Analysis of complex random systems that evolve in space and time

我是一个概率论工作的基础上随机过程的问题，使用工具从分析。这些问题的动机是物理学或金融学中的应用，例如随机扰动下弦的振动，或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，用于分析复杂随机现象的系统建模的行为。我建议的研究计划侧重于两个不同领域的问题：（一）随机分析;（二）重尾时间序列。 (I)随机偏微分方程（英语：Stochastic partial differential equations，缩写为SPDE）是一种数学对象，用于模拟在空间和时间中同时演化的物理现象的行为，并且受到随机扰动（噪声）的影响。他们的研究需要随机分析工具（伊藤演算或马利亚文演算）。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色噪声（布朗运动的时空推广）的扰动，并且只有在空间维度1上才有无规场解。我的研究计划的目标是发现和研究新的属性的解决方案，波和热方程在更高的维度，扰动更一般的类噪声过程，作为更灵活的替代高斯白色噪声。这些结果将提供新的观点之间的动态相互作用的规律性的噪声和随机场的解决方案所表现出的属性，从而更深入地了解的效果的噪声的解决方案的行为。这些调查将构成重大进展的理论SPDE，提供了坚实的数学理由，某些物理现象。 (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模型。引入了多元正则变分的概念来描述高维中的类似行为。当我们在一个固定的时间间隔（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，我将推进与各种时间序列模型相关的点过程的渐近理论，这些模型的值在无限维函数空间中，并将应用此理论导出有关此类序列中变量的部分和或部分最大值的新结果。这些结果将提供重要的新见解的极值理论的时间序列模型，在时间和空间上的演变，并可用于各种应用，如预测的时刻和位置，在臭氧水平超过一个给定的阈值。

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.