Boundary trace of reflected diffusions on uniform domains

Time

Fri., 16:00-17:00

Oct. 18, 2024

Venue

Lecture Hall C548

Shuangqing Complex Building A

清华大学双清综合楼A座C548报告厅

Online

Zoom Meeting ID: 271 534 5558

Passcode: YMSC

Speaker

Zhen-Qing Chen 陈振庆

University of Washington

陈振庆, 美国华盛顿大学(西雅图)数学系终身教授。他于1992年在位于圣路易斯的华盛顿大学获得博士学位。主要研究领域为概率论及其应用，具体方向包含随机分析、马氏过程、狄氏型理论、位势理论、扩散和反常亚扩散过程、随机环境下的随机过程、随机最优控制、分析和偏微分方程中的概率方法及其应用等。在包括《欧洲数学学会杂志》(Journal of the European Mathematical Society)、《概率年刊》(Annals of Probability)、《概率论及其相关领域》(Probability Theoryand Related Fields)等学术期刊上发表论文200余篇，著有两本专著。

陈振庆教授分别于2007年和2014年当选为国际数理统计学会会士和美国数学学会会士。2019年获伯努利国际概率统计学会的伊藤奖。自2015年起任国际数学综合性期刊《位势理论》的主编, 是《美国数学学会通讯》(Proceedings of the American Mathematical Society)应用数学和概率统计方向的协调编辑(Coordinating Editor) 及多个国际数学和概率期刊的编委。

About the lecture 

Abstract

Starting with a transient irreducible diffusion process $X^0$ on a locally compact separable metric space $(D, d)$, one can construct a canonical symmetric reflected diffusion process $\bar X$ on a completion $D^*$ of $(D, d)$ through the theory of reflected Dirichlet spaces.

The boundary trace process $\check X$ of $X$ on the boundary $\partial D:=D^*\setminus D$ is the reflected diffusion process $\bar X$ time-changed by a smooth measure $\nu$ having full quasi-support on $\partial D$. The Dirichlet form of the trace process $\check X$ is called the trace Dirichlet form.

In the first part of the paper, we give a Besov space type characterization of the domain of the trace Dirichlet form for any good smooth measure $\nu$ on the boundary $\partial D$. In the second part of this paper, we study properties of the harmonic measure of $\bar X$ on the boundary $\partial D$. In particular, we provide a condition equivalent to the doubling property of the harmonic measure. Finally, we characterize and provide estimates of the jump kernel of the trace Dirichlet form under the doubling condition of the harmonic measure on $\partial D$.

This is based on a joint work with Shiping Cao.