## The Annals of Probability

- Ann. Probab.
- Volume 21, Number 1 (1993), 433-452.

### Transience/Recurrence and Central Limit Theorem Behavior for Diffusions in Random Temporal Environments

Mark Pinsky and Ross G. Pinsky

#### Abstract

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.

#### Article information

**Source**

Ann. Probab., Volume 21, Number 1 (1993), 433-452.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176989410

**Digital Object Identifier**

doi:10.1214/aop/1176989410

**Mathematical Reviews number (MathSciNet)**

MR1207232

**Zentralblatt MATH identifier**

0773.60076

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J60: Diffusion processes [See also 58J65]

Secondary: 60H25: Random operators and equations [See also 47B80]

**Keywords**

Diffusion processes random environment transience and recurrence central limit theorem random parabolic operators

#### Citation

Pinsky, Mark; Pinsky, Ross G. Transience/Recurrence and Central Limit Theorem Behavior for Diffusions in Random Temporal Environments. Ann. Probab. 21 (1993), no. 1, 433--452. doi:10.1214/aop/1176989410. https://projecteuclid.org/euclid.aop/1176989410