Reflected stable subordinators for fractional Cauchy problems

Boris Baeumer, Mihály Kovács, Mark M. Meerschaert, René L. Schilling, Peter Straka

Published 2013-01-23Version 1

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.