STABILITY IN DISTRIBUTION FOR A CLASS OF SINGULAR DIFFUSIONS

• 成果类型: Article
• 署名作者: BASAK, GK; BHATTACHARYA, RN
• 署名单位: Indiana University System; Indiana University Bloomington
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176989928
• 发表日期: 1992
• 页码: 312-321
• 关键词: invariant-measures; recurrence

摘要

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 --> infinity, to a unique invariant probability. For this we establish the following: (i) tightness of {p(t; x, dy): t greater-than-or-equal-to 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(.) is Lipschitzian, sigma(0) not-equal 0? If not, what additional conditions must be imposed?