Fluctuations in the homogenization of semilinear equations with random potentials. Issue 12 (1st December 2016)

• Record Type: Journal Article
• Authors: Bal, Guillaume; Jing, Wenjia
• Is Part Of: Communications in partial differential equations. Volume 41:Issue 12(2016)
• Page Start: 1839
• Page End: 1859
• Publication Date: 2016-12-01
• Subjects: Probability measure in Hilbert spaces -- random fields -- semilinear elliptic equation -- stochastic homogenization -- variational problem; 35B27 -- 35J61 -- 60F05; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2016.1238482
• Languages: English
• ISSNs: 0360-5302

Abstract

ABSTRACT: We study the stochastic homogenization and obtain a random fluctuation theory for semilinear elliptic equations with a rapidly varying random potential. To first order, the effective potential is the average potential and the nonlinearity is not affected by the randomness. We then study the limiting distribution of the properly scaled homogenization error (random fluctuations) in the space of square integrable functions, and prove that the limit is a Gaussian distribution characterized by homogenized solution, the Green's function of the linearized equation around the homogenized solution, and by the integral of the correlation function of the random potential. These results enlarge the scope of the framework that we have developed for linear equations to the class of semilinear equations.