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The Maximum of the Local Time of a Diffusion Process in a Drifted Brownian Potential

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Book cover Séminaire de Probabilités XLVIII

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2168))

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Abstract

We consider a one-dimensional diffusion process X in a (−κ∕2)-drifted Brownian potential for κ ≠ 0. We are interested in the maximum of its local time, and study its almost sure asymptotic behaviour, which is proved to be different from the behaviour of the maximum local time of the transient random walk in random environment. We also obtain the convergence in law of the maximum local time of X under the annealed law after suitable renormalization when κ ≥ 1. Moreover, we characterize all the upper and lower classes for the hitting times of X, in the sense of Paul Lévy, and provide laws of the iterated logarithm for the diffusion X itself. To this aim, we use annealed technics.

AMS Classification (2010): 60K37, 60J60, 60J55, 60F15

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Acknowledgements

I would like to thank Zhan Shi for many helpful discussions. I am also grateful to an anonymous referee for a very careful reading of the first version of the paper and for valuable comments which helped improve the presentation of the paper, including a recommendation to merge the first version of this paper with [19].

This research was partially supported by the French ANR project MEMEMO2 2010 BLAN 0125.

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  1. Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 78035, Versailles, France

    Alexis Devulder

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  1. Alexis Devulder
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Correspondence to Alexis Devulder .

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Editors and Affiliations

  1. Laboratoire de Mathématiques, Université de Versailles-St Quentin, Versailles , France

    Catherine Donati-Martin

  2. Campus scientifique, IECL, Vandoeuvre-les-Nancy CEDEX, France

    Antoine Lejay

  3. Laboratoire de Mathématiques, Université de Versailles-St Quentin, Versailles, France

    Alain Rouault

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Devulder, A. (2016). The Maximum of the Local Time of a Diffusion Process in a Drifted Brownian Potential. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLVIII. Lecture Notes in Mathematics(), vol 2168. Springer, Cham. https://doi.org/10.1007/978-3-319-44465-9_5

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