Subordination in the sense of Bochner of L p -semigroups and associated Markov processes

We show that the subordination induced by a convolution semigroup (subordination in the sense of Bochner) of a C ₀-semigroup of sub-Markovian operators on an L ^p space is actually associated to the subordination of a right (Markov) process. As a consequence, we solve the martingale problem associate with the L ^p -infinitesimal generator of the subordinate semigroup. We also prove quasi continuity properties for the elements of the domain of the L ^p -generator of the subordinate semigroup. It turns out that an enlargement of the base space is necessary. A main step in the proof is the preservation under such a subordination of the property of a Markov process to be a Borel right process. We use several analytic and probabilistic potential theoretical tools.