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

报告人： 赵国焕 研究员 中科院数学与系统科学研究院

报告时间： 9月11日(周一) 10:00-11:00

报告地点：管理楼1418

摘要：

We consider the linear non-local operator $\mathcal{L}$ denoted by \[\mathcal{L} u (x) = \int_{\mathbb{R}^d} \left(u(x+z)-u(x)\right) a(x,z)J(z)\,\mathrm{d} z.\] 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 $\mathcal{L}$, 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. This is based on joint work with Eryan Hu (Tianjin University).