Homogenization for chaotic dynamical systems David Kelly Ian Melbourne Department of Mathematics / Renci Mathematics Institute UNC Chapel Hill University of Warwick November 3, 2013 Duke/UNC Probability Seminar. David Kelly (Warwick/UNC) Homogenization for chaotic dynamical systems November 3, 2013 1 / 26
Outline of talk • Invariance principles (turning chaos into Brownian motion) • Homogenization of chaotic slow-fast systems • Why rough path theory is useful David Kelly (Warwick/UNC) Homogenization for chaotic dynamical systems November 3, 2013 2 / 26
Invariance principles
Donsker’s Invariance Principle I Let { ξ i } i ≥ 0 be i.i.d. random variables with E ξ i = 0 and E ξ 2 i < ∞ . Let S n = � n − 1 j =0 ξ i and define the path 1 W ( n ) ( t ) = √ nS ⌊ nt ⌋ . Then Donsker’s invariance principle * states that W ( n ) → w W in cadlag space, where W is a multiple of Brownian motion. It’s called an invariance principle because the result doesn’t care what random variables you use. David Kelly (Warwick/UNC) Homogenization for chaotic dynamical systems November 3, 2013 4 / 26
Donsker’s Invariance Principle II (Young 98, Melbourne, Nicol 05,08) We can even replace { ξ i } i ≥ 0 with iterations of a chaotic map. That is, let T : Λ → Λ be a “sufficiently chaotic” map, with T -invariant ergodic measure µ on probability space (Λ , M ), and let v : Λ → R d satisfy � Λ v d µ = 0 . If ⌊ nt ⌋− 1 v ◦ T j , � W ( n ) ( t ) = n − 1 / 2 j =0 then W ( n ) → w W in the cadlag space, where W is Brownian motion with covariance ∞ ∞ � � � Σ αβ = v γ ( v β ◦ T n ) d µ + v β ( v γ ◦ T n ) d µ � � v α v β d µ + Λ Λ Λ n =1 n =1 David Kelly (Warwick/UNC) Homogenization for chaotic dynamical systems November 3, 2013 5 / 26
Donsker’s Invariance Principle III We can do the same in continuous time, with a chaotic flow. That is, let { φ t } be a “sufficiently chaotic” flow on Λ, with invariant measure µ . Let v : Λ → R d satisfy � Λ v d µ = 0 . If � ε − 2 t W ( n ) ( t ) = ε v ◦ φ s ds , 0 then W ( n ) → w W in the sup-norm topology, where W is Brownian motion with covariance � ∞ � ∞ � � � Σ αβ = v γ ( v β ◦ φ s ) d µ ds + v β ( v γ ◦ φ s ) d µ ds v α v β d µ + Λ 0 Λ 0 Λ David Kelly (Warwick/UNC) Homogenization for chaotic dynamical systems November 3, 2013 6 / 26
What does “sufficiently chaotic” mean? In the discrete time case sufficiently chaotic ≈ decay of correlations More precisely, for the above v ∈ L 1 (Λ) and all w ∈ L ∞ (Λ), we have that � � � � � � w � ∞ n − τ , � v w ◦ T n d µ � � � � Λ for τ big enough. This holds for • Uniformly expanding or uniformly hyperbolic • Non-uniformly hyperbolic maps modeled by “Young towers”. Eg. Henon-like attractors, Lorenz attractors (flows) David Kelly (Warwick/UNC) Homogenization for chaotic dynamical systems November 3, 2013 7 / 26
Invariance principle: sketch of proof I The continuous invariance principle follows from the discrete invariance principle. David Kelly (Warwick/UNC) Homogenization for chaotic dynamical systems November 3, 2013 8 / 26
Invariance principle: sketch of proof II The idea is to use a known invariance principle for martingales. Namely, suppose m 1 , m 2 , . . . is a stationary, ergodic, martingale difference sequence. If ⌊ nt ⌋− 1 n − 1 � n − 1 / 2 � m i is a martingale, then m i → w BM i =0 i =0 i =0 v ◦ T i were a martingale then we’d be in business. So if � n − 1 David Kelly (Warwick/UNC) Homogenization for chaotic dynamical systems November 3, 2013 9 / 26
Invariance principle: Idea of proof III Actually, it is only a semi-martingale, with respect to the “filtration” T − 1 M , T − 2 M , T − 3 M , . . . where M is the sigma algebra from the original measure space. Moreover, we can write v = m + a where n − 1 � m ◦ T i M n := is a martingale i =0 and n − 1 � a ◦ T i A n := is bounded uniformly in n i =0 This is called a martingale approximation. David Kelly (Warwick/UNC) Homogenization for chaotic dynamical systems November 3, 2013 10 / 26
Invariance principle: Idea of proof IIII So if we write W ( n ) ( t ) = M ( n ) ( t ) + A ( n ) ( t ) ⌊ nt ⌋− 1 ⌊ nt ⌋− 1 m ◦ T i + n − 1 / 2 = n − 1 / 2 � � a ◦ T i i =0 i =0 Then we clearly have that W ( n ) → w W . However ... the world isn’t quite so nice, since in fact T − 1 M ⊃ T − 2 M ⊃ T − 3 M ⊃ . . . So we need to reverse time. David Kelly (Warwick/UNC) Homogenization for chaotic dynamical systems November 3, 2013 11 / 26
Using invariance principles for slow-fast systems
Slow-Fast systems in continuous time This idea can be applied to the homogenisation of slow-fast systems. For example dX ( ε ) = ε − 1 h ( X ( ε ) ) v ( Y ( ε ) ( t )) + f ( X ( ε ) , Y ( ε ) ) dt dY ( ε ) = ε − 2 g ( Y ( ε ) ) , dt where the fast dynamics Y ( ε ) ( t ) = Y ( ε − 2 t ) with ˙ Y = g ( Y ) describing a � chaotic flow, with ergodic measure µ and again v d µ = 0. We can re-write the equations as dX ( ε ) = h ( X ( ε ) ) dW ( ε ) + f ( X ( ε ) , Y ( ε ) ) dt where � t � ε − 2 t W ( ε ) ( t ) def = ε − 1 v ( Y ( ε ) ( s )) ds = ε v ( Y ( s )) ds 0 0 David Kelly (Warwick/UNC) Homogenization for chaotic dynamical systems November 3, 2013 13 / 26
Fast-Slow systems in discrete time We can do the same for discrete time systems. For example, define X : N → R d and Y : N → Λ by X ( n + 1) = X ( j ) + ε h ( X ( n )) v ( Y ( n )) + ε 2 f ( X ( n ) , Y ( n )) Y ( n + 1) = TY ( n ) , where T : Λ → Λ is a chaotic map. If we let X ( ε ) ( t ) = X ( ⌊ ε − 2 t ⌋ ) and Y ( ε ) = Y ( ⌊ ε − 2 t ⌋ ) then we have dX ( ε ) = h ( X ( ε ) ) dW ( ε ) + f ( X ( ε ) , Y ( ε ) ) dt where ⌊ ε − 2 t ⌋− 1 W ( ε ) ( t ) def � v ◦ T j = ε j =0 and where the integral is computed as a left Riemann sum. David Kelly (Warwick/UNC) Homogenization for chaotic dynamical systems November 3, 2013 14 / 26
For simplicity, we will focus on the more natural continuous time homogenization.
What is known? (Melbourne, Stuart ‘11) If the flow is chaotic enough so that � ε − 2 t W ( ε ) ( t ) = ε v ( y ( s )) ds → w W , 0 and either d = 1 or h = Id then we have that X ( ε ) → X , where dX = h ( X ) ◦ dW + F ( X ) dt , where the stochastic integral is of Stratonovich type and where � F ( · ) = f ( · , v ) d µ ( v ). David Kelly (Warwick/UNC) Homogenization for chaotic dynamical systems November 3, 2013 16 / 26
Continuity with respect to noise (Sussmann ‘78) The crucial fact that allows these results to go through is continuity with respect to noise. That is, let dX = h ( X ) dU + F ( X ) dt , where U is a smooth path. If d = 1 or h ( x ) = Id for all x , then Φ : U → X is continuous in the sup-norm topology. Therefore, if W ( ε ) → w W then X ( ε ) = Φ( W ( ε ) ) → w Φ( W ) . David Kelly (Warwick/UNC) Homogenization for chaotic dynamical systems November 3, 2013 17 / 26
This famously falls apart when the noise is both multidimensional and multiplicative . That is, when d > 1 and h � = Id . This fact is the main motivation behind rough path theory
Continuity with respect to rough paths (Lyons ‘97) As above, let dX = h ( X ) dU + F ( X ) dt , where U is a smooth path. Let U : [0 , T ] → R d × d be defined by � t U αβ ( t ) def U α ( s ) dU β ( s ) . = 0 Then the map Φ : ( U , U ) �→ X is continuous with respect to the “ d γ topology”. This is known as continuity with respect to the rough path ( U , U ). David Kelly (Warwick/UNC) Homogenization for chaotic dynamical systems November 3, 2013 19 / 26
The d γ topology The space of γ -rough paths is a metric space (but not a vector space). Objects in the space are pairs of the form ( U , U ) where U is a γ -H¨ older � path and where U is a natural “candidate” for the iterated integral UdU . On the space we define the metric � | U ( s , t ) − V ( s , t ) | + | U ( s , t ) − V ( s , t ) | � d γ ( U , U , V , V ) = sup | s − t | γ | s − t | γ s , t ∈ [0 , T ] where � t U βγ ( s , t ) = U β ( s , r ) dU γ ( r ) U ( s , t ) = U ( t ) − U ( s ) and s David Kelly (Warwick/UNC) Homogenization for chaotic dynamical systems November 3, 2013 20 / 26
Continuity with respect to rough paths Thus, we set � t W ( ε ) ,αβ ( t ) = W ( ε ) ,α ( s ) dW ( ε ) ,β ( s ) , 0 (which is defined uniquely). If we can show that ( W ( ε ) , W ( ε ) ) → w ( W , W ) where W is some identifiable type of iterated integral of W , then we have X ( ε ) → X = Φ( W , W ) . David Kelly (Warwick/UNC) Homogenization for chaotic dynamical systems November 3, 2013 21 / 26
Convergence of the rough path We have the following result Theorem (Kelly, Melbourne ‘13) If the fast dynamics are ”sufficiently chaotic”, then ( W ( ε ) , W ( ε ) ) → w ( W , W ) where W is a Brownian motion and � t W α ( s ) ◦ dW β ( s ) + 1 W αβ ( t ) = 2 D αβ t 0 where � ∞ � D β,γ = ( v β v γ ◦ φ s − v γ v β ◦ φ s ) d µ ds , 0 Λ and φ is the flow generated by the chaotic dynamics ˙ y = f ( y ) . David Kelly (Warwick/UNC) Homogenization for chaotic dynamical systems November 3, 2013 22 / 26
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