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We investigate the stationary diffusion equation with a coefficient given bya (transformed) L\'evy random field. L\'evy random fields are constructed bysmoothing L\'evy noise fields with kernels from the Mat\'ern class. We showthat L\'evy noise naturally extends Gaussian white noise within Minlos' theoryof generalized random fields. Results on the distributional path spaces ofL\'evy noise are derived as well as the amount of smoothing to ensure suchdistributions become continuous paths. Given this, we derive results on thepathwise existence and measurability of solutions to the random boundary valueproblem (BVP). For the solutions of the BVP we prove existence of moments (inthe $H^1$-norm) under adequate growth conditions on the L\'evy measure of thenoise field. Finally, a kernel expansion of the smoothed L\'evy noise fields isintroduced and convergence in $L^n$ ($n\geq 1$) of the solutions associatedwith the approximate random coefficients is proven with an explicit rate.

Integrability and Approximability of Solutions to the Stationary Diffusion Equation with L\'evy Coefficient