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Potential Analysis

, Volume 26, Issue 1, pp 1–29 | Cite as

Chernoff's Theorem and Discrete Time Approximations of Brownian Motion on Manifolds

  • Oleg G. Smolyanov
  • Heinrich v. Weizsäcker
  • Olaf Wittich
Article

Abstract

Let \((S(t))_{t \ge 0}\) be a one-parameter family of positive integral operators on a locally compact space \(L\). For a possibly non-uniform partition of \([0,1]\) define a finite measure on the path space \(C_L[0,1]\) by using a) \(S(\Delta t)\) for the transition between any two consecutive partition times of distance \(\Delta t\) 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 \(L\) be a closed smooth submanifold of a manifold \(M\). We prove convergence of Brownian motion on \(M\), conditioned to visit \(L\) at all partition times, to a process on \(L\) whose law has a density with respect to Brownian motion on \(L\) which contains scalar, mean and sectional curvatures terms. Various approximation schemes for Brownian motion on \(L\) are also given.

Key words

approximation of Feller semigroups Brownian bridge conditional process geodesic interpolation infinite dimensional surface measure (mean, scalar, sectional) curvature pseudo-Gaussian kernels Wick's formula. 

Mathematics Subject Classifications (2000)

Primary 58J65 43C40 47D06 60D06 

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Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  • Oleg G. Smolyanov
    • 1
  • Heinrich v. Weizsäcker
    • 2
  • Olaf Wittich
    • 3
  1. 1.Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia
  2. 2.Fachbereich MathematikTechnische Universität KaiserslauternKaiserslauternGermany
  3. 3.Mathematisches InstitutUniversität TübingenTübingenGermany

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