Localization of favorite points for diffusion in a random environment

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