马氏过程在生物和统计物理中的若干应用

The main aim of this project is to study the applications of Markov processes in the natural selection and fluctuation dissipation theorem. (1) Natural selection is one of the core concepts in the evolutionary biology. It is a key mechanism of evolution. How to generate the ancestral selection graph and the probability properties of the ancestral distribution are the main problems in the study of the natural selection. There are many studies on the natural selection which is independent of the gene frequency of the population. However, there exist few works on the natural selection depending on the gene frequency of the population, which is much closer to the biological reality. We will use Markov jump processes in the random environment generated by diffusion processes to study the algorithm of generating the ancestral selection graph and the probabilistic representations of the ancestral distribution for the natural selection depending on the gene frequency of the population. (2) The fluctuation dissipation theorem is one of the most important results in the statistical physics. It expresses the response of an observation to an external perturbation by a correlation function of this observation and another one that is conjugate to it. Using the fluctuation dissipation theorem one can predict the expectation of the changes of the system when it suffers from an external perturbation. The fluctuation dissipation theorem has been deeply studied in the physics. However, very few rigorous mathematical results have been derived and most of the theoretical results are based on the formal deduction. The project will use the theory of Markov processes to establish the mathematical framework of the fluctuation dissipation theorem.

本项目主要研究马氏过程在自然选择和涨落耗散定理中的应用。（1）自然选择是进化生物学中最核心的概念之一，它是导致生物进化的关键因素。如何生成祖先选择图和祖先分布的概率性质是研究自然选择的核心问题。人们对自然选择不依赖种群的基因频率的情形已有很多研究，但是对自然选择依赖种群的基因频率这种更加符合生物实际的研究还很少。本项目拟用以扩散过程为随机环境的马氏跳过程来研究当自然选择依赖种群的基因频率时生成祖先选择图的随机算法和祖先分布的概率表示。（2）涨落耗散定理是统计物理中最重要的结果之一，它把一个物理观测量关于外部的扰动表示成这个观测量和另一个共轭观测量的相关函数。利用涨落耗散定理我们可以预测系统关于外部扰动在期望意义下的变化。涨落耗散定理在物理中的研究已经非常深入，但是其理论推导是基于形式上的推导，没有严格的数学理论做支撑。本项目拟用马氏过程理论建立涨落耗散定理的数学框架。

本项目主要研究马氏过程在生物、统计物理和随机控制中的若干应用，具体如下：.(1)马氏过程在生物中的应用:利用鞅问题和藕合的方法证明随机基因调控网络中的广义密度依赖马氏链的极限是逐段决定马氏过程，从而建立基因调控中两类模型的联系。 .(2)马氏过程在统计物理中的应用:通过Schauder估计和弱连续半群理论建立非时齐扩散过程的Agarwal型涨落耗散定理和时齐扩散的Seifert&Speck型涨落耗散定理的数学理论。.(3)马氏过程在随机控制中的应用:i)证明离散时间马氏决策过程和连续时间随机博弈风险灵敏性平均准则相应的最优策略和纳什均衡的存在性。ii)证明离散时间和连续时间不可数状态空间非零和随机博弈期望平均准则存在几乎平稳纳什均衡点。