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.
"Convergence of time-inhomogeneous geodesic random walks and its application to coupling methods." Ann. Probab. 40 (5) 1945 - 1979, September 2012. https://doi.org/10.1214/11-AOP676