Integrability and Approximability of Solutions to the Stationary Diffusion Equation with Lévy Coefficient

Abstract

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

On properties and applications of Gaussian subordinated Lévy fields

“…[38] for some applications in finance). For higher-dimensional parameter domains, some extensions of the Gaussian model have been proposed in the literature: The authors of [19] consider (smoothed) Lévy noise fields with high distributional flexibility and continuous realizations in the context of random PDEs. In [10], the authors propose a general random field model which allows for spatial discontinuities with flexible jump geometries.…”

Multilevel Monte Carlo estimators for elliptic PDEs with Lévy-type diffusion coefficient

“…Furthermore, Gaussian random fields with Matérn-type covariance operators have P-almost surely spatial continuous paths. There are some extensions in the literature (see, for example, [11], [29] and [17]).…”