无穷维随机微分方程

A large number of models were found that could be described by partial differential equations with random parameters, such as the coefficients or the forcing term. As a result, the study of SDE in infinite dimensional space has begun to attract a lot of attention of many researchers. Recently, since the non-local operators are widely used, there are many researchers studying the partial differential equations with non-local operators. Taking into account of the uncertainties in the actual and the observation error, to study the stochastic partial differential equations with non-local operators is a very important research topic. The project applicant wishes to carry out the study of infinite dimensional stochastic differential equations, especially stochastic partial differential equations with non-local operators. Applicant wants to apply the so-called "strong approach", i.e. through the underlying infinite dimensional stochastic differential equation to study the various properties of the solutions, including existence and uniqueness of solutions, ergodicity, stochastic dynamical systems and related properties. This requires us to improve and refine techniques from the theory of infinite dimensional ordinary differential equations. On the other hand, the applicant wants to use the "weak approach", i.e. through the infinite dimensional Kolmogorov equation associated with the stochastic partial differential equations and the corresponding Fokker-Planck equation to obtain the existence and uniqueness of the solutions to the stochastic partial differential equaitons when the noise is space-time white noise or with singular drift. This requires for the "weak approach" to develop a theory of partial differential operators / equations in infinitely many variables for functions (i.e. Kolmogorov equations) or for measures (i.e. Fokker-Planck equations) to cover realistic and interesting examples of applications, where typically coefficients are very singular.

无穷维空间上的随机微分方程，是一个重要的研究课题, 是目前国际上概率论领域的一个热门方向。最近几年由于非局部算子的广泛出现， 因此有很多工作研究带有非局部算子的确定性方程。 考虑到实际中的不确定性以及观测的误差，研究在随机扰动下带有非局部算子的偏微分方程是很重要的研究课题。项目申请人希望开展无穷维随机微分方程的研究尤其是带有非局部算子的随机偏微分方程. 申请人准备通过"强方法"即是对无穷维随机微分方程本身来研究解的各种性质,包括解的存在唯一性,遍历性以及随机动力系统的相关性质。这要求我们改进优化随机偏微分方程中的技巧。另一方面，申请人希望通过研究与随机微分方程对应的无穷维Kolmogorov方程以及相应的Fokker-Planck方程得到当噪声是时空白噪声或漂移项比较奇异时随机微分方程解的存在唯一性。这要求发展对无穷多个变量的函数和测度的偏微分方程的理论来应用到更多现实中的例子。

无穷维空间上的随机微分方程，是一个重要的研究课题, 是目前国际上概率论领域的一个热门方向。最近几年由于非局部算子的广泛出现， 因此有很多工作研究带有非局部算子的确定性方程。 我们通过"强方法"即是对无穷维随机微分方程本身来研究解的各种性质, 通过改进优化随机偏微分方程中的技巧，得到了次临界条件下随机quasi-geostrophic方程解的遍历性，随机动力系统的相关性质，超临界条件下，局部解的存在唯一性,以及在特殊噪声下不爆炸的结果。另一方面，我们通过研究与随机微分方程对应的无穷维Kolmogorov方程以及相应的Fokker-Planck方程得到当噪声是时空白噪声或漂移项比较奇异时随机微分方程解的存在唯一性。. 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。我们学习了这两套理论研究了一类带有奇异噪声的随机偏微分方程。我们得到了由时空白噪声驱动的三维Navier-Stokes方程局部解的存在唯一性。