FRACTIONAL CAUCHY PROBLEMS ON BOUNDED DOMAINS

• 成果类型: Article
• 署名作者: Meerschaert, Mark M.; Nane, Erkan; Vellaisamy, P.
• 署名单位: Michigan State University; Auburn University System; Auburn University; Indian Institute of Technology System (IIT System); Indian Institute of Technology (IIT) - Bombay
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/08-AOP426
• 发表日期: 2009
• 页码: 979-1007
• 关键词: time random-walks; brownian-motion; diffusion; equation

摘要

Fractional Cauchy problems replace the usual first-order time derivative by a fractional derivative. This paper develops classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain D subset of R-d with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse stable subordinator whose scaling index corresponds to the order of the fractional time derivative. Dirichlet problems corresponding to iterated Brownian motion in a bounded domain are then solved by establishing a correspondence with the case of a half-derivative in time.