Aspects of Stochastic Differential Geometry in Function Space

这个职业发展奖是为了使PI能够研究带记忆的随机微分系统，几何约束下的随机偏微分方程组，以及它们的数值方面。这样的模型出现在工程和物理应用中。该奖项有几个目标：第一个目标是为PI提供获得微分几何和数值分析方面的新专业知识的机会。这项活动将采取研究(随机)微分几何、随机偏微分方程式和随机数值学的现有文献的形式。国际和平研究所将与这些快速发展领域的领先专家进行磋商。第二个目标是从微分几何和数值学中引入新的技术，以发展具有记忆和几何约束的随机系统理论。除了与物理应用有很大的相关性外，带约束的无限维随机系统的研究将导致与当前快速增长的随机微分几何研究的非常有趣的联系。第三个目标是在现有的和快速发展的随机偏微分方程论中引入动力和微分几何分量。PI将研究由这类系统产生的随机流的遍历理论。第四个目标是开发有效的算法，用于对机械、控制和电气工程、金融和经济学中出现的具有记忆的随机模型进行数值模拟，例如金融证券的期权定价，其演变受其过去历史的支配。第五个目标是将调查结果汇编成一门研究生课程，目标是数学、工程和/或金融专业的学生。

This career advancement award is to enable the PI to study stochastic differential systems with memory, stochastic partial differential equations under geometric constraints, and their numerical aspects. Such models arise in engineering and physical applications. The award has several objectives:The first objective is to offer the PI the opportunity to acquire new expertise in differential geometry and numerical analysis. This activity will take the form of a study of the current literature on (stochastic) differential geometry, stochastic partial differential equations and stochastic numerics. The PI will consult with leading experts in these rapidly-developing areas. The second objective is to introduce new techniques from differential geometry and numerics in order to develop a theory of stochastic systems with memory and geometric constraints. In addition to its great relevance to physical applications, it is expected that the study of infinite-dimensional stochastic systems with constraints would lead to very interesting connections with current and fast-growing research in stochastic differential geometry. The third objective is introduce a dynamical and differential geometric component into the existing and fast-growing theory of stochastic partial differential equations. The PI will study the ergodic theory of stochastic flows generated by such systems. The fourth objective is to develop efficient algorithms for the numerical simulation of stochastic models with memory that arise in mechanical, control and electrical engineering, finance and economics, e.g. the option-pricing of financial securities whose evolution is governed by their past history.The fifth objective is to compile the results of the investigations into a graduate course targeting students majoring in mathematics, engineering and/or finance.