Higher order PDE's and iterated Processes

Abstract

We introduce a class of stochastic processes based on symmetric $α$-stable processes. These are obtained by taking Markov processes and replacing the time parameter with the modulus of a symmetric $α$-stable process. We call them $α$-time processes. They generalize Brownian time processes studied in \cite{allouba1, allouba2, allouba3}, and they introduce new interesting examples. We establish the connection of $α-$time processes to some higher order PDE's for $α$ rational. We also study the exit problem for $α$-time processes as they exit regular domains and connect them to elliptic PDE's. We also obtain the PDE connection of subordinate killed Brownian motion in bounded domains of regular boundary.