Cylindrical Lévy Processes in the Lévy White Noise Approach

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.