Diffusion Processes and Partial Differential Equations

扩散过程和偏微分方程拟议研究的摘要尼古拉·V·克里洛夫该项目涉及扩散过程和偏微分方程理论中的一些重要的现代课题。它们源于实际应用，包括随机过程的最优控制、扩散过程的最优滤波、种群遗传学中出现的白噪声驱动随机偏微分方程组(SPDEs)、完全非线性偏微分方程组理论。该项目包括对求解的数值方法的研究。在最优质量传输问题和几何中出现完全非线性的偏微分方程组。用这种方程描述了各种船体的刚度等特性。在工程、目标跟踪、模式识别等许多领域中也会出现控制问题和完全非线性偏微分方程组。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，轨迹通常只在有一定误差或噪声的情况下才能观察到。因此，第一个问题是滤除观测中的噪声。卡尔曼和布西最先解决了这些问题，他们在阿波罗计划期间构造和使用了他们的过滤器。目前一个主要的问题是在天气预报等问题上取得更好的结果，这是SPDES和随机过程的过滤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."