Mean-field models in the theory of random media. I

It is the first in the series of works treating the problems of the theory of random media on the basis of the mean field (nonlocal) diffusion approximation with the corresponding operator $\overline\Delta_V$, $V\subset\mathbf Z^d$. The general introduction to the whole cycle is presented including a brief survey of problems in the theory of random media. The localization problem for the operator $H_V=\overline\Delta_V+\xi(x)$ is also considered, where $\{\xi(x)\}$ are i. i. d. continious random variables, $|V|\to\infty$. It is proved that the localization in the average (uniformly in $V$) takes place.