We investigate the stationary diffusion equation with a coefficient given by a (transformed) Levy random field. Levy random fields are constructed by smoothing Levy noise fields with kernels from the Matern class. We show that Levy noise naturally extends Gaussian white noise within Minlos theory of generalized random fields. Results on the distributional path spaces of Levy noise are derived as well as the amount of smoothing to ensure such distributions become continuous paths. Given this, we derive results on the pathwise existence and measurability of solutions to the random boundary value problem (BVP). For the solutions of the BVP we prove existence of moments (in the $H^1$-norm) under adequate growth conditions on the Levy measure of the noise field. Finally, a kernel expansion of the smoothed Levy noise fields is introduced and convergence in $L^n$ ($ngeq 1$) of the solutions associated with the approximate random coefficients is proven with an explicit rate.
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