Limiting behavior of a diffusion in an asymptotically stable environment

Let $V$ be a two sided random walk and let $X$ denote a real valued diffusion process with generator ${1/2}e^{V([x])}\frac{d}{dx}(e^{-V([x])}\frac{d}{dx})$. This process is known to be the continuous equivalent of the one dimensional random walk in random environment with potential $V$. Hu and Shi (1997) described the L\'evy classes of $X$ in the case where $V$ behaves approximately like a Brownian motion. In this paper, based on some fine results on the fluctuations of random walks and stable processes, we obtain an accurate image of the almost sure limiting behavior of $X$ when $V$ behaves asymptotically like a stable process. These results also apply for the corresponding random walk in random environment.