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 1, 216 Statistical properties of dynamical systems, ESI Vienna. David Kelly (CIMS) Fast-slow May 1, / 3

2 Weak invariance principles Suppose we have a deterministic flow φ t : Λ Λ with ergodic invariant measure ν and an observable v : Λ R n. Define ε 2 t W ε (t)(ω) = ε v φ s (ω)ds where ω ν. A weak invariance principle (WIP) describes the convergence result W n w W in C, where W is a multiple of Brownian motion in R n. WIPs are known for large classes of uniformly and non-uniformly hyperbolic flows. David Kelly (CIMS) Fast-slow May 1, / 3

3 Can we use WIPs for homogenization of deterministic fast-slow systems? David Kelly (CIMS) Fast-slow May 1, / 3

4 Fast-slow with additive noise Consider the fast-slow system x ε = ε 1 v(y ε ) + f (x ε ) y ε = ε 2 g(y ε ) where y ε (t) = y(ε 2 t), y() ν and ẏ = g(y) generates a flow φ t that has a WIP. Thm (Melbourne, Stuart 11): x ε w X (in the sup-norm topology) where X satisfies the SDE dx = dw + f (X )dt Can be extended to multiplicative noise x ε = ε 1 h(x ε )v(y ε ) + f (x ε ) provided that h = V, and extended to discrete-time formulations (Gottwald, Melbourne 14). David Kelly (CIMS) Fast-slow May 1, / 3

5 Proof Simple fact: Suppose that W is a uniformly continuous path and that X solves the integral ODE X (t) = X () + W (t) + f (X (s))ds Then the map Φ : W X, (C C ) is continuous. The slow variables satisfy the integral equation x ε (t) = x() + ε 1 v(y ε (s))ds + f (x ε (s))ds = x() + W ε (t) + f (x ε (s))ds By cts mapping thm, if W ε w W then x ε = Φ(W ε ) w Φ(W ) = X. David Kelly (CIMS) Fast-slow May 1, / 3

6 More complicated fast-slow systems? 1 Continuous time with non-trivial products x ε = ε 1 h(x ε )v(y ε ) + f (x ε ) 2 Continuous time with non-products x ε = ε 1 h(x ε, y ε ) + f (x ε, y ε ) 3 Discrete-time with non-trivial products and let x n (t) := x (n) nt x (n) j+1 = x (n) j + n 1/2 h(x (n) j )v(y j ) + n 1 f (x (n) j ) y j+1 = T y j David Kelly (CIMS) Fast-slow May 1, / 3

7 Non-trivial product case x ε = ε 1 h(x ε )v(y ε ) (v : Λ R n, h : R d R d n, take f = ) David Kelly (CIMS) Fast-slow May 1, / 3

8 Since W ε (t) = ε 1 v(y ε(s))ds, we can write the slow variables in the integral form x ε (t) = x() + h(x ε (s))dw ε (s) where dw ε denotes Lebesgue-Stieltjes integration (dw ε = Ẇ ε ds). Can we apply the same continuity trick? The map Φ is defined on BV paths, but W ε converges in a much weaker topology, so we must extend Φ to a bigger space of paths. David Kelly (CIMS) Fast-slow May 1, / 3

9 Attempted construction of Φ First define the map Φ : W X where W is smooth and X (t) = X () + h(x (s))dw (s), with the integral interpreted in Lebesgue-Stieltjes sense. Extend the map Φ to the closure of the space of smooth paths under the C γ Hölder topology, for some γ (1/3, 1/2]. ie. Φ(W ) = lim n Φ(W n ) where W n is a smooth approx of W C γ. This map is not well-posed, the limiting X depends on the choice of sequence W n. David Kelly (CIMS) Fast-slow May 1, / 3

10 Counter-example to well-posedness Let X = (X 1, X 2, X 3 ) be defined by X 1 (t) = dz 1 X 2 (t) = X 3 (t) = X 2 dz 1 dz 2 X 1 dz 2 where Z = (,, ). Since Z is smooth, we should clearly have X = (,, ). But we could equally take the approximation Z n = (n 1 cos(n 2 t), n 1 sin(n 2 t)) for t 2π. Clearly Z n in sup-norm topology (actually in C γ for all γ < 1/2). But X 3 (t) = Z 2 ndz 1 n Thus X n (2π) (,, 2π). Z 1 ndz 2 n = 2 area inside the curve (Z 1 n, Z 2 n) David Kelly (CIMS) Fast-slow May 1, / 3

11 Rough path theory (Lyons 98) Consider the space of pairs (W, W) : [, T ] R n R n n where W is a smooth path and W αβ (t) = W α (s)dw β (s) is the iterated integral of W. The space of γ-rough paths C γ is the closure of the above space under the γ-hölder metric ρ γ (W, W; W, W) = sup s,t where W (s, t) = W (t) W (s) and W(s, t) = s W (s, t) W (s, t) W(s, t) W(s, t) s t γ + sup s,t s t 2γ W (s, r)dw (r). David Kelly (CIMS) Fast-slow May 1, / 3

12 Rough path theory (Lyons 98) Define Φ on the space of smooth pairs by Φ(W, W) = X where X (t) = X () + h(x (s))dw (s) Then extend Φ to rough paths C γ as Φ(W, W) = lim n Φ(W n, W n ), where (W n, W n ) is a smooth approx of (W, W) in C γ. Thm: The map Φ : C γ C is well-defined and continuous. David Kelly (CIMS) Fast-slow May 1, / 3

13 Rough paths for fast-slow systems Corollary: Let x ε solve x ε (t) = x() + h(x ε (s))dw ε (s) where W ε is a smooth and let W ε = W ε dw ε. Suppose that (i) (W ε, W ε ) w (W, W) in C, where W is a BM and W(t) = W dw + λt, where the integral is of Itô type. (ii) (E W ε (s, t) q ) 1/q t s 1/2 and ( E W ε (s, t) 2q) 1/(2q) t s uniformly in s, t, ε with q > 3. Then x ε w X in C where X satisfies the SDE dx = h(x )dw + α,β,κ λ αβ κ h α (X )h βκ (X )dt. David Kelly (CIMS) Fast-slow May 1, / 3

14 Proof of Corollary The iterated WIP + the Kolmogorov estimates yield (W ε, W ε ) w (W, W) in the rough path topology. Since Φ is continuous wrt this topology, we have x ε = Φ(W ε, W ε ) w Φ(W, W). Standard result from rough path theory: if W BM and W(t) = W dw + λt then X = Φ(W, W) satisfies the SDE dx = h(x )dw + α,β,κ λ αβ κ h α (X )h βκ (X )dt. David Kelly (CIMS) Fast-slow May 1, / 3

15 Iterated WIP Suppose that φ t : Ω Ω is a suspension flow, with Poincaré map T that is modeled by a mixing Young tower whose return times have sufficiently decaying tails. Thm (Kelly, Melbourne 16 ): (W ε, W ε ) w (W, W) in C, where W is a BM with covariance Σ, W = W dw + λt and Σ αβ = lim ε E ν W α ε (1)W β ε (1) = and λ αβ = lim ε E ν W αβ ε (1) = ( = holds if the integrals exist) E ν (v α v β φ s + v β v α φ s )ds E ν v α v β φ s ds David Kelly (CIMS) Fast-slow May 1, / 3

16 Corollary (Kelly, Melbourne 16 ) Under the above hypotheses x ε w X in C, where dx = h(x )dw + α,β,κ λ αβ κ h α (X )h βκ (X )dt and λ is as above. Or in Stratonovich form dx = h(x ) dw + α,β,κ θ αβ κ h α (X )h βκ (X )dt where θ αβ = 2λ αβ Σ αβ = E ν (v α v β φ s v β v α ) φ s ds David Kelly (CIMS) Fast-slow May 1, / 3

17 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): If (U ε, M ε ) w (U, W ) then ( ) ( ) 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. David Kelly (CIMS) Fast-slow May 1, / 3

18 Proof II : Martingale approximation Let τ j be the j-th return time for the Poincaré map T, then W ε (t) = ε N(ε 2 t) 1 τj+1 j= τ j N(ε 2 t) 1 = ε ṽ T j + A ε,1 (t) j= where A ε,1 = O(ε) in probability. v φ s ds + A ε,1 (t) Apply martingale-coboundary decomposition ṽ = m + χ T χ N(ε 2 t) 1 W ε (t) = ε m T j + ε(χ T N(ε 2t) χ) + A ε,1 (t) j= = M ε (t) + A ε (t) David Kelly (CIMS) Fast-slow May 1, / 3

19 Proof III: Compute the integrals Decompose the iterated integral W ε dw ε = W ε dm ε + M ε da ε + A ε da ε As a good martingale seqeuence dm ε converges to the Itô integral W dw. By ergodic thm, we can compute A ε dm ε A ε da ε λt Even though A ε uniformly, its iterated integral does not. David Kelly (CIMS) Fast-slow May 1, / 3

20 Non-product case ẋ ε = ε 1 h(x ε, y ε ) + f (x ε, y ε ) h, f : R d Ω R d David Kelly (CIMS) Fast-slow May 1, / 3

21 Non-product = infinite dimensional product Let B be some Banach space of functions ϕ : R d R d. Define the paths W ε, V ε : [, T ] B W ε (t) = ε 1 h(, y ε (s))ds V ε (t) = f (, y ε (s))ds and define the linear functional H : B R d by H(x)ϕ = ϕ(x) (ie. H(x) corresponds to integration against δ x ). Then we have x ε (t) = x() + H(x ε (s))dw ε (s) + H(x ε (s))dv ε (s) and rough path theory works for Banach space valued paths! David Kelly (CIMS) Fast-slow May 1, / 3

22 Thm (Kelly, Melbourne 15). Under the same assumptions on φ t, x ε w X where dx = σ(x )db + b(x )dt where B is a standard BM on R d and b(x) = f (x, y)dν(y) + d B(h k (x, ), k h(x, )) k=1 σσ T (x) = B(h i (x, ), h j (x, )) + B(h j (x, ), h i (x, )) and B is the integrated autocorrelation of the fast dynamics B(v, w) = E ν W ε,v (1)W ε,w (1) = and W ε,v = ε ε 2 t v φ s ds, v : Ω R E ν v w φ s ds David Kelly (CIMS) Fast-slow May 1, / 3

23 Discrete time case x (n) j+1 = x (n) j + n 1/2 h(x (n) j )v(y j ) y j+1 = T y j where T : Λ Λ, ergodic invariant measure µ and let x n (t) := x (n) nt David Kelly (CIMS) Fast-slow May 1, / 3

24 Notice that x n satisfies the summation equation nt 1 x n (t) = x() + h(x n (j/n))n 1/2 v T j j= which we can write as a Left-Riemann sum x n (t) = x() + where W n (t) = n 1/2 nt 1 j= v T j. h(x n (s ))dw n (s) There are large classes of maps for which W n w W, where W is a multiple of BM. Can we use rough path theory? David Kelly (CIMS) Fast-slow May 1, / 3

25 Define the iterated sum W αβ n (t) = n 1 = nt 1 j= j 1 i= W α (s )dw β (s) v α T i v β T j Can we use an iterated WIP for the discrete-time rough path (W n, W n ) to obtain homogenization for x n? David Kelly (CIMS) Fast-slow May 1, / 3

26 Thm (Kelly 15): Let x n solve x n (t) = x() + h(x n (s ))dw n (s) where (W n, W n ) are the cadlag step functions as above. Suppose that (i) (W n, W n ) w (W, W) in the Skorokhod topology, where W is a BM and W(t) = W dw + λt, where the integral is of Itô type. (ii) (E W n (s, t) q ) 1/q t s 1/2 and ( E W n (s, t) 2q) 1/(2q) t s uniformly in s, t {j/n : j n 1} and n 1 with q > 3. Then x ε w X in the Skorokhod topology, where X satisfies the SDE dx = h(x )dw + α,β,κ λ αβ κ h α (X )h βκ (X )dt. David Kelly (CIMS) Fast-slow May 1, / 3

27 Proof ideas I Let W n be a smooth interpolation of the step-function W n (for instance, linear interpolation) and let W n = W n d W n. If we define x n (t) = x() + h( x n (s))d W n (s) + κ h α ( x n (s))h βκ d Z αβ n (s). where Z n = W n W n, then one can show x n x n = o(1) in probability. Similar to backward error analysis for numerical methods. David Kelly (CIMS) Fast-slow May 1, / 3

28 Proof ideas II The assumptions on (W n, W n ) guarantee that ( W n, W n ; Z n ) w (W, W; λt) in the rough path topology, where W = W λt = W dw + (λ λ)t. We obtain x n = Φ( W n, W n ; Z n ) w Φ(W, W λt; λt) = X and hence dx = h(x )dw + α,β,κ(λ λ) αβ κ h α (X )h βκ (X )dt + α,β,κ λ αβ κ h α (X )h βκ (X )dt David Kelly (CIMS) Fast-slow May 1, / 3

29 Suppose T is modeled by a mixing Young tower whose return times have sufficiently decaying tails. Thm (Kelly, Melbourne 16 ) x n w X in C, where dx = h(x )dw + α,β,κ λ αβ κ h α (X )h βκ (X )dt and λ αβ = n 1 E ν v α v β T n David Kelly (CIMS) Fast-slow May 1, / 3

30 References 1 - D. Kelly & I. Melbourne. Smooth approximations of SDEs. Ann. Probab. (216). 2 - D. Kelly & I. Melbourne. Deterministic homogenization of fast slow systems with chaotic noise. arxiv (215). 3 - D. Kelly. Rough path recursions and diffusion approximations. Ann. App. Probab. (215). All my slides and lecture notes from last week s course are/will be on my website ( Thank you! David Kelly (CIMS) Fast-slow May 1, / 3