Stochastic Partial Differential Equations, Fractional Noises and Limit Theorems

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This proposal deals with different topics in stochastic analysis, which is a part of probability theory that studies dynamical systems under the action of random impulses. A basic objective is the study of stochastic partial differential equations driven by rough noises, which are more singular than the classical white noise. These equations provide random field models that play a fundamental role in a wide range of areas of applied mathematics and mathematical physics, such as growth models for interfaces, turbulence in fluid dynamics and polymer models. The proposed research will focus, among other topics, on intermittency and chaotic properties of the solutions, which are related to observed characteristics in particular physical models. A second objective of the proposal is to broaden the range of the 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 proved to be a powerful tool in deriving rates of convergence in central limit theorems. In this direction, the proposal aims to establish new asymptotic results in a variety of frameworks including numerical schemes for fractional diffusions, where the noise exhibits long-range dependence. Getting exact rates of convergence in this context is of great relevance in practical applications of these models.A first working block of the proposal is the study of the stochastic heat and wave equations perturbed by a Gaussian noise which is white in time and it is a fractional Brownian motion in space with Hurst parameter less than 1/2. The roughness of the noise creates new difficulties and important challenges in the analysis of these equations. We plan to develop an in-depth study of such equations in a wide range of directions including existence and uniqueness of solutions, modulus of continuity in space and time, rough initial data, regularity of the density of the solution, moment estimates, sharp tail estimates, intermittency properties, chaotic behavior and asymptotic analysis when the space or time parameters are large. A second working block deals with the applications of Malliavin calculus in variety of open problems, including the smoothness of the joint density for the solution to a general class of stochastic partial differential equations at a fixed number of spatial points, the analysis of the fractional Bessel processes and the eigenvalues of a matrix-valued fractional Brownian motion, and the computation of the p-variation of divergence integrals with respect to the fractional Brownian motion. The research will focus in the rough case where the Hurst parameter is less than 1/2, and new challenging difficulties appear. A third working block of the proposal is to establish new central limit theorems where the limit is a mixture of Gaussian laws and to obtain rates of convergence and asymptotic error distributions, using techniques of Malliavin calculus. Particular asymptotic problems to be investigated include approximation schemes for stochastic differential equations driven by a fractional Brownian motion, weighted q-variations of the fractional Brownian motion and limit theorems for intersection local times.