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

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.