Pathwise regularity of solutions for a class of elliptic SPDEs with symmetric Lévy noise

2025/07/16 by Juan José de la Vega Jiménez, Jiménez, Juan J.
Economics, Econometrics and Finance · #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2507.12656

openalex publication_date 2025/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, we investigate the existence and uniqueness of random-field solutions to the elliptic SPDE -Lu=ξ on a bounded domain D with Dirichlet boundary conditions u=0 on ∂ D, driven by symmetric Lévy noise ξ. Under general sufficient conditions on the coefficients of the second-order operator L, we prove the existence of a mild solution via the corresponding Green's function and show that the same framework applies to the spectral fractional Laplacian of power γ∈ (0,∞). In particular, whenever γ>\tfracd2, the solution admits a continuous modification.

Citations

• Second order elliptic partial differential equations driven by L 'evy\n white noise
• Power variations in fractional Sobolev spaces for a class of parabolic stochastic PDEs
• Random field solutions to linear SPDEs driven by symmetric pure jump Lévy space-time white noises
• A general framework for SPDE-based stationary random fields
• Systems of stochastic Poisson equations: hitting probabilities
• Stochastic integrals for spde's: a comparison
• Uniform Modulus of Continuity of Random Fields
• The Green function for uniformly elliptic equations

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• Isolated Singularities and Measure Data Problems for Semilinear Equations Driven by Stable Lévy Operators