On stable Markov processes

Necessary conditions are given for a symmetric [alpha]-stable (S[alpha]S) process, 1 < [alpha] < 2, to be Markov. These conditions are then applied to find Markov or weakly Markov processes within certain important classes of S[alpha]S processes: time changed Lévy motion, scale mixed Gaussian processes, moving averages and harmonizable processes. Two stationary S[alpha]S Markov processes are introduced, the right and the left S[alpha]S Ornstein-Uhlenbeck processes. Some of the results are in sharp contrast to the Gaussian case [alpha] = 2.