Localization of favorite points for diffusion in a random environment

For a diffusion Xt in a one-dimensional Wiener medium W, it is known that there is a certain process (br(W))r>=0 that depends only on the environment, so that Xt-blogt(W) converges in distribution as t-->[infinity]. The paths of b are step functions. Denote by FX(t) the point with the most local time for the diffusion at time t. We prove that, modulo a relatively small time change, the paths of the processes (br(W))r>=0, (FX(er))r>=0 are close after some large r.