On the integral representation of excessive functions under Bochner subordination

Let P be a semigroup of kernels on a Lusin space E with associated resolvent U, let beta be a Bochner subordinator and let P^beta be the subordinate semigroup of P by means of beta. In this paper we give sufficient conditions to have an integral representation of P^beta-excessive functions in terms of U-exit laws and beta. As application, if P is the semigroup of a transient right Markov process X, we derive a probabilistic representation of P^beta$-excessive functions in terms of additive functionals of X and beta.