随机环境的概率模型研究

Probability models with random environments, due to their theoretical significance and wide applications, have attracted the attention of more and more researchers in recent years. These models are among the most important topics and fascinating hot spots in the international probability research. The present project aims to merge the modern researches in several important probability areas to investigate and analyze important probability models with random environments arising from mathematics and physics, which include: Markov chains and branching processes in random environments, branching random walks and Mandelbrot’s multiplicative cascades, random walks in random scenery or with random conductance, directed polymers in random environment, and infinite interacting particle systems. Although these models are of different origins, they all have a wide range of applications in physics, biology and computer science. Their common feature is the strong dependence among the involved random variables. This dependence leads to many new, interesting and significant but challenging problems. The goal of the project is, by combining our strengths and by applying and further developing the existing probability techiques and results, especially those on large deviations theory, Gaussian approximation, random matrices and stochastic differential equations, and via a careful study of concrete problems, to develop novel methods, and to gain deep understanding of the models and their asymptotic behaviors, which allows us to catch the essence that the models describe. In the meanwhile we expect that the new methods and results to be developed will promote and enrich significantly the probability theory.

随机环境的概率模型， 因其重要的理论意义和广泛的应用背景，近年来引起了越来越多的学者的关注，是国际概率界十分重要的研究热点之一。本项目旨在融合概率论几个重要领域的前沿研究探索分析随机环境的重要数学物理模型，包括: 随机环境中的马氏链和分枝过程，分枝随机游动和Mandelbrot 乘积瀑布，随机风景和随机电导中的随机游动，随机环境中有向聚合模型和无穷交互作用粒子系统。这些模型的起源多种多样，但都在物理、生物和计算机科学等有广泛的应用。其共同特点是有关随机变量间的强依赖性，由此带来了许多新的有意义而富有挑战的问题。项目目的在于联合优势，借鉴和发展已有的概率工具和结果，尤其是大偏差理论、高斯逼近、随机矩阵和随机(偏)微分方程的新方法和新成果，通过对一系列具体问题的研究, 开拓新的研究技巧，深刻理解相关模型的本质性质，透过概率模型认识其描述的自然界本质规律，同时发展和丰富概率论的内容。

项目围绕随机环境中的聚合模型和点过程、交互作用粒子系统、马氏链与分枝过程、分枝随机游动和Mandelbrot乘积瀑布以及在非可加概率下的随机序列的等重要概率模型开展研究，取得了系列成果。项目对Poisson点过程最长增加子列、多元点过程的随机积分、β系综的线性统计量、Airy过程及与之相关的单调函数非参数估计、随机矩阵乘积、具有长程相关性随机环境下有向聚合模型等随机聚合模型，建立了中心极限定理与弱收敛及其收敛速度、大偏差、中篇差、Cramer中篇差、集中不等式等精细的极限性质；项目对几类重要的交互作用粒子系统给出大尺度极限行为、平均场极限行的精细刻画，包括为大偏差、中偏差、极限解等；项目发现了随机环境中多型分枝过程的基本鞅及其性质，解决了50年以来的一个困难问题，为建立随机环境中多型分枝过程的基本极限定理开辟了道路，项目给出随机环境下带随机迁移的上临界分枝过程的Berry-Essen界、上临界猝灭分枝过程加权矩存在的判别法则以及一般分枝机制的连续状态非线性分枝过程的灭绝和爆炸的判别准则以及爆炸速度等的重要性质的精确刻画；项目发现了Mandelbrot瀑布的自然鞅之极限随机变量的加权矩存在的充要条件, 建立了自正则化鞅与平稳序列的Cramér型中偏差展开式，给出了随机环境中的分枝随机游动矩收敛性及极限矩存在的判别条件；项目建立了反应模型不确定性的次线性期望框架下独立和鞅差型变量最大部分和的指数不等式的上下界、Borel-Cantelli引理等研究工具，获得了Lindeberg中心极限定理、Donsker不变原理、重对数律的充分必要条件等普适性定理。项目发表和录用论文84篇，一些发表在《中国科学》、《数学学报》、《应用数学学报》、《Ann. Probab.》、《Ann. Applied. Probab.》、《Stoch. Process. Appl.》、《Bernoulli》、《Ann. Statist.》等有影响的杂志上，研究成果已引起了学者们的关注。