Higher order PDE's and iterated processes

We introduce a class of stochastic processes based on symmetric $\alpha$-stable processes, for $\alpha \in (0,2]$. These are obtained by taking Markov processes and replacing the time parameter with the modulus of a symmetric $\alpha$-stable process. We call them $\alpha$-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 $\alpha$-time processes to some higher order PDE's for $\alpha$ rational. We also obtain the PDE connection of subordinate killed Brownian motion in bounded domains of regular boundary.