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

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  • Diel, Roland
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    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.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 121 (2011)
    Issue (Month): 10 (October)
    Pages: 2303-2330

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    Handle: RePEc:eee:spapps:v:121:y:2011:i:10:p:2303-2330

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    Keywords: Diffusion in Brownian environment Local time;

    References

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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. 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.
    3. 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.
    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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