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.