Excursions from Hyperplanes for the α-stable Lévy Process

Abstract

This thesis advances the theoretical understanding of isotropic α-stable Lévy processes, Brownian motion, and self-similar Markov processes (ssMps) through excursion theory and skew-product representations. New results include deep Wiener-Hopf factorisations and explicit ladder potential measures, as well as the establishment of Hunt-Nagasawa duality for isotropic α-stable processes and transient Brownian motion. An excursion framework relative to hyperplanes is developed, characterizing key distributions at first passage times and exploring conditioned processes. A cylindrical Lamperti-Kiu transform is introduced to represent ssMps as Markov additive processes (MAPs), facilitating the study of fluctuations and excursions relative to hyperplanes and slabs. Additionally, first-entry and first-exit problems for isotropic α-stable processes in infinite strips are analysed, with numerical methods such as Monte Carlo algorithms devised to address high-dimensional integral equations.

Date of Award	7 May 2025
Original language	English
Awarding Institution	University of Bath
Supervisor	Andreas Kyprianou (Supervisor), Juan Carlos Pardo (Supervisor) & Alex Cox (Supervisor)