Spatial regularity of semigroups generated by Lévy type operators

Abstract

We apply the probabilistic coupling approach to establish the spatial regularity of semigroups associated with Lévy type operators, by assuming that the martingale problem of Lévy type operators is well posed. In particular, we can prove the Lipschitz continuity of the semigroups under Hölder continuity of coefficients, even when the Lévy kernel corresponding to Lévy type operators is singular.

18 pages