  We investigate the stationary diffusion equation with a coefficient given by
a (transformed) L\'evy random field. L\'evy random fields are constructed by
smoothing L\'evy noise fields with kernels from the Mat\'ern class. We show
that L\'evy noise naturally extends Gaussian white noise within Minlos' theory
of generalized random fields. Results on the distributional path spaces of
L\'evy 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 L\'evy measure of the
noise field. Finally, a kernel expansion of the smoothed L\'evy noise fields is
introduced and convergence in $L^n$ ($n\geq 1$) of the solutions associated
with the approximate random coefficients is proven with an explicit rate.
