广义分枝过程与带跳随机积分方程

Branching processes are models for the evolution of populations of particles. Those processes constitute an important subclass of Markov processes.In most realistic situations, however, this property is unlikely to be appropriate. This consideration has motivated the study of generalized branching processes, which may not satisfy branching property. Such a process is constructed as the strong solution of a stochastic equation driven by a Gauss noise and a Poisson noise. Special cases of this general model include the so-called stochastic logistic growth model and the CIR model studied recently by several authors. Hence, the study of stochastic equations with jumps is of urgent demands by the society. At the same time, the study of stochastic equations with jumps can also bring new methods to the research of generalized continuous branching processes...On the theoretical side, based on existing research we will study the boundary classification, the speed of coming down from infinity, the long-time behavior and ergodicity of the process. On the application side, we are going to construct mathematical model for some biology systems by using stochastic integral equations with jumps.

分枝过程通常用于描述人口演化，是一类非常重要的马尔科夫过程。分枝过程满足的一个基本假设是群体内每个个体的行为相互独立。然而在实际中这一假设很难满足。为了去掉这一假设，我们引入了广义分枝过程。这类过程可通过由Gauss白噪声和Poisson随机测度驱动的带跳随机积分方程来构造。该方程涵盖生态学中常见的logistic增长模型和金融学中常见的CIR利率模型。因此研究带跳随机积分方程不仅对广义分枝过程有重要理论意义，还有迫切的社会需求。..本项目将会研究广义分枝过程和相关的带跳随机积分方程解的边界分类、从无穷远点出发的速度估计、过程的长时间行为以及遍历性等问题。我们还将建立带跳随机积分方程在生物学中的应用。

分枝过程通常用于描述人口演化，是一类非常重要的马尔科夫过程。分枝过程满足的一个基本假设是群体内每个个体的行为相互独立。然而在实际中这一假设很难满足。为了去掉这一假设，我们引入了广义分枝过程。这类过程可通过由Gauss白噪声和Poisson随机测度驱动的带跳随机积分方程来构造。该方程涵盖生态学中常见的logistic增长模型和金融学中常见的CIR利率模型。本项目研究了广义分枝过程和相关的带跳随机积分方程解的边界分类、从无穷远点出发的速度估计、过程的长时间行为以及遍历性等问题。