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

We consider d-dimensional diffusion processes in multi-parameter random environments which are given by values at different d points of one-dimensional α-stable or (r, α)-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.