Higher order PDE’s and iterated processes

Abstract

We introduce a class of stochastic processes based on symmetric α \alpha -stable processes, for α ∈ ( 0 , 2 ] \alpha \in (0,2] . These are obtained by taking Markov processes and replacing the time parameter with the modulus of a symmetric α \alpha -stable process. We call them α \alpha -time processes. They generalize Brownian time processes studied in Allouba and Zheng (2001), Allouba (2002), (2003), and they introduce new interesting examples. We establish the connection of α \alpha -time processes to some higher order PDE’s for α \alpha rational. We also obtain the PDE connection of subordinate killed Brownian motion in bounded domains of regular boundary.