Tightness of localization and return time in random environment

Consider a class of diffusions with random potentials which behave asymptotically as Brownian motion. We study the tightness of localization around the bottom of some Brownian valley, and determine the limit distribution of the return time to the origin after a typical time. Via the Skorokhod embedding in random environment, we also solve the return time problem for Sinaiʹs walk.