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Limit Law of the Local Time for Brox’s Diffusion

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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).

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  1. Laboratoire MAPMO–C.N.R.S. UMR 6628–Fédération Denis-Poisson, Université d’Orléans, Orléans, France

    Pierre Andreoletti & Roland Diel

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  1. Pierre Andreoletti
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  2. Roland Diel
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Correspondence to Roland Diel.

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Andreoletti, P., Diel, R. Limit Law of the Local Time for Brox’s Diffusion. J Theor Probab 24, 634–656 (2011). https://doi.org/10.1007/s10959-010-0314-7

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  • DOI: https://doi.org/10.1007/s10959-010-0314-7

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