L\'evy-type operators with low singularity kernels: regularity estimates and martingale problem

arXiv2304.14056MaRDI QIDQ6509842

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Abstract: We consider the linear non-local operator $cL$ denoted by [ mathcal{L} u (x) = int_{mathbb{R}^d} left(u(x+z)-u(x) ight) a(x,z)J(z),d z. ] Here $a(x,z)$ is bounded and $J(z)$ is the jumping kernel of a L'evy process, which only has a low-order singularity near the origin and does not allow for standard scaling. The aim of this work is twofold. Firstly, we introduce generalized Orlicz-Besov spaces tailored to accommodate the analysis of elliptic equations associated with $mathcalL$, and establish regularity results for the solutions of such equations in these spaces. Secondly, we investigate the martingale problem associated with $mathcalL$. By utilizing analytic results, we prove the well-posedness of the martingale problem under mild conditions. Additionally, we obtain a new Krylov-type estimate for the martingale solution through the use of a Morrey-type inequality for generalized Orlicz-Besov spaces.

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