We study the sample path asymptotics of a class of recurrent diffusion processes with random potentials, including examples of Sinai’s simple random walk in random environment and Brox’s diffusion process with Brownian potential. The main results consist of several integral criteria which completely characterize all the possible Lévy classes, therefore providing a very precise image of the almost sure asymptotic behaviors of these processes.
"The limits of Sinai's simple random walk in random environment." Ann. Probab. 26 (4) 1477 - 1521, October 1998. https://doi.org/10.1214/aop/1022855871