  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
($H= \frac 12$).
