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Limit Law of the Local Time for Brox’s Diffusion

Author

Listed:
  • Pierre Andreoletti

    (Université d’Orléans)

  • Roland Diel

    (Université d’Orléans)

Abstract

We consider Brox’s model: a one-dimensional diffusion in a Brownian potential W. We show that the normalized local time process (L(t,m log t +x)/t, x∈R), where m log t is the bottom of the deepest valley reached by the process before time t, behaves asymptotically like a process which only depends on W. As a consequence, we get the weak convergence of the local time to a functional of two independent three-dimensional Bessel processes and thus the limit law of the supremum of the normalized local time. These results are discussed and compared to the discrete time and space case for which the same questions have been answered recently by Gantert, Peres, and Shi (Ann. Inst. Henri Poincaré, Probab. Stat. 46(2):525–536, 2010).

Suggested Citation

  • Pierre Andreoletti & Roland Diel, 2011. "Limit Law of the Local Time for Brox’s Diffusion," Journal of Theoretical Probability, Springer, vol. 24(3), pages 634-656, September.
    • Handle: RePEc:spr:jotpro:v:24:y:2011:i:3:d:10.1007_s10959-010-0314-7
    DOI: 10.1007/s10959-010-0314-7
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    References listed on IDEAS

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    1. Cheliotis, Dimitris, 2008. "Localization of favorite points for diffusion in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 118(7), pages 1159-1189, July.
    2. Bertoin, Jean, 1993. "Splitting at the infimum and excursions in half-lines for random walks and Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 47(1), pages 17-35, August.
    3. Yueyun Hu & Zhan Shi, 1998. "The Local time of Simple Random Walk in Random Environment," Journal of Theoretical Probability, Springer, vol. 11(3), pages 765-793, July.
    4. Shi, Zhan, 1998. "A local time curiosity in random environment," Stochastic Processes and their Applications, Elsevier, vol. 76(2), pages 231-250, August.
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    Cited by:

    1. Grégoire Véchambre, 2023. "Almost Sure Behavior for the Local Time of a Diffusion in a Spectrally Negative Lévy Environment," Journal of Theoretical Probability, Springer, vol. 36(2), pages 876-925, June.

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