Iterated Brownian motion in an open set

• 成果类型: Article
• 署名作者: DeBlassie, RD
• 署名单位: Texas A&M University System; Texas A&M University College Station
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051604000000404
• 发表日期: 2004
• 页码: 1529-1558
• 关键词: heat-type equations; biharmonic functions; wiener-processes; pde connection; time processes; exit times; distributions; logarithm; modulus; cones

摘要

Suppose a solid has a crack filled with a gas. If the crack reaches the surrounding medium, how long does it take the gas to diffuse out of the crack? Iterated Brownian motion serves as a model for diffusion in a crack. If tau is the first exit time of iterated Brownian motion from the solid, then P(tau > t) can be viewed as a measurement of the amount of contaminant left in the crack at time t. We determine the large time asymptotics of P(tau > t) for both bounded and unbounded sets. We also discuss a strange connection between iterated Brownian motion and the parabolic operator 1/8 Delta(2) - partial derivative/partial derivativet.