We study a limiting behavior of a one-dimensional diffusion process with a random potential. The potential consists of two independent contracted self-similar processes with different indices for the right and the left hand sides of the origin. Brox (1986) and Schumacher (1985) studied a diffusion process with a Brownian potential, and showed, roughly speaking, after a long time with high probability the process is at the bottom of a valley. Their result was extended to a diffusion process in an asymptotically self-similar random environment by Kawazu, Tamura and Tanaka (1989). Our model is a variant of their models. But we show, roughly speaking, after a long time it is possible that our process is not at the bottom of a valley. We also study asymptotic behaviors of the minimum process and the maximum process of our process.
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