传输噪声驱动的随机分数阶quasi-geostrophic方程

随机偏微分方程在物理、生物、金融等众多领域有着广泛的应用，目前已成为概率论中发展最为迅速的分支领域之一。本项目主要研究由传输噪声驱动的带分数阶耗散项的随机quasi-geostrophic方程。Quasi-geostrophic方程能很好地描述海洋和大气的大尺度运动，在地球物理流体动力学具有重要意义。本项目首先拟建立带分数阶微分项的随机过程的Lp范数的Itô公式。该Itô公式是研究随机分数阶偏微分方程的基本工具之一。利用该公式本项目拟分别研究由传输噪声驱动的随机分数阶quasi-geostrophic方程和由Marcus型Lévy噪声驱动的随机分数阶quasi-geostrophic方程。我们拟研究上述两个方程解的存在性、轨道唯一性、不变测度存在性和大偏差原理。期望本项目的成果不仅能丰富现有的随机分析的理论结果，还能为随机分数阶偏微分方程的研究提供新的思路。

Due to its enormous richness of applications ranging from physics and biology to finance, the subject of stochastic partial differential equations has become one of the most rapidly developing areas of probability theory. The broad aim of this project is to investigate fractionally dissipative stochastic quasi-geostrophic equations with transport noise. Quasi-geostrophic model is of great significance in geophysical fluid dynamics as it describes some basic aspects of large-scale flows in the oceans and atmosphere very well. One of three main aims of this project is to establish an Itô formula for the Lp norm of processes with fractional differentials which is one of the most fundamental tools in the study of stochastic fractional partial differential equations. By using this Itô formula we will study stochastic fractional quasi-geostrophic equations driven by transport noise and also stochastic fractional quasi-geostrophic equations driven by Lévy noise in the Marcus canonical form. We plan to investigate the existence, pathwise uniqueness of solutions, existence of invariant measures and large deviations of both equations. We hope that the study of this project could not only enrich the existing theories on stochastic analysis but also provide some new insights into the analysis of stochastic fractional partial differential equations.