Generating diffusions with fractional Brownian motion

Abstract

We study fast / slow systems driven by a fractional Brownian motion $B$ with Hurst parameter $H\in (\frac 13, 1]$. Surprisingly, the slow dynamic converges on suitable timescales to a limiting Markov process and we describe its generator. More precisely, if $Y^\varepsilon$ denotes a Markov process with sufficiently good mixing properties evolving on a fast timescale $\varepsilon \ll 1$, the solutions of the equation $$ dX^\varepsilon = \varepsilon^{\frac 12-H} F(X^\varepsilon,Y^\varepsilon)\,dB+F_0(X^\varepsilon,Y^\varepsilon)\,dt\; $$ converge to a regular diffusion without having to assume that $F$ averages to $0$, provided that $H< \frac 12$. For $H > \frac 12$, 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 ($H= \frac 12$).