Couplings of the Brownian motion via discrete approximation under lower Ricci curvature bounds

Along an idea of von Renesse, couplings of the Brownian motion on a Riemannian manifold and their extensions are studied. We construct couplings as a limit of coupled geodesic random walks whose components approximate the Brownian motion respectively. We recover Kendall and Cranston's result under lower Ricci curvature bounds instead of sectional curvature bounds imposed by von Renesse. Our method provides applications of coupling methods on spaces admitting a sort of singularity.