Additive functionals as rough paths

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 correction the Lévy area can be described terms asymmetry (nonreversibility) underlying process. apply this abstract result three model problems: First, we study random walks conductances annealed law. If Itô path, then see iterated integrals even though process is reversible. there no correction. The second example nonreversible Ornstein–Uhlenbeck process, while last diffusion periodic environment. As technical step, prove an estimate for p-variation stochastic respect martingales viewed as extension Burkholder–Davis–Gundy inequality local martingale paths (In Séminaire de Probabilités XLI (2008) 421–438 Springer; In Probability Analysis Interacting Physical Systems (2019) 17–48 J. Differential Equations 264 (2018) 6226–6301) case where only integrator martingale.