The Annals of Probability

Lévy processes and Lévy white noise as tempered distributions

Robert C. Dalang and Thomas Humeau

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We identify a necessary and sufficient condition for a Lévy white noise to be a tempered distribution. More precisely, we show that if the Lévy measure associated with this noise has a positive absolute moment, then the Lévy white noise almost surely takes values in the space of tempered distributions. If the Lévy measure does not have a positive absolute moment of any order, then the event on which the Lévy white noise is a tempered distribution has probability zero.

Article information

Ann. Probab., Volume 45, Number 6B (2017), 4389-4418.

Received: September 2015
Revised: October 2016
First available in Project Euclid: 12 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G51: Processes with independent increments; Lévy processes
Secondary: 60G60: Random fields 60G20: Generalized stochastic processes 60H40: White noise theory

Lévy white noise Lévy process Lévy random field tempered distribution positive absolute moment


Dalang, Robert C.; Humeau, Thomas. Lévy processes and Lévy white noise as tempered distributions. Ann. Probab. 45 (2017), no. 6B, 4389--4418. doi:10.1214/16-AOP1168.

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