Diffusion Processes and Partial Differential Equations

这个项目集中在现代偏微分方程(PDE)和扩散过程理论中的一些中心主题。这些问题来自实际应用，用数学术语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偏微分方程组(SPDE)。这些问题是用完全非线性偏微分方程组的理论来描述的，该项目包括寻找其解的数值方法的研究。完全非线性偏微分方程组出现在多种背景下，包括控制理论、最优质量传输问题和几何，仅举三例。所有船体(例如舰船或导弹)的刚度和其他特性都是用这种方程描述的。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。有许多随机过程既需要控制，也需要控制(例如，股票投资组合的表现，导弹的弹道)。例如，在目标跟踪中，需要强调的是，通常情况下，观察到的弹丸轨迹带有一定的误差或噪声。因此，控制弹道的首要问题是滤除观测值中的噪声。这些问题最初是由卡尔曼和布西解决的，他们在阿波罗计划期间建造和使用了他们的过滤器。不用说，为了取得更准确的结果，还需要做更多的工作。改进的过滤器将与天气预报(或更广泛地说，气候变化预报)等努力相关，这是随机偏微分方程和随机过程过滤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.