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      Reflected stable subordinators for fractional Cauchy problems

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          Abstract

          In a fractional Cauchy problem, the first time derivative is replaced by a Caputo fractional derivative of order less than one. If the original Cauchy problem governs a Markov process, a non-Markovian time change yields a stochastic solution to the fractional Cauchy problem, using the first passage time of a stable subordinator. This paper proves that a spectrally negative stable process reflected at its infimum has the same one dimensional distributions as the inverse stable subordinator. Therefore, this Markov process can also be used as a time change, to produce stochastic solutions to fractional Cauchy problems. The proof uses an extension of the D. Andr\'e reflection principle. The forward equation of the reflected stable process is established, including the appropriate fractional boundary condition, and its transition densities are explicitly computed.

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          Finite difference approximations for fractional advection–dispersion flow equations

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            Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints

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              Boundary value problems for fractional diffusion equations

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                Author and article information

                Journal
                2013-01-23
                Article
                1301.5605
                86137bb2-91fe-4b78-81d0-9af142d6aa69

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
                Custom metadata
                60G52, 60J99
                Trans. Amer. Math. Soc., 368(1) (2016), 227-248
                math.PR math.AP

                Analysis,Probability
                Analysis, Probability

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