Summary
The limiting behavior of one-dimensional diffusion process in an asymptotically self-similar random environment is investigated through the extension of Brox's method. Similar problems are then discussed for a random walk in a random environment with the aid of optional sampling from a diffusion model; an extension of the result of Sinai is given in the case of asymptotically self-similar random environments.
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Kawazu, K., Tamura, Y. & Tanaka, H. Limit theorems for one-dimensional diffusions and random walks in random environments. Probab. Th. Rel. Fields 80, 501–541 (1989). https://doi.org/10.1007/BF00318905
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DOI: https://doi.org/10.1007/BF00318905