Fluctuations from a random fractional averaging limit

We consider a system of multiscale stochastic differential equations whose slow component is driven by a fractional Brownian motion with Hurst parameter H > 1 2. Under ergodic assumptions ensuring the applicability of the fractional averaging and fractional homogenization theorems [HL20, HL22], we establish a fluctuation result. The deviation of the slow motion, scaled by ε 1 2 −H , from its effective, time-dependent random limit converges, as the time-separation scale ε → 0, to the solution of a stochastic differential equation driven by a fractional Brownian motion and influenced by an additional space-time Gaussian field. Since the averaging principle and the fractional homogenization hold in different modes of convergence, obtaining the required joint convergence is a delicate matter. Moreover, neither the continuity of the Itô-Lyons solution map nor the martingale method is directly applicable for our purposes, so the proof requires several innovations. To establish the fluctuation theorem, we combine cumulant methods with a residue lemma and formulate the enlarged system as a rough differential equation in a suitable space.