Analysis of complex random systems that evolve in space and time

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I am a probabilist working on problems based on stochastic processes, using tools from analysis. These problems are motivated by applications in physics or finance, for example the vibration of a string under random perturbations, or the evolution of stock prices in markets which exhibit extreme behaviour. ******My long-term objective is to develop and implement novel techniques for analyzing the behaviour of systems modelling complex random phenomena. My proposed research program focuses on problems in two distinct areas: (I) stochastic analysis; and (II) heavy-tailed time series.*******(I) Stochastic partial differential equations (SPDEs) are mathematical objects used for modeling the behaviour of physical phenomena that evolve simultaneously in space and time, and that are subject to random perturbations (noise). Their study requires tools from stochastic analysis (Ito calculus or Malliavin calculus). Fundamental examples are the wave equation and the heat equation. In the classical theory, these equations are perturbed by Gaussian white noise (a space-time generalization of Brownian motion) and have random field solutions only in spatial dimension 1. The goal of my research program is to discover and study new properties of the solutions to the wave and heat equations in higher dimensions, perturbed by more general classes of noise processes, as more flexible alternatives to Gaussian white noise. These results will offer new perspectives on the dynamical interplay between the regularity of the noise and the properties exhibited by the random field solution, leading to a deeper understanding of the effect of the noise on the behaviour of solution. These investigations will constitute significant advances to the theory of SPDEs, offering a solid mathematical justification for certain physical phenomena.*******(II) Variables with heavy (or regularly varying) tails are encountered frequently in applications in finance, insurance and environmental studies, as models for perturbations that exhibit extreme behaviour. The concept of multivariate regular variation was introduced to describe a similar behaviour in higher dimensions. When we observe processes continuously over a fixed interval of time (or a region in space), we need an infinite-dimensional theory analogous to the theory of multivariate regular variation. In this program, I will advance the asymptotic theory for point processes associated with various time series models with values in an infinite-dimensional space of functions, and will apply this theory for deriving new results about the partial sum or partial maximum of the variables in such series. These results will give important new insights into the extreme value theory for time series models which evolve in time and space, and could be used in a variety applications, such as predicting the moment and location at which the ozone level exceeds a given threshold.******