Almost sure asymptotics for the maximum local time in Brownian environment

2010/03/02 by Diel, Roland
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1003.0661

Abstract

We study the asymptotic behaviour of the maximum local time L*(t) of the Brox's process, the diffusion in Brownian environment. Shi proved that the maximum speed of L*(t) is surprisingly, at least t log(log(log t)) whereas in the discrete case it is t. We show here that t log(log(log t)) is the proper rate and we prove that for the minimum speed the rate is the same as in the discrete case namely t/log(log(log t)).

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• The maximum of the local time of a diffusion process in a drifted Brownian potential
• A density for the local time of the Brox diffusion
• Stochastic Differential Equation for Brox Diffusion
• Local time of Martin-Lof Brownian motion