Open Access
January, 1992 Stability in Distribution for a Class of Singular Diffusions
Gopal K. Basak, Rabi N. Bhattacharya
Ann. Probab. 20(1): 312-321 (January, 1992). DOI: 10.1214/aop/1176989928


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?


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Gopal K. Basak. Rabi N. Bhattacharya. "Stability in Distribution for a Class of Singular Diffusions." Ann. Probab. 20 (1) 312 - 321, January, 1992.


Published: January, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0749.60073
MathSciNet: MR1143422
Digital Object Identifier: 10.1214/aop/1176989928

Primary: 60J60

Keywords: Asymptotic flatness , Stochastic stability , Unique invariant probability

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 1 • January, 1992
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