本项目是随机偏微分方程和分数阶偏微分方程的交叉课题，主要研究高斯噪音及Lévy噪音驱动的等离子体中的随机分数阶偏微分方程组解的适定性、不变测度的遍历性、不变流形的存在性及不同的逼近形式、随机吸引子和随机惯性流形的存在性等问题。项目围绕随机Hasegawa-Mima方程、随机可压量子Navier-Stokes方程组、等离子体非线性光学方程及其对应的分数阶模型展开。重点研究:(1)退化和非退化高斯噪音驱动的随机整数阶及相应分数阶Hasegawa-Mima方程解的适定性、正则性以及生成的随机动力系统的长时间行为、不变测度的存在唯一性、不变流形的存在性及其逼近;(2)高斯噪音驱动的随机可压量子N-S方程弱解的适定性及随机动力系统的动力学行为，以及随机可压量子N-S方程收敛到随机N-S方程的极限问题;(3)具有随机初值的等离子体非线性光学方程解的适定性、正则性、相应动力系统的动力学行为。

This project is an intersect project between stochastic PDE and fractional PDE. We mainly consider the stochastic fractional partial differential equations in plasma driven by Gaussian noise and Lévy noise, and pay attention to the well-posedness of solutions、ergodicity of invariant measure、existence and various approximations of invariant manifolds、existence of random attractors and inertial manifold and so on. We will mainly focus on the researches of stochastic Hasegawa-Mima equation, stochastic compressible quantum Navier-Stokes equations, stochastic nonlinear optical equations in plasma, and their corresponding fractional models. (1)For the stochastic fractional Hasegawa-Mima equations driven by Gaussian noise and degenerated Gaussian noise, we study the well-posedness and the regularity of solution, the long time behaviors of the associated random dynamical systems, such as the existence and uniqueness of invariant measure, the existence of invariant manifolds and their approximations.(2)We investigate the stochastic compressible quantum Navier-Stokes equations driven by Gaussian noise, and pay attention to the well-posedness and the semiclassical limits of solution.(3) We study the stochastic equations in plasma and nonlinear optical with random initial value. Firstly, we consider the well-posedness and regularity of the solution. Secondly, we discuss dynamic behaviors of the associated dynamical system.