Recurrence and transience properties of multi-dimensional diffusion processes in selfsimilar and semi-selfsimilar random environments

Abstract

We consider $d$-dimensional diffusion processes in multi-parameter random environments which are given by values at different $d$ points of one-dimensional $\alpha $-stable or $(r, \alpha )$-semi-stable Lévy processes. From the model, we derive some conditions of random environments that imply the dichotomy of recurrence and transience for the $d$-dimensional diffusion processes. The limiting behavior is quite different from that of a $d$-dimensional standard Brownian motion. We also consider the direct product of a one-dimensional diffusion process in a reflected non-positive Brownian environment and a one-dimensional standard Brownian motion. For the two-dimensional diffusion process, we show the transience property for almost all reflected Brownian environments.