Stochastic solutions of a class of Higher order Cauchy problems in $\rd$

Details Stochastic solutions of a class of Higher order Cauchy problems in $\rd$ Stochastic solutions of a class of Higher order Cauchy problems in $\rd$

2008-09-29

arxiv.org/abs/0809.4824v2

Stochastics and Dynamics, Volume: 10, Issue: 3(2010) , 341-366

Mathematics

Probability

26 pages

Scientific paper

10.1142/S021949371000298X

We study solutions of a class of higher order partial differential equations in bounded domains. These partial differential equations appeared first time in the papers of Allouba and Zheng \cite{allouba1}, Baeumer, Meerschaert and Nane \cite{bmn-07}, Meerschaert, Nane and Vellaisamy \cite{MNV}, and Nane \cite{nane-h}. We express the solutions by subordinating a killed Markov process by a hitting time of a stable subordinator of index $0<\beta <1$, or by the absolute value of a symmetric $\alpha$-stable process with $0<\alpha\leq 2$, independent of the Markov process. In some special cases we represent the solutions by running composition of $k$ independent Brownian motions, called $k$-iterated Brownian motion for an integer $k\geq 2$. We make use of a connection between fractional-time diffusions and higher order partial differential equations established first by Allouba and Zheng \cite{allouba1} and later extended in several directions by Baeumer, Meerschaert and Nane \cite{bmn-07}.

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