Averaging of symmetric diffusion in random medium

Abstract

Let a~j(y, ~)), y ~ B d, ~ , be random fields, homogeneous with respect to integral shifts, let a~j = aj~, i, ]= i~ d and let, with probability I, almost everywhere in R d, the condition hold: A_J~l~ <~ a,j(y, ~o ) ~,~ <~ A+ 8 ~ I ~ ( 0 . 1 ) for any ~{~}~R ~ where A+ are positive nonrandom constanLs; here and throughout, repetition of indices is understood as summation from I to d; i~E 2 = ~i~iWe know [I-3] that operators L~u ~t /2 . D,(a,j(g, o ) Dju)', (0 .2) D~ = O/Ox~, g -= x/e, approximate as s + 0 to an "averaged" operator with constant coefficients