Brownian Occupation Measures on Compact Manifolds

  • Gunnar A. Brosamler
Part of the Progress in Probability and Statistics book series (PRPR, volume 12)


Let M be a compact C manifold of dimension d. A C metric g and a C vector field V on M determine a “Brownian motion” on M, i.e. a strong Markov process with continuous sample paths and generator \( L = \frac{1}{{2\,}} {\Delta_g} + V, \) where Δ g stands for the Laplace-Beltrami operator, associated with g. (If necessary we indicate by the subscript g, that an object is associated with the metric g.) Notice that the metric and the vector field can be recovered from the Brownian motion, from its generator L, to be precise.


Brownian Motion Iterate Logarithm Invariant Probability Measure Vector Field Versus Strong Markov Process 
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  1. [1]
    J. R. Baxter and G. A. Brosamler, Recurrence of Brownian Motions on Compact Manifolds, Colloque en l’Honneur de Laurent Schwartz, Asterisque 132, 15–46 (1985).MathSciNetGoogle Scholar
  2. [2]
    G. A. Brosamler, Laws of the Iterated Logarithm for Brownian Motions on Compact Manifolds, Z. Wahrscheinlichkeitstheorie verw. Gebiete 65, 99–114 (1983).MathSciNetCrossRefGoogle Scholar
  3. [3]
    Z. Ciesielski and S. J. Taylor, First Passage Times and Sojourn Times for Brownian Motion in Space and the Exact Hausdorff Measure of the Sample Path, Trans. Amer. Math. Soc. 103, 434–450 (1962).MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    L. Elie, Equivalent de la densité d’une diffusion en temps petits. Cas des points process, Astérisque 84–85, 55–71 (1981).Google Scholar
  5. [5]
    J. Kuelbs and R. LePage, The Law of the Iterated Logarithm for Brownian Motion in a Banach Space, Trans. Amer. Math. Soc. 185, 253–264 (1973).MathSciNetCrossRefGoogle Scholar
  6. [6]
    J. Kuelbs and W. Philipp, Almost Sure Invariance Principles for Partial Sums of Mixing B-Valued Rnadom Variables, Ann. Prob. 8, 1003–1036 (1980).MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    H. P. McKean, Stochastic Integrals, Academic Press, New York (1969).MATHGoogle Scholar
  8. [8]
    E. Nelson, The Adjoint Markoff Process, Duke Math. J. 25, 671–690 (1958).MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univeristy Press, Princeton (1970).MATHGoogle Scholar
  10. [10]
    S. J. Taylor, The Exact Hausdorff Measure of the Sample Path for Planar Brownian Motion, Proc. Cambridge Phil. Soc. 60, 253–258 (1964).MATHCrossRefGoogle Scholar
  11. [11]
    D. Williams, Markov Properties of Brownian Local Time, Bull. Amer. Math. Soc. 75, 1035–1036 (1969).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston, Inc. 1986

Authors and Affiliations

  • Gunnar A. Brosamler
    • 1
    • 2
  1. 1.Fachbereich MathematikUniversitaet des SaarlandesSaarbrueckenWest Germany
  2. 2.Department of MathematicsThe University of British ColumbiaVancouverCanada

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