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 H¨older 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 result on the Lipschitz continuity of an auxiliary map. The proofs rely on basic properties of Brownian motion.
Brownian motion and exposed solutions of differential inclusions
COLOMBO, GIOVANNI;
2013
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 H¨older 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 result on the Lipschitz continuity of an auxiliary map. The proofs rely on basic properties of Brownian motion.Pubblicazioni consigliate
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