Additive functionals as rough paths

Abstract

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 Lévy area that can be described in terms of the asymmetry (nonreversibility) 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 Itô 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 nonreversible 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 inequality for local martingale rough paths of (In Séminaire de Probabilités XLI (2008) 421–438 Springer; In Probability and Analysis in Interacting Physical Systems (2019) 17–48 Springer; J. Differential Equations 264 (2018) 6226–6301) to the case where only the integrator is a local martingale.