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

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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.