Pointwise two-scale expansion for parabolic equations with random   coefficients
  We investigate the first-order correction in the homogenization of linear parabolic equations with random coefficients. In dimension $3$ and higher and for coefficients having a finite range of dependence, we prove a pointwise version of the two-scale expansion. A similar expansion is derived for elliptic equations in divergence form. The result is surprising, since it was not expected to be true without further symmetry assumptions on the law of the coefficients. 
Yu Gu and Jean-Christophe Mourrat
math.PR math.AP