Ballistic random walks in random environment as rough paths: convergence and area anomaly

Annealed functional CLT in the rough path topology is proved for standard class of ballistic random walks environment. Moreover, `area anomaly', i.e. a deterministic linear correction second level iterated integral rescaled path, identified terms stochastic area on regeneration interval. The main theorem formulated more general settings, namely any discrete process with uniformly bounded increments which admits structure where times have finite moments. Here largest moment translates into degree regularity topology. In particular, convergence holds $\alpha$-Holder all $\alpha<1/2$ whenever moments are finite, case latter may be compared to special Dirichlet environments away from $1/2$, explicitly corresponding trap parameter.