Stochastic Partial Differential Equations: Theory and Applications

University Of Southern California, Los Angeles CA

Investigators

Abstract

The research goals of this project are to develop new analytical, statistical, and numerical methods for stochastic partial differential equations. There are three directions in the proposed research: Wiener Chaos decomposition for linear and nonlinear equations, analysis of stochastic partial differential equations in domains, and statistical inference for stochastic partial differential equations. Wiener Chaos decomposition is a stochastic analog of the classical Fourier method and is a powerful tool in the study of stochastic equations, from both analytical and numerical points of view. In particular, Wiener Chaos decomposition is the key component of several new numerical algorithms for nonlinear filtering and the stochastic Navier-Stokes equation. Analysis of stochastic partial differential equations in domains is necessary for development of a complete theory, similar to the one for deterministic partial differential equations. In addition to purely theoretical results about existence and uniqueness of solutions, this analysis leads to new numerical methods for such equations. Methods of statistical inference make it possible to relate an abstract equation to a concrete problem from applications. Unlike finite-dimensional systems, a consistent estimation in a stochastic partial differential equation is possible even when both the observation time and the amplitude of noise are fixed. Stochastic partial differential equations are both an interesting mathematical object and an effective modelling tool in various branches of applied science, such as ecology, finance, meteorology, and navigation. Equations studied in this project can describe, for example, the term structure of interest rates, the propagation of an oil spill in the ocean, and the motion of the target on the radar screen. The research objectives of the project are to study mathematical properties of stochastic partial differential equations and to develop effective computational and statistical methods for relating an equation to a model. The educational objective of the project is to increase understanding of stochastic partial differential equations among the students and researchers both inside and outside of the mathematical community. This educational objective will be achieved by developing several basic and special topics graduate courses aimed at helping graduate students make a transition to independent research.