Brownian motion and exposed solutions of differential inclusions

Abstract

We present a research program designed by A. Bressan and some partial results related to it. First, we construct a probability measure supported on the space of solutions to a planar differential inclusion, where the right hand side is a Lipschitz continuous segment. Such measure assigns probability one to solutions having derivatives a.e. equal to one of the endpoints of the segment. Second, for a class of planar differential inclusions with Hölder continuous right hand side F, we prove existence of solutions whose derivatives are exposed points of F. Finally, we complete the research program if the right hand side of the differential inclusion does not depend on the state and prove a related result on the Lipschitz continuity of an auxiliary map. The proofs rely on basic properties of the scalar Brownian motion.