Recurrence of direct products of diffusion processes in random media having zero potentials

Abstract

In this paper, we consider the recurrence of some multi-dimensional diffusion processes in random environments including zero potentials. Previous methods on diffusion processes in random environments are not applicable to the case of such environments. In main theorems, we obtain a sufficient condition to be recurrent for the product of a multi-dimensional diffusion process in semi-selfsimilar random environments and one-dimensional Brownian motion, and also more explicit sufficient conditions in the case of Gaussian random environments and random environments generated by Lévy processes. To prove them, we introduce an index which measures the strength of recurrence of symmetric Markov processes, and give some sufficient conditions for recurrence of direct products of symmetric diffusion processes. The index is given by the Dirichlet forms of the Markov processes.