Asymptotic expansion of the invariant measure for ballistic random walk in the low disorder regime

We consider a random walk in random environment in the low disorder regime on Zd, that is, the probability that the random walk jumps from a site x to a nearest neighboring site x+e is given by p(e)+..(x,e), where p(e) is deterministic, {{.(x,e):|e|1=1}:x.Zd} are i.i.d. and .>0 is a parameter, which is eventually chosen small enough. We establish an asymptotic expansion in . for the invariant measure of the environmental process whenever a ballisticity condition is satisfied. As an application of our expansion, we derive a numerical expression up to first order in . for the invariant measure of random perturbations of the simple symmetric random walk in dimensions d=2.