On Generators of Subordinate Semigroups

Abstract

Let $X$ be a standard Markov process with semigroup $(P_t)$. We show how to compute the infinitesimal generators (weak and strong) of the semigroup $Q_tf(x) = E^x\{m_tf(X_t)\}$ with $m_t = \exp(-\tau_t)$ and $\tau_t$ a right continuous, increasing strong additive functional; the computation is in terms of the infinitesimal operators of $(P_t)$ and the Levy system of the joint process $(X, \tau)$.