Rough path analysis and non-linear stochastic systems

自然界中的许多现象在时间上表现为混沌的复杂演化。数学家经常用微分方程来研究这些随机运动。一般来说，这些方程非常复杂，有许多未知变量，并且经常包含噪声。在过去的十年中，由主要研究者与合作研究者和其他同事开发的粗糙路径分析是准确描述混沌复杂系统所控制的演化的机器。粗糙路径理论的一个关键观点是观察到嵌入在复杂进化系统中的网络信息可以完全由其签名描述。当使用的时刻相比，签名代表了一个巨大的进步，作为一种手段来描述非线性混沌系统。与矩相反，它很容易捕捉事件的顺序，因此可以测量整体速度，旋转等等。在这个项目中，我们进一步发展粗糙路径的分析，并建立一个微分方程的理论，输入涉及复杂的合奏随机，相互作用的粒子。我们打算应用该理论来研究随机演化系统模拟湍流的签名。在项目进展过程中开展的研究是为研究高维复杂混沌系统（例如湍流）提供尖端技术。

Many phenomena in nature appear as chaotic complex evolutions in time. Mathematicians often study these random movements by means of differential equations. In general these equations are very complicated with many unknown variables, and very often include noise. The rough path analysis which has been developed over the last decade by the principle investigator, with the Co-investigator and other co-workers, is the machinery which accurately describes the evolutions governed by chaotic complex systems. A key perspective in this rough path theory is the observation that the net information embedded in a complex evolution system can be completely described by its signature. When compared with the use of moments, the signature represents a huge step forward as a means for describing non-linear chaotic systems. In contrast to moments, it easily captures the order of events, and hence measures bulk velocity, rotation and much more besides. In this project we further develop the analysis of rough paths and establish a theory of differential equations with inputs involving complicated ensembles of random, interacting particles. We intend to apply the theory to study the signatures of turbulent flows modelled by stochastic evolution systems. The research developed during the progress of the project is to provide cutting-edge technologies for the study of high dimensional, complicated chaotic systems, for example turbulent flows.