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Almost sure asymptotics for the local time of a diffusion in Brownian environment

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  • Diel, Roland

Abstract

Here, we study the asymptotic behavior of the maximum local time of the diffusion in Brownian environment. Shi (1998) [17] proved that, surprisingly, the maximum speed of is at least tlog(log(logt)); whereas in the discrete case, it is t. We show that tlog(log(logt)) is the proper rate and that for the minimum speed the rate is the same as in the discrete case (see Dembo et al. (2007) [6]) namely t/log(log(logt)). We also prove a localization result: almost surely for large time, the diffusion has spent almost all the time in the neighborhood of four points which only depend on the environment.

Suggested Citation

  • Diel, Roland, 2011. "Almost sure asymptotics for the local time of a diffusion in Brownian environment," Stochastic Processes and their Applications, Elsevier, vol. 121(10), pages 2303-2330, October.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:10:p:2303-2330
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    References listed on IDEAS

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    1. Hu, Yueyun, 2000. "Tightness of localization and return time in random environment," Stochastic Processes and their Applications, Elsevier, vol. 86(1), pages 81-101, March.
    2. Kesten, Harry, 1986. "The limit distribution of Sinai's random walk in random environment," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 138(1), pages 299-309.
    3. 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.
    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.
    2. Hu, Yaozhong & Lê, Khoa & Mytnik, Leonid, 2017. "Stochastic differential equation for Brox diffusion," Stochastic Processes and their Applications, Elsevier, vol. 127(7), pages 2281-2315.

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