Series Expansions for Stochastic Partial Differential Equations with Symmetric $$\alpha $$ α -Stable Lévy Noise

Raluca M. Balan () and Juan J. Jiménez ()
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Raluca M. Balan: University of Ottawa
Juan J. Jiménez: University of Ottawa

Journal of Theoretical Probability, 2025, vol. 38, issue 3, 1-63

Abstract: Abstract In this article, we examine a stochastic partial differential equation (SPDE) driven by a symmetric $$\alpha $$ α -stable (S $$\alpha $$ α S) Lévy noise that is multiplied by a linear function $$\sigma (u)=u$$ σ ( u ) = u of the solution. The solution is interpreted in the mild sense. For this model, in the case of Gaussian noise, the solution has an explicit Wiener chaos expansion and is studied using tools from Malliavin calculus. These tools cannot be used for an infinite-variance Lévy noise. In this article, we provide sufficient conditions for the existence of a solution, and we give an explicit series expansion of this solution. To achieve this, we use multiple stable integrals, which were developed in Samorodnitsky and Taqqu (J Theoret Probab 3:267–287, 1990; and in: Cambanis, Samorodnitsky, Taqqu (eds) Stable Stochastic Processes and Related Topics, Progress in Probability, vol 25, Birkhäuser, Boston, 1991) and originate from the LePage series representation of the noise. To give a meaning to the stochastic integral which appears in the definition of the solution, we embed the space–time Lévy noise into a Lévy basis and use stochastic integration theory (Bichteler in Stochastic Integration with Jumps, Cambridge University Press, Cambridge, 2002; and in: Kallianpur (ed) Theory and Applications of Random Fields, Springer, Berlin, pp 1–18, 1983) with respect to this object, as in other studies of SPDEs with heavy-tailed noise (Chong in J Theoret Probab 30:1014–1058, 2017, Stoch Proc Appl 127:2262–2280, 2017; Chong et al. in Stoch PDEs Anal Comput 7:123–168, 2019). As applications, we consider heat and wave equations with linear multiplicative noise, also called parabolic/hyperbolic Anderson models. Chong in J Theoret Probab 30:1014–1058, 2017, Stoch Proc Appl 19 127:2262–2280, 2017, Stoch PDEs Anal Comput 7:123–168, 2019)

Keywords: Stochastic partial differential equations; Random fields; $$\alpha $$ α -stable random measure; Lévy basis; Primary 60H15; Secondary 60G60; 60G52 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10959-025-01433-w

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