Nonlinear effects within invariance principles

We study invariance principles in the rough path topology for stationary discrete and continuous time processes. Simple second moment computations give explicit Green-Kubo-type formulas for both the covariance and the linear correction, called the area anomaly, of the iterated integral of the limiting Brownian motion. A key observation is that the area anomaly admits a structural description: it is the antisymmetric part of the relevant Green–Kubo expression. This provides a unified explanation of area corrections that appear in several different frameworks, including Markovian, regenerative, and deterministic fast-slow models, and highlights a common mechanism underlying results that previously arose from model-specific arguments.