分支随机游动及其相关问题的研究

The main goals of this project are to study various generalizations of the classical branching random walks and to further explore some of the known results on branching random walks. The target questions in this project are based on the recent exciting developments in the areas of or related to branching random walks, and they can be divided into the following four directions. 1. Recently some interesting phenomena, somewhat contradicting the classical intuition, were discovered in several types of time inhomogeneous branching random walks. However, the behavior of the more general time inhomogeneous branching random walks remains unknown, which is subject to further study in this project. Also, since reseachers have studied the time inhomogeneous and space inhomogeneous branching random walks, respectively, this project will also study the behavior of the time-space inhomogeneous branching random walks as a natural extension of those two types of models. 2. The applicant wants to discuss some properties of branching random walks when the movements of the particles are locally dependent within a finite time range. 3. The leftmost particle of some special type of time inhomogeneous branching random walks has a very interesting correction term to the linear term. The result is similar to the leftmost path in branching random walks. But, in both problems, the higher order correction terms are elusive. The applicant wants to find finer asymptotics through finer probability estimates. 4. The technique and intuition on branching random walks are used to prove results in other models. The applicant wants to explore more connection between branching random walks and other models and to study those models, which will either benefit from or benefit the study of branching random walks.

本项目主要研究经典的分支随机游动的一些推广和一些现有结果的更深入的探索。其中拟解决的问题都是基于最近几年来申请者和其他研究者在分支随机游动及其相关模型上取得的一些新的研究进展，具体分为以下四个方向： 1．基于已知的几类特殊的时间非齐次环境下的分支随机游动的结果，探索更为一般的时间非齐次环境下的分支随机游动的性质；基于现有的时间非齐次环境和空间非齐次环境下的分支随机游动的研究，探索时空（同时）非齐次的环境下的分支随机游动的性质； 2．在经典的分支随机游动的模型中，引入一些局部的相依性，探索在失去一部分独立性后的分支随机游动的性质； 3．在现有的关于分支随机游动的最右路径和一类时间非齐次环境下的分支随机游动的结果上，继续探索其渐进性质中高阶项的大小； 4．探索与分支随机游动密切相关的一些随机模型的性质。