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

TL;DR

This paper investigates solutions to higher order PDEs in bounded domains, expressing them via subordinated Markov processes, iterated Brownian motions, and fractional-time diffusions, extending previous theoretical frameworks.

Contribution

It introduces stochastic representations of higher order PDE solutions using subordinated processes and iterated Brownian motions, expanding the understanding of fractional and higher order PDEs.

Findings

Solutions expressed via subordinated Markov processes

Representation using iterated Brownian motions

Connection established between fractional diffusions and higher order PDEs

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 $0<\alpha\leq 2$, 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}.