Fast-slow systems with chaotic noise

Fast-slow systems Let Y ̇ = g(Y) be some chaotic ODE with state space Λ and ergodic invariant measure µ. We consider fast-slow systems of the form dX (ε) dt = ε−1h(X (ε),Y (ε)) + f (X (ε),Y (ε)) dY (ε) dt = ε−2g(Y (ε)), where ε 1 and h, f: Re × Λ → Re and ∫ h(·, y) µ(dy) = 0. Also assume that Y (0) ∼ µ. The aim is to characterize the distribution of X (ε) as ε → 0. Fast-slow systems as SDEs Consider the simplified slow equation dX (ε) dt = ε−1h(X (ε))v(Y (ε)) + f (X (ε)) where h: Re → Re×d and v: Λ → Rd with ∫ v(y)µ(dy) = 0. If we write W (ε)(t) = ε−1 ∫ t 0 v(Y (ε)(s))ds then X (ε)(t) = X (ε)(0) + ∫ t 0 h(X (ε)(s))dW (ε)(s) + ∫ t 0 f (X (ε)(s))ds where the integral is of Riemann-Lebesgue type. Invariance principle for W (ε) We can write W (ε) as W (ε)(t) = ε ∫ t/ε2 0 v(Y (s))ds = ε bt/ε2c−1∑ j=0 ∫ j+1 j v(Y (s))ds The assumptions on Y lead to decay of correlations for the sequence ∫ j+1 j v(Y (s))ds. One can show that W (ε) ⇒W in the sup-norm topology, where W is a multiple of Brownian motion. What about the SDE