Higher order PDE's and iterated Processes
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.
math.PR↗