测度值马氏过程及相关非线性方程性质研究

本项目主要研究测度值马氏过程及其对应非线性方程，其中测度值马氏过程主要包括分支扩散、超扩散、分支-稳定过程与超-稳定过程,非线性方程包括非线性偏微分方程、 非线性积分-微分方程及非线性反应-扩散系统。 我们的研究内容主要由四部分构成：（一）测度值马氏过程中心极限定理的研究、（二）多物种分支扩散与多物种超扩散强大数定理研究、（三）测度值马氏过程对应的非线性偏微分方程及非线性积分-微分方程波形解的研究、（四）超过程与非线性积分-微分方程联系的研究。我们将采用概率与分析相结合的手段，将最近Lévy过程的精细研究结果应用到测度值过程性质的研究中，在测度值过程对应的各种非线性方程及测度值马氏过程的概率位势理论方面得到突破性研究成果。

This project mainly investigates measure-valued Markov processes and related nonlinear equations. The measure-valued Markov processes we are going to investigate include, among others, branching diffusions, superdiffusions, branching stable processes and super-stable processes. The related nonlinear equations include nonlinear partial differential equations, nonlinear integral-differential equations and systems of reaction-diffusion equations. Our research consists of four parts: (1) central limit theorems for measure-valued Markov processes; (2) strong law of large numbers for multi-type branching diffusions and multi-type superdiffusions; (3) traveling wave solutions of related nonlinear partial differential equations and nonlinear integral-differential equations; and (4) relations between superprocesses and nonlinear equations. We will combine probabilistic and analytical methods, apply the recent deep results on Lévy processes to investigate properties of measure-valued Markov processes. We aim to make some breakthrough results in the research on various nonlinear equations related to measure-valued Markov processes and on probabilistic potential theory of measure-valued markov processes.