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

TL;DR

This paper introduces an index based on Dirichlet forms to measure recurrence in symmetric Markov processes and applies it to prove recurrence of multi-dimensional diffusions in random environments with zero potentials.

Contribution

It presents a new index for recurrence and provides sufficient conditions for the recurrence of direct product diffusion processes, including in random environments.

Findings

Established an index for recurrence strength using Dirichlet forms.

Provided conditions ensuring recurrence of multi-dimensional diffusions.

Proved recurrence in random environments with zero potentials.

Abstract

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.