Localization of diffusion processes in one-dimensional random environment
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.
diffusion process in a random environment
scientific article; zbMATH DE number 98462