $(X,d_{t},m_{t})_{t\in I}$ satisfy the regularity assumptions (21) and (LABEL:assumption4). Assume that the pointwise dynamic Bochner inequality (25) holds at time $t\in I$. Then for every $u\in\mathit{Dom}(\Delta_{t})$ with $u,\Gamma_{t}(u)\in L^{\infty}(X,m_{t})$

1. i)

$\Gamma_{t}(u)\in\mathcal{F}$* with*

[TABLE] 2. ii)

There exists a finite nonnegative Borel measure $\mu_{+}=\mu_{+}(t)$ such that every $\mathcal{E}_{t}$-polar set is $\mu_{+}$-negligible and for each $g\in\mathcal{F}$ the $\mathcal{E}_{t}$-q.c. representative $\tilde{g}\in L^{1}(X,\mu_{+})$ with

[TABLE]

In particular $\Gamma_{2,t}(u)$ is a finite Borel measure with

[TABLE]

Proof.

Let $u_{\varepsilon}=h_{\varepsilon}^{t}u$. Choosing $\psi=\Gamma_{t}(u_{\varepsilon})$ and $\varphi=-(\partial_{t}\Gamma_{t})(u_{\varepsilon})-2\Gamma_{t}(u_{\varepsilon},\Delta_{t}u_{\varepsilon})$ in Lemma 4.7 and applying the pointwise dynamic Bochner inequality together with the Leibniz rule yields

[TABLE]

Applying the Leibniz rule once again we obtain

[TABLE]

Note that as $\varepsilon\to 0$, $\Gamma(u_{\varepsilon})\to\Gamma(u)$ pointwise, in $L^{1}$ and in the weak∗ $L^{\infty}$ topology. The latter is due to the fact that $\Gamma(u_{\varepsilon}-u)$ is uniformly bounded and converges to [math] in $L^{1}$. Moreover by the uniform boundedness of $\Gamma(u_{\varepsilon})$ in $L^{\infty}$ we obtain that $\Gamma(u_{\varepsilon})\to\Gamma(u)$ in $L^{2}$. Hence we find

[TABLE]

and by Proposition 4.2

[TABLE]

while

[TABLE]

In order to show that

[TABLE]

we show that $\Gamma_{t}(u_{\varepsilon},\Gamma_{t}(u_{\varepsilon}))$ weakly converges to $\Gamma_{t}(u,\Gamma_{t}(u))$ in $L^{2}$. Take a sufficiently smooth testfunction $\varphi$ ($\varphi\in\mathcal{F}\cap L^{\infty}$), then we easily deduce

[TABLE]

by the strong $L^{2}$ convergence of $\Delta_{t}u_{\varepsilon}$, the weak∗-$L^{\infty}$ convergence of $\Gamma(u_{\varepsilon})$ and the $L^{1}$ convergence of $\Gamma(u_{\varepsilon},\varphi)$. Moreover $||\Gamma_{t}(u_{\varepsilon},\Gamma_{t}(u_{\varepsilon}))||_{2}$ is uniformly bounded in $\varepsilon$ since

[TABLE]

where we used [14, Corollary 3.3.9]. Consequently we obtain that $\Gamma_{t}(u_{\varepsilon},\Gamma_{t}(u_{\varepsilon}))$ weakly converges to $\Gamma_{t}(u,\Gamma_{t}(u))$ in $L^{2}$ since $\mathcal{F}\cap L^{\infty}$ is dense in $L^{2}$ [1, Theorem 4.5].

We conclude

[TABLE]

We show the second claim again by using the semigroup mollification $u_{\varepsilon}:=h^{t}_{\varepsilon}u$. By Lemma 4.7 we deduce that

[TABLE]

where $\Delta_{t}^{*}$ is the measure valued Laplacian, and $\mu_{+}(u_{\varepsilon})$ the nonnegative Borel measure with $\mu_{+}(u_{\varepsilon})(X)\leq\int(\Delta_{t}u_{\varepsilon})^{2}+\frac{1}{2}(\partial_{t}\Gamma_{t})(u_{\varepsilon})\,dm_{t}$. Hence, since $g=\tilde{g}$ q.e.

[TABLE]

Note that the right hand side converges as $\varepsilon\to 0$ since $\Gamma(u_{\varepsilon})\to\Gamma(u)$ weakly in $\mathcal{F}$. Indeed, take a test function $\varphi\in\mathit{Dom}(\Delta_{t})$. Then

[TABLE]

Since $\mathcal{E}_{t}(\Gamma_{t}(u_{\varepsilon}))$ is uniformly bounded in $\varepsilon$ by the first claim and $\mathit{Dom}(\Delta_{t})$ is dense in $\mathcal{F}$ we deduce that

[TABLE]

Define the linear functional $\tilde{\mu}_{+}(u)\colon\mathcal{F}\cap L^{\infty}\to\mathbb{R}$ by

[TABLE]

Note that if $g\geq 0$ we have $\tilde{\mu}_{+}(u)(g)\geq 0$ by the pointwise dynamic Bochner inequality. The Hahn-Banach theorem implies that there exists a linear functional $M\colon\mathcal{F}\to\mathbb{R}$ such that $M(g)=\mu_{+}(u)(g)$ for all $g\in\mathcal{F}\cap L^{\infty}$ and $M(g)\geq 0$ for all $g\in\mathcal{F}$ such that $g\geq 0$ a.e.. Moreover, if $g\in\mathcal{F}$ with $g\leq 1$ $m$-a.e.

[TABLE]

Thus by Proposition 2.5 in [28] there exists a unique finite and nonnegative Borel measure $\mu_{+}$ in $X$ such that every $\mathcal{E}_{t}$-polar set is $\mu_{+}$-negligible and for each $g\in\mathcal{F}$ the $\mathcal{E}_{t}$-q.c. representative $\tilde{g}\in L^{1}(X,\mu_{+})$ with

[TABLE]

Consequently

[TABLE]

and hence $\Gamma_{2,t}$ is measure valued with $2\Gamma_{2,t}(u)=(\partial_{t}\Gamma_{t})(u)\,m_{t}+\,\mu_{+}$. ∎

By virtue of Lebesgue’s decomposition theorem we denote by $\gamma_{2,t}(u)\in L^{1}(X,m_{t})$ the density wrt $m_{t}$

[TABLE]

and thus by Proposition 4.9

[TABLE]

We define for $u,h\in\mathit{Dom}(\Delta_{t})$ such that $\Gamma_{t}(u),\Gamma_{t}(h)\in L^{\infty}(X,m_{t})$

[TABLE]

where $g\in\mathcal{F}\cap L^{\infty}$. Note that the right-hand side is well-defined by (29) and

[TABLE]

Similarly,

[TABLE]

The following Lemma is an adaptation of Lemma 3.3 in [28].

Lemma 4.10**.**

Let $\bar{u}=(u_{i})_{i=1}^{n}$ with $u_{i}\in\mathit{Dom}(\Delta_{t})$ such that $u,\Gamma_{t}(u)\in L^{\infty}(X,m_{t})$ and let $\Psi\in\mathcal{C}^{3}(\mathbb{R}^{n})$ with $\Psi(0)=0$. Then

[TABLE]

In particular $m_{t}$-a.e.

[TABLE]

Proof.

Note that $\Psi(\bar{u})\in\mathit{Dom}(\Delta_{t})$ with $\Gamma_{t}(u)\in L^{\infty}$ since

[TABLE]

Thus by definition for each $g\in\mathcal{F}\cap L^{\infty}$

[TABLE]

We calculate using the notation $\psi=\Psi(\bar{u}),$ $\psi_{i}=\partial_{i}\Psi(\bar{u})$ and $\psi_{ij}=\partial_{ij}\Psi(\bar{u})$ for the first term

[TABLE]

where

[TABLE]

and

[TABLE]

On the other hand

[TABLE]

Adding up and collecting terms yields

[TABLE]

for each $g\in\mathcal{F}\cap L^{\infty}$.

For arbitrary $g\in\mathcal{F}$, set $g^{n}:=g\wedge n$. Then, by dominated convergence (recall that $\tilde{g}\in L^{1}(X,\mu_{+})$)

[TABLE]

Similarly we can pass to the limit for the other integrals and obtain for all $g\in\mathcal{F}$

[TABLE]

and hence the result. ∎

Proposition 4.11**.**

Let $(X,d_{t},m_{t})_{t\in I}$ satisfy the regularity assumptions (21) and (LABEL:assumption4). Assume that the pointwise dynamic Bochner inequality (25) holds at time $t\in I$. Then for every $u\in\mathit{Dom}(\Delta_{t})\cap L^{\infty}(X,m_{t})$ such that $\Gamma_{t}(u)\in L^{\infty}(X,m_{t})$

[TABLE]

Proof.

We choose the same polynomial $\Psi\colon\mathbb{R}^{3}\to\mathbb{R}$ as in [28] by

[TABLE]

where $\bar{u}=(u_{1},u_{2},u_{3})$, where each $u_{i}\in\mathit{Dom}(\Delta_{t})\cap L^{\infty}(X,m_{t})$ with $\Gamma_{t}(u_{i})\in L^{\infty}(X,m_{t})$. We apply Proposition 4.9 and obtain

[TABLE]

where both sides of the inequality depend on $\lambda,a,b\in\mathbb{R}$. Choosing $\lambda,a,b$ in a dense and countable subset $D$ of $\mathbb{R}$ yields that (31) holds $m$-a.e. for all $\lambda,a,b$ in $D$. Since

[TABLE]

and

[TABLE]

cf. [28, Lemma 3.3], both sides are continuous in $\lambda,a,b$, and hence we conclude that (31) holds for all $\lambda,a,b$ in $\mathbb{R}$.

Thus, for $m$-a.e. $x\in X$ we may set $a:=u_{2}(x)$, $b:=u_{3}(x)$ so that

[TABLE]

$m$-a.e., and exploiting (31) yields

[TABLE]

Using Cauchy-Schwartz inequality $\Gamma_{t}(u_{2},u_{3})^{2}\leq\Gamma_{t}(u_{2})\Gamma_{t}(u_{3})$ this can be transformed into

[TABLE]

and since $\lambda$ is arbitrary [14, Lemma 3.3.6] we obtain

[TABLE]

From the definition of the Hessian we deduce that

[TABLE]

and consequently

[TABLE]

We obtain (32) for arbitrary $u_{3}\in\mathcal{F}\cap L^{\infty}(X,m_{t})$ by approximating $u_{3}$ by a sequence $u_{3}^{n}$ converging in energy with

[TABLE]

pointwise and in $L^{1}(X,m_{t})$, cf. Theorem 3.4 in [28] Hence we may choose $u_{3}=\Gamma_{t}(u_{1},u_{2})$, and obtain the result choosing $u_{1}=u_{2}$.

∎

Now we are ready to prove Theorem 4.5.

Proof of Theorem 4.5.

Define for each $\varepsilon>0$ the concave and smooth function $\omega_{\varepsilon}(\cdot):=(\varepsilon+\cdot)^{\alpha}-\varepsilon^{\alpha}$. Note that this function satisfies

[TABLE]

For each $s,t\in(0,T)$ under consideration as well as $u\in\mathrm{Lip}(X)$ and $g\in\mathcal{F}\cap L^{\infty}$ with $g\geq 0$, we set $u_{r}=P_{r,s}u$, $g_{r}=P^{*}_{t,r}g$ for $r\in[s,t]$. Note that for a.e. $r\in[s,t]$ $u_{r}\in\mathit{Dom}(\Delta_{r})$ and $u,\Gamma_{r}(u)\in L^{\infty}(X,m_{r})$.

We consider the function

[TABLE]

Choose $s\leq\sigma<\tau\leq t$ and $\delta>0$ sufficiently small that $\sigma\leq\tau-\delta$ such that

[TABLE]

Note that by Lebesgue’s density theorem, this is true at least for a.e.  $\sigma\geq s$ and for a.e. $\tau\leq t$. Then from

[TABLE]

and the concavity of $\omega_{\varepsilon}$ we deduce

[TABLE]

Let us denote with a slight abuse of notation $\hat{g}_{r}=g_{r}\omega_{\varepsilon}^{\prime}(\Gamma_{r}(u_{r}))$. Note that $\hat{g}\in L^{1}\cap L^{\infty}(X)$ and $\hat{g}\in\mathcal{F}$. Each of the four terms will be considered separately. Since $r\mapsto\mu_{r}$ is a solution to the dual heat equation, we obtain

[TABLE]

due Lebesgue’s density theorem applied to $r\mapsto\Delta_{r}g_{r}e^{-f_{r}}$. Note that the latter function is in $L^{2}$ (Theorem 2.4) and the function $r\mapsto\omega_{\varepsilon}(\Gamma_{r}(u_{r}))$ is in $L^{\infty}$ thanks to (26).

The second term can estimated according to Proposition 4.2:

[TABLE]

The term $(III^{\prime})$ is transformed as follows

[TABLE]

Here again we used Lebesgue’s density theorem (applied to $r\mapsto\Delta_{r}u_{r}$) and the ‘nearly continuity’ of $r\mapsto\hat{g}_{r}$ as map from $(s,t)$ into $L^{2}(X,m)$ and as map into $\mathcal{F}$ (Lusin’s theorem). Moreover, we used the boundedness (uniformly in $r$ and $x$) of $g_{r}$ and of $\Gamma_{r}(u_{r})$ as well as the square integrability of $\Delta_{r}u_{r}$.

Similarly, the term $(III^{\prime\prime})$ will be transformed:

[TABLE]

We therefore obtain

[TABLE]

Applying (30), Proposition 4.11, (33) and taking into account the concavity of $\omega_{\varepsilon}$ we further deduce for a.e. $r\in[s,t]$,

[TABLE]

Hence we showed that, given $u$ and $g$, there exists exceptional sets (which are null sets) for $\tau$ and $\sigma$ outside of these sets

[TABLE]

holds. Choosing $g$’s from a dense countable set one may achieve that the exceptional sets for $\sigma$ and $\tau$ in (35) do not depend on $g$. Next we may assume that $\sigma,\tau\in[s,t]$ with $\sigma<\tau$ is chosen such that (35) simultaneously holds for all $u$ from a dense countable set ${\mathcal{C}}_{1}$ in $\mathrm{Lip}(X)$. We approximate arbitrary $u\in\mathrm{Lip}(X)$ by $u_{n}\in{\mathcal{C}}_{1}$ in energy and in $L^{2}$ such that $\sqrt{\Gamma_{\tau}(P_{\tau,\sigma}u_{n})}\rightharpoonup G$ in $L^{2}$, for some $G\in L^{2}(X)$. This is possible since $||\sqrt{\Gamma_{\tau}(P_{\tau,\sigma}u_{n})}||_{L^{2}(X)}$ is uniformly bounded. Then we have on the one hand

[TABLE]

since

[TABLE]

On the other hand we find

[TABLE]

Indeed, since $P_{\tau,\sigma}u_{n}\to P_{\tau,\sigma}u$ and $\sqrt{\Gamma(P_{\tau,\sigma}u_{n})}\rightharpoonup G$ in $L^{2}(X)$ we know $\Gamma(P_{\tau,\sigma}u)\leq G^{2}$ $m$-a.e. and hence

[TABLE]

where $\tilde{\omega}(r)=\omega(r^{2})$, which is convex and monotone. Combining (35), (36) and (37) yields

[TABLE]

Letting $\varepsilon\to 0$ we showed that

[TABLE]

Since $\mathrm{Lip}(X)$ is dense in $\mathcal{F}$ we can extend (38) to arbitrary $u\in\mathcal{F}$. Since $g$ is arbitrary we obtain the result. ∎

4.2 From $L^{2}$-transport estimates to Bochner’s inequality

For the proof of Theorem 4.3 it is still left to show that super-Ricci flow implies the pointwise dynamic Bochner inequality (25) at each time $t$. In more detail we show the following.

Theorem 4.12**.**

Let $(X,d_{t},m_{t})_{t\in I}$ be a one-parameter family of geodesic Polish metric measure spaces satisfying (6), (7), (21) and (LABEL:assumption4) such that each $(X,d_{t},m_{t})$ is a RCD$(K,N)$ space. If the transport estimate (1) holds, then the pointwise dynamic Bochner inequality (25) holds at all $t\in I$. Moreover the regularity assumption (26) is satisfied.

For the proof of Theorem 4.12 we follow the argumentation in the proof of Theorem 5.13 in [18]. The argumentation in [18] is inspired by [10], where the authors prove the equivalence between Wasserstein contraction estimates and Bochner’s inequality in the static setting.

Proof of Theorem 4.12.

Since the transport estimate (1) implies the $L^{2}$-gradient estimate and the heat flow satisfies the maximum principle, the regularity assumption is clearly satisfied.

Define $u=h_{\varepsilon}^{t}u_{0}$, where $u_{0}\in L^{\infty}(X,m_{t})\cap L^{2}(X,m_{t})$ and $h_{\varepsilon}^{t}$ the static semigroup mollification

[TABLE]

Here, $(H_{r}^{t})_{r\geq 0}$ denotes the (static) semigroup associated to $\mathcal{E}_{t}$ and $\kappa\in\mathcal{C}_{c}^{\infty}((0,\infty))$ with $\kappa\geq 0$ and $\int_{0}^{\infty}\kappa_{r}\,dr=1$. Recall that $u,\Delta_{t}u\in\mathit{Dom}(\Delta_{t})\cap\mathrm{Lip}_{b}(X)$.

Let $g\in\mathcal{F}\cap L^{\infty}(X,m_{t})$ such that $g\geq 0$. Then, the transport estimate (1) together with Lemma 5.10 and Lemma 5.11 in [18] eventually yields

[TABLE]

see [18, Section 5]. Then following the lines in [18], we subtract $\frac{1}{2}\int\Gamma_{t}(u)gdm_{t}$ on each side and divide by $t-s$ obtaining

[TABLE]

We decompose the first term on the left-hand side into the following two terms

[TABLE]

Recall that $\Gamma_{t}(u)\in\mathcal{F}$ [28, Lemma 3.2] and thus we can apply Lemma 2.3, which gives us

[TABLE]

while, since $|\frac{\Gamma_{s}(u)-\Gamma_{t}(u)}{(t-s)}|\leq 2L\Gamma_{t}(u)\in L^{\infty}(X,m_{t})$,

[TABLE]

where we used Proposition 4.2 in the last inequality and that $P^{*}_{t,s}ge^{-f_{s}}\to ge^{-f_{t}}$ in $L^{1}(X,m)$ as $s\to t$.

Regarding the second term on the left-hand side of (LABEL:decomposition), note that the Leibniz rule and the integration by parts formula is applicable and we get

[TABLE]

Subtracting $\int\Gamma_{t}(u)gdm_{t}$ and applying (42)

[TABLE]

Letting $s\nearrow t$ we have since $g\in\mathcal{F}\cap L^{\infty}(X,m_{t})$ and $\Delta_{t}u\in\mathrm{Lip}_{b}(X)$, $g\Delta_{t}u\in\mathcal{F}\cap L^{1}(X,m_{t})$

[TABLE]

by virtue of Lemma 2.3. In order to determine

[TABLE]

we need to argue whether $\Gamma_{t}(g,u)\in\mathcal{F}$. But this is the case, since, due to our static RCD$(K,\infty)$ assumption, we may apply Theorem 3.4 in [28] and obtain

[TABLE]

where $\gamma_{2}(u),\gamma_{2}(g)\in L^{1}(X,m_{t})$. Our regularity assumptions on $u$ and $g$ provide that the right hand side is in $L^{1}(X,m_{t})$ and consequently Lemma 2.3 implies

[TABLE]

Combining these observations we find

[TABLE]

Hence from (LABEL:decomposition), (40), (LABEL:decomp1.2) and (LABEL:decomp2)

[TABLE]

Let now $g\in\mathit{Dom}(\Delta_{t})\cap L^{\infty}(X,m)$ with $g\geq 0$ and $u\in\mathit{Dom}(\Delta_{t})\cap L^{\infty}(X,m_{t})$ with $\Gamma_{t}u\in L^{\infty}(X,m_{t})$. Then from the above argumentation we obtain

[TABLE]

Since $(\partial_{t}\Gamma_{t})(u)(x)=-2H_{t}(x)\Gamma_{t}(u)(x)$ for $m$-a.e. $x\in X$ and $|H_{t}(x)|\leq C$, we obtain the assertion by letting $\varepsilon\to 0$ with taking into account that

[TABLE]

∎

4.3 Proof of the equivalence result

It remains to finalize the proof of Theorem 4.3. Before we do so let us quickly prove the following Lemma.

Lemma 4.13**.**

Suppose that $(X,d_{t},m_{t})_{t\in I}$ is a super-Ricci flow satisfying (21) and (LABEL:assumption4). Then

1. i)

for every $u\in\mathcal{F}\cap L^{\infty}(X,m_{s})$ and every $\beta\in[1,2]$

[TABLE] 2. ii)

for every $\mu,\nu\in\mathcal{P}(X)$ and every $p\in[1,\infty]$

[TABLE]

Proof.

Note that, taking into account $\Gamma(u)=|\nabla u|_{*}^{2}$ due to our static Riemannian curvature bound, (44) holds at least for a.e. $s\leq t$ by Definition 2.5, Theorem 4.12 and Theorem 4.5. Then applying Kuwada’s duality [19, Theorem 2.2] implies that (19) holds at all these time instances. Indeed, (44) implies that for all $u\in\mathrm{Lip}_{b}(X)$, $|\nabla_{t}P_{t,s}u|_{*}\leq P_{t,s}(|\nabla_{s}u|_{*}^{\beta})^{1/\beta}$ and thus by Proposition 3.11 in [3] $\mathrm{lip}_{t}P_{t,s}u\leq P_{t,s}(|\nabla_{s}u|_{*}^{\beta})^{1/\beta}$. We obtain

[TABLE]

by virtue of $|\nabla u|_{*}=\mathrm{lip}\,u$ (Theorem 6.1 in [11] and Theorem 6.2 in [1]). We deduce from Theorem 2.2 in [19] for a.e. $s\leq t$

[TABLE]

where $p$ is the Hölder conjugate of $\beta$; $1/p+1/\beta=1$. Since both sides of the above equation are continuous in $s$ and $t$ (see Lemma 3.3), we obtain that it holds for all times $s\leq t$ and thus also (46) holds for all times by Theorem 2.2 in [19]. Following the same argumentation as above we find that (44) holds for all $s\leq t$ and all $u\in\mathrm{Lip}_{b}(X)$. By density of $\mathrm{Lip}(X)\subset\mathcal{F}$ we find $i)$. The same applies to $p=1$ in (45) by noting that $\mathrm{lip}_{t}(P_{t,s}u)\leq P_{t,s}(\mathrm{lip}_{s}(u)^{\beta})^{1/\beta}$ for all $\beta\geq 2$ by virtue of Jensen’s inequality. ∎

Proof of Theorem 4.3.

By virtue of Lemma 4.13, Theorem 3.8 and Definition 2.5 the only implication left to show is $iii)$ implies one of the other assertions. We show that $iii)$ implies $ii)$. This can be seen by choosing an $W_{p,t}$-optimal transport plan $\gamma$ between $\mu,\nu$. Let $q_{t,s}$ be the transition kernel of the coupled process. Then $q_{t,s}(z_{1},dx,z_{2},dy)\gamma(dz_{1},dz_{2})$ is a coupling of $\hat{P}_{t,s}\mu$ and $\hat{P}_{t,s}\nu$. Hence

[TABLE]

where we used $iii)$ in the last inequality. ∎