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 Levy 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.
Highlights
Introduction and resultsIt is well-known that a multi-dimensional standard Brownian motion, consisting of d independent one-dimensional standard Brownian motions, is recurrent if d = 1 or 2, and transient otherwise
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
We derive some conditions of random environments that imply the dichotomy of recurrence and transience for the d-dimensional diffusion processes
Summary
In the case where the environment is Lévy’s Brownian motion W (x) with a multi-dimensional time, Tanaka showed the recurrence of the diffusion process for almost all environments in any dimension in [20]. These results were shown by Ichihara’s recurrent test introduced in [8]. Mathieu studied asymptotic behavior of multi-dimensional diffusion processes in random environments by using the Dirichlet form and showed the convergence theorem for the case where the environment is a nonnegative reflected Lévy’s Brownian motion in [11]. Showed the convergence theorem in the case where the random environment consists of d independent one-dimensional reflected non-negative Brownian motions, which is a model studied in [17]. (ii) If either {−Wk(−xk), xk ≥ 0, Qk} or {−Wk(xk), xk ≥ 0, Qk} is a subordinator for some k, XW is transient for almost all environments in any dimension
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