Lévy-type operators with low singularity kernels: regularity estimates and martingale problem
Abstract
We consider the linear non-local operator $\mathcal{L}$ denoted by \[ \mathcal{L} u (x) = \int_{\mathbb{R}^d} \left(u(x+z)-u(x)\right) a(x,z)J(z)\,d z. \] Here $a(x,z)$ is bounded and $J(z)$ is the jumping kernel of a Lévy 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 $\mathcal{L}$, and establish regularity results for the solutions of such equations in these spaces. Secondly, we investigate the martingale problem associated with $\mathcal{L}$. 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.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2023
- DOI:
- 10.48550/arXiv.2304.14056
- arXiv:
- arXiv:2304.14056
- Bibcode:
- 2023arXiv230414056H
- Keywords:
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- Mathematics - Probability
- E-Print:
- 43 pages