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

In this paper, 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. Moreover, as an application, we prove the recurrence of some multi-dimensional diffusion processes in random environments including zero potentials.