Additive functionals as rough paths

TL;DR

This paper demonstrates that additive functionals of stationary Markov processes converge to a rough path version of Brownian motion, with correction terms influenced by the process's reversibility, and applies this to various models.

Contribution

It establishes convergence of additive functionals to rough paths with explicit correction terms, extending rough path theory to non-reversible Markov processes and providing new estimates for stochastic integrals.

Findings

Convergence of additive functionals to rough paths with correction for non-reversibility.

Application to random walks with random conductances, Ornstein-Uhlenbeck, and periodic diffusions.

Extension of rough path inequalities for martingale integrators.

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 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.