Strong convergence to the homogenized limit of elliptic equations with random coefficients II

Abstract

Consider a discrete uniformly elliptic divergence form equation on the d⩾3 dimensional lattice Zd with random coefficients. In Conlon and Spencer [Trans. Amer. Math. Soc., http://www.math.lsa.umich.edu/~conlon/paper/hom10.pdf], rate of convergence results in homogenization and estimates on the difference between the averaged Green's function and the homogenized Green's function for random environments which satisfy a Poincaré inequality were obtained. Here, these results are extended to certain environments in which correlations can have arbitrarily small power law decay. These environments are simply related via a convolution to environments which do satisfy a Poincaré inequality.

Strong convergence to the homogenized limit of parabolic equations with random coefficients

“…Here these results are extended to certain environments in which correlations can have arbitrarily small power law decay. Similar results for discrete elliptic equations were obtained in [12].…”

“…In the present paper we shall prove rate of convergence results in homogenization of the parabolic PDE (1.1) for certain environments that include some Gaussian environments in which Γ is not integrable. To do this we extend the method introduced in [12] for elliptic PDE in divergence form to the parabolic case. The idea is to consider environments defined by a(ω) = ã(ω(0, 0)) where ω :…”

Long range correlation inequalities for massless Euclidean fields

“…In the present paper, we shall prove rate of convergence results in homogenization of the parabolic PDE (1.16) for certain environments that include some Gaussian environments in which Γ is not integrable. To do this, we extend the method introduced in [13] for elliptic PDE in divergence form to the parabolic case. The idea is to consider environments defined by a(ω) =ã(ω(0, 0)) where ω :…”