Schauder estimates for degenerate L\'evy Ornstein-Ulhenbeck operators

TL;DR

This paper proves Schauder estimates for a class of degenerate Le9vy Ornstein-Uhlenbeck operators in both elliptic and parabolic cases, enabling well-posedness results for related integro-partial differential equations.

Contribution

It introduces a novel approach to establish Schauder estimates for degenerate Le9vy Ornstein-Uhlenbeck operators without symmetry or dilation invariance assumptions.

Findings

Established global Schauder estimates in anisotropic Hf6lder spaces.

Proved well-posedness of the associated IPDEs.

Extended results to non-linear, space-time dependent drifts.

Abstract

We establish global Schauder estimates for integro-partial differential equations (IPDE) driven by a possibly degenerate L\'evy Ornstein-Uhlenbeck operator, both in the elliptic and parabolic setting, using some suitable anisotropic H\"older spaces. The class of operators we consider is composed by a linear drift plus a L\'evy operator that is comparable, in a suitable sense, with a possibly truncated stable operator. It includes for example, the relativistic, the tempered, the layered or the Lamperti stable operators. Our method does not assume neither the symmetry of the L\'evy operator nor the invariance for dilations of the linear part of the operator. Thanks to our estimates, we prove in addition the well-posedness of the considered IPDE in suitable functional spaces. In the final section, we extend some of these results to more general operators involving non-linear, space-time dependent drifts.