## Electronic Journal of Probability

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

#### Abstract

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

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

Dates
Accepted: 7 May 2019
First available in Project Euclid: 21 June 2019

https://projecteuclid.org/euclid.ejp/1561082669

Digital Object Identifier
doi:10.1214/19-EJP317

Zentralblatt MATH identifier
07088998

#### Citation

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

#### References

• [1] Balan, R. M. SPDEs with $\alpha$-stable Lévy noise: a random field approach. Int. J. Stoch. Anal., 22 pp. Art. ID 793275, 22, 2014.
• [2] Barndorff-Nielsen, O. E. and Basse-O’Connor, A. Quasi Ornstein-Uhlenbeck processes. Bernoulli, 17(3):916–941, 2011.
• [3] Chong, C. Stochastic PDEs with heavy-tailed noise. Stochastic Process. Appl., 127(7):2262–2280, 2017.
• [4] Chong, C. and Klüppelberg, C. Integrability conditions for space-time stochastic integrals: theory and applications. Bernoulli, 21(4):2190–2216, 2015.
• [5] Cohn, Measurable choice of limit points and the existence of separable and measurable processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 22:161–165, 1972.
• [6] Conus, D. The Non-linear Stochastic Wave Equation in High Dimensions: Existence, Hölder-continuity and Itô-Taylor Expansion. PhD thesis, EPFL, 2008.
• [7] Dalang, R., Khoshnevisan, D., Mueller, C., Nualart, D., and Xiao, Y. A minicourse on stochastic partial differential equations. Held at the University of Utah, Salt Lake City, UT, May 8–19, 2006. Edited by D. Khoshnevisan and F. Rassoul-Agha. Vol. 1962 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2009.
• [8] Dalang, R. C. Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s. Electron. J. Probab., 4:no. 6, 29 pp. (electronic), 1999.
• [9] Fageot, J. and Humeau, T. Unified View on Lévy White Noises: General Integrability Conditions and Applications to Linear SPDE. ArXiv e-prints, (1708.02500), August 2017.
• [10] Gel’fand, I. M. and Vilenkin, N. Y. Generalized functions. Vol. 4. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977].
• [11] Humeau, Th. Stochastic partial differential equations driven by Lévy white noises: Generalized random processes, random field solutions and regularity. PhD. Thesis no.8223, École Polytechnique Fédérale de Lausanne, Switzerland (2017).
• [12] Kallenberg, O. Foundations of modern probability. Springer-Verlag, New York, 2nd edition, 2002.
• [13] Khoshnevisan, D. Analysis of stochastic partial differential equations, volume 119 of CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, RI, 2014.
• [14] Kumar, U. and Riedle, M. The stochastic Cauchy problem driven by a cylindrical Lévy process. Preprint (2018). arXiv:1803.04365v1
• [15] Lebedev, V. A. Fubini’s theorem for parameter-dependent stochastic integrals with respect to $L^{0}$-valued random measures. Teor. Veroyatnost. i Primenen., 40(2):313–323, 1995.
• [16] Mizohata, S. The theory of partial differential equations. Cambridge University Press, 1973.
• [17] Peszat, S. and Zabczyk, J. Stochastic partial differential equations with Lévy noise. An evolution equation approach. Encyclopedia of Mathematics and its Applications, 113. Cambridge University Press, Cambridge, 2007.
• [18] Rajput, B. S. and Rosiński, J. Spectral representations of infinitely divisible processes. Probab. Theory Related Fields, 82(3):451–487, 1989.
• [19] Rosiński, J. On path properties of certain infinitely divisible processes. Stochastic Process. Appl., 33(1):73–87, 1989.
• [20] Samorodnitsky, G. and Taqqu, M. S. Stable non-Gaussian random processes: Stochastic models with infinite variance. Stochastic Modeling. Chapman & Hall, 1994.
• [21] Sato, K.-I. Lévy processes and infinitely divisible distributions. Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999.
• [22] Schwartz, L. Théorie des distributions. Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. IX-X. Hermann, Paris, 1966.
• [23] Stein, E. M. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.
• [24] Trèves, F. Topological vector spaces, distributions and kernels. Academic Press, New York-London, 1967.
• [25] Walsh, J. B. An introduction to stochastic partial differential equations. In École d’été de probabilités de Saint-Flour, XIV—1984, volume 1180 of Lecture Notes in Math., pp. 265–439. Springer, Berlin, 1986.
• [26] Wheeden, R. L. and Zygmund, A. Measure and integral. An introduction to real analysis. Pure and Applied Mathematics, Vol. 43. Marcel Dekker, Inc., New York-Basel, 1977.