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Sample path properties of diffusion processes on compact manifolds

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1452)

Abstract

For a diffusion process on a compact manifold M ϕ and ψ-mixing properties are established. A law of the iterated logarithm for the convergence of sample paths to the invariant measure of the diffusion with respect to the Vasserstein metric is deduced.

Keywords

  • Invariant Measure
  • Fundamental Solution
  • Compact Manifold
  • Iterate Logarithm
  • Sample Path Property

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Azencott R., Diffusion sur les varietés differentiables, Comptes Rendues Acad. Sc. Paris Sér. A 274 (1972), 651–654.

    MATH  Google Scholar 

  2. Billingsley, “Convergence of Probability Measures,” Wiley, New York, 1968.

    MATH  Google Scholar 

  3. Blümlinger M., Drmota M., Tichy R., A uniform law of the iterated logarithm for Brownian motion on compact Riemannian manifolds, Math. Zeitschr. 201, 495–507.

    Google Scholar 

  4. Chung K., “Lectures from Markoff processes to Brownian motion,” Springer, New York, 1982.

    CrossRef  MATH  Google Scholar 

  5. Doob J., “Stochastic processes,” Wiley, New York, 1953.

    MATH  Google Scholar 

  6. Dynkin E., “Markoff Processes Vol. I,” Springer, New York, 1968.

    Google Scholar 

  7. Friedman A., “Partial differential equations of parabolic type,” Prentice Hall, Englewood Cliffs, N.J., 1964.

    MATH  Google Scholar 

  8. Hlawka E., Beiträge zur Theorie der Gleichverteilung I–V, Sitzungsber. Öst. Akad. Wiss. Math.—nat. Kl. 197, 1–154, 209–289.

    Google Scholar 

  9. Itô S., The fundamental solution of the parabolic equation in a differentiable manifold, I, Osaka Math. J. 5 (1953), 75–92.

    MathSciNet  MATH  Google Scholar 

  10. _____, The fundamental solution of the parabolic equation in a differentiable manifold, II, Osaka Math. J. 6 (1954), 167–185.

    MathSciNet  MATH  Google Scholar 

  11. _____, A boundary value problem of partial differential equations of parabolic type, Duke Math. J. 24 (57), 299–312.

    Google Scholar 

  12. Nirenberg L., Maximum principle for parabolic equations, Comm. pure a. appl. Math. 6 (1953), 167–177.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. Phillip W., “Mixing sequences of random variables and probabilistic number theory,” AMS Memoirs 114, Providence, 1971.

    Google Scholar 

  14. Rachev S.T., The Monge-Kantorovich mass transference problem and its stochastic applications, Th. Prob. a. Applic. 29 (1984), 647–676.

    CrossRef  MATH  Google Scholar 

  15. Reznik M., The law of the iterated logarithm for stationary processes satisfying mixing conditions, Th. Prob. a. Applic. 13 (1968), 606–621.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. Stackelberg O., A uniform law of the iterated logarithm for functions C-uniformly distributed mod 1, Indiana Univ. Math. J. 21 (1971), 515–528.

    CrossRef  MathSciNet  MATH  Google Scholar 

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© 1990 Springer-Verlag

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Blümlinger, M. (1990). Sample path properties of diffusion processes on compact manifolds. In: Hlawka, E., Tichy, R.F. (eds) Number-Theoretic Analysis. Lecture Notes in Mathematics, vol 1452. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0096977

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  • DOI: https://doi.org/10.1007/BFb0096977

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-53408-2

  • Online ISBN: 978-3-540-46864-6

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