奇异的随机偏微分方程：强方法和弱方法

The study of stochastic partial differential equations has been attracted a lof of attention of many researchers. Recently the theory of singular stochastic partial differential equations has made great progress. In this project, we will study singular stochastic partial differential equations. This project consists of two major parts: strong approach and weak approach. We will use the "strong approach", i.e. exploit the power of the regularity structure approach and the paracontrolled distribution approach for existence/uniqueness of solutions to stochastic conservative Cahn-Hilliard equation and the random dynamical systems associated with singular stochastic partial differential equations. On the other hand, we will use weak approach, i.e. combine the Dirichlet form approach and the energy solutions approach to develop probabilistically weak solutions theory to singular stochastic partial differential equations, including stochastic quantization equation, singular stochastic partial differential equations on loop space and so on. Also we want to compare the results from strong and weak approach, and combine them to study the properties of singular stochastic partial differential equations.

随机偏微分方程是一个重要的研究课题，是目前国际概率论领域一个热门方向。最近几年奇异的随机偏微分方程的理论有了很大发展。 在这个项目中我们计划研究奇异的随机偏微分方程。我们计划运用强方法和弱方法进行研究。一方面，我们计划采用强方法即通过正则结构和拟控制分布的方法研究随机保守Cahn-Hilliard方程以及与奇异随机偏微分方程联系的随机动力系统的性质等等。另一方面，我们计划运用弱方法即结合狄氏型和能量解的方法发展奇异的随机偏微分方程的弱解理论，具体包括随机量子化方程，loop空间上奇异的随机偏微分方程等等。同时，我们计划比较强方法和弱方法的结果，结合这两种方法的优势进一步研究以上方程的性质。

奇异随机偏微分方程的理论近年来有了重大发展，Hairer提出了正则结构理论可以给出奇异随机偏微分方程的良定性。同时，由 Gubinelli, Imkeller, Perkowski提出了paracontrolled distribution的方法也可以用于研究奇异随机偏微分方程。本项目通过正则结构和拟控制分布（强方法）和狄氏型理论（弱方法）研究了带有奇异噪声随机偏微分方程，得到了带有奇异噪声的随机偏微分方程的全局适定性、普适性、遍历性等。具体主要研究了以下几个问题：.1.构造了三维随机量子化方程对应的狄氏型.2.用狄氏型理论构造了有限体积、无穷体积几何随机热方程的鞅解、运用泛函不等式研究了解的.长时间行为。.3. 得到了三维随机欧拉方程的鞅解的全局存在性，不唯一，强马氏选择的结果。.4. 得到了全空间上Phi43模型的弱普适性结果。.相关成果得到Fields奖得主Hairer的多次引用。