随机偏微分方程，特别是具有物理背景的流体力学方程已成为国际概率论界非常活跃的研究领域之一。本项目主要研究两类重要流体力学方程的遍历性和大偏差，包括有粘随机Primitive方程和无粘随机标量守恒律方程。这是两类不同的方程。对于前者，由于该方程在结构上具有高度非线性性，需要新的随机分析和偏微分方程中的技巧来处理一些非平凡的高阶Sobolev范数。对于后者，因为该方程没有粘性项，正则性低，偏微分方程理论中的经典定义和方法不再适用，所以无论从理论还是方法上都需要创新。本项目将研究以下三部分：（1）由可乘噪声驱动的三维Primitive方程的遍历性（2）由可乘噪声驱动的二维Primitive方程的占位时大偏差（3）由可乘噪声驱动的无粘随机标量守恒律方程的Freidlin-Wentzell大偏差。上述研究成果可以帮助我们更好的了解海洋流体的运动并做出天气预测。

Stochastic partial differential equations, especially the hydrodynamic equations with physical background, have become one of the active research fields in the international probability theory. This project mainly studies the ergodic and large deviation principle of two important stochastic partial differential equations in fluid mechanics, including viscous stochastic primitive equations and inviscid stochastic scalar conservation laws. They are two completely different equations. For the former, we need new techniques in stochastic analysis and partial differential equations to deal with some non-trivial high order Sobolev norms because of high nonlinearity in structure. For the latter, because the equation has no viscous term and its regularity is low, the classical definition and method of partial differential equation theory is no longer applicable, so it needs innovation both in theory and method. This project will study the following three parts : (1) Ergodicity of three-dimensional primitive equations driven by multiplicative noise. (2) Large deviation principle of occupation measure for two-dimensional primitive equations driven by multiplicative noise. (3) The Freidlin-Wentzell’s large deviations for stochastic scalar conservation laws driven by multiplicative noise. The above research results can help us better understand the motion of large scale ocean and make weather prediction.