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

TL;DR

This paper studies the stationary diffusion equation with a Le9vy-based random coefficient, establishing existence, regularity, and convergence results for solutions, and introduces a kernel expansion for approximation.

Contribution

It provides new theoretical insights into the integrability and approximability of solutions to diffusion equations with Le9vy random coefficients, including convergence rates.

Findings

Existence of solutions with moments in the $H^1$-norm.

Pathwise existence and measurability of solutions.

Convergence of approximate solutions with explicit rates.

Abstract

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.