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Limit theorem for one-dimensional diffusion process in Brownian environment. (English) Zbl 0662.60082

Stochastic analysis, Proc. Jap.-Fr. Semin., Paris/France 1987, Lect. Notes Math. 1322, 156-172 (1988).
[For the entire collection see Zbl 0635.00012.]
The large time behaviour of a diffusion analogue of Sinai’s random walk is considered. First, it is proved by an argument of Th. Brox [Ann. Probab. 14, 1206-1218 (1986; Zbl 0608.60072)] that the diffusion process under consideration can be approximated by appropriate reflecting diffusions.
Using Pitman’s theorem it is shown that the law of a so-called standard valley of the environment can be described by a Bessel process of index 3.
The main result of the paper states that the process, without scaling but only by centering, has a limit distribution as \(t\to \infty\).
Reviewer: P.Kröger

MSC:

60J60 Diffusion processes
60G17 Sample path properties