Localization of favorite points for diffusion in a random environment

For a diffusion X t in a one-dimensional Wiener medium W , it is known that there is a certain process ( b r ( W ) ) r ≥ 0 that depends only on the environment, so that X t − b log t ( W ) converges in distribution as t → ∞ . The paths of b are step functions. Denote by F X ( 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 ( b r ( W ) ) r ≥ 0 , ( F X ( e r ) ) r ≥ 0 are close after some large r .