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Gradient Estimates on Alexandrov Surfaces
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Alexandrov spaces arclength asymptotic Br(x Brownian motion comparison triangle continuous convergence Corollary coupling probability Cr+(X Curv(X Cut(M defined definition denote diffusion process dimr Dirichlet form Dr+(X equation equivalent Euclidean follows formula geodesic segments geodesic triangle gluing gradient estimates harmonic functions heat kernel Hence implies inequality integral lemma Let X,d Lipschitz Lipschitz continuous locally uniformly lower curvature bound lower Ricci curvature lower sectional curvature manifolds with lower marginal processes Markov Markov property martingale martingale problem measure metric tensor n-dimensional Riemannian nonnegative obtain patches Pi path space points proposition regularity respect Ricci curvature Ricci curvature bounds Riemannian manifold right hand side rough dimension satisfies sectional curvature sectional curvature bound semigroup sequence space with lower stochastic subset sufficiently small tangent cone tending to zero topological uniform weak limit weak Riemannian structure weakly yields