Invariant measures of Lévy-type operators and their associated Markov processes

Abstract

A distributional equation as a criterion for invariant measures of Markov processes associated to Lévy-type operators is established. This is obtained via a characterization of infinitesimally invariant measures of the associated generators. Particular focus is put on the one-dimensional case where the distributional equation becomes a Volterra-Fredholm integral equation, and on solutions to Lévy-driven stochastic differential equations. The results are accompanied by various illustrative examples.