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Some remarks and examples concerning the transience and recurrence of random diffusions. (English) Zbl 0766.60098
This paper deals with diffusion processes in random environment, where the environment is given by an ergodic, finite state Markov chain [see Pinsky and 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.
Reviewer: W.Kliemann (Ames)

##### MSC:
 60J60 Diffusion processes
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