Cylindrical Lévy Processes in the Lévy White Noise Approach

本项目致力于研究无限维空间中柱面Lévy过程的性质，并发展分析随机偏微分方程(SPDEs)的技术，其中驾驶噪声是用这样的过程建模的。研究的内容将是概率领域的新数学。这些技术的应用包括研究复杂和/或有噪声的物理系统。本项目的具体研究重点如下：首先，确定柱面Lévy过程与SPDEs中常用的其他形式的Lévy噪声，即独立散布随机测量、Lévy单和Lévy白噪声之间的关系。然后，分析了柱形Lévy过程的“跳跃”，因为柱形过程不存在与经典随机过程相同意义上的跳跃，并研究了对这些“跳跃”的理解会产生什么后果，例如是否有可能得到柱形过程的Lévy-Ito分解。此外，研究了哪些框架可以用来分析柱形Lévy过程的正则性，进而研究了以柱形Lévy过程为驱动噪声的SPDEs解的正则性，并发展了一种正则性分析方法。最后，针对上述问题所开发的方法将被进一步应用于柱形Lévy过程和由此类过程驱动的SPDEs的研究。回答这些数学研究问题并开发新的数学结果的方法将是建立在柱形Lévy过程和SPDEs的现有理论的基础上，利用概率论、随机分析、泛函分析和点集拓扑学的方法。

This project is concerned with researching the properties of cylindrical Lévy processes in infinite dimensional spaces and developing techniques for analysing stochastic partial differential equations (SPDEs) where the driving noise is modelled using such processes. The content of the research will be novel mathematics in the area of probability. Applications of these techniques include the study of complex and/or noisy physical systems.The particular research focus of this project is as follows: first, to determine the relationship between cylindrical Lévy processes and other forms of Lévy noises which are commonly in use in SPDEs, namely independently scattered random measures, Lévy sheets and Lévy-white-noise. Then, to analyse the 'jumps' of a cylindrical Lévy process, as the cylindrical process does not have jumps in the same sense as a classical stochastic process, and to investigate what consequences an understanding of these 'jumps', for example whether it is possible to derive a Lévy-Ito decomposition for the cylindrical process. Furthermore, to investigate which frameworks can be used to analyse the regularity of cylindrical Lévy processes, and in turn the regularity of solutions to SPDEs with a cylindrical Lévy process as the driving noise, and to develop a methodology for regularity analysis. Finally, further applications of the methods developed for the above questions in the study of cylindrical Lévy processes and SPDEs driven by such processes will be sought.The approach that will be taken to answer these mathematical research questions and develop novel mathematical results will be to build upon the current theory of cylindrical Lévy processes and SPDEs utilising the methods of Probability Theory, Stochastic Analysis, Functional Analysis and Point-set Topology.