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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.