报告人：李利平 青年副研究员 复旦大学数学科学学院
报告时间：2024年10月29日 15：30-16：30
报告地点：红瓦楼723室
报告内容简介：The birth-death process is a special type of continuous-time Markov chain with index set $\mathbb{N}$. Its resolvent matrix can be fully characterized by a set of parameters $(\gamma, \beta, \nu)$, where $\gamma$ and $\beta$ are non-negative constants, and $\nu$ is a positive measure on $\mathbb{N}$. By employing the Ray-Knight compactification, the birth-death process can be realized as a c\`adl\`ag process with strong Markov property on the one-point compactification space $\overline{\mathbb{N}}_{\partial}$, which includes an additional cemetery point $\partial$. In a certain sense, the three parameters that determine the birth-death process correspond to its killing, reflecting, and jumping behaviors at $\infty$ used for the one-point compactification, respectively.
In general, providing a clear description of the trajectories of a birth-death process, especially in the pathological case where $|\nu|=\infty$, is challenging. This talk aims to address this issue by studying the birth-death process using approximation methods. Specifically, we will approximate the birth-death process with simpler birth-death processes that are easier to comprehend. For two typical approximation methods, our main results establish the weak convergence of a sequence of probability measures, which are induced by the approximating processes, on the space of all c\`adl\`ag functions. This type of convergence is significantly stronger than the convergence of transition matrices typically considered in the theory of continuous-time Markov chains.
报告人简介：2015年在复旦大学获博士学位，导师是应坚刚教授。博士毕业后进入中国科学院数学与系统科学研究院进行博士后研究，合作导师是马志明院士。2017年7月之后留院工作。2021年7月至2023年7月作为洪堡学者访问德国比勒菲尔德大学。2022年4月转入上海复旦大学工作。主要研究领域是概率论与随机分析，尤其是马氏过程和狄氏型理论的研究。在Annals of Probability，Transactions of the AMS，Annals of Applied Probability, Annales de l’Institut Henri Poincaré，Stochastic processes and their applications等国内外学术期刊上已发表学术论文近三十篇。曾主持2015年度率先行动联合资助优秀博士后项目，中国博士后科学基金第59批面上项目, 国家自然科学基金青年基金项目, 国家自然科学基金面上项目, 参与国家自然科学基金重点项目“非局部狄氏型和随机偏微分方程若干问题研究”（260万）等。
报告邀请人：何萍