Motivation Exit problem The irreversible case & periodic orbits Exit through an unstable periodic orbit Equadiff 2015 Small eigenvalues and mean transition times for irreversible diffusions Barbara Gentz (Bielefeld) & Nils Berglund (Orl´ eans) Lyon, France, 7 July 2015
Motivation Exit problem The irreversible case & periodic orbits Exit through an unstable periodic orbit Motivation: Two coupled oscillators 1 / 29
Motivation Exit problem The irreversible case & periodic orbits Exit through an unstable periodic orbit Synchronization of two coupled oscillators First observed by Huygens; see e.g. [Pikovsky, Rosenblum, Kurths 2001] Motion of pendulums x i = ( θ i , ˙ θ i ) � x 1 = f 1 ( x 1 ) ˙ x 2 = f 2 ( x 2 ) ˙ For a good parametrisation φ i of the limit cycles � ˙ φ 1 = ω 1 ˙ φ 2 = ω 2 where ω i denotes the natural frequencies 2 / 29
Motivation Exit problem The irreversible case & periodic orbits Exit through an unstable periodic orbit Synchronization of two coupled oscillators First observed by Huygens; see e.g. [Pikovsky, Rosenblum, Kurths 2001] Motion of pendulums x i = ( θ i , ˙ θ i ) with coupling � x 1 = f 1 ( x 1 ) + ε h 1 ( x 1 , x 2 ) ˙ x 2 = f 2 ( x 2 ) + ε h 2 ( x 1 , x 2 ) ˙ For a good parametrisation φ i of the limit cycles � ˙ φ 1 = ω 1 + ε g 1 ( x 1 , x 2 ) ˙ φ 2 = ω 2 + ε g 2 ( x 1 , x 2 ) where ω i denotes the natural frequencies 2 / 29
Motivation Exit problem The irreversible case & periodic orbits Exit through an unstable periodic orbit Coupled oscillators with slightly different frequencies � ˙ � ψ = φ 1 − φ 2 ψ = − ν + ε q ( ψ, ϕ ) with ν = ω 2 − ω 1 = ⇒ ϕ = φ 1 + φ 2 with ω = ω 1 + ω 2 ϕ = ω + O ( ε ) ˙ 2 2 ψ/ 2 Assume 2 � ⊲ Detuning ν = ω 2 − ω 1 small ⊲ Coupling strength ε � ε 0 � Observation ⊲ Synchronization ϕ 0 0 � 2 � 3 / 29
Motivation Exit problem The irreversible case & periodic orbits Exit through an unstable periodic orbit Coupled oscillators subject to noise Averaging ω d ψ d ϕ ≃ − ν + ε ¯ q ( ψ ) Adler equation (special choice of coupling) q ( ψ ) = sin ψ ¯ Observations ⊲ Fixed points at sin ψ = ν ε ⊲ Synchronization 4 / 29
Motivation Exit problem The irreversible case & periodic orbits Exit through an unstable periodic orbit Coupled oscillators subject to noise Averaging ω d ψ d ϕ ≃ − ν + ε ¯ q ( ψ ) + noise noise Adler equation (special choice of coupling) q ( ψ ) = sin ψ ¯ Observations ⊲ Fixed points at sin ψ = ν ε ⊲ Synchronization ⊲ In the presence of noise: occasional transitions ( → phase slips) 4 / 29
Motivation Exit problem The irreversible case & periodic orbits Exit through an unstable periodic orbit Without averaging ψ � ˙ unstable ψ = − ν + ε q ( ψ, ϕ ) + noise ϕ = ω + O ( ε ) + noise ˙ stable ' Observations ⊲ Synchronization ⊲ In the presence of noise: occasional transitions ( → phase slips) ⊲ Phase slips correspond to passage through unstable orbit Question ⊲ Distribution of phase ϕ when crossing unstable periodic orbit? To tackle ⊲ Stochastic exit problem 5 / 29
Motivation Exit problem The irreversible case & periodic orbits Exit through an unstable periodic orbit Exit problem: Wentzell–Freidlin theory and beyond 6 / 29
Motivation Exit problem The irreversible case & periodic orbits Exit through an unstable periodic orbit Transition probabilities and generators x ∈ R n d x t = f ( x t ) d t + σ g ( x t ) d W t , ⊲ Transition probability density p t ( x , y ) ⊲ Markov semigroup T t : For measurable ϕ ∈ L ∞ , � ( T t ϕ )( x ) = E x { ϕ ( x t ) } = p t ( x , y ) ϕ ( y ) d y ⊲ Infinitesimal generator L ϕ = d d t T t ϕ | t =0 of the diffusion: + σ 2 ( gg T ) ij ( x ) ∂ 2 ϕ f i ( x ) ∂ϕ � � ( L ϕ )( x ) = ∂ x i 2 ∂ x i ∂ x j i , j i ⊲ Adjoint semigroup: For probability measures µ � ( µ T t )( y ) = P µ { x t = d y } = p t ( x , y ) µ (d x ) with generator L ∗ 7 / 29
Motivation Exit problem The irreversible case & periodic orbits Exit through an unstable periodic orbit Stochastic exit problem ⊲ D ⊂ R n bounded domain x τ D ⊲ First-exit time τ D = inf { t > 0: x t �∈ D} ⊲ First-exit location x τ D ∈ ∂ D ⊲ Harmonic measure µ ( A ) = P x { x τ D ∈ A } D Facts (following from Dynkin’s formula – see textbooks on stochastic analysis) ⊲ u ( x ) = E x { τ D } satisfies � Lu ( x ) = − 1 for x ∈ D u ( x ) = 0 for x ∈ ∂ D ⊲ For ϕ ∈ L ∞ ( ∂ D , R ), h ( x ) = E x { ϕ ( x τ D ) } satisfies � Lh ( x ) = 0 for x ∈ D h ( x ) = ϕ ( x ) for x ∈ ∂ D 8 / 29
Motivation Exit problem The irreversible case & periodic orbits Exit through an unstable periodic orbit Wentzell–Freidlin theory x ∈ R n d x t = f ( x t ) d t + σ g ( x t ) d W t , ⊲ Large-deviation rate function / action funtional � T I ( γ ) = 1 γ t − f ( γ t )] T D ( γ t ) − 1 [˙ where D = gg T [˙ γ t − f ( γ t )] d t , 2 0 ⊲ Large-deviation principle: For a set Γ of paths γ : [0 , T ] → R n P { ( x t ) 0 � t � T ∈ Γ } ≃ e − inf Γ I /σ 2 Consider first exit from D contained in basin of attraction of an attractor A ⊲ Quasipotential V ( y ) = inf { I ( γ ): γ connects A to y in arbitrary time } , y ∈ ∂ D 9 / 29
Motivation Exit problem The irreversible case & periodic orbits Exit through an unstable periodic orbit Wentzell–Freidlin theory V ( y ) = inf { I ( γ ): γ connects A to y in arbitrary time } , y ∈ ∂ D Facts σ → 0 σ 2 log E { τ D } = V = inf ⊲ lim y ∈ ∂ D V ( y ) [Wentzell, Freidlin 1969] ⊲ If infimum is attained in a single point y ∗ ∈ D then σ → 0 P {� x τ D − y ∗ � > δ } = 0 lim ∀ δ > 0 [Wentzell, Freidlin 1969] ⊲ Minimizers of I are optimal transition paths; found from Hamilton equations ⊲ Limiting distribution of τ D is exponential σ → 0 P { τ D > s E { τ D }} = e − s lim [Day 1983; Bovier et al 2005] 10 / 29
Motivation Exit problem The irreversible case & periodic orbits Exit through an unstable periodic orbit The reversible case x ∈ R n d x t = − ∇ V ( x t ) d t + σ d W t , 2 e 2 V /σ 2 ∇ · e − 2 V /σ 2 ∇ is self-adjoint in ⊲ L = σ 2 2 ∆ − ∇ V ( x ) · ∇ = σ 2 L 2 ( R n , e − 2 V /σ 2 d x ) ⊲ Reversibility (detailed balance): e − 2 V ( x ) /σ 2 p t ( x , y ) = e − 2 V ( y ) /σ 2 p t ( y , x ) Facts Assume V has N local minima ⊲ − L has N exponentially small ev’s 0 = λ 0 < · · · < λ N − 1 + spectral gap ⊲ Precise expressions for the λ i (Kramers’ law) ⊲ λ − 1 are the expected transition times between neighbourhoods of minima, i i = 1 , . . . , N − 1 (in specific order) Methods Large deviations [Wentzell, Freidlin, Sugiura, . . . ]; Semiclassical analysis [Mathieu, Miclo, Kolokoltsov, . . . ]; Potential theory [Bovier, Gayrard, Eckhoff, Klein]; Witten Laplacian [Helffer, Nier, Le Peutrec, Viterbo]; Two-scale approach, using transport techniques [Menz, Schlichting 2012] 11 / 29
Motivation Exit problem The irreversible case & periodic orbits Exit through an unstable periodic orbit The irreversible case 12 / 29
Motivation Exit problem The irreversible case & periodic orbits Exit through an unstable periodic orbit Irreversible case If f is not of the form −∇ V ⊲ Large-deviation techniques still work, but . . . ⊲ L not self-adjoint, analytical approaches harder ⊲ not reversible, standard potential theory does not work Nevertheless, ⊲ Results exist on the Kramers–Fokker–Planck operator L = σ 2 ∂ x − σ 2 y − σ 2 y + σ 2 2 y ∂ 2 V ′ ( x ) ∂ ∂ y + γ � ∂ �� ∂ � 2 2 ∂ y 2 ∂ y [H´ erau, Hitrik, Sj¨ ostrand, . . . ] ⊲ Question What is the harmonic measure for the exit through an unstable periodic orbit? 13 / 29
Motivation Exit problem The irreversible case & periodic orbits Exit through an unstable periodic orbit Random Poincar´ e maps Near a periodic orbit, in appropriate coordinates d ϕ t = f ( ϕ t , x t ) d t + σ F ( ϕ t , x t ) d W t ϕ ∈ R x ∈ E ⊂ R n − 1 d x t = g ( ϕ t , x t ) d t + σ G ( ϕ t , x t ) d W t ⊲ All functions periodic in ϕ (e.g. period 1) ⊲ f � c > 0 and σ small ⇒ ϕ t likely to increase ⊲ Process may be killed when x leaves E x X 1 E X 2 X 0 ϕ 1 2 Random variables X 0 , X 1 , . . . form (substochastic) Markov chain 14 / 29
Motivation Exit problem The irreversible case & periodic orbits Exit through an unstable periodic orbit Random Poincar´ e map and harmonic measures x E X 1 X 0 ϕ 1 − M ⊲ First-exit time τ of z t = ( ϕ t , x t ) from D = ( − M , 1) × E ⊲ µ z ( A ) = P z { z τ ∈ A } is harmonic measure (w.r.t. generator L ) ⊲ µ z admits (smooth) density h ( z , y ) w.r.t. arclength on ∂ D (under hypoellipticity condition) [Ben Arous, Kusuoka, Stroock 1984] ⊲ Remark: Lh ( · , y ) = 0 (kernel is harmonic) ⊲ For Borel sets B ⊂ E � P X 0 { X 1 ∈ B } = K ( X 0 , B ) := K ( X 0 , d y ) B where K ( x , d y ) = h ((0 , x ) , (1 , y )) d y =: k ( x , y ) d y 15 / 29
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