Electronic Journal of Probability

Random field solutions to linear SPDEs driven by symmetric pure jump Lévy space-time white noises

Robert C. Dalang and Thomas Humeau

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We study the notions of mild solution and generalized solution to a linear stochastic partial differential equation driven by a pure jump symmetric Lévy white noise, with symmetric $\alpha $-stable Lévy white noise as an important special case. We identify conditions for existence of these two kinds of solutions, and, together with a new stochastic Fubini theorem, we provide conditions under which they are essentially equivalent. We apply these results to the linear stochastic heat, wave and Poisson equations driven by a symmetric $\alpha $-stable Lévy white noise.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 60, 28 pp.

Received: 14 September 2018
Accepted: 7 May 2019
First available in Project Euclid: 21 June 2019

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Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 60G60: Random fields 60G51: Processes with independent increments; Lévy processes

linear stochastic partial differential equation Lévy white noise generalized stochastic process mild solution stochastic Fubini theorem $\alpha $-stable noise

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Dalang, Robert C.; Humeau, Thomas. Random field solutions to linear SPDEs driven by symmetric pure jump Lévy space-time white noises. Electron. J. Probab. 24 (2019), paper no. 60, 28 pp. doi:10.1214/19-EJP317. https://projecteuclid.org/euclid.ejp/1561082669

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