Averaging and fluctuations for parabolic equations with rapidly oscillating random coefficients

In this paper we consider the parabolic equation with random coefficients: $$D_t u^\varepsilon (t,x) = \sum\limits_{ij} {a_{ij} } \left( {\frac{t}{\varepsilon }\user2{,}x,\omega } \right)D_{x_i x_j } u^\varepsilon (t\user2{,}x) + \sum\limits_i {b_i } \left( {\frac{t}{\varepsilon }\user2{,}x,\omega } \right)D_{x_i } u^\varepsilon (t\user2{,}x).$$ We show that ue(t,x) converges to the solution uo(t,x) of the averaging equation: $$D_t u^0 (t,x) = \sum\limits_{ij} E \left( {a_{ij} \left( {\frac{t}{\varepsilon }\user2{,}x,\omega } \right)} \right)D_{x_i x_j } u^0 (t\user2{,}x) + \sum\limits_i {E\left( {b_i \left( {\frac{t}{\varepsilon }\user2{,}x,\omega } \right)} \right)} D_{x_i } u^0 (t\user2{,}x).$$ Also, the fluctuation process \(y^\varepsilon (t,x)\left( { \equiv \left( {u^\varepsilon (t,x)--u^0 (t\user2{,}x)} \right)/\sqrt \varepsilon } \right)\) converges weakly to a generalized Ornstein-Uhlenbeck process on L′.