退化跳过程驱动的偏微分方程的遍历性

In this project, we shall develop a coupling technique to investigate the ergodicity of stochastic partial differential equations driven by degenerate jump processes. We shall start from four concrete models, i.e., Lipschitz type nonlinear PDEs, Reaction-diffusion equations, Burgers equations and Navier-Stokes equations, to build a framework for applying our coupling technique. We shall put these four models into this framework to get their ergodicity systematically. ..Our coupling technique is roughly as follows: (1). Sampling a Markov chain from the original stochastic systems according to their characteristics; (2). Building a coupling (e.g. maximal coupling, mirror coupling,etc) for the Markov chain; (3). Building the ergodicity of the Markov chain by the coupling; (4). Building the ergodicity of the original system from that of Markov chain. ..We hope that the developed coupling technique in this project will have a lot of applications and solve new problems, and that the built framework will systematically solve the ergodicity problems for a family of stochastic partial differential equations driven by degenerate jump processes.

本项目将发展出一套耦合技术来研究退化纯跳过程驱动的随机偏微分方程的遍历性。从四个具体的模型（即Lipschitz非线性模型，反应扩散方程，Burgers方程及Navier-Stokes方程）出发，采取由易到难，由浅入深，层层推进的解决思路，从中提炼出一个统一框架，然后从这个框架出发，系统解决这四个问题。我们耦合方法大致如下：(1).根据随机动力系统的自身特性，从原随机过程中抽取马氏链；(2).在马氏链上建立耦合，耦合的选择可以根据问题本身来定，如极大耦合，镜像耦合等；(3).运用以上的耦合，我们建立马氏链的遍历性；(4).比较马氏链和原随机过程，得到原随机过程的遍历性。..我们发展出来的耦合方法将会有广泛的应用，解决新的问题，对随机过程理论发展有很大的推动作用。我们建立的框架将系统地解决一类退化跳过程驱动的随机系统的遍历性问题，对随机过程理论和遍历理论都有很大的推动作用。

在这个研究项目的支持下，我们系统地研究了一类具有重尾特性Levy过程驱动的随机（偏）微分方程的极限特性，得到了一系列重要随机（偏）微分方程（如随机实Ginzburg-Landau方程，随机反应扩散方程，随机Porous medium方程）的遍历性和Donsker-Varadhan型大偏差和中偏差。相关成果发表在Bernoulli, Stochatic Processes and Their Applications, Potential Analysis, Science China Mathematics等概率一流杂志上。另一方面，我们还把相关成果应用到了重尾分布极限定理收敛速度问题，解决了两个重要的概率逼近问题，相关成果发表在概率顶级期刊Probability Theory and Related Fields和Annals of Applied Probability上或者统计一流刊物Electronic Journal of Statistics上。主持人在’Annals of Applied Probability’发表的论文‘Approximation of stable law in Wasserstein-1 distance by Stein's method’被认为是Stein方法的近年来的突破性进展，开辟了新的研究问题 （it ‘breaks new grounds in Stein’s famous method, and also opens up many new interesting research problems’, from a 5 pages’ referee’s report.）。