Intrinsic Coupling on Riemannian Manifolds and Polyhedra
Max-K. von Renesse, Technical University Berlin
Abstract
Starting from a central limit theorem for geometric random walks
we give an elementary construction of couplings between Brownian
motions on Riemannian manifolds. This approach shows that cut lo-
cus phenomena are indeed inessential for Kendall's and Cranston's
stochastic proof of gradient estimates for harmonic functions on Rie-
mannian manifolds with lower curvature bounds. Moreover, since the
method is based on an asymptotic quadruple inequality and a central
limit theorem only it may be extended to certain non smooth spaces
which we illustrate by the example of Riemannian polyhedra. Here we
also recover the classical heat kernel gradient estimate which is well
known from the smooth setting.
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