Averaging of random operators

For the equation with rapidly oscillating coefficients in a bounded domain $\mathscr O\subset\mathbf R^n$
$$ \sum_{i,j=1}^n\frac\partial{\partial x_i}a_{ij}\biggl(\frac x\varepsilon\biggr),\qquad u_\varepsilon(x)|_{\partial\mathscr O}=f_1(x), $$
where $(a_{ij}(y))$ form homogeneous random fields, an averaged equation of the form
$$ \sum_{i,j=1}^nq_{ij}\frac{\partial^2} {\partial x_i\partial x_j}u_0(x)=f(x),\qquad u_0(x)|_{\partial\mathscr O}=f_1(x), $$
is constructed with coefficients $q_{ij}$ which do not depend on $x$; various applications of this result are also obtained.
Bibliography: 22 titles.