Stochastic Calculus of Variations and Limit Theorems

The goal of this project is to investigate a variety of problems in stochastic analysis, which is a part of probability theory that studies dynamical systems under the action of random impulses. A central objective is the analysis of stochastic partial differential equations, such as the heat and wave equations, perturbed by random noises. These equations provide mathematical models in a wide range of areas, such as growth models for interfaces, turbulence in fluid dynamics and polymer models.

The proposed research will focus on the ergodicity and random fluctuations of spatial averages, which are related to observed characteristics in particular physical models. A second objective of the project is to broaden the range of applications of the stochastic calculus of variations, also called Malliavin calculus. The Malliavin calculus is a mathematical theory that extends the classical calculus of variations from functions to stochastic processes.

It has proven to be a powerful tool in deriving rates of convergence in central limit theorems, which are of great relevance in statistical inference. Particular emphasis will be put in the analysis of random processes with long memory which are useful to handle data coming from finance, telecommunications and other areas. The project provides research training opportunities for graduate students.

A first working block of the project consists in establishing quantitative central limit theorems for spatial averages of a wide class of stochastic partial differential equations driven by a Gaussian noise which has homogeneous covariance. Challenging problems are the case of the three dimensional wave equation driven by a noise which is white in time and it has a Riesz covariance in space, and also the case of noises which are rougher that the white noise.

Establishing the rate for probability densities using techniques of Malliavin calculus is a central goal of the project. A second working block deals with deriving the asymptotic behavior of functionals of the fractional Brownian motion related to local times. An innovative methodology based on the Clark-Ocone formula will be developed.

In a third working block we plan to address several open problems in the applications of the stochastic calculus of variation in limit problems including local asymptotic expansions of densities and rates of convergence for Euler approximations in stochastic Volterra equations.

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.