Analysis of complex random systems that evolve in space and time

我是一个概率学家，研究基于随机过程的问题，使用分析工具。这些问题是由物理学或金融学的应用程序引起的，例如随机扰动下弦的振动，或者市场中表现出极端行为的股票价格的演变。******我的长期目标是开发和实施新的技术来分析系统模拟复杂随机现象的行为。我提出的研究计划侧重于两个不同领域的问题：(I)随机分析；（II）重尾时间序列。*******(I)随机偏微分方程（SPDEs）是一种数学对象，用于模拟在空间和时间上同时进化的物理现象的行为，并受到随机扰动（噪声）的影响。他们的研究需要随机分析工具（伊藤演算或马利文演算）。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，并且仅在空间维度1上具有随机场解。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，受更一般类型的噪声过程的扰动，作为高斯白噪声的更灵活的替代品。这些结果将为噪声的规律性与随机场解所表现出的性质之间的动力学相互作用提供新的视角，从而更深入地理解噪声对溶液行为的影响。这些研究将构成spde理论的重大进展，为某些物理现象提供坚实的数学依据。*******(II)具有重尾（或规则变化）的变量在金融、保险和环境研究的应用中经常遇到，作为表现出极端行为的扰动模型。引入了多变量正则变分的概念来描述高维的类似行为。当我们在固定的时间间隔（或空间区域）内连续观察过程时，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，我将提出与无限维函数空间中具有值的各种时间序列模型相关的点过程的渐近理论，并将该理论应用于该序列中变量的部分和或部分最大值的新结果。这些结果将为在时间和空间中演化的时间序列模型的极值理论提供重要的新见解，并可用于各种应用，例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.******