概率统计系列讲座六：
报告题目：【Potential Theory of Markov Processes with Jump Kernels Decaying at the Boundary】
时间：2020年9月23日 (星期三)上午 09:00
地点：腾讯会议（会议ID： 511 3889 6273）
主讲：美国伊利诺伊大学教授，宋仁明
主办：数学与信息学院
参加对象：统计系老师与学生
报告摘要：In this talk I will present some recent results on the potential theory of Markov processes with jump kernels decaying at the boundary of its state space $\mathbb R^d_+$, the upper half space of $\mathbb R^d$. The jump kernel is of the form $J^D(x, y)=|x-y|^{-d-\alpha}B(x, y)$, where $\alpha\in (0, 2)$ and $B(x, y)$, which involves three parameters $\beta_1, \beta_2$ and $\beta_2$, tends to 0 when $x$ or $y$ tends to the boundary. We assume that the killing function is of the form $\kappa(x)=cx_d^{-\alpha}$. Our main result is a boundary Harnack principle which says that, for any $p>(\alpha-1)_+$, there are values of $\beta_1, \beta_2, \beta_3$ and the constant $c$ such that nonnegative harmonic functions of the process must decay at the rate $x^p_d$ if they vanish near a certain portion of the boundary. We also have examples showing that there are values of $\beta_1, \beta_2, \beta_3$ for which the boundary Harnack principle fails.
This talk is based on a joint paper with Panki Kim and Zoran Vondracek.
