Summary: A detailed study of various distinguished limits of the Green-Kubo formula for the self-diffusion coefficient is presented in this paper. First, an alternative representation of the Green-Kubo formula in terms of the solution of a Poisson equation is derived when the microscopic dynamics is Markovian. Then the techniques developed in [M. Avellaneda and A. Majda, Commun. Math. Phys. 146, No. 1, 139–204 (1992; Zbl 0754.76046)] are used to obtain a Stieltjes integral representation formula for the symmetric and antisymmetric parts of the diffusion tensor. The effect of irreversible microscopic dynamics on the diffusion coefficient is analysed and various asymptotic limits of physical interest are studied. Several examples are presented that confirm the findings of our theory.