The Limiting Behavior And Large Deviation Of The Right-most Particle In The Branching Process

There is an important branch in the study of probability theory-the branching process,which has been a research hotspot for a long time and is currently fruitful.The branching process describes a random evolution of particle populations in a closed or open system.In theory,the branching process is deeply related to many important processes in the stochastic process.Because the Brownian motion has very good properties,the branching Brownian motion is widely studied,such as the expectation of the number of particles,asymptotic behavior of the rightmost particle and large deviation of the rightmost particle position,etc.Poisson process,Brownian motion,?-stable process(0<??2)are all basic Levy processes.When ?-2,the ?-stable process is Brownian motion.The main research content of this article is the branching symmetric ?-stable process.The first chapter of this paper mainly introduces the research background and significance of the branching process,the current status of the research,and the content and innovation of this research.The second chapter is some basic concepts related to stochastic processes that will be used in the article,as well as some basic theorems that will be used when proving in the article.The first section of Chapter 3 first introduces the model of the 2-branching Brownian motion;the second section gives the large deviations for the rightmost position in a k branching Brownian motion.The fourth chapter is the key part of this paper.The first section gives the model of the branching symmetric ?-stable process,Then use spine decomposition and Girsanov measure transformation as tools,with the help of many-to-one lemma to transform the relevant proof of the particle trajectory into ?-stable process into simple problems that we can solve;the second,third,and fourth sections are the proof process and conclusion of the particle number expectation,the asymptotic behavior of the particle number,and the rightmost particle position speed;the last section is about summary and prospect of the research on branching symmetric ?-stable process.Chapter 5 studies the numerical solutions of stochastic It(?)-Volterra integral equations related to Brownian motion.