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

## TL;DR

This paper establishes the equivalence of synthetic Ricci curvature bounds, gradient estimates, and pathwise Brownian coupling on metric measure spaces with variable Ricci bounds, linking curvature conditions to transport costs and stochastic couplings.

## Contribution

It proves the equivalence between curvature-dimension conditions, gradient estimates, and a new characterization via coupled Brownian motions with variable Ricci bounds.

## Key findings

- Equivalence of Ricci curvature bounds and gradient estimates in variable curvature spaces.
- Characterization of curvature bounds through nonincreasing perturbed p-transport costs.
- Existence of coupled Brownian motions satisfying exponential contraction properties.

## Abstract

Given a metric measure space $(X,\mathsf{d},\mathfrak{m})$ and a lower semicontinuous, lower bounded function $k\colon X\to\mathbb{R}$, 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-\'Emery estimate $\Delta\Gamma(f)/2 - \Gamma(f,\Delta f) \geq k\,\Gamma(f)$ 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}}(\mu_1,\mu_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 $s\leq t$, 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*}