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Large Deviation Principle for Empirical Measures of Once-reinforced Random Walks on Finite Graphs

发布时间:2022-11-09 浏览量:54

时   间:  2022-11-09 14:00 — 15:00

地   点:  腾讯会议 APP()
报告人:  刘勇 教授
单   位:  北京大学
邀请人:  张登
备   注:  腾讯会议:577-755-864,会议密码:123456。报告人介绍:刘勇教授,1999年在北京大学数学学院获得博士学位,在中国科学与数学与系统科学研究院作博士后,目前是北京大学数学学院教授。主要研究兴趣是大偏差理论,随机分析和随机偏微分方程。
报告摘要:  

The once-reinforced random walk (ORRW) is a kind of non-Markov process with the transition probability only depending on the current weights of all edges. The weights are set to be 1 initially. At the first time an edge is traversed, its weight is changed to a positive parameter δ at once, and it will remain in δ.  We introduce a log-transforms of exponential moments of restricted empirical measure functionals, and prove a variational formula for the limit of the functionals through a variational representation given by a novel dynamic programming equation associated with these functionals. As a corollary, we deduce the large deviation principle for the empirical measure of the ORRW. Its rate function is decreasing in δ, and is not differentiable at δ=1. Moreover, we characterize the critical exponent for the exponential integrability of a class of stopping times including the cover time and the hitting time. For the critical exponent, we show that it is continuous and strictly decreasing in δ, and describe a relationship between its limit (as δ→0) and the structure of the graph. This is a joint work with Dr. Xiangyu Huang and Professor Kainan Xiang.