Fast-slow systems with chaotic noise

Fast-slow systems We consider fast-slow systems of the form dX dt = εh(X,Y) + ε2f (X,Y) dY dt = g(Y), where ε 1. dY dt = g(Y) be some mildly chaotic ODE with state space Λ and ergodic invariant measure µ. (eg. 3d Lorenz equations.) 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 dXε dt = ε−1h(Xε,Yε) + f (Xε,Yε) dYε dt = ε−2g(Yε), We turn Xε into a random variable by taking Y (0) ∼ µ. The aim is to characterise the distribution of the random path Xε as ε → 0. Why is model reduction important? 1- The reduced model is lower dimensional and less stiff than the original fast-slow system. 2- Helps the user make informed guess when the model is unknown. Fast-slow systems as SDEs Consider the simplified slow equation dXε dt = ε−1h(Xε)v(Yε) + f (Xε) where h: Rn → Rn×d and v: Λ → Rd with ∫ v(y)µ(dy) = 0. If we write Wε(t) = ε −1 ∫ t 0 v(Yε(s))ds then Xε(t) = Xε(0) + ∫ t 0 h(Xε(s))dWε(s) + ∫ t 0 f (Xε(s))ds where the integral is of Riemann-Stieltjes type (dWε = dWε ds ds). Invariance principle for Wε We can write Wε as Wε(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 Wε ⇒W in the sup-norm topology, where W is a multiple of Brownian motion. We will call this class of Y mildly chaotic. What about the SDE