Almost sure asymptotics for the maximum local time in Brownian environment

TL;DR

This paper investigates the asymptotic behavior of the maximum local time in Brox's diffusion process within a Brownian environment, establishing precise growth rates and contrasting them with discrete cases.

Contribution

It determines the exact asymptotic growth rates of the maximum local time in Brox's process, clarifying the speed of its growth and comparing it to discrete analogs.

Findings

Maximum local time grows at rate t log(log(log t))

Minimum speed of local time growth is t/log(log(log t))

Results differ from discrete case, highlighting unique continuous environment behavior

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)).