Coupling by reflection of diffusion processes via discrete approximation under a backward Ricci flow

TL;DR

This paper develops a coupling by reflection for time-inhomogeneous diffusion processes on manifolds under a natural extension of Ricci curvature bounds, including backward Ricci flow, leading to gradient estimates and strong Feller properties.

Contribution

It introduces a novel coupling construction for diffusion processes under backward Ricci flow using discrete approximation methods.

Findings

Established convergence of geodesic random walks in law.

Derived gradient estimates for the diffusion semigroup.

Proved the strong Feller property under the given conditions.

Abstract

A coupling by reflection of a time-inhomogeneous diffusion process on a manifold are studied. The condition we assume is a natural time-inhomogeneous extension of lower Ricci curvature bounds. In particular, it includes the case of backward Ricci flow. As in time-homogeneous cases, our coupling provides a gradient estimate of the diffusion semigroup which yields the strong Feller property. To construct the coupling via discrete approximation, we establish the convergence in law of geodesic random walks as well as a uniform non-explosion type estimate.