Numerics and Theory for Stochastic Evolution Equations University of Bielefeld, 22–24 November 2006 Barbara Gentz, University of Bielefeld http://www.math.uni-bielefeld.de/ ˜ gentz Desynchronisation of coupled bistable oscillators perturbed by additive white noise Joint work with Nils Berglund & Bastien Fernandez, CPT, Marseille
Metastability in stochastic lattice models ⊲ Lattice: Λ ⊂ Z d ⊲ Configuration space: X = S Λ , S finite set (e.g. {− 1 , 1 } ) ⊲ Hamiltonian: H : X → R (e.g. Ising model or lattice gas) ⊲ Gibbs measure: µ β ( x ) = e − βH ( x ) /Z β ⊲ Dynamics: Markov chain with invariant measure µ β (e.g. Metropolis such as Glauber or Kawasaki dynamics) 1
Metastability in stochastic lattice models ⊲ Lattice: Λ ⊂ Z d ⊲ Configuration space: X = S Λ , S finite set (e.g. {− 1 , 1 } ) ⊲ Hamiltonian: H : X → R (e.g. Ising model or lattice gas) ⊲ Gibbs measure: µ β ( x ) = e − βH ( x ) /Z β ⊲ Dynamics: Markov chain with invariant measure µ β (e.g. Metropolis such as Glauber or Kawasaki dynamics) Results (for β ≫ 1) on ⊲ Transition time between empty and full configuration ⊲ Transition path ⊲ Shape of critical droplet ⊲ Frank den Hollander, Metastability under stochastic dynamics , Stochastic Process. Appl. 114 (2004), 1–26 ⊲ Enzo Olivieri and Maria Eul´ alia Vares, Large deviations and metastability , Cambridge University Press, Cambridge, 2005 1-a
Metastability in reversible diffusions dx σ ( t ) = −∇ V ( x σ ( t )) dt + σ dB ( t ) ⊲ V : R d → R : potential, growing at infinity ⊲ B ( t ): d -dimensional Brownian motion Invariant measure: µ σ ( dx ) = e − 2 V ( x ) /σ 2 dx Z σ 2
Metastability in reversible diffusions dx σ ( t ) = −∇ V ( x σ ( t )) dt + σ dB ( t ) ⊲ V : R d → R : potential, growing at infinity ⊲ B ( t ): d -dimensional Brownian motion Invariant measure: µ σ ( dx ) = e − 2 V ( x ) /σ 2 dx Z σ 2-a
Metastability in reversible diffusions dx σ ( t ) = −∇ V ( x σ ( t )) dt + σ dB ( t ) ⊲ V : R d → R : potential, growing at infinity ⊲ B ( t ): d -dimensional Brownian motion Invariant measure: Mont Col de la Gineste µ σ ( dx ) = e − 2 V ( x ) /σ 2 Puget dx Z σ Luminy Cassis 2-b
Metastability in reversible diffusions dx σ ( t ) = −∇ V ( x σ ( t )) dt + σ dB ( t ) ⊲ V : R d → R : potential, growing at infinity ⊲ B ( t ): d -dimensional Brownian motion Invariant measure: Mont Col de la Gineste µ σ ( dx ) = e − 2 V ( x ) /σ 2 Puget dx Z σ Luminy Cassis Transition time τ between potential wells (first-hitting time): ⊲ Large deviations (Wentzell & Freidlin): lim σ → 0 σ 2 log E τ ⊲ Subexponential asymptotics (Bovier, Eckhoff, Gayrard, Klein; Helffer, Nier, Klein) 2-c
Metastability in reversible diffusions ⊲ Stationary points: S = { x : ∇ V ( x ) = 0 } ⊲ Saddles of index k ∈ N 0 : S k = { x ∈ S : Hess V ( x ) has k negative eigenvalues } Graph G = ( S 0 , E ): x ↔ y x, y belong to unstable manifold of some s ∈ S 1 iff x σ ( t ) resembles Markovian jump process on G Mont Col de la Gineste Puget Luminy Cassis 3
Metastability in reversible diffusions ⊲ Stationary points: S = { x : ∇ V ( x ) = 0 } ⊲ Saddles of index k ∈ N 0 : S k = { x ∈ S : Hess V ( x ) has k negative eigenvalues } ⊲ (Multi-)Graph G = ( S 0 , E ): x ↔ y x, y belong to unstable manifold of some s ∈ S 1 iff ⊲ x σ ( t ) resembles Markovian jump process on G Col de la Gineste Mont Col de la Gineste Puget Mont Puget Luminy Cassis Luminy Cassis 3-a
The model ⊲ Lattice: Λ = Z /N Z , N � 2 dx i ( t ) = Interacting diffusions (Dawson, G¨ artner, Deuschel, Cox, Greven, Shiga, Klenke, Fleischmann; M´ el´ eard; Kondratiev, R¨ ockner, Carmona, Xu; . . . ) √ Gradient system: dx σ ( t ) = −∇ V γ ( x σ ( t )) dt + σ N dB ( t ) � � U ( x i ) + γ ( x i +1 − x i ) 2 Global potential: V γ ( x ) = 4 i ∈ Λ i ∈ Λ 4
The model ⊲ Lattice: Λ = Z /N Z , N � 2 ⊲ Position x i of i th particle: Λ ∋ i �→ x i ∈ R ⊲ Configuration space: X = R Λ dx i ( t ) = Interacting diffusions (Dawson, G¨ artner, Deuschel, Cox, Greven, Shiga, Klenke, Fleischmann; M´ el´ eard; Kondratiev, R¨ ockner, Carmona, Xu; . . . ) √ Gradient system: dx σ ( t ) = −∇ V γ ( x σ ( t )) dt + σ N dB ( t ) � � U ( x i ) + γ ( x i +1 − x i ) 2 Global potential: V γ ( x ) = 4 i ∈ Λ i ∈ Λ 4-a
The model ⊲ Lattice: Λ = Z /N Z , N � 2 ⊲ Position x i of i th particle: Λ ∋ i �→ x i ∈ R ⊲ Configuration space: X = R Λ 4 x 4 − 1 ⊲ Bistable local potential: U ( x ) = 1 2 x 2 ⊲ Nonlinear local drift term: f ( x ) = − U ′ ( x ) = x − x 3 dx i ( t ) = f ( x i ( t )) dt Interacting diffusions (Dawson, G¨ artner, Deuschel, Cox, Greven, Shiga, Klenke, Fleischmann; M´ el´ eard; Kondratiev, R¨ ockner, Carmona, Xu; . . . ) √ Gradient system: dx σ ( t ) = −∇ V γ ( x σ ( t )) dt + σ N dB ( t ) � � U ( x i ) + γ ( x i +1 − x i ) 2 Global potential: V γ ( x ) = 4 i ∈ Λ i ∈ Λ 4-b
The model ⊲ Lattice: Λ = Z /N Z , N � 2 ⊲ Position x i of i th particle: Λ ∋ i �→ x i ∈ R ⊲ Configuration space: X = R Λ 4 x 4 − 1 ⊲ Bistable local potential: U ( x ) = 1 2 x 2 ⊲ Nonlinear local drift term: f ( x ) = − U ′ ( x ) = x − x 3 ⊲ Coupling between sites: discretised Laplacian ⊲ Coupling strength γ ≥ 0 � � dx i ( t ) = f ( x i ( t )) dt + γ x i +1 ( t ) − 2 x i ( t ) + x i − 1 ( t ) dt 2 Interacting diffusions (Dawson, G¨ artner, Deuschel, Cox, Greven, Shiga, Klenke, Fleischmann; M´ el´ eard; Kondratiev, R¨ ockner, Carmona, Xu; . . . ) √ Gradient system: dx σ ( t ) = −∇ V γ ( x σ ( t )) dt + σ N dB ( t ) � � U ( x i ) + γ ( x i +1 − x i ) 2 Global potential: V γ ( x ) = 4 i ∈ Λ i ∈ Λ 4-c
The model ⊲ Lattice: Λ = Z /N Z , N � 2 ⊲ Position x i of i th particle: Λ ∋ i �→ x i ∈ R ⊲ Configuration space: X = R Λ 4 x 4 − 1 ⊲ Bistable local potential: U ( x ) = 1 2 x 2 ⊲ Nonlinear local drift term: f ( x ) = − U ′ ( x ) = x − x 3 ⊲ Coupling between sites: discretised Laplacian ⊲ Coupling strength γ ≥ 0 ⊲ Independent Gaussian white noise dB i ( t ) acting on each site � � √ dx i ( t ) = f ( x i ( t )) dt + γ x i +1 ( t ) − 2 x i ( t ) + x i − 1 ( t ) dt + σ NdB i ( t ) 2 Interacting diffusions (Dawson, G¨ artner, Deuschel, Cox, Greven, Shiga, Klenke, Fleischmann; M´ el´ eard; Kondratiev, R¨ ockner, Carmona, Xu; . . . ) √ Gradient system: dx σ ( t ) = −∇ V γ ( x σ ( t )) dt + σ N dB ( t ) � � U ( x i ) + γ ( x i +1 − x i ) 2 Global potential: V γ ( x ) = 4 i ∈ Λ i ∈ Λ 4-d
The model ⊲ Lattice: Λ = Z /N Z , N � 2 ⊲ Position x i of i th particle: Λ ∋ i �→ x i ∈ R ⊲ Configuration space: X = R Λ 4 x 4 − 1 ⊲ Bistable local potential: U ( x ) = 1 2 x 2 ⊲ Nonlinear local drift term: f ( x ) = − U ′ ( x ) = x − x 3 ⊲ Coupling between sites: discretised Laplacian ⊲ Coupling strength γ ≥ 0 ⊲ Independent Gaussian white noise dB i ( t ) acting on each site � � √ dx i ( t ) = f ( x i ( t )) dt + γ x i +1 ( t ) − 2 x i ( t ) + x i − 1 ( t ) dt + σ NdB i ( t ) 2 ⊲ Interacting diffusions (Dawson, G¨ artner, Deuschel, Cox, Greven, Shiga, Klenke, Fleischmann; M´ el´ eard; Kondratiev, R¨ ockner, Carmona, Xu; . . . ) √ Gradient system: dx σ ( t ) = −∇ V γ ( x σ ( t )) dt + σ N dB ( t ) � � U ( x i ) + γ ( x i +1 − x i ) 2 Global potential: V γ ( x ) = 4 i ∈ Λ i ∈ Λ 4-e
The model ⊲ Lattice: Λ = Z /N Z , N � 2 ⊲ Position x i of i th particle: Λ ∋ i �→ x i ∈ R ⊲ Configuration space: X = R Λ 4 x 4 − 1 ⊲ Bistable local potential: U ( x ) = 1 2 x 2 ⊲ Nonlinear local drift term: f ( x ) = − U ′ ( x ) = x − x 3 ⊲ Coupling between sites: discretised Laplacian ⊲ Coupling strength γ ≥ 0 ⊲ Independent Gaussian white noise dB i ( t ) acting on each site � � √ dx i ( t ) = f ( x i ( t )) dt + γ x i +1 ( t ) − 2 x i ( t ) + x i − 1 ( t ) dt + σ NdB i ( t ) 2 ⊲ Interacting diffusions (Dawson, G¨ artner, Deuschel, Cox, Greven, Shiga, Klenke, Fleischmann; M´ el´ eard; Kondratiev, R¨ ockner, Carmona, Xu; . . . ) √ Gradient system: dx σ ( t ) = −∇ V γ ( x σ ( t )) dt + σ N dB ( t ) � U ( x i ) + γ � ( x i +1 − x i ) 2 Global potential: V γ ( x ) = 4 i ∈ Λ i ∈ Λ 4-f
Weak coupling For γ = 0: S = {− 1 , 0 , 1 } Λ , S 0 = {− 1 , 1 } Λ , G = hypercube 5
Weak coupling For γ = 0: S = {− 1 , 0 , 1 } Λ , S 0 = {− 1 , 1 } Λ , G = hypercube Theorem ∀ N ∃ γ ⋆ ( N ) > 0 s.t. ⊲ All x ⋆ ( γ ) ∈ S k ( γ ) depend continuously on γ ∈ [0 , γ ⋆ ( N )) �� √ √ � ⊲ 1 N � 2 γ ⋆ ( N ) � γ ⋆ (3) = 1 4 � inf 3 + 2 3 − 3 = 0 . 2701 . . . 3 5-a
Weak coupling For γ = 0: S = {− 1 , 0 , 1 } Λ , S 0 = {− 1 , 1 } Λ , G = hypercube Theorem ∀ N ∃ γ ⋆ ( N ) > 0 s.t. ⊲ All x ⋆ ( γ ) ∈ S k ( γ ) depend continuously on γ ∈ [0 , γ ⋆ ( N )) �� √ √ � ⊲ 1 N � 2 γ ⋆ ( N ) � γ ⋆ (3) = 1 4 � inf 3 + 2 3 − 3 = 0 . 2701 . . . 3 For 0 < γ ≪ 1: � V γ ( x ⋆ ( γ )) = V 0 ( x ⋆ (0)) + γ i (0)) 2 + O ( γ 2 ) ( x ⋆ i +1 (0) − x ⋆ 4 i ∈ Λ 5-b
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