Diffusive and rough homogenisation in fractional noise field

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.