Invariant measures related with Poisson driven stochastic differential equation

A Poisson driven stochastic differential equation generates a semigroup of operators (Pt)t[greater-or-equal, slanted]0 describing the evolution of measures along trajectories and a Markov operator P corresponding to the change of measures from a jump to jump. We show that the semigroup (Pt)t[greater-or-equal, slanted]0 has a finite invariant measure if and only if the operator P has the same property. The main result is applied to problems related with the existence and the dimension of invariant measures.