Invariant measure for diffusions with jumps

Our purpose is to study an ergodic linear equation associated to diffusion processes with jumps in the whole space. This integro-differential equation plays a fundamental role in ergodic control problems of second order Marko v processes. The key result is to prove the existence and uniqueness of an invariant density function for a jump diffusion, whose lower order coeffici ents are only Borel measurable. Based on this invariant probability, existe nce and uniqueness (up to an additive constant) of solutions to the ergodic linear equation are established.