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The limits of Sinai’s simple random walk in random environment. (English) Zbl 0936.60088

Summary: We study the sample path asymptotics of a class of recurrent diffusion processes with random potentials, including examples of Ya. G. Sinaj’s simple random walk in random environment [Theory Probab. Appl. 27, 256-268 (1982); translation from Teor. Veroyatn. Primen. 27, 247-258 (1982; Zbl 0497.60065)] and Th. Brox’s diffusion process with Brownian potential [Ann. Probab. 14, 1206-1218 (1986; Zbl 0608.60072)]. The main results consist of several integral criteria which completely characterize all the possible Lévy classes, therefore providing a very precise image of the almost sure asymptotic behaviors of these processes.

MSC:

60K37 Processes in random environments
60J60 Diffusion processes
60F15 Strong limit theorems
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