Approximation of Random Differential Equations Driven by Physical Brownian Motion with Fast Oscillating Noise

Abstract

We investigate approximation of random differential equations driven by semimartingales satisfying a singularly perturbed Langevin equation with scaled mixing random force. By a diffusion approximation approach, we explore the limit of the rough path lift of this semimartingale, and a universal limit theorem is applied to identify the limit of random differential equation. A structurally parallel proof also applies to establish an iterated weak invariance principle for the mixing random force, which is itself an independent interesting result. We find that, the limit of both of the second-level processes, have the form of iterated integral of Stratonovich form plus an anti-symmetric part which is proportional to the time increment.