Differential Equations and Rough Path Theory

常微分方程式常被用来模拟粒子的运动。同样，偏微分方程式可以用来描述整个粒子分布的演化。在这类模型中加入随机性是很自然的：有时是因为这是一个考虑了随机噪声的更现实的描述，有时是因为这种随机性是模型本身的基础，就像股票市场的情况一样。一个好的随机噪声模型，如著名的布朗运动，不可能在时间上平滑演化(否则，噪声在小范围内是可预测的！)。因此，微分方程的随机扰动本质上是不规则的，需要新的方法和理论。它的开创性贡献使这一切成为可能，这是20世纪数学的伟大成就之一。尽管它有很多好处，但它的理论可能是脆弱的，一些在应用中自然出现的问题需要付出很大努力，或者根本无法处理。本质上，这种限制来自这样一个事实，即随机积分是公平博弈(将布朗运动视为无偏或公平随机游走的极限)和被积函数(=赌博策略)的变换。如果一个人能够解决噪声情景，然后应用(非随机)微分方程的标准理论，生活将会容易得多。直到1998年才意识到这确实是可能的，相应的理论被贴上了粗糙路径理论的标签。这在概念上是要付出代价的：我们熟悉的欧几里德空间上的布朗运动必须被一个随机过程所取代，这个随机过程的值在所谓的李群中。研究的第一部分旨在了解布朗运动(在李群上)可以在多大程度上被其他高斯过程取代，并给出这种高斯粗糙路径的完整刻画。将分数布朗运动和所谓的Volterra过程等许多特殊情况结合在一起的统一理论是一个真正的机会。潜在的应用包括数学金融模型。第二部分研究的对象是带噪声的偏微分方程。(这样的方程出现在应用科学的许多领域。)这种带有粗略噪声的方程的确定性理论将为随机偏微分方程解提供一种稳健的、构造性的方法，改善我们对这些方程的理解以及如何用数值方法来处理它们。

An ordinary differential equation is often used to model the movement of a particle. Similarly, partial differential equation can be used to described the evolution of an entire distribution of particles. It is natural to add randomness to such models: sometimes because this is a more realistic description which takes into account random noise, sometimes because this randomness is fundamental to the model itself as is the case for the stock market. A good model of random noise, such as the celebrated Brownian motion, cannot evolve smoothly in time (otherwise the noise would be predictable on a small scale!). As a result, stochastic perturbations of differential equation are intrinsically irregular and require fundamentally new methods and theories. The ground-breaking contributions of It, which allowed to make all this possible, is one of the great achievements of 20th century mathematics. With all its benefits It's theory can be fragile and some questions that arise naturally in applications require major effort or cannot be treated at all. In essence, this restrictions come from the fact that stochastic integrals are transforms of fair games (think of Brownian motion as the limit of an unbiased or fair random walk) and the integrand (= the gambling strategy ). Life would be much easier if one could fix a noise-scenario and then apply a standard theory of (non-random) differential equations. It was only realized in 1998 that this is indeed possible and the corresponding theory has been labelled rough path theory . There is a conceptual price to pay: Brownian motion on the familiar Euclidean space has to be replaced by a stochastic process with values in a so-called Lie group. The first part of the proposed research aims to understand to what extent Brownian motion (on the Lie-group) may be replaced by other Gaussian processes and to give a full characterization of such Gaussian rough paths . There is a real chance for a unified theory which brings together many particular cases such as fractional Brownian motion and so-called Volterra processes. Potential applications include models of mathematical Finance. The second part of the research is aimed at partial differential equations with noise. (Such equations arise in many fields of applied science.) A deterministic theory of such equations with noise in the rough sense would provide a robust and constructive approach to stochastic partial differential equations, improving both our understanding of these equations as well as how to treat them numerically.