Multiscale stochastic homogenization of monotone operators

Multiscale stochastic homogenization is studied for divergence structure parabolic problems. More specifically we consider the asymptotic behaviour of a sequence of realizations of the form $\frac{\partial u^\omega_\varepsilon}{\partial t}- $div$(a(T_1(\frac{x}{\varepsilon_1})\omega_1, T_2(\frac{x}{\varepsilon_2})\omega_2 ,t, D u^\omega_\varepsilon))=f.$ It is shown, under certain structure assumptions on the random map $a(\omega_1,\omega_2,t,\xi)$, that the sequence $\{u^\omega_\e}$ of solutions converges weakly in $ L^p(0,T;W^{1,p}_0(\Omega))$ to the solution $u$ of the homogenized problem $ \frac{\partial u}{\partial t} - $div$( b( t,D u )) = f$.