Fast-slow systems with chaotic noise

Fast-slow systems Let dYdt = g(Y) be some ‘mildly chaotic ’ ODE with state space Λ and ergodic invariant measure µ. (eg. 3d Lorenz equations.) We consider fast-slow systems of the form dX dt = εh(X,Y) + ε2f (X,Y) dY dt = g(Y), where ε 1 and h, f: Rn × Λ → Rn and ∫ h(x, y) µ(dy) = 0. Our aim is to find a reduced equation dX̄dt = F (X ̄ ) with X ̄ ≈ X. Fast-slow systems If we rescale to large time scales we have dX (ε) dt = ε−1h(X (ε),Y (ε)) + f (X (ε),Y (ε)) dY (ε) dt = ε−2g(Y (ε)), We turn X (ε) into a random variable by taking Y (0) ∼ µ. The aim is to characterise the distribution of the random path 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: Rn → Rn×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 (dW (ε) = dW (ε) ds ds). 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. For very general classes of chaotic Y, it is known that W (ε) ⇒W in the sup-norm topology, where W is a multiple of Brownian motion. What about the SDE