Cylindrical Lévy Processes in the Lévy White Noise Approach

该项目涉及研究无限维空间中圆柱Lévy过程的性质，并开发分析随机偏微分方程（SPDE）的技术，其中使用这种过程模拟驾驶噪声。研究的内容将是概率领域的新数学。这些技术的应用包括复杂和/或噪声物理系统的研究，本项目的具体研究重点如下：第一，确定柱Lévy过程与SPDE中常用的其他形式的Lévy噪声，即独立散射随机测度，Lévy片和Lévy白噪声之间的关系。然后，分析一个圆柱Lévy过程的“跳跃”，因为圆柱过程不具有与经典随机过程相同意义上的跳跃，并研究理解这些“跳跃”的后果，例如是否有可能推导出圆柱过程的Lévy-Ito分解。此外，研究哪些框架可以用来分析圆柱Lévy过程的正则性，进而分析以圆柱Lévy过程为驱动噪声的SPDE解的正则性，并发展正则性分析的方法。最后，本文将进一步探讨这些方法在圆柱Lévy过程和由其驱动的随机微分方程研究中的应用，并将利用概率论，随机分析，泛函分析与点集拓扑

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.