Diffusion Processes and Partial Differential Equations

这个项目的重点是在偏微分方程（PDE）和扩散过程的现代理论的一些中心议题。问题产生于实际应用，在数学上，被描述为随机过程的最优控制，扩散过程的最优滤波，和白噪声驱动的随机PDE（SPDE）。这些问题的语言制定的完全非线性偏微分方程的理论，该项目包括调查的数值方法，找到他们的解决方案。完全非线性偏微分方程出现在多种情况下，包括控制理论，最优质量运输问题，几何，仅举三例。所有类型的船体（比如船只或导弹）的刚度和其他特性都用这样的方程来描述。控制问题和完全非线性方程也出现在工程、目标跟踪、模式识别和许多其他应用领域。有许多随机过程需要控制并且控制起来很重要（例如，股票投资组合的表现，导弹的轨迹）。例如，在目标跟踪中，重要的是要强调，通常，在观察到射弹的轨迹时，存在一定的误差或噪声。因此，控制轨迹的第一个问题是从观测中滤除噪声。这些问题最初是由卡尔曼和布西解决的，他们在阿波罗计划中构建并使用了他们的滤波器。不用说，为了取得更准确的结果，还需要做更多的工作。改进的滤波器将与天气预报（或者更一般地说，气候变化预报）等工作有关，这是随机偏微分方程以及随机过程的滤波和预测理论的可能具体应用之一。

This project focuses on some of the central topics in the modern theory of partial differential equations (PDE) and diffusion processes. The problems arise from practical applications and, in mathematical terms, are described as optimal control of random processes, optimal filtering of diffusion processes, and white-noise-driven stochastic PDE (SPDE). The problems are formulated in the language of the theory of fully nonlinear PDE, and the project includes investigation of numerical methods for finding their solutions.Fully nonlinear partial differential equations arise in a multitude of contexts, including control theory, optimal mass transportation problems, and geometry, to name just three. Rigidity and other characteristics of all kinds of hulls (of, say, ships or missiles) are described in terms of such equations. Control problems and fully nonlinear equations also turn up in engineering, target tracking, pattern recognition, and a host of other applied areas. There are many random processes that it is both desirable and important to control (e.g., the performance of a stock portfolio, the trajectory of a missile). In target tracking, for instance, it is important to emphasize that the trajectory of a projectile is observed, in general, with certain errors or noises. Therefore, the first problem in controlling the trajectory is to filter the noise out of the observations. Such problems were initially solved by Kalman and Bucy, who constructed and used their filter during the Apollo program. Needless to say, much more work needs to be done in order to achieve more accurate results. Improved filters would be relevant to endeavors such as weather forecasting (or, more generally, climate change forecasting), which is one of the possible concrete applications of stochastic partial differential equations and the theory of filtering and prediction of random processes.