  We consider a system of differential equations in a fast long range dependent
random environment and prove a homogenization theorem involving multiple
scaling constants. The effective dynamics solves a rough differential equation,
which is `equivalent' to a stochastic equation driven by mixed It\^o integrals
and Young integrals with respect to Wiener processes and Hermite processes.
Lacking other tools we use the rough path theory for proving the convergence,
our main technical endeavour is on obtaining an enhanced scaling limit theorem
for path integrals (Functional CLT and non-CLT's) in a strong topology, the
rough path topology, which is given by a H\"older distance for stochastic
processes and their lifts. In dimension one we also include the negatively
correlated case, for the second order / kinetic fractional BM model we also
bound the error.
