Let $\sigma(t)$ be an ergodic Markov chain on a finite state space $E$ and for each $\sigma \in E$, define on $\mathbb{R}^d$ the second-order elliptic operator $L_\sigma = \frac{1}{2} \sum^d_{i,j = 1} a_{ij}(x; \sigma)\frac{\partial^2}{\partial x_i\partial x_j} + \sum^d_{i = 1} b_i(x;\sigma)\frac{\partial}{\partial x_i}.$ Then for each realization $\sigma(t) = \sigma(t, \omega)$ of the Markov chain, $L_{\sigma(t)}$ may be thought of as a time-inhomogeneous diffusion generator. We call such a process a diffusion in a random temporal environment or simply a random diffusion. We study the transience and recurrence properties and the central limit theorem properties for a class of random diffusions. We also give applications to the stabilization and homogenization of the Cauchy problem for random parabolic operators.
Publié le : 1993-01-14
Classification:
Diffusion processes,
random environment,
transience and recurrence,
central limit theorem,
random parabolic operators,
60J60,
60H25
@article{1176989410,
author = {Pinsky, Mark and Pinsky, Ross G.},
title = {Transience/Recurrence and Central Limit Theorem Behavior for Diffusions in Random Temporal Environments},
journal = {Ann. Probab.},
volume = {21},
number = {4},
year = {1993},
pages = { 433-452},
language = {en},
url = {http://dml.mathdoc.fr/item/1176989410}
}
Pinsky, Mark; Pinsky, Ross G. Transience/Recurrence and Central Limit Theorem Behavior for Diffusions in Random Temporal Environments. Ann. Probab., Tome 21 (1993) no. 4, pp. 433-452. http://gdmltest.u-ga.fr/item/1176989410/