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The domain of definition of the Lévy white noise

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  • Fageot, Julien
  • Humeau, Thomas

Abstract

It is possible to construct Lévy white noises as generalized random processes in the sense of Gel’fand and Vilenkin, or as an independently scattered random measures introduced by Rajput and Rosinski. In this article, we unify those two approaches by extending the Lévy white noise Ẋ, defined as a generalized random process, to an independently scattered random measure. We are then able to give general integrability conditions for Lévy white noises, thereby maximally enlarging their domain of definition. Based on this connection, we provide new criteria for the practical determination of the domain of definition, including specific results for the subfamilies of Gaussian, symmetric-α-stable, generalized Laplace, and compound Poisson white noises. We also apply our results to formulate a general criterion for the existence of generalized solutions of linear stochastic partial differential equations driven by a Lévy white noise.

Suggested Citation

  • Fageot, Julien & Humeau, Thomas, 2021. "The domain of definition of the Lévy white noise," Stochastic Processes and their Applications, Elsevier, vol. 135(C), pages 75-102.
    • Handle: RePEc:eee:spapps:v:135:y:2021:i:c:p:75-102
    DOI: 10.1016/j.spa.2021.01.007
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    References listed on IDEAS

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    1. Fageot, Julien & Fallah, Alireza & Unser, Michael, 2017. "Multidimensional Lévy white noise in weighted Besov spaces," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1599-1621.
    2. Berger, David, 2020. "Lévy driven CARMA generalized processes and stochastic partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 130(10), pages 5865-5887.
    3. Adler, R. J. & Monrad, D. & Scissors, R. H. & Wilson, R., 1983. "Representations, decompositions and sample function continuity of random fields with independent increments," Stochastic Processes and their Applications, Elsevier, vol. 15(1), pages 3-30, June.
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    Cited by:

    1. Mendes, R. Vilela, 2024. "On a family of Lévy processes without support in S′," Statistics & Probability Letters, Elsevier, vol. 208(C).

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