Large And Moderate Deviations For A Real Valued Branching Random Walk With A Random Environment In Time

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.