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

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>d/2, the solution admits a continuous modification.