Speaker：Guohuan Zhao (Academy of Mathematics Systems Science, CAS)

Time：2023-5-11, 15:00

Location：Tencent Meeting ID：402-263-988（No Password）

Abstract:

The main focus of this talk is on the linear non-local operator $L$ defined by \begin{align*} L u (x) = \int_{\mathbb{R}^d} (u(x+z)-u(x)) a(x,z)J(z)~d z.\end{align*} Here $J$ is the jumping kernel of a L\'evy process, which exhibits only a low-order singularity near the origin and does not permit standard scaling. To analyze elliptic equations associated with $L$, I will introduce generalized Orlicz-Besov spaces that are specifically tailored for this purpose. Moreover, I will establish certain regularity properties of the solutions to such equations in these spaces. Additionally, I intend to introduce the martingale problem associated with $L$. By exploiting analytic results, we demonstrate the well-posedness of the martingale problem under mild conditions, and establish a new Krylov-type estimate for the corresponding Markov processes. This is based on joint work with Eryan Hu from Tianjin University.