Normal approximation for a random elliptic equation

• 成果类型: Article
• 署名作者: Nolen, James
• 署名单位: Duke University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-013-0517-9
• 发表日期: 2014
• 页码: 661-700
• 关键词: homogenization; fluctuations; conductance; variance

摘要

We consider solutions of an elliptic partial differential equation in with a stationary, random conductivity coefficient that is also periodic with period . Boundary conditions on a square domain of width 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 , 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 is large. We quantify this approximation by an error estimate in total variation. The error estimate relies on a second order Poincar, inequality developed recently by Chatterjee.