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

Here, we study the asymptotic behavior of the maximum local time L ∗ ( t ) of the diffusion in Brownian environment. Shi (1998) [17] proved that, surprisingly, the maximum speed of L ∗ ( t ) is at least t log ( log ( log t ) ) ; whereas in the discrete case, it is t . We show that t log ( log ( log t ) ) 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 ( log t ) ) . 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.