FIRST EXIT TIMES FOR LEVY-DRIVEN DIFFUSIONS WITH EXPONENTIALLY LIGHT JUMPS

• 成果类型: Article
• 署名作者: Imkeller, Peter; Pavlyukevich, Ilya; Wetzel, Torsten
• 署名单位: Humboldt University of Berlin
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/08-AOP412
• 发表日期: 2009
• 页码: 530-564
• 关键词: metastability; CONVERGENCE; asymptotics; flights

摘要

We consider a dynamical system described by the differential equation (Y) over dot(t) = -U'(Y-t) with a unique stable point at the origin. We perturb the system by the Levy noise of intensity E to obtain the stochastic differential equation dX(t)(epsilon) = -U'(X-t-(epsilon)) dt + epsilon dL(t). The process L is a symmetric Levy process whose jump measure nu has exponentially light tails, nu([u, infinity)) similar to exp(- u(alpha)), alpha > 0, u -> infinity. We study the first exit problem for the trajectories of the solutions of the stochastic differential equation from the interval (-1, 1). In the small noise limit epsilon -> 0, the law of the first exit time sigma(x), x is an element of (-1, 1), has exponential tail and the mean value exhibiting an intriguing phase transition at the critical index alpha = 1, namely, In E sigma similar to epsilon(-alpha) for 0 < alpha < 1, whereas In E sigma similar to epsilon(-1)vertical bar ln epsilon vertical bar(1-1/alpha) for alpha > 1.