  With recently developed tools, we prove a homogenisation theorem for a random
ODE with short and long-range dependent fractional noise. The effective
dynamics are not necessarily diffusions, they are given by stochastic
differential equations driven simultaneously by stochastic processes from both
the Gaussian and the non-Gaussian self-similarity universality classes. A key
lemma for this is the `lifted' joint functional central and non-central limit
theorem in the rough path topology.
