Metric Measure Spaces with Variable Ricci Bounds and Couplings of Brownian Motions

TL;DR

This paper extends Ricci curvature bounds to variable lower bounds in metric measure spaces and studies coupled Brownian motions, establishing new inequalities and stability results for these generalized curvature conditions.

Contribution

It introduces the $CD(k, abla)$ condition for variable Ricci bounds, extending existing curvature-dimension conditions, and proves existence of coupled Brownian motions with contraction properties under these bounds.

Findings

Extension of $CD(K, abla)$ to variable Ricci bounds

Equivalence of $CD(k, abla)$ with $EVI_k$ inequalities

Existence of coupled Brownian motions with exponential contraction

Abstract

The goal of this paper is twofold: we study metric measure spaces $(X,d,m)$ with variable lower bounds for the Ricci curvature and we study pathwise coupling of Brownian motions. Given any lower semicontinuous function $k:X\to \mathbb R$ we introduce the curvature-dimension condition $CD(k,\infty)$ which canonically extends the curvature-dimension condition $CD(K,\infty)$ of Lott-Sturm-Villani for constant $K\in \mathbb R$. For infinitesimally Hilbertian spaces we prove i) its equivalence to an evolution-variation inequality $EVI_k$ which in turn extends the $EVI_K$-inequality of Ambrosio-Gigli-Savar\'e; ii) its stability under convergence and its local-to-global property. For metric measure spaces with uniform lower curvature bounds $K$ we prove that for each pair of initial distributions $\mu_1,\mu_2$ on $X$ there exists a coupling $B_t=(B_t^1,B_t^2)$, $t\ge0$, of two Brownian motions on $X$ with the given initial distributions such that a.s. for all $s,t\ge0$ $$d(B^1_{s+t},B^2_{s+t})\le e^{-K t/2}\cdot d(B_s^1,B_s^2).$$