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Modelling Lévy space-time white noises

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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
Accepted/In press5 Apr 2021
E-pub ahead of print6 May 2021
PublishedOct 2021

Bibliographical note

Funding Information: The authors thank the referees for some valuable comments and helpful suggestions to improve the?presentations. Publisher Copyright: © 2021 The Authors. Journal of the London Mathematical Society is copyright © London Mathematical Society. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

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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.

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