带有奇异噪声的随机偏微分方程

Recently Hairer developed a theory of regularity structures which gave a meaning to a class of singular SPDEs. Another successful approach to these equations called paracontrolled distribution was introduced by Gubinelli, Imkeller, Perkowski. We want to use these two approaches to study existence and uniqueness of solutions to SPDEs driven by space-time white noise. Moreover, approximation results for a class of equations exhibiting similar features as the KPZ equation are very important. This is related to the universality of the KPZ equation. It is also related to the open problem that the three dimensional dynamic Ising–Kac model converges to the $\Phi^4_3$ model. The long term goal is to study existence and uniqueness of solutions and more general approximations to parabolic singular partial differential equations. Moreover，uniqueness and properties of the solutions to SPDEs are also very important, especially the new regularity of the solutions to SPDEs driven by space-time white noise. We would like to study the uniqueness and the properties of the solutions to SPDEs driven by space-time white noise.

最近Hairer发展了正则结构理论成功用于研究奇异随机偏微分方程。另外一个由 Gubinelli, Imkeller, Perkowski提出的称为paracontrolled distribution的方法也可以用于研究奇异随机偏微分方程。本项目拟用正则结构理论以及paracontrolled distribution的方法对一类带有奇异噪声的随机偏微分方程进行研究。更进一步，关于这样一类方程的逼近问题也非常重要。这和KPZ方程的普适性相关。也对应于公开问题三维Ising-Kac模型收敛到$\Phi^4_3$模型。本项目长期的目标是研究奇异随机偏微分方程解的存在唯一性和更加一般的逼近。另外，研究随机偏微分方程的唯一性及其性质仍是十分重要的研究课题，特别是由时空白噪声驱动的随机微分方程可能表示出新的正则性结果。本项目同时拟对带有时空白噪声的随机偏微分方程唯一性及其性质进行研究。

2014年Hairer提出正则结构理论成功用于研究奇异随机偏微分方程。同时，由 Gubinelli, Imkeller, Perkowski提出的称为paracontrolled distribution的方法也可以用于研究奇异随机偏微分方程。本项目研究了带有奇异噪声随机偏微分方程，通过正则结构和拟控制分布和狄氏型理论研究了带有奇异噪声的随机偏微分方程的全局适定性、逼近结果、遍历性等。具体主要研究了以下几个问题：.1.随机量子化方程和狄氏型理论的关系，运用狄氏型理论得到了二维随机量子化方程的鞅解唯一性；同时构造了三维随机量子化方程对应的狄氏型.2.发展拟控制分布理论证明了三维随机量子化方程的格点逼近问题；.3.得到了带有保守噪声的二维Cahn-Hilliard方程的全局适定性、遍历性；.4.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。.相关成果得到Fields奖得主Hairer的多次引用。