We study fast / slow systems driven by a fractional Brownian motion B with Hurst parameter H∈(13,1]. Surprisingly, the slow dynamic converges on suitable timescales to a limiting Markov process and we describe its generator. More precisely, if Yε denotes a Markov process with sufficiently good mixing properties evolving on a fast timescale ε≪1, the solutions of the equation
dXε=ε12−HF(Xε,Yε)dB+F0(Xε,Yε)dt
converge to a regular diffusion without having to assume that F averages to 0, provided that H<12. For H>12, a similar result holds, but this time it does require F to average to 0. We also prove that the n-point motions converge to those of a Kunita type SDE.
One nice interpretation of this result is that it provides a continuous interpolation between the homogenisation theorem for random ODEs with rapidly oscillating right-hand sides (H=1) and the averaging of diffusion processes (H=12).