Modelling Lévy space-time white noises

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Abstract

In this work, we compare Lévy space-time white noises and cylindrical Lévy processes. Lévy space-time white noises are defined as infinitely divisible independently scattered random measures and cylindrical Lévy processes are defined by means of the theory of cylindrical processes.
It is shown that Lévy space-time white noises correspond to an entire sub-class 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. For the latter case, we show that this embedding is possible if and only if a certain integrability condition is satisfied. We establish that both embedded cylindrical processes are induced by genuine Lévy processes in the space of general or tempered distributions. We complete the picture by establishing Lévy space-time white noise as the weak derivative of Lévy additive sheets.
Original languageEnglish
Pages (from-to)1452-1474
Number of pages23
JournalJOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES
Volume104
Issue number3
Early online date6 May 2021
DOIs
Publication statusPublished - Oct 2021

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