Some remarks and examples concerning the transience and recurrence of random diffusions
type: Item
instance of: scholarly article
title: Some remarks and examples concerning the transience and recurrence of random diffusions
zbMATH Open document ID: 0766.60098
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1993-02-04T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77443
review text: This paper deals with diffusion processes in random environment, where the environment is given by an ergodic, finite state Markov chain [see \textit{Pinsky} and \textit{Pinsky}, Ann. Probab. 21, 433 (1993)]. The authors provide (nontrivial) examples showing that the random diffusion may be transient, although the components (for fixed environment) are positive recurrent, and, vice versa, for transient components the random diffusion can be positive recurrent. In cases where the joint process is reversible, a more regular behavior is observed: If at least one component is transient, then the random diffusion is transient; all components are positive recurrent iff the random diffusion is positive recurrent; if the averaged operator is recurrent, then the random diffusion is recurrent. However, even in the reversible case, it is possible that the components are null recurrent, while the random diffusion is transient, and the averaged operator may be transient, although the random diffusion is recurrent. Furthermore, a limit theorem is proved for the case, where the jump rate of the Markov chain goes to zero.
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 110500
zbMATH Keywords: ergodicity
zbMATH Keywords: diffusion processes in random environment
zbMATH Keywords: random diffusion
zbMATH Keywords: positive recurrent
zbMATH Keywords: reversible case
author: Ross G. Pinsky
author: Michael Scheutzow
reviewed by: Wolfgang Kliemann
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 110500