Spatial regularity of semigroups generated by L\'{e}vy type operators

TL;DR

This paper uses probabilistic coupling to prove spatial regularity of semigroups generated by Lévy type operators, demonstrating Lipschitz continuity under Hölder continuous coefficients even with singular Lévy kernels.

Contribution

It introduces a novel approach applying probabilistic coupling to establish regularity of Lévy type semigroups under minimal kernel assumptions.

Findings

Proves Lipschitz continuity of semigroups with Hölder continuous coefficients.

Handles singular Lévy kernels in the regularity analysis.

Establishes spatial regularity assuming well-posedness of the martingale problem.

Abstract

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