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 $\mathbb Z^d$. That is, the probability that the random walk jumps from a site $x$ to a nearest neighboring site $x+e$ is given by $p(e)+\epsilon \xi(x,e)$, where $p(e)$ is deterministic, $\{\{\xi(x,e):|e|_1=1\}:x\in\mathbb Z^d\}$ are i.i.d. and $\epsilon>0$ is a parameter which is eventually chosen small enough. We establish an asymptotic expansion in $\epsilon$ 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 $\epsilon$ for the invariant measure of random perturbations of the simple symmetric random walk in dimensions $d=2$.