Diffusion Processes and Partial Differential Equations

扩散过程和偏微分方程拟议研究摘要尼古拉·V·克雷洛夫该项目涉及扩散过程和偏微分方程（PDE）理论中的一些重要的现代主题。它们来自实际应用，包括随机过程的最优控制，扩散过程的最优滤波，群体遗传学中出现的白色噪声驱动的随机偏微分方程（SPDE），完全非线性偏微分方程理论。该项目包括一个调查的数值方法找到解决方案。完全非线性偏微分方程出现在最佳质量运输问题和几何。各种船体的刚度和其它特性都可用这些方程来描述。 控制问题和完全非线性偏微分方程也出现在工程、目标跟踪、模式识别和许多其他领域。我们想要控制的随机过程有很多，例如，投资组合的表现或导弹的轨迹。在目标跟踪中，需要强调的是，轨迹通常只在存在一定误差或噪声的情况下才被观测到。因此，第一个问题是从观测中滤除噪声。这些问题首先由卡尔曼和布西解决，他们在阿波罗计划中构建并使用了他们的滤波器。目前的一个主要问题是在天气预报等问题中获得更好的结果，这是随机过程滤波和预测理论的一个非常实际的应用。"

Diffusion Processes and Partial Differential equations Abstract of Proposed Research Nicolai V KrylovThe project relates to some important modern topics in the theories of Diffusion Processes and Partial Differential Equations (PDEs). They arise from practical applications and include optimal control of random processes, optimal filtering of diffusion processes, white noise driven stochastic PDEs (SPDEs) arising in population genetics, the theory of fully nonlinear PDEs. The project includes an investigation of the numerical methods of finding solutions.Fully nonlinear PDEs arise in the optimal mass transportation problem and in geometry. Rigidity and other characteristics of all kinds of hulls are described in terms of such equations. Control problems and fully nonlinear PDEs also arise in engineering, target tracking, pattern recognition, and many other areas. There are many random processes which we want to control, for instance, the performance of a portfolio or the trajectory of a missile. In target tracking it is important to emphasize that the trajectory is usually only observed with certain errors or noises. Thus the first problem is to filter the noise out of the observations. Such problems were first solved by Kalman and Bucy, who constructed and used their filter during the Apollo program. Currently a major problem is to obtain better results in such problems as weather forecasting, which is a very practical application of SPDEs and the theory of filtering and prediction of random processes."