Diffusion Processes with One-sided Selfsimilar Random Potentials

Abstract

Long-time behavior of diffusion processes with one-sided random potentials starting from the origin is studied. As random potentials, some strictly stable processes are given just on the negative side in the real line. This model is an extension of the diffusion process with a one-sided Brownian potential studied by Kawazu, Suzuki and Tanaka (Tokyo J. Math. 24, 211–229 2001) and Kawazu and Suzuki (J. Appl. Probab. 43, 997–1012 2006). In this paper, we analyze our model by different methods from theirs. We use the theory concerning the convergence of a sequence of bi-generalized diffusion processes studied by Ogura (J. Math. Soc. Japan 41, 213–242 1989) and Tanaka (Comm. Pure Appl. Math. 47, 755–766 1994). For diffusion processes with one-sided random potentials, the limit theorems introduced by them cannot be used. We improve their limit theorems and apply the improved limit theorem to examining the long-time behavior of our model. As a result, we show that limit distributions exist under the Brownian scaling with some probability, and under a sub-diffusive scaling with the remaining probability.