A note on invariant measures for Filippov systems

Abstract

We are interested in Filippov systems which preserve a probability measure on a compact manifold. Using the formalism coming from the theory of differential inclusions, we define a measure to be invariant for a Filippov system as the natural analogous definition of invariant measure for flows. Our first main result states that if a differential inclusion admits an invariant probability measure, this measure does not see the trajectories where there is a break of uniqueness. Our second main result provides a necessary and sufficient condition in order to exist an invariant probability measure preserved by a Filippov system. Our third main result concerns Filippov systems which preserve a probability measure equivalent to the volume measure. As a corollary the volume preserving Filippov systems are the refractive ones. Then, in light of our previous results, we analyze the existence of invariant measures for many examples.