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) + εξ(x, e), where p(e) is deterministic, {{ξ(x, e) : |e|1 = 1} : x ∈ ℤd} are i.i.d. and ε > 0 is a parameter, which is eventually chosen small enough. We establish an asymptotic expansion in ε 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 ε for the invariant measure of random perturbations of the simple symmetric random walk in dimensions d = 2.