Fast-slow systems with chaotic noise

Transcription

1 Fast-slow systems with chaotic noise David Kelly Ian Melbourne Courant Institute New York University New York NY May 12, 215 Averaging and homogenization workshop, Luminy.

2 Fast-slow systems We consider fast-slow systems of the form where ε 1. dy dt dx dt = εh(x, Y ) + ε2 f (X, Y ) dy dt = g(y ), = g(y ) be some mildly chaotic ODE with state space Λ and ergodic invariant measure µ. (eg. 3d Lorenz equations.) h, f : R n Λ R n and h(x, y) µ(dy) =. Our aim is to find a reduced equation for X.

3 Fast-slow systems If we rescale to large time scales ( ε 2 ) we have dx ε dt dy ε dt = ε 1 h(x ε, Y ε ) + f (X ε, Y ε ) = ε 2 g(y ε ), We turn X ε into a random variable by taking Y () µ. The aim is to characterise the distribution of the random path X ε as ε.

4 Fast-slow systems as SDEs Consider the simplified slow equation dx ε dt = ε 1 h(x ε )v(y ε ) + f (X ε ) where h : R n R n d and v : Λ R d with v(y)µ(dy) =. If we write W ε (t) = ε 1 v(y ε(s))ds then X ε (t) = X ε () + h(x ε (s))dw ε (s) + f (X ε (s))ds where the integral is of Riemann-Stieltjes type (dw ε = dwε ds ds).

5 Invariance principle for W ε We can write W ε as /ε 2 W ε (t) = ε v(y (s))ds = ε t/ε 2 1 j+1 j= 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. We will call this class of Y mildly chaotic.

6 What about the SDE? Since X ε (t) = X ε () + h(x ε (s))dw ε (s) + f (X ε (s))ds This suggest a limiting SDE X (t) = X () + h( X (s)) dw (s) + f ( X (s))ds But how should we interpret dw? Stratonovich? Itô? neither?

7 For additive noise h(x) = I the answer is simple.

8 Continuity with respect to noise (Sussmann 78) Consider X (t) = X () + du(s) + f (X (s))ds, where U is a uniformly continuous path. The above equation is well defined and moreover Φ : U X is continuous in the sup-norm topology. Also works in the multiplicative noise case (h(x )du) but only when U is one dimensional.

9 The simple case (Melbourne, Stuart 11) If the flow is mildly chaotic (W ε W ) then X ε X in the sup-norm topology, where d X = dw + f ( X )ds. In the multiplicative 1d noise case, the limit is Stratonovich d X = h( X ) dw + f ( X )ds.

10 The strategy The solution map takes irregular path space to solution space Φ : W ε X ε If this map were continuous then we could lift W ε W to X ε X.

11 When the noise is both multidimensional and multiplicative, this strategy fails.

12 Ito, Stratonovich and family SDEs are very sensitive wrt approximations of BM. Suppose and define an approximation dx = h(x )dw + f (X )dt dx n = h(x n )dw n + f (X n )dt with some approximation W n of W. Taking n, X n might converge to something completely different to X. It all depends on the approximation W n.

13 Eg. 1 If W n is a step function approximation of W, then X n converges to the Ito SDE dx = h(x )dw + f (X )dt Eg. 2 (Wong-Zakai) If W n is a linear interpolation of W, then X n converges to the Stratonovich SDE dx = h(x ) dw + f (X )dt Eg. 3 (McShane, Sussman) If W n is a higher order interpolation of W, we can get limits which are neither Ito nor Stratonovich.

14 It is not enough to know that W n BM. We need more information.

15 Rough path theory (Lyons 97) Provides a unified definition of a DE driven by a noisy path X (t) = X () + h(x (s))du(s) + h(x (s))ds when the du integral is not well defined. In addition to U we must be given another path U : [, T ] R d d which is (formally) an iterated integral U ij (t) def = U i (s)du j (s). These extra components tells us how to interpret the method of integration.

16 Rough path theory (Lyons 97) Given a rough path U = (U, U) we can construct a solution X (t) = X () + h(x (s))du(s) + h(x (s))ds Eg. 1 If U = W and U = W dw is the Ito iterated integral, then the constructed X is the solution to the Ito SDE. Eg. 2 If U = W and U = W dw is the Stratonovich iterated integral, then the constructed X is the solution to the Stratonovich SDE.

17 Rough path theory (Lyons 97) Most importantly (for us) the map Φ : (U, U) X is an extension of the classical solution map and is continuous with respect to the rough path topology.

18 Convergence of fast-slow systems Returning to the slow variables X ε (t) = X ε () + h(x ε (s))dw ε (s) + f (X ε (s))ds If we let then X ε = Φ(W ε, W ε ). W ij ε (t) = W i ε(r)dw j ε(r) Due to the continuity of Φ, if (W ε, W ε ) (W, W), then X ε X, where X (t) = X () + h( X (s))dw(s) + h( X (s))ds with W = (W, W).

19 Theorem (K. & Melbourne 14) If the fast dynamics are mildly chaotic, then (W ε, W ε ) (W, W) where W is a Brownian motion and W ij (t) = where the integral is Itô type and Cov ij (W ) = λ ij = W i (s)dw j (s) + λ ij t E µ {v i (Y ()) v j (Y (s))} ds. E µ {v i (Y ())v j (Y (s))+v j (Y ()) v i (Y (s)))} ds

20 Homogenized equations Corollary Under the same assumptions as above, the slow dynamics X ε X where d X = h( X )dw + f ( X ) + λ ij k h i ( X )h kj ( X ) dt. i,j,k in Itô form, with λ ij = E µ {v i (Y ()) v j (Y (s))} ds d X = h( X ) dw + f ( X ) + λ ij k h i ( X )h kj ( X ) dt i,j,k in Stratonovich form, with λ ij = E µ {v i (Y ()) v j (Y (s)) v j (Y ()) v i (Y (s))} ds.

21 Proof I : Find a martingale The strategy is to decompose W ε (t) = M ε (t) + A ε (t) where M ε is a good martingale sequence (Kurtz-Protter 92) ( ) ( ) U ε, M ε, U ε dm ε U, W, UdW where the integrals are of Itô type. And A ε uniformly, but oscillates rapidly. Hence A ε is like a corrector.

22 Proof II : Martingale approximation (Gordin 69) Introduce a Poincaré section Λ with Poincaré map T and return times τ j. Write W ε (t) = ε N(ε 2 t) 1 τj+1 j= τ j v(y (s))ds N(ε 2 t) 1 = ε ṽ(t j Y ()) = ε j= N(ε 2 t) 1 j= V j. We have a CLT sum for a stationary random sequence {V j } with natural filtration F j = T j M (where M is the σ-algebra for the Y () probability space )

23 Proof II : Martingale approximation (Gordin 69) Use a martingale approximation to show that ε N ε 1 j V j W. Write V j = M j + (Z j Z j+1 ) where E(M j F j ) =. A good choice (if it converges) is the series Z j = E(V j+k F j ). k= Convergence of this series is guaranteed by decay of correlations for the Poincaré map.

24 Proof II : Martingale approximation (Gordin 69) The good martingale is M ε (t) = ε N ε 1 j= M j and the corrector is A ε (t) = ε(z Z Nε 1). We then get N(ε 2 t) 1 W ε (t) = ε M j + ε(z Z Nε 1) W (t) + j= by Martingale CLT and boundedness of Z. We are sweeping a lot under the rug here since F j F j+1. Need to reverse the martingales.

25 Proof III: Computing the iterated integral To compute W ε we decompose it W ε dw ε = M ε dm ε + M ε da ε + A ε dm ε + A ε da ε Since M ε is a good martingale sequence M ε dm ε W dw A ε dm ε. Even though A ε = O(ε), the iterated term A ε da ε does not vanish. The last two terms are computed as ergodic averages M ε da ε + A ε da ε λt (a.s)

26 Extensions + Future directions The general fast-slow system (with h(x, y)) can be treated with infinite dimensional rough paths (or alternatively, rough flows - Bailleul+Catellier ) Rough path tools can be adapted to address discrete-time fast-slow maps. Fast-slow systems with feedback. Ergodic properties of Y X are poorly understood. Stochastic PDE limits; regularity structures.

27 References 1 - D. Kelly & I. Melbourne. Smooth approximations of SDEs. To appear in Ann. Probab. (214). 2 - D. Kelly & I. Melbourne. Deterministic homogenization of fast slow systems with chaotic noise. arxiv (214). 3 - D. Kelly. Rough path recursions and diffusion approximations. To appear in Ann. App. Probab. (214). All my slides are on my website ( Thank you!