The authors study the limit behaviour (as \(\varepsilon \to 0\)) of the solutions \(u_{\varepsilon}\) of the following Cauchy problems \[ {{\partial u_{\varepsilon}}\over{\partial t}} (x,t) - \text{div} \big( a (\varepsilon^{-1}x, \xi_{\varepsilon^{-2}t}) \cdot \nabla_x u_{\varepsilon}(x,t)\big) - \varepsilon^{-1} b(\varepsilon^{-1}x, \xi_{\varepsilon^{-2}t}) \cdot \nabla_x u_{\varepsilon}(x,t) = 0, \quad u_{\varepsilon}(x,0) = u_0(x), \] where \(\xi_{\cdot}\) is an ergodic stationary random process. About the coefficients the authors suppose that \(a_{ij}(z,y)\) and \(b_i(z,y)\) are periodic with respect to the variable \(z\), the matrix \(a\) is definite positive and there is a constant \(C>0\) such that \(| a_{ij}(z,y)| +| \nabla_z a_{ij}(z,y)| +| \nabla_y a_{ij}(z,y)| \leqslant C\) and \(| b_i(z,y)| +| \nabla_z b_{i}(z,y)| +| \nabla_y b_{i}(z,y)| \leqslant C\). Moreover an assumption about the process is made (a summability assumption about the strong mixing coefficient, uniform mixing coefficient, maximum correlation coefficient). The main result is a convergence result, in properly chosen moving coordinates, of the distributions of the solutions \(u_{\varepsilon}\) to the solution of a stochastic partial differential equation. Under other properly chosen moving coordinates, the solutions converge almost surely, in a weak topology, to the solution of the equation \({{\partial u^0}\over{\partial t}} - \sum_{i,j=1}^n \overline{a}_{ij} {{\partial^2 u^0}\over{\partial x_i \partial x_j}} = 0\) for some constant coefficients \(a_{ij}\). At the end operators with diffusion coefficients are considered.