Localization of diffusion processes in one-dimensional random environment
type: Item
instance of: scholarly article
title: Localization of diffusion processes in one-dimensional random environment
zbMATH Open document ID: 0761.60072
author: Kiyoshi Kawazu
author: Yozo Tamura
author: Hiroshi Tanaka
published in: Journal of the Mathematical Society of Japan
publication date: 1993-01-17T00:00:00Z
review text: The authors consider the asymptotic behavior of a one-dimensional diffusion process in a random environment, that is, given a stochastic process \(\{W(x), x\in\mathbb{R}\}\) as an environment, they consider the limit theorem of a diffusion process \(X_ t=X(t,W)\), which is formally characterized by the following stochastic differential equation \[ dX_ t=\text{Brownian differential}-{1\over 2} W'(X_ t)dt. \] When \(\{W(x), x\in\mathbb{R}\}\) is a Brownian motion, \textit{Th. Brox} [Ann. Probab. 14, 1206- 1218 (1986; Zbl 0608.60072)] showed that \((\log t)^{-2} X(t,\cdot)- b(t,\cdot)\) tends to 0 in probability as \(t\to\infty\) where \(b(t,W)\) is a suitable function only depending on \(t\) and the environment. This results correspond to those by \textit{Ya. G. Sinaj} [Theory Probab. Appl. 27, 256- 268 (1982); translation from Teor. Veroyatn. Primen. 27, 247-258 (1982; Zbl 0497.60065)] for random walk in random environment. A more refined result (localization) is obtained by \textit{A. O. Golosov} [Commun. Math. Phys. 92, 491-506 (1984; Zbl 0534.60065)] in the case of Sinaj's reflecting random walk in random environment. The authors consider the localization property of diffusions in a wider class of environment. Their main result is stated as follows: The process \(\{X(e^ \lambda+t,\cdot)-b_ \lambda(\cdot), t\geq 0\}\) converges as \(\lambda\to\infty\) to a stationary process \(\{\widetilde\omega(t), t\geq 0\}\) in the sense of weak convergence. --- They characterize completely the limit process \(\{\widetilde\omega(t), t\geq 0\}\). In order to obtain the result, they study certain time reversals of random walks, conditioned random walks and their relations for the study of the shape of random environment processes.
Mathematics Subject Classification ID: 60J70
zbMATH DE Number: 98462
zbMATH Keywords: time reversal
zbMATH Keywords: diffusion process in a random environment
zbMATH Keywords: stochastic differential equation
zbMATH Keywords: random walk in random environment
zbMATH Keywords: Sinaj's reflecting random walk
zbMATH Keywords: localization property of diffusions
zbMATH Keywords: shape of random environment processes
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04430515
OpenAlex ID: W1972872345
DOI: 10.2969/JMSJ/04430515
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description: scientific article; zbMATH DE number 98462