Fluctuations in the Homogenization of Semilinear Equations with Random Potentials

TL;DR

This paper develops a stochastic homogenization theory for semilinear elliptic equations with random potentials, characterizing the fluctuations around the homogenized solution as Gaussian and extending previous linear frameworks.

Contribution

It introduces a fluctuation theory for semilinear equations with random potentials, expanding the linear case analysis to nonlinear equations.

Findings

Homogenized potential equals the average potential.

Fluctuations follow a Gaussian distribution.

Limit distribution characterized by Green's function and correlation of the potential.

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 the 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.