Convergence of time-inhomogeneous geodesic random walks and its application to coupling methods

TL;DR

This paper investigates how time-inhomogeneous geodesic random walks approximate diffusion processes on manifolds with evolving metrics, extending Ricci curvature bounds, and applies this to construct effective coupling methods for gradient estimates.

Contribution

It introduces a new convergence result for time-discretized geodesic random walks under time-dependent Ricci curvature conditions, including backward Ricci flow, without extra curvature bounds.

Findings

Established convergence of geodesic random walks to diffusions under time-inhomogeneous conditions

Constructed a reflection coupling method for gradient estimates

Provided estimates for coupling time in evolving geometric settings

Abstract

We study an approximation by time-discretized geodesic random walks of a diffusion process associated with a family of time-dependent metrics on manifolds. The condition we assume on the metrics is a natural time-inhomogeneous extension of lower Ricci curvature bounds. In particular, it includes the case of backward Ricci flow, and no further a priori curvature bound is required. As an application, we construct a coupling by reflection which yields a nice estimate of coupling time, and hence a gradient estimate for the associated semigroups.