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

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 $s,t\ge0$ $$d(B^1_{s+t},B^2_{s+t})\le e^{-K t/2}\cdot d(B_s^1,B_s^2).$$