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

Abstract

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 $0<\alpha \le 2$, 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 $k\ge 2$. 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}.

26 pages

References in corpus (10)

• Fractional Cauchy problems on bounded domains
• Fractional diffusion equations and processes with randomly varying time
• Brownian subordinators and fractional Cauchy problems
• Correlated continuous time random walks
• Iterated Brownian motion in an open set
• Brownian-Time Processes: The PDE Connection II and the Corresponding Feynman-Kac Formula
• Higher order PDE's and iterated Processes
• Iterated Brownian Motion in Parabola-Shaped Domains
• Iterated Brownian motion in bounded domains in R^n
• Lifetime asymptotics of iterated Brownian motion in R^{n}

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