  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.
