Spatial regularity of semigroups generated by L\'evy type operators
Abstract
We apply the probabilistic coupling approach to establish the spatial regularity of semigroups associated with L\'evy type operators, by assuming that the martingale problem of L\'evy type operators is well posed. In particular, we can prove the Lipschitz continuity of the semigroups under H\"older continuity of coefficients, even when the L\'evy kernel corresponding to L\'evy type operators is singular.
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