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


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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  1. 1.
    Andersson, L., Driver, B.K.: Finite dimensional approximations to Wiener measure and path integral formulas on manifolds. J. Funct. Anal. 165, 430–498 (1999)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    doCarmo, M.P.: Riemannian Geometry. Birkhäuser, Boston (1992)Google Scholar
  3. 3.
    Dudley, R.M.: Real Analysis and Probability. Wadsworth and Brooks/Cole, Pacific Grove, California (1989)MATHGoogle Scholar
  4. 4.
    Ethier, S.N., Kurtz, T.G.: Markov Processes, Characterizations and Convergence. John Wiley, New York (1986)Google Scholar
  5. 5.
    Jost, J.: Riemannian Geometry and Geometric Analysis, 2nd edn. Springer, Berlin Heidelberg New York (1998)MATHGoogle Scholar
  6. 6.
    Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin Heidelberg New York (1980)MATHGoogle Scholar
  7. 7.
    Minakshisundaram, S., Pleijel, A.: Some properties of the Eigenfunctions of the Laplace operator on Riemannian manifolds. Canad. J. Math. 1, 242–256 (1949)MATHMathSciNetGoogle Scholar
  8. 8.
    Roe, J.: Elliptic Operators, Topology and Asymptotic Methods. Longman, London (1988)MATHGoogle Scholar
  9. 9.
    Sidorova, N.: The Smolyanov surface measure on trajectories in a Riemannian manifold. Infinite Dimensional Anal. Quantum Probab. Related Topics 7, 461–471 (2004)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Sidorova, N., Smolyanov, O.G., Weizsäcker, H.v., Wittich, O.: Conditioning Brownian motion to small tubular neighborhoods. In preparation (2005)Google Scholar
  11. 11.
    Sidorova, N., Smolyanov, O.G., Weizsäcker, H.v., Wittich, O.: The surface limit of Brownian motion in tubular neighbourhoods of an embedded Riemannian manifold. J. Funct. Anal. 206, 391–413 (2004)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Smolyanov, O.G., Weizsäcker, H.v., Wittich, O.: Brownian motion on a manifold as limit of stepwise conditioned standard Brownian motions. In: Stochastic Processes, Physics and Geometry: New Interplays. II: A Volume in Honour of S. Albeverio, volume 29 of Canad. Math. Soc. Conf. Proc., pp. 589–602. Amer. Math. Soc., Providence, Rhode Island (2000)Google Scholar
  13. 13.
    Smolyanov, O.G., Weizsäcker, H.v., Wittich, O.: Chernoff's theorem and the construction of Semigroups. In: Ianelli, M., Lumer, G. (eds.) Evolution Equations: Applications to Physics, Industry, Life sciences and Economics – EVEQ 2000, pp. 355–364. Birkhäuser, Cambridge, MA (2003)Google Scholar
  14. 14.
    Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer, Berlin Heidelberg New York (1979)MATHGoogle Scholar
  15. 15.
    Tokarev, A.G.: Unpublished Notes. Moscow (2001)Google Scholar
  16. 16.
    Wittich, O.: An explicit local uniform large deviation bound for Brownian bridges. Stat. Probab. Lett. 73, 51–56 (2005)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Wittich, O.: Effective dynamics on small tubular neighbourhoods. In preparation (2005)Google Scholar

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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