Cylindrical Lévy Processes in the Lévy White Noise Approach

Student thesis: Doctoral ThesisDoctor of Philosophy


We study the regularity properties of cylindrical Lévy processes and Lévy space-time white noises, by examining their embeddings on the one hand in the spaces of general and tempered (Schwartz) distributions, and on the other hand in weighted Besov spaces. In this manner we analyse when the embedded Lévy object possesses a regularised version in the sense of Itô and Nawata.

Lévy space-time white noises are defined as independently scattered random measures and cylindrical Lévy processes are defined by means of the theory of cylin-drical processes. It is shown that Lévy space-time white noises correspond to an entire subclass of cylindrical Lévy processes, which is completely characterised by the characteristic functions of its members. We embed the Lévy space-time white noise, or the corresponding cylindrical Lévy process, in the space of general and tempered distributions and establish that in each case the embedded cylindrical processes are induced by (genuine) Lévy processes in the corresponding space.

We use wavelet analysis to characterise the Lévy measures in weighted Besov spaces. Then we characterise the ranges of such Besov spaces in which L^2(R^d) is or is not embedded continuously and the embedding is or is not Radonifying. We apply these results, given a cylindrical Lévy process L in L^2(R^d), to characterise when L is and is not induced by a Lévy process in a given Besov space. These results are then applied to give sharp Besov regularity analysis to two important classes of cylindrical Lévy processes, the canonical stable cylindrical process, and ‘hedgehog’ processes constructed as a P-a.s. weakly convergent infinite random sum.
Date of Award1 Nov 2022
Original languageEnglish
Awarding Institution
  • King's College London
SupervisorMarkus Riedle (Supervisor) & Eugene Shargorodsky (Supervisor)

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