Growth rate of the scalar product for a supercritical branching process in a random environment and for products of positive random matrices
科研大讨论系列报告
报告题目(Title)：Growth rate of the scalar product for a supercritical branching process in a random environment and for products of positive random matrices
报告人(Speaker)：Quansheng Liu （刘全升）Universite Bretagne Sud (University of South Brittany), France
地点(Place)：后主楼 1220
时间(Time)： 2023年12月21日（周四）下午2:00-3:00
邀请人(Inviter)：陈昕昕
报告摘要
Let $Z_n =(Z_n (1), \cdots, Z_n (d))$ be a supercritical d-type branching process in an independent and identically distributed random environment $\xi= (\xi_0, \xi_1, \cdots )$, starting with $Z_0=x$. Let $M_n$ be the mean matrix of the offspring distribution at time $n$: its $(i,j)$-th entry $M_n(i,j)$ is the conditional expectation of the number of type $j$ particles produced by a type $i$ particle of generation $n$ (so that $M_n$ depends only on $\xi_n$). We establish a Kesten-Stigum type theorem for the scalar product $\langle Z_n, y \rangle $ for any non-negative vector $y $: we prove that under suitable conditions, $W_n^x (y):= \langle Z_n, y \rangle / \langle x M_0 \cdots M_{n-1}, y \rangle $ converges in probability to some r.v. $W^x$ (which does not depend on $y$), and we give a criterion for $W^x$ to be non-degenerate. For the proof, we introduce a martingale which has the same limit as $W_n(y)$, and we establish a Perron-Frobenius type theorem for the products of positive random matrices: we define some positives vectors $u_n,v_n >0$ and positive scalars $\lambda_n, a_n >0$ depending only on the environment sequence $\xi$, such that $(W_n^x (u_n))$ is a martingale which converges a.s. to $W^x$, that $M_n u_{n+1}^T= \lambda_n u_n^T$ for all $n$, and that uniformly in $x$ and $y$, $\langle x M_0 \cdots M_{n}, y \rangle \sim a_n \langle u_0, x \rangle \langle v_{n}, y \rangle $. (Based on a joint work with Ion Grama and Thi Trang Nguyen)
主讲人简介
刘全升，法国特级教授,就职于南布列塔尼大学,享受法国优秀科研津贴(PES/PEDR)。1980-1989于武汉大学数学系学习，连续经历本科、硕士和博士研究生阶段。1989年11月进入巴黎第六大学深造，1993年获得该校概率论专业博士文凭。1993至2000年任法国雷恩大学讲师、副教授。2000年9月起任法国南布列塔尼大学教授。2007至2013及2017至2022，任南布列塔尼大学数学系主任；2012至2013及2017至2022任大西洋布列塔尼数学实验室副主任。多年任南布列塔尼大学行政议会成员，科技议会成员和专家评委，国际交流专员，数学与应用数学专业研究生工作负责人。多次任中国教育部长江学者通讯评审专家,新世纪数学奖通讯评审专家，法国国家基金委评审专家，和波兰国家研究中心基金委评审专家。研究课题涉及概率统计，分形几何和数字图像处理。近年主要研究随机环境中的概率统计问题，尤其是关于大偏差理论、随机矩阵乘积，几类重要的随机环境的数学物理和应用概率模型，例如分枝过程、分枝随机游动和图像去噪。在《J. Eur. Math. Soc.》、《Probab. Th. Rel. Fields》、《Annals of Probability》、《Annals of Applied Probability》、《IEEE Trans. Image Processing》、《SIAM J. Imaging Sciences》、《Inverse Problems and Imaging》、《J. Scientific Computing》等期刊上发表100余篇论文。
