Grant: Discovery Grants Program - Individual

This research program will focus on problems in two distinct areas: (I) Stochastic Partial Differential Equations (SPDEs); (II) Heavy-Tailed Time Series. (I) SPDEs are mathematical objects that are used for modeling the behavior of physical phenomena which are subject to random perturbations. Two fundamental examples are the wave equation and the heat equation. In the classical theory, these equations are perturbed by a Gaussian white noise. When the noise is colored in time (for instance when the noise is fractional in time with Hurst index greater than 1/2), the existence of solutions of the nonlinear equations is an open problem, except in the case of the parabolic/hyperbolic Anderson model (pAm/hAm). The goals of this research program are to develop a unified theory for the study of random models driven by general classes of noise processes. When the noise is white in time and rough in space (i.e. fractional in space with Hurst index less than 1/2), we will prove a new quantitative central limit theorem for the spatial integral of the solution, using tools from Malliavin calculus. Another component of the program will be dedicated to the study of SPDEs driven by colored Levy noise with finite variance, or with heavy-tailed Levy noise. These investigations will constitute significant advancements of the theory of SPDEs and will lead to a deeper understanding of the effect of the noise on the behavior of the solution. Training of HQP is an essential component of this proposal. Current PhD student Liang studies the continuity in law of the solutions of pAm and hAm driven by space-time homogeneous Gaussian noise, with respect to the parameters of the noise. Current PhD student Jimenez investigates SPDEs driven by Levy noise. A future PhD student will investigate the exact asymptotic behavior of the moments of the solution of hAm with colored noise in time. A future MSc student will study the applications of Malliavin calculus in finance. A future undergraduate student will study Black-Scholes formula for the price of an European call option. (II) Distributions with heavy (i.e. regularly varying) tails are encountered frequently in many applications. In higher dimensions, this leads to the concept of multivariate regular variation. This theory has been extended to the infinite-dimensional framework, by considering regular variation for processes with values in the Skorohod space D of right-continuous functions with left limits. A future PhD student will investigate in depth the asymptotic theory for the sum and maximum associated with various time series models based on regularly varying processes with values in D, such as stochastic recurrence models, or bilinear models. A future MSc student will study the asymptotic behavior of the maximum of independent identically distributed random vectors using the concept of multivariate regular variation, and will provide an infinite-dimensional extension of this result.