## The Annals of Probability

- Ann. Probab.
- Volume 20, Number 1 (1992), 312-321.

### Stability in Distribution for a Class of Singular Diffusions

Gopal K. Basak and Rabi N. Bhattacharya

#### Abstract

A verifiable criterion is derived for the stability in distribution of singular diffusions, that is, for the weak convergence of the transition probability $p(t; x, dy)$, as $t \rightarrow \infty$, to a unique invariant probability. For this we establish the following: (i) tightness of $\{p(t; x, dy): t \geq 0\}$; and (ii) asymptotic flatness of the stochastic flow. When specialized to highly nonradial nonsingular diffusions the results here are often applicable where Has'minskii's well-known criterion fails. When applied to traps, a sufficient condition for stochastic stability of nonlinear diffusions is derived which supplements Has'minskii's result for linear diffusions. We also answer a question raised by L. Stettner (originally posed to him by H. J. Kushner): Is the diffusion stable in distribution if the drift is $Bx$ where $B$ is a stable matrix, and $\sigma(\cdot)$ is Lipschitzian, $\sigma(\underline{0}) \neq 0$? If not, what additional conditions must be imposed?

#### Article information

**Source**

Ann. Probab., Volume 20, Number 1 (1992), 312-321.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176989928

**Digital Object Identifier**

doi:10.1214/aop/1176989928

**Mathematical Reviews number (MathSciNet)**

MR1143422

**Zentralblatt MATH identifier**

0749.60073

**JSTOR**

links.jstor.org

**Subjects**

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

**Keywords**

Unique invariant probability asymptotic flatness stochastic stability

#### Citation

Basak, Gopal K.; Bhattacharya, Rabi N. Stability in Distribution for a Class of Singular Diffusions. Ann. Probab. 20 (1992), no. 1, 312--321. doi:10.1214/aop/1176989928. https://projecteuclid.org/euclid.aop/1176989928