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

Understand

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 $k\geq 2$.