Analysis of complex random systems that evolve in space and time

我是一个概率论者，使用分析中的工具，研究基于随机过程的问题。这些问题是由物理学或金融学中的应用引起的，例如，弦在随机扰动下的振动，或者表现出极端行为的市场中股票价格的演变。*我的长期目标是开发和实施新的技术，用于分析模拟复杂随机现象的系统的行为。我建议的研究计划集中于两个不同领域的问题：(I)随机分析；和(Ii)重尾时间序列。*(I)随机偏微分方程(SPDEs)是用于模拟物理现象的行为的数学对象，这些现象在空间和时间上同时演化，并受到随机扰动(噪声)的影响。他们的研究需要随机分析(伊藤微积分或Malliavin微积分)的工具。基本的例子是波动方程和热方程。在经典理论中，这些方程受到高斯白噪声(布朗运动的时空推广)的扰动，并且只有空间维1的随机场解。我的研究计划的目标是发现和研究高维波动和热方程解的新性质，这些方程受到更一般类别的噪声过程的扰动，作为高斯白噪声的更灵活的替代。这些结果将为噪声的规律性和随机场解所表现出的性质之间的动态相互作用提供新的视角，从而加深对噪声对解行为的影响的理解。这些研究将是对SPDEs理论的重大进步，为某些物理现象提供了坚实的数学证明。*(Ii)在金融、保险和环境研究的应用中，经常会遇到具有重尾(或规则变化)的变量，作为表现出极端行为的扰动的模型。引入了多元正则变差的概念来描述高维上的相似行为。当我们在一段固定的时间间隔(或空间中的一个区域)上连续观察过程时，我们需要一个类似于多元正则变分理论的无限维理论。在这个程序中，我将提出与各种时间序列模型有关的点过程的渐近理论，这些模型的值在无穷维函数空间中，并将应用该理论来推导关于这类序列中变量的部分和或部分最大值的新结果。这些结果将为时间序列模型的极值理论提供重要的新见解，这些模型在时间和空间上演变，并可用于各种应用，如预测臭氧水平超过给定阈值的时刻和位置。

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.******