Stochastic Analysis and Asymptotic Problems

University Of Kansas Center For Research Inc, Lawrence KS

Investigators

Abstract

This research project investigates questions in stochastic analysis, a part of probability theory that studies dynamical systems under the action of random impulses. A central objective of the project is the study of partial differential equations perturbed by rough noises; such equations provide mathematical models for a wide range of phenomena, including interface growth, turbulence in fluid dynamics, and polymer structure. The research will investigate, among other topics, the intermittency and chaotic properties of solutions, which are related to important characteristics of physical systems. A second objective of the project is to broaden the range of applications of the stochastic calculus of variations, also called Malliavin calculus. Malliavin calculus is a mathematical theory that extends the classical calculus of variations from functions to stochastic processes. It has proved to be a powerful tool in deriving rates of convergence in central limit theorems, which are of great relevance in statistical inference. Emphasis will be placed on analysis of random processes with long memory, which are useful for analysis of data coming from finance, telecommunications, and other areas. This project addresses questions in three topical areas. The first topic is the study of the stochastic heat equation driven by a Gaussian noise that is white in time and has homogenous spatial covariance. A challenging goal is to derive a change-of-variable formula for functionals of the solution and investigate applications of this formula to a range of questions, including large-time asymptotics and intermittency properties. It is also planned to further develop the analysis of stochastic partial differential equations driven by rough noises and time-independent noises. A second topic concerns applications of Malliavin calculus to a variety of open questions, including rates of convergence and asymptotic expansions of densities, when the target distribution is a mixture of Gaussian laws and limit theorems for geometric Gaussian functionals. A third topic addresses questions in the analysis of fractional Brownian motion and related self-similar Gaussian processes, including the dynamics of eigenvalues of matrix-valued fractional Brownian motions and the exponential integrability of self-intersection local times. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.