Chernoff's theorem and discrete time approximations of Brownian motion on manifolds

O.G. Smolyanov, H. Weizsäcker, von, O. Wittich

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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.
    Original languageEnglish
    Pages (from-to)1-29
    JournalPotential Analysis
    Volume26
    Issue number1
    DOIs
    Publication statusPublished - 2007

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