Levy过程驱动的随机偏微分方程的遍历性及相关问题

随机偏微分方程是随机分析研究的重点和热点问题之一，尤其对涉及流体力学等有深刻物理背景的随机偏微分方程的研究，既有重要的理论价值，又有实际意义。本项目主要研究流体同时受连续和间断两类噪声影响系统的动力学行为，这对流体力学研究有重要意义，对理解和研究无穷维随机动力系统也有帮助。本项目研究的问题有：（1）随机微分方程的Malliavin分析，为研究随机偏微分方程遍历性作准备。（2）Levy过程驱动的偏微分方程的遍历性、泛函不等式和大偏差等。（3）Levy过程驱动的Burgers方程和二维Navier-Stokes方程的遍历性、指数遍历性、泛函不等式、Malliavin分析和大偏差等。（4）Levy过程驱动的三维Navier-Stokes方程弱解的存在性、Markov选择和遍历性等；三维随机Tamed Navier-Stokes方程解的存在唯一性及性质。（5）发展随机可积系统的求解法。

Stochastic partial differential equation (SPDE) is one of the most important and hottest research topics of stochastic analysis, espically those dealing with hydrodynamics. Research on this kind of SPDEs is of great importance for not only pure mathenatics but also for solving practical problems. In this project, we are going to investigate the dynamic behaviors of fluid affected by both continuous and discontinous noise. This reseach is not only important for hydrodynamics itself, but also for better understanding and hence further reseach on infinite dimensional dynamics. Specifically, following problems will be investigated in this project: (1) Malliavin calculus for stochastic differential equations. (2) Ergodicity, functional inequalities and large deviation principles for SPDEs driven by Levy noise. (3) Ergodicity, exponent ergodicity, functional inequalities, Malliavin calculus and large deviation principles for stochastic Burgers equations and 2D stochastic Navier-Stokes equations driven by Levy noise. (4) The existence of weak solutions, Markov solutions and ergodicity of the solutions for 3D stochastic Navier-Stokes equations by Levy noise, and the existence, uniqueness and the properties of the solution for 3D stochastic tamed Navier-Stokes equations by Levy noise. (5) Looking for white noise functional solutions for stochastic integrable systems.