  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 \cite{golden2, AvelMajda91} 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 analyzed and various
asymptotic limits of physical interest are studied. Several examples are
presented that confirm the findings of our theory.
