Aspects of Stochastic Differential Geometry in Function Space

Southern Illinois University At Carbondale, Carbondale IL

Investigators

Abstract

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.