This article corresponds to a course given by F. Otto at the summer school CEMRACS 2013 in Luminy, France. It is based on a contribution by the authors [GO], which is a continuum version of [Gloria & Otto, Annals of Probability, 2011]. It slightly differs from [GO] because the present analysis does not rely on Green's functions and treats periodic approximations. As opposed to [GO], we also treat non-symmetric coefficients.