Localization of favorite points for diffusion in random environment

For a diffusion X_t in a one-dimensional Wiener medium W, it is known that there is a certain process b_x(W) that depends only on the environment W, so that X_t-b_{logt}(W) converges in distribution as t goes to infinity. We prove that, modulo a relatively small time change, the process {b_x(W):x>0}is followed closely by the process {F_X(e^x): x>0}, with F_X(t) denoting the point with the most local time for the diffusion at time t.