Symposium on Probability models with random environments随机环境概率模型学术研讨会

Symposium on Probability models with random environments

随机环境概率模型学术研讨会

Time  Schedule

Titles and Abstracts

Favorite points of randomly biased walks on trees 

 Yueyun HU

Département de Mathématiques (LAGA CNRS-UMR 7539)

Université Paris 13, France

yueyun@math.univ-paris13.fr

Considering a class of recurrent, randomly biased walks on trees, we present some recent results on the problem of favorite points in both slow-movement case and sub-diffusive case. This talk is based on joint works with Dayue Chen (Peking University), Loic de Raphelis (Lyon 1), and with Zhan Shi (Paris 6).

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 Limit theorems for branching processes in random environments

Quansheng LIU

Univ. Bretagne -Sud,  France

quansheng.liu@univ-ubs.fr

 I will present some current main research interests and some recent progress on branching processes in random environments. Particular attention will be paid on large deviation results.

Directed polymer in Random Environment with Correlation

      Guanglin RANG

                 College of Mathematics and Statistics, Wuhan University

                      glrang.math@whu.edu.cn

We consider the limit behavior of partition function of directed polymers in random
environment represented by linear model instead of a family of i.i.d.variables in 1 + 1
dimensions. Under the assumption that the correlation decays algebraically, using the method developed in [Ann. Probab., 42(3):1212-1256, 2014], under a new scaling we show the scaled partition function as a process defined on [0,1] × R, converges weakly to the solution to some stochastic heat equations driven by fractional Brownian field. The Hurst parameter is determined by the correlation exponent of the random environment. Here multiple Itô integral with respect to fractional Gaussian field and spectral representation of stationary process are heavily involved

Some New Advances in Random Growth Models

Zhonggen SU

School of Mathematical Sciences, Zhejiang University

suzhonggen@zju.edu.cn

随机增长过程用于描述一类随机时间(t)推移而在空间位置(x)上发生变化(增长)的随机现象.随机性源于以下几方面： (1) 初始状态随机； (2)生存环境随机； (3)变化方式随机；(4)研究角度随机---赋予概率空间. 刻画随机增长过程的最基本数量， 包括：大小、长度、高度等,它们是随机变量，主要参数为时间t和位置x. {最基本问题}: 揭示随机增长过程的增长方式(规律)，如给定某时刻t某位置x,其大小(长度或高度)的概率分布是什么？当t趋于0或无穷时，极限规律 (极限分布)是什么？不同时刻不同位置上，其增长方式如何相互影响？随机增长过程形式多样，应用广泛，已成为概率论、数理统计、统计物理、化学、生物等学科中描述随机现象的基本过程.  经典例子：随机游动(Random Walks)；Poisson过程； Galton-Watson分支过程；无穷交互粒子系统(Infinite Interacting Particle Systems)。 现代例子：随机环境下随机游动(Random Walks in Random Environment)； 渗流模型(Last Passage Percolation Time)；最长增加增列长度(Length of Longest Increasing Subsequences)；角落增长模型(Corner Growth Models)； 有向聚合模型(Directed Polymer Models)； Dyson布朗运动(Dyson Brownian Motion)； 随机粘性落体(Random Ballistic Depositions)； Kadrdar-Parisi-Zhang 增长界面 (KPZ Growth Interfaces).

本报告将简要介绍随机增长过程的研究在过去20年里所取得的进展，特别和Tracy-Widom分布及KPZ普适性类有关有关的主要结果，也介绍一些有趣的问题。

A continuous-state nonlinear branching process

Xiaowen Zhou

Concordia University, Canada

xiaowen.zhou@concordia.ca

A general continuous-state branching process can be  identified as the unique solution to a SDE driven by a Brownian motion and a compensated Poisson random measure; see Bertoin and Le Gall (2006) and Dawson and Li (2012). Adapting this SDE, we can interpret solution to the modified SDE as a continuous-state branching process with branching rates depending on the current population size. Using a martingale approach, we study its survival/extinction behaviors and find respective sufficient conditions on the branching parameters under which the process either survives with probability one or dies out with a positive probability. Similarly, we can also discuss the explosion behaviors for the nonlinear continuous-state branching process. We will show that those conditions are quite sharp.

（This is a collaborative work with Peisen Li and Xu Yang）

References:

[1] J. Bertoin and J.-F. Le Gall (2006): Stochastic flows associated to coalescent processes III: Limit theorems.  Illinois J. Math., 50, 147--181.

[2] D. Dawson and Z. Li (2012): Stochastic equations, flows and measure-valued processes.  Ann. Probab., 40,  813--857.