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May 2009 Fractional Cauchy problems on bounded domains
Mark M. Meerschaert, Erkan Nane, P. Vellaisamy
Ann. Probab. 37(3): 979-1007 (May 2009). DOI: 10.1214/08-AOP426


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⊂ℝd 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.


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Mark M. Meerschaert. Erkan Nane. P. Vellaisamy. "Fractional Cauchy problems on bounded domains." Ann. Probab. 37 (3) 979 - 1007, May 2009.


Published: May 2009
First available in Project Euclid: 19 June 2009

zbMATH: 1247.60078
MathSciNet: MR2537547
Digital Object Identifier: 10.1214/08-AOP426

Primary: 35C10 , 60G99

Keywords: boundary value problem , bounded domain , Brownian subordinator , Caputo derivative , Cauchy problem , Fractional diffusion , iterated Brownian motion , uniformly elliptic operator

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 3 • May 2009
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