  We consider additive functionals of stationary Markov processes and show that
under Kipnis-Varadhan type conditions they converge in rough path topology to a
Stratonovich Brownian motion, with a correction to the Levy area that can be
described in terms of the asymmetry (non-reversibility) of the underlying
Markov process. We apply this abstract result to three model problems: First we
study random walks with random conductances under the annealed law. If we
consider the Ito rough path, then we see a correction to the iterated integrals
even though the underlying Markov process is reversible. If we consider the
Stratonovich rough path, then there is no correction. The second example is a
non-reversible Ornstein-Uhlenbeck process, while the last example is a
diffusion in a periodic environment.
  As a technical step we prove an estimate for the p-variation of stochastic
integrals with respect to martingales that can be viewed as an extension of the
rough path Burkholder-Davis-Gundy inequalities for local martingale rough paths
of Friz et al to the case where only the integrator is a local martingale.
