Lévy Processes and Infinitely Divisible Measures in the Dual of a Nuclear Space

C. A. Fonseca-Mora ()
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C. A. Fonseca-Mora: Universidad de Costa Rica

Journal of Theoretical Probability, 2020, vol. 33, issue 2, 649-691

Abstract: Abstract Let $$\Phi $$Φ be a nuclear space and let $$\Phi '_{\beta }$$Φβ′ denote its strong dual. In this work, we prove the existence of càdlàg versions, the Lévy–Itô decomposition and the Lévy–Khintchine formula for $$\Phi '_{\beta }$$Φβ′-valued Lévy processes. Moreover, we give a characterization for Lévy measures on $$\Phi '_{\beta }$$Φβ′ and provide conditions for the existence of regular versions to cylindrical Lévy processes in $$\Phi '$$Φ′. Furthermore, under the assumption that $$\Phi $$Φ is a barrelled nuclear space we establish a one-to-one correspondence between infinitely divisible measures on $$\Phi '_{\beta }$$Φβ′ and Lévy processes in $$\Phi '_{\beta }$$Φβ′. Finally, we prove the Lévy–Khintchine formula for infinitely divisible measures on $$\Phi '_{\beta }$$Φβ′.

Keywords: Lévy processes; Infinitely divisible measures; Cylindrical Lévy processes; Dual of a nuclear space; Lévy–Itô decomposition; Lévy–Khintchine formula; Lévy measure; 60B11; 60G51; 60E07; 60G20 (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-019-00972-3

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