Limit theorems for a diffusion process with a one-sided brownian potential

We consider a diffusion process X(t) with a one-sided Brownian potential starting from the origin. The limiting behavior of the process as time goes to infinity is studied. For each t > 0, the sample space describing the random potential is divided into two parts. Ã, and B̃_t, both having probability 1/2, in such a way that our diffusion process X(t) exhibits quite different limiting behavior depending on whether it is conditioned on Ã_t or on B̃_t (t → ∞). The asymptotic behavior of the maximum process of X(t) is also investigated. Our results improve those of Kawazu, Suzuki, and Tanaka (2001).