【作者】 王欣；

【导师】 黄春茂；

【作者基本信息】 哈尔滨工业大学 ， 应用数学， 2019， 硕士

【摘要】 随机环境中的分枝随机游动是随机过程研究方向的一个重要分支,受到学者们的广泛关注。自然界中很多问题,如植物繁衍、细胞分裂等,都可以用随机环境中的分枝随机游动来描述。该模型可以用于自然科学中物种演变过程的研究,在生物学、生态学、物理学等诸多科目的发展中起到了十分重要的作用,因此具有深厚的现实背景和巨大的应用价值。在前人的工作基础上,本文研究了依时随机环境中实值分枝随机游动的相关极限性质,主要包括鞅的收敛性、关于个体数目计数测度的大偏差和中偏差。在分枝随机游动的课题研究中,主要兴趣之一是关于个体数目计数测度的极限定理,而该测度的极限性质依赖于相关自然鞅的收敛情况。针对依时随机环境中的实值分枝随机游动,本文详细研究了模型中自然鞅的收敛性。借助于Doob鞅收敛定理和关于鞅的不等式,对p∈(1,2],给出了鞅分别在quenched概率和annealed概率下的逐点L~p收敛条件,并利用Cauchy公式和鞅不等式找到了鞅的一致收敛区域。利用鞅的收敛结论,本文着重考察了关于第n代个体数目计数测度的大偏差和中偏差。其中对大偏差的研究采用了直接计算大偏差上界与下界的技术路线:上界利用G?rtner-Ellis定理给出;下界则借助于多重分形分析,结合鞅的收敛计算水平集的Hausdorff维数所得。对中偏差的研究表现在为相关计数测度建立了中偏差原理,其解决途径是考虑测度的对数矩函数,通过精细计算并利用鞅的一致收敛性得到对数矩函数的极限,然后再应用G?rtner-Ellis定理。对于实值分枝随机游动而言,即使不考虑环境的影响,其许多极限行为尤其是在高维空间中的情况尚不明确。而依时随机环境的加入尽管增强了模型的应用性,但与此同时也大幅度增加了探究过程极限性质的难度。本文的研究丰富和发展了分枝随机游动这一大课题的研究内容,同时也可在对随机环境和高维实数空间的处理方面,为其他相关随机过程的极限理论研究提供参考。更多还原

【Abstract】 Branching random walk with a random environment is an important branch of the research direction of stochastic processes,and has attracted extensive attention from scholars.Many problems in nature,such as plant reproduction,cell division and so on,can be described by branching random walks in random environments.This model can be used in the study of species evolution process in the natural sciences,and plays a very important role in the development of many subjects such as biology,ecology and physics,therefore,it has a profound realistic background and great application value.Based on the work of predecessors,this paper studies the related limit properties of a real valued branching random walk with a random environment in time,including the convergence of martingale,large and moderate deviations associated to counting measures of the number of individuals.In the research of branching random walk,one of the main interests is the limit theorem for the counting measures of the number of individuals,and the limit properties of the measure depends on the convergence of the relevant natural martingale.Aiming at the real valued branching random walk with a random environment in time,the convergence of natural martingale in the model is studied in detail in this paper.By means of Doob’s convergence theorem and the inequality about martingale,for p∈(1,2],the point-by-point L~p convergence conditions of martingale with respect to quenched law and annealed law are given respectively,and the uniform convergence region of martingale is found by Cauchy’s formula and the inequality for martingale.Using the convergence conclusion of martingale,this paper focuses on the large and moderate deviations of the counting measure of the number of individuals of generation n.Among them,the study of large deviations uses the technical route of directly calculating the upper and lower bounds of large deviations: the upper bounds are given by the G?rtner-Ellis theorem;the lower bounds are obtained by means of multifractal analysis combined with the convergence of martingale to calculate the Hausdorff dimension of the level sets.The study of the moderate deviations shows establishing a moderate deviation principle for the related counting measures.The solution is to consider the logarithmic moment functions of the corresponding measures,and obtain their limits by finely calculating and using the uniform convergence of martingale,and then apply the G?rtner-Ellis theorem.For a real valued branching random walk,even if the environmental impact is not considered,many of its limit behaviors,especially in high-dimensional space,are not clear.The addition of a random environment in time,while enhancing the applicability of the model,greatly increases the difficulty of exploring the limit nature of the process at the same time.The research in this paper enriches and develops the research content of branching random walks,and also provides reference for the limit theory research of other related stochastic processes in the treatment of random environment and high-dimensional real space.更多还原