Boundary estimates and Green function's expansion for elliptic systems with random coefficients

Abstract

We investigate boundary estimates for elliptic operators with stationary random coefficients exhibiting integrable correlations, arising from stochastic homogenization theory. As practical applications, we establish decay estimates for Green functions in both quenched and annealed senses. Furthermore, we derive notable annealed estimates for boundary correctors, including central limit theorem (CLT)-scaling type estimates. By extending the lemma of Bella, Giunti, and Otto [10] to accommodate boundary conditions, we ultimately obtain error estimates for the two-scale expansion of Green functions at the level of mixed derivatives, thereby establishing connections to other related fields.