  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.
