A sharp bound on the l² norm of the solution of a random elliptic difference equation
Tomasz Komorowski, Lenya Ryzhik · Communications in Mathematical Sciences · 2011
We consider a stationary solution of the Poisson equation (λ + L ω )φ λ (x;ω) = -∂ * b(x;ω), where λ > 0 and L ω is a random, discrete, elliptic operator given by L ω u(x) := ∂ * [a(x;ω)∂u(x)], x ∈ Z.Here ∂f (x) := f (x + 1)f (x) and ∂ * f (x) := f (x -1)f (x) for an arbitrary function f : Z → R. The coefficients {(a(x;ω),b(x;ω)), x ∈ Z} form a stationary random field over a probability space (Ω,F ,P).We prove that if the field of coefficients is sufficiently strongly mixing then φ λ (0) P -the L 2 norm of with respect to the probability measure P -behaves as Ĉλ -1/4 , as λ ≪ 1 for some constant Ĉ > 0. In addition ∂φ λ (0) -∂φ 0 (0) P ≤ Cλ 1/4 for λ ∈ (0,1] and some constant C > 0. These results complement those of [A.Gloria, F. Otto, preprint, 2010] and [J.C.Mourrat, preprint, 2010] that hold for an analogous problem in the multidimensional setting.