Normal approximation for a random elliptic equation

• Published in: Probability theory and related fields Vol. 159; no. 3-4; pp. 661 - 700
• Main Author: Nolen, James
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2014; Springer Nature B.V
• Subjects: Approximation; Boundary conditions; Domains; Economics; Errors; Estimates; Finance; Homogenization; Insurance; Management; Mathematical analysis; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Partial differential equations; Probability; Probability Theory and Stochastic Processes; Quantitative Finance; Random variables; Statistics for Business; Studies; Texts; Theoretical; Vibration; 35B27; 35J15; 60F05; 60H25
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-013-0517-9

We consider solutions of an elliptic partial differential equation in R d with a stationary, random conductivity coefficient that is also periodic with period L . Boundary conditions on a square domain of width L are arranged so that the solution has a macroscopic unit gradient. We then consider the average flux that results from this imposed boundary condition. It is known that in the limit L → ∞ , this quantity converges to a deterministic constant, almost surely. Our main result is that the law of this random variable is very close to that of a normal random variable, if the domain size L is large. We quantify this approximation by an error estimate in total variation. The error estimate relies on a second order Poincaré inequality developed recently by Chatterjee.