Tanaka, Hiroshi 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 Cited in 1 ReviewCited in 7 Documents MSC: 60J60 Diffusion processes 60G17 Sample path properties Keywords:large time behaviour of a diffusion; reflecting diffusions; Bessel process Citations:Zbl 0635.00012; Zbl 0608.60072 PDFBibTeX XML