On Homogenization of Elliptic Equations with Random Coefficients

In this paper, we investigate the rate of convergence of the solution uε of the random elliptic partial difference equation (∇ε∗a(x/ε, ω)∇ε + 1)uε(x, ω) = f(x) to the corresponding homogenized solution. Here x ∈ εZd, and ω ∈ Ω represents the randomness. Assuming that a(x)’s are independent and uniformly elliptic, we shall obtain an upper bound εα for the rate of convergence, where α is a constant which depends on the dimension d ≥ 2 and the deviation of a(x, ω) from the identity matrix. We will also show that the (statistical) average of uε(x, ω) and its derivatives decay exponentially for large x. Submitted to EJP on December 2, 1999. Final version accepted on April 3, 2000.