From stochastic resonance to stationary measures of McKean-Vlasov - PowerPoint PPT Presentation

From stochastic resonance to stationary measures of McKean-Vlasov type equations S. Herrmann Institut de Mathématiques de Bourgogne joint work with P. Imkeller (Berlin - D), D. Peithmann (Berlin - D), J. Tugaut (Bielefeld - D) October 6, 2011 S. Herrmann (IMB) University of Burgundy, Dijon October 6, 2011 1 / 15

Introduction Outline 1 Introduction : link between exit problem and stochastic resonance 2 Exit time for a self-stabilizing process (McKean-Vlasov) living in a convex landscape: the inertia of a particular process attracted by its own law... 3 Exit time for a self-stabilizing diffusion in a double-well landscape S. Herrmann (IMB) University of Burgundy, Dijon October 6, 2011 2 / 15

Introduction The stochastic resonance framework ➟ to understand how a weak deterministic and periodic input of a given dynamical system can be amplified by noisy perturbations. In particular, there exists an optimal relation between the period log ( T ) (deterministic input) and the noise intensity ǫ so that the stochastic paths look like the most periodic as possible (quality measures are needed) dX t = √ ǫ dW t − V ′ ( X t ) dt + A 0 cos ( 2 π t / T ) dt , t ≥ 0 , where ( W t , t ≥ 0 ) is a Brownien motion, the diffusion coefficient is constant and V is a double-well potential. S. Herrmann (IMB) University of Burgundy, Dijon October 6, 2011 3 / 15

Introduction The stochastic resonance framework ➟ to understand how a weak deterministic and periodic input of a given dynamical system can be amplified by noisy perturbations. In particular, there exists an optimal relation between the period log ( T ) (deterministic input) and the noise intensity ǫ so that the stochastic paths look like the most periodic as possible (quality measures are needed) dX t = √ ǫ dW t − V ′ ( X t ) dt + A 0 cos ( 2 π t / T ) dt , t ≥ 0 , where ( W t , t ≥ 0 ) is a Brownien motion, the diffusion coefficient is constant and V is a double-well potential. S. Herrmann (IMB) University of Burgundy, Dijon October 6, 2011 3 / 15

Introduction The stochastic resonance framework ➟ to understand how a weak deterministic and periodic input of a given dynamical system can be amplified by noisy perturbations. In particular, there exists an optimal relation between the period log ( T ) (deterministic input) and the noise intensity ǫ so that the stochastic paths look like the most periodic as possible (quality measures are needed) dX t = √ ǫ dW t − V ′ ( X t ) dt + A 0 cos ( 2 π t / T ) dt , t ≥ 0 , where ( W t , t ≥ 0 ) is a Brownien motion, the diffusion coefficient is constant and V is a double-well potential. S. Herrmann (IMB) University of Burgundy, Dijon October 6, 2011 3 / 15

Introduction The stochastic resonance framework ➟ to understand how a weak deterministic and periodic input of a given dynamical system can be amplified by noisy perturbations. In particular, there exists an optimal relation between the period log ( T ) (deterministic input) and the noise intensity ǫ so that the stochastic paths look like the most periodic as possible (quality measures are needed) dX t = √ ǫ dW t − V ′ ( X t ) dt + A 0 cos ( 2 π t / T ) dt , t ≥ 0 , where ( W t , t ≥ 0 ) is a Brownien motion, the diffusion coefficient is constant and V is a double-well potential. Study of the exit problem (from one well): joint work with P. Imkeller and D. Peithmann. S. Herrmann (IMB) University of Burgundy, Dijon October 6, 2011 3 / 15

Introduction Several physicists (Jung, Behn, Pantazelou, Moss ’92) introduced the stochastic resonance phenomenon for globally coupled systems ( N individuals in interaction, for instance, neural networks): N t = √ ǫ dW i t ) dt − 1 � t − X j dX i t − V ′ ( X i F ′ ( X i t ) dt + A 0 cos ( 2 π t / T ) dt , N j = 1 where W i are indep. BM, V is a double-well potential, F ′ is linear. The quality measures used in practice depends on invariant measures of simplified systems. ➟ Difficulties if ǫ is small and N large (mean field). Propagation of chaos � N The empirical measure 1 j = 1 δ X j t converges towards u t which N corresponds to the distribution of the McKean-Vlasov solution: t = √ ǫ dW t − V ′ ( X ǫ dX ǫ � R F ′ ( X ǫ t − x ) du ǫ t ) dt − t ( x ) dt + A 0 cos ( 2 π t / T ) dt . References: Sznitman ’91, Dawson & Gärtner ’87, McKean ’66 ’67, Stroock & Varadhan ’79, Oelschläger ’85, Funaki ’84, Tamura ’84 ’87, Benachour, Roynette, Talay & Vallois ’98, Benachour, Roynette & Vallois ’98 S. Herrmann (IMB) University of Burgundy, Dijon October 6, 2011 6 / 15

Introduction In order to analyse stochastic resonance for the limit process, it suffices to describe the exit problem (transitions between the metastables states of the dynamical system) for the nonlinear stochastic process: t = √ ǫ dW t − V ′ ( X ǫ � dX ǫ F ′ ( X ǫ t − x ) du ǫ t ) dt − t ( x ) dt + A 0 cos ( 2 π t / T ) dt . R First step: asymptotic behavior as ǫ << 1 without the periodic perturbation. Difficulties: the process is nonlinear the drift term depends on the time (non homogeneous) the drift term depends on the small parameter ǫ . The study concerns a fonction V which represents: either a convex landscape or a double-well landscape Assumption: V & F loc. Lipschitz, even with polyn. growth. S. Herrmann (IMB) University of Burgundy, Dijon October 6, 2011 7 / 15

Convexe case 2. Exit problem: self-stabilizing process living in a convex landscape t = √ ǫ dW t − V ′ ( X ǫ dX ǫ t ) dt − b ǫ ( t , X ǫ t ) dt , � b ǫ ( t , x ) = F ′ ( x − y ) du ǫ t ( y ) dt = E [ F ′ ( x − X ǫ t )] . R Asymptotic behaviour of the exit time ( ǫ → 0): Find the Large Deviations Principle associated to X ǫ on the time interval [ 0 , T ] , then study on R + . Convergence & Large Deviations The self-stabilising process starting from x 0 converges in distribution towards the solution: ˙ ψ t = − V ′ ( ψ t ) , ψ 0 = x 0 . X ǫ satisfies LDP in ( C ([ O , T ]) , � · � ∞ ) with the associated rate function: � T T ( ϕ ) = 1 I x 0 ϕ t + V ′ ( ϕ t ) + F ′ ( ϕ t − ψ t ( x 0 )) � 2 dt , si ϕ ∈ H 1 � ˙ x 0 . 2 0 S. Herrmann (IMB) University of Burgundy, Dijon October 6, 2011 8 / 15

Convexe case If the SDE is observed on [ s , s + T ] then the LDP is associated with � T ϕ t + V ′ ( ϕ t ) + F ′ ( ϕ t − ψ s + t ( x )) � 2 dt . I T ( ϕ ) = 1 0 � ˙ 2 ✔ Exit time for diffusions in convex landscapes. Intuitive idea: splitting the real axis in large time intervals of length L . τ denotes the exit time of D which con- tains the stable point of V : x stable . 0 L 2L 3L 4L 5L 6L 1 On each interval time dependent LDP describing the probability to exit the domain D in this interval. 2 After a large number of intervals: the exit probability is close to a particular p associated with � L L ( ϕ ) = 1 ϕ t + V ′ ( ϕ t ) + F ′ ( ϕ t − x stable ) � 2 dt I ∞ � ˙ 2 0 Minimal cost E ∞ = inf { I ∞ t ( ϕ ) : ϕ cont. , ϕ 0 = x stable , ϕ t ∈ D c , t > 0 } S. Herrmann (IMB) University of Burgundy, Dijon October 6, 2011 9 / 15

Double-well case ➽ The convexity is essential : what happens in a double-well landscape ? Self-stabilizing process in a double-well landscape V t ( x ) dt + √ ǫ dW t . � dX ǫ t = − V ′ ( X ǫ F ′ ( X ǫ t − x ) du ǫ t ) dt − R The minima of V are reached at x = − a and a . ➽ Aim: to describe the transitions in the small noise limit. ➽ Simplification: stationary regime. We replace u ε t by an invariant u ε . Invariant measures and asymptotic behavior as ε → 0 In order to find an invariant measure we solve ǫ � ′ = 0 . u ǫ ( x )( V ′ ( x ) + F ′ ⋆ u ǫ ( x )) 2 u ′′ � ǫ ( x ) + S. Herrmann (IMB) University of Burgundy, Dijon October 6, 2011 12 / 15

Double-well case Each invariant measure takes the exponential form �� x 1 � − 2 �� F ′ ⋆ u ǫ ( y ) dy + V ( x ) u ǫ ( x ) = λ ( u ǫ ) exp = A ( u ǫ ) . ǫ 0 We have to solve a fixed point problem associated with the application A ➥ simplification in the particular linear F ′ ( x ) = α x . Linear case If F ′ ( x ) = α x and V ′′ convex, there exist exactly: one symmetric invariant measure two asymmetric measures close to δ a et δ − a If m is the mean of the invariant measure then m = Ψ ǫ ( m ) with V ( x )+ α x 2 � � �� − 2 � R x exp 2 − α mx dx Ψ ǫ ( m ) = ǫ � � V ( x )+ α x 2 �� − 2 � R exp 2 − α mx dx ǫ There exist exactly 3 solutions corresponding to 3 invariant measures. S. Herrmann (IMB) University of Burgundy, Dijon October 6, 2011 13 / 15