Fluctuation of Solutions to Linear Elliptic Equations with Noisy Diffusion Coefficients

Abstract

We consider a linear elliptic equation in divergence form on a bounded domain (or on ℝ d ) in dimension d ≥ 2, whose coefficients are perturbed by a stationary noise of correlation length ϵ > 0. We give estimates on the fluctuation of the solution in function of the correlation length ϵ of the noise, both in terms of strong L 2 and weak L 1 norms. This result can be seen as a quantification of the propagation of uncertainties in linear elliptic partial differential equations.