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Chernoff approximations of Feller semigroups in Riemannian manifolds
Chernoff approximations of Feller semigroups and the associated diffusion pro-cesses in Riemannian manifolds are studied. The manifolds are assumed to beof bounded geometry, thus including all compact manifolds and also a widerange of non-compact manifolds. Sufficient conditions are established for a classof second order elliptic operators to generate a Feller semigroup on a (gener-ally non-compact) manifold of bounded geometry. A construction of Chernoffapproximations is presented for these Feller semigroups in terms of shift oper-ators. This provides approximations of solutions to initial value problems forparabolic equations with variable coefficients on the manifold. It also yields weakconvergence of a sequence of random walks on the manifolds to the diffusionprocesses associated with the elliptic generator. For parallelizable manifolds thisresult is applied in particular to the representation of Brownian motion on themanifolds as limits of the corresponding random walks.