Optimal transport, gradient estimates, and pathwise Brownian coupling on spaces with variable Ricci bounds

Abstract

Given a metric measure space $(X,\mathsf{d},\mathfrak{m})$ and a lower semicontinuous, lower bounded function $k:X\to \mathbb{R}$, we prove the equivalence of the synthetic approaches to Ricci curvature at $x\in X$ being bounded from below by $k(x)$ in terms of $\bullet$ the Bakry-Émery estimate $\Delta \Gamma (f)/2-\Gamma (f,\Delta f)\ge k\Gamma (f)$ in an appropriate weak formulation, and $\bullet$ the curvature-dimension condition $\mathrm{CD}(k,\infty )$ in the sense Lott-Sturm-Villani with variable $k$. Moreover, for all $p\in (1,\infty )$, these properties hold if and only if the perturbed $p$-transport cost \begin{equation*} W_p^{\underline{k}}(μ_1,μ_2,t):=\inf_{(\mathsf{b}^1,\mathsf{b}^2)} \mathbb{E}\Big[\mathrm{e}^{\int_0^{2t} p \underline{k}\left(\mathsf{b}^1_{r}, \mathsf{b}^2_{r}\right)/2\,\mathrm{d} r} \mathsf{d}^p\!\left(\mathsf{b}^1_{2t},\mathsf{b}^2_{2t} \right)\!\Big]^{1/p} \end{equation*} is nonincreasing in $t$. The infimum here is taken over pairs of coupled Brownian motions ${\mathsf{b}}^1$ and ${\mathsf{b}}^2$ on $X$ with given initial distributions ${\mu }_1$ and ${\mu }_2$, respectively, and $\underline{k}(x,y):={\inf }_{\gamma }{\int }_0^{1}k({\gamma }_s)\mathrm{d}s$ denotes the "average" of $k$ along geodesics $\gamma$ connecting $x$ and $y$. Furthermore, for any pair of initial distributions ${\mu }_1$ and ${\mu }_2$ on $X$, we prove the existence of a pair of coupled Brownian motions ${\mathsf{b}}^1$ and ${\mathsf{b}}^2$ such that a.s. for every $s,t\in [0,\infty )$ with $s\le t$, we have \begin{equation*} \mathsf{d}\!\left(\mathsf{b}_t^1,\mathsf{b}_t^2\right)\leq \mathrm{e}^{-\int_s^t \underline{k}\left(\mathsf{b}_r^1,\mathsf{b}_r^2\right)/2\,\mathrm{d} r} \mathsf{d}\!\left(\mathsf{b}_s^1,\mathsf{b}_s^2\right)\!. \end{equation*}

38 pages