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 tlog(log(logt))tlog(log(logt)); whereas in the discrete case, it is tt. We show that tlog(log(logt))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))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.

► Almost sure results for the diffusion in Brownian environment.
► We prove a law of the iterated logarithm for the maximum local time.
► The diffusion spends almost all the time in the neighborhood of four points.