Occupation time theorems for one-dimensional random walks and diffusion processes in random environments
Already at the beginning of the paper the authors develop a general theory for the growth in time, in the log scale, of a class of one-dimensional Brownian additive functionals. These results they apply to obtain asymptotic growth in time of occupation times of a family of one-dimensional diffusion processes. In the next step the authors achieve the long time asymptotics of occupation times for diffusions (and random walks) in random environments. They fundamentally assume that the environment is self-similar in the case of diffusion and asymptotically self-similar in the case of random walks. The environments on the positive and negative sides are independent symmetric stable processes which may have different exponents. The case of random walks is studied by imbedding them in birth-and-death processes with asymptotically self-similar environments.
scientific article; zbMATH DE number 5527423