Chernoff's Theorem and Discrete Time Approximations of Brownian Motion on Manifolds |
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Authors: | Oleg G. Smolyanov Heinrich v. Weizsäcker Olaf Wittich |
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Affiliation: | 1.Faculty of Mechanics and Mathematics,Moscow State University,Moscow,Russia;2.Fachbereich Mathematik,Technische Universit?t Kaiserslautern,Kaiserslautern,Germany;3.Mathematisches Institut,Universit?t Tübingen,Tübingen,Germany |
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Abstract: | Let be a one-parameter family of positive integral operators on a locally compact space . For a possibly non-uniform partition of define a finite measure on the path space by using a) for the transition between any two consecutive partition times of distance and b) a suitable continuous interpolation scheme (e.g. Brownian bridges or geodesics). If necessary normalize the result to get a probability measure. We prove a version of Chernoff's theorem of semigroup theory and tightness results which yield convergence in law of such measures as the partition gets finer. In particular let be a closed smooth submanifold of a manifold . We prove convergence of Brownian motion on , conditioned to visit at all partition times, to a process on whose law has a density with respect to Brownian motion on which contains scalar, mean and sectional curvatures terms. Various approximation schemes for Brownian motion on are also given. |
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Keywords: | approximation of Feller semigroups Brownian bridge conditional process geodesic interpolation infinite dimensional surface measure (mean, scalar, sectional) curvature pseudo-Gaussian kernels Wick's formula. |
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