Random Dynamical Systems with Non-Gaussian Fluctuations Some Advances & Perspectives

Random Dynamical Systems with Non-Gaussian Fluctuations Some Advances & Perspectives Jinqiao Duan Illinois Institute of Technology Department of Applied Mathematics Chicago, IL 60616, USA www.iit.edu/~duan

An Overview on Random Dynamical Systems Probabilistic methods for deterministic dynamical systems Analytic Methods Invariant manifolds, asymptotic behavior, special invariant sets (random periodic orbits) Topological Methods Conley index, topological degree, topological equivalence, bifurcation Numerical Methods Algorithms providing dynamical information or preserving dynamical properties Stochastic Modeling of Multiscale Systems Quantifying missing mechanisms Parameterizing unresolved scales

Outline 1 Motivation 2 Dynamical Systems driven by Non-Gaussian noises 3 Exit time 4 Bifurcation 5 Invariant manifolds 6 Conclusions

Gaussian vs. Non-Gaussian random variables Gaussian random variable X(ω): Probability density function 1 2π e u2 /2 P(X x) = x 1 e u2 /2 du 2π Non-Gaussian random variable X(ω): Probability density function f(u) 0, P(X x) = x f(u) du R f(u)du = 1. Gaussian process & non-gaussian process

Brownian motion B t : a Gaussian process Independent increments: B t2 B t1 and B t3 B t2 independent Stationary increments with B t B s N(0, t s) In particular, B t N(0, t) Continuous sample paths, but nowhere differentiable Reference: I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus

Brownian motion B t : PDF is exponentially decaying At given time t, B t has probability density function (PDF): 1 2πt e u2 /2t Exponentially decaying (Gaussian distribution).

A sample path for Brownian motion B t (ω) 2 A Scalar Brownian Motion 1.5 1 0.5 W t 0 0.5 1 1.5 2 0 5 10 15 20 25 30 t Figure: Continuous path, but nowhere differentiable

A math model for Gaussian white noise d dt B t(ω) Generalized time derivative of Brownian motion Reason: Fourier transform of correlation of d dt B t(ω) is constant Spectrum is constant: White light (Therefore: white noise) L. Arnold: Stochastic Differential Equations with Applications, 1974

Lévy motion L t (ω): a non-gaussian process Independent increments: L t2 L t1 and L t3 L t2 independent Stationary increments L t L s non-gaussian distribution (but depends only on t s) Stochastically continuous sample paths (continuous in probability): P( L t L s > δ) 0 when t s, for all δ > 0. Note: There exists a modification of L t whose paths are continuous from the right and have the left limits at every time. Reference: D. Applebaum Lévy Processes and Stochastic Calculus

Lévy motion L t (ω): a non-gaussian process Independent increments: L t2 L t1 and L t3 L t2 independent Stationary increments L t L s non-gaussian distribution (but depends only on t s) Sample paths are right continuous with left limits at every time ( Càdlàg").

Lévy-Khintchine formula: It is known that any Lévy motion is completely determined by the Lévy-Khintchine formula. This says that for any one-dimensional Lévy motion L t, there exists a a R, d > 0 and a measure ν such that Ee iλl t = exp{i aλt dt λ2 2 +t R\{0} (e iλu 1 iλu I{ u < 1}) ν(du)} I(S) is the indicator function of the set S, i.e., it takes value 1 on this set and takes zero value otherwise.

Lévy-Ito decomposition: L t = at + B t + { L n (t) + L 0 (t)} n=1 Lévy motion = drift motion + Brownian motion + jump motion"

Lévy jump measure ν(u): The jumps are described by Lévy jump measure ν(du) on R: (u 2 1) ν(du) <, R\{0} or equivalently, R\{0} u 2 ν(du) <. 1 + u2

Intuition for Lévy jump measure ν(u) ν(a): Number of jumps of size" A, for A R ν(r {0}): Intensity of jumps (how often" it jumps) Facts: ν(r {0}) = countable and dense jumps on any finite time interval ν(r {0}) < countable jumps on the whole time axis

Examples of Lévy motions Brownian motion B t, with jump measure ν(du) = 0 Poisson process N t of intensity λ > 0: P(N t = n) = (λt)n n! e λt with jump measure ν(du) = λδ 1 (du) α-stable symmetric Lévy motion: Lévy jump measure: ν(du) = 1 u 1+α (du), with 0 < α < 2 A family of symmetric Lévy motion: Lévy jump measure: ν(du) = f(ln u ) 1 u 1+α (du)

α-stable symmetric Lévy Noise Lévy jump measure: ν(du) = 1 u 1+α (du), with 0 < α < 2 α = 2: Brownian motion B t Heavy tail for 0 < α < 2: Power law P( L t > u) 1 u α Light tail for α = 2: Exponential law P( B t > u) e u2 /2 2πu

A sample path for Lévy motion L t (ω): Jumps 0.3 A Scalar Alpha Stable Levy Motion with Alpha=0.25 0.25 0.2 0.15 L t 0.1 0.05 0 0.05 0 5 10 15 t Figure: α stable Lévy motion: α = 0.25

A sample path for Lévy motion L t (ω): Jumps 0.1 A Scalar Alpha Stable Levy Motion with Alpha=0.75 0 0.1 0.2 L t 0.3 0.4 0.5 0.6 0 5 10 15 t Figure: α stable Lévy motion: α = 0.75

Lévy motion: How bad are the jumps? On any finite time intervals, Lévy motion jumps at most countable number of times Sample paths are right-continuous with left limit at every time

Lévy motion: Power law" distribution Lévy motion can be decomposed into a few component processes Lévy motion has a component whose probability density function (PDF) is not exponentially decaying (non-gaussian). In particular: PDFs of power law" 1 u k (instead of e u2 )

Why Lévy motion L t (ω): Jumps or flights Abrupt climate change such as Dansgaard-Oeschger events. Ditlevsen 1999: Ice record for temperature Diffusion of tracers in rotating annular flows: Pauses near coherent structures & jumps or flights" in between PDF for flight times: Power laws Swinney et al. 1995 Data in biology and other areas Shlesinger et al.: Lévy Flights and Related Topics in Physics, 1995 Financial data

A math model for non-gaussian white noise d dt L t(ω) Generalized time derivative of Lévy motion Reason: Fourier transform of correlation of d dt L t(ω) is constant Spectrum is constant: White light (Therefore: white noise)

Stochastic Differential Equations driven by Brownian or by Lévy noises Gaussian or non-gaussian noises

Deterministic Dynamical Systems x = f(x), x(0) = x 0 Solution map: ϕ(t, x 0 ) Flow" property: ϕ(t + s, x 0 ) = ϕ(t,ϕ(s, x 0 ))

What is a random dynamical system? Driving system ( noise") Stochastic flow property ( Cocycle")

Definition of Random Dynamical Systems Driving system & Cocycle property Model for noise: Driving system (Ω, F, P) a probability space. {θ t } t IR be a measurable flow on Ω: θ : IR Ω Ω such that (1) θ is (B(IR) F, F)-measurable; (2) θ 0 = id Ω, θ t1 θ t2 = θ t1 +t 2, t 1, t 2 IR. Driving system: P-preserving measurable flow.

Model for evolution: Cocycle property ( stochastic flow") A measurable random dynamical system on the measurable space (H, B) over a driven flow (Ω, F, P,θ t ) is a mapping ϕ : IR Ω H H with Measurability: ϕ is (B(IR) F B(H), F)-measurable, and Cocycle property: ϕ(0,ω, x) = x H, ϕ(t + s,ω, x) = ϕ(t,θ s ω,ϕ(s,ω, x)) (1) for t, s IR, ω Ω, and x H.

Visualizing the cocycle property

Stochastic Differential Equations with Brownian motion Drift: f(x t ) Diffusion: g(x t ) Solution map: ϕ(t,ω, x) dx t = f(x t )dt + g(x t )db t, X 0 = x Does this generate a random dynamical system?

SDEs with Brownian motion Model for noise: Wiener shift θ t Canonical sample space Ω := C 0 (R, R n ) (set of continuous functions) Identifying sample paths: ω(t) B t (ω) θ t : Ω Ω θ t ω(s) := ω(t + s) ω(t) Model for evolution: solution map ϕ(t, ω, x) has cocycle property

SDEs with Brownian Motion: Generate a random dynamical system Under very general smoothness conditions on the coefficients, a SDE driven by Brownian motion generates a random dynamical system. Reference: L. Arnold Random Dynamical Systems

SDEs with Lévy motion Drift: f(x t ) Diffusion: g(x t ) Solution map: ϕ(t,ω, x) dx t = f(x t )dt + g(x t )dl t, X 0 = x Does this generate a random dynamical system? Random dynamical systems driven by non-gaussian fluctuations Reference: Kunita 1996

Dynamical systems issues: Exit time Bifurcation Invariant manifolds

Exit from a domain D x Figure:

Exit time dx t = f(x t )dt + g(x t )db t, X 0 given (2) where f is an n dimensional vector function, g is an n m matrix function, and B t (ω) is an m dimensional Brownian motion. The generator is local: Av = ( v) T f(x) + 1 2 Tr[g(x)gT (x)d 2 (v)], (3) where D 2 is the Hessain differential matrix and Tr denotes the trace. Exit time from a domain D: σ x (ω) := inf{t : X t D}

How to compute mean exit time? Mean exit time u(x) := Eσ x (ω): Au = 1, u D = 0. Refs: Naeh, Klosek, Matkowsky and Schuss 1990

2D Example ẋ ẏ = y + εḃ1 t = x y 3 + εḃ2 t

y Phase portrait for unperturbed system: ẋ = y, ẏ = x y 3 x = y y = x y 3 2 1.5 1 0.5 0 0.5 1 1.5 2 2 1.5 1 0.5 0 0.5 1 1.5 2 x Figure:

Mean exit time from a domain: ε = 0.6 120 100 80 u(x,y) 60 40 20 0 1 0.5 1 0 y 0.5 1 1 0.5 x 0 0.5 Figure: u(x, y): Mean exit time for orbit staring at (x, y)

Theorem: Exit time under Brownian motion Theorem ẋ = U (x) + εḃt, Mean exit time from a domain of an unperturbed stable equilibrium: E x σ(ε) O(e 1 ε 2 ) Ref: Freidlin-Wentzell 1979

SDE with non-gaussian Lévy noise or in 1D: dx ε t dx ε t = b(x ε t )dt + εdl t, X 0 = x = U (X ε t )dt + εdl t, X 0 = x Potential function U( ) has a minimum at 0 Unperturbed system has a stable equilibrium at 0 Lévy process L t (ω): Stationary and independent increments, with jumps. Goal: Exit time from a bounded domain containing a stable equilibrium

Generator of Lévy motion Lévy motion L t : Markov process Distribution of L t : µ t (probability measures) Semigroup of L t : P t f(x) := f(x + u)µ t (du) Generator A of L t : Af(x) := d dt [P tf(x)] t=0

α-stable symmetric Lévy Noise Lévy jump measure: ν(du) = 1 u 1+α (du), with 0 < α 2 The generator is nonlocal (non-gaussian): Af(x) = R\{0} [f(x + u) f(x) I{ u < 1} uf (x)] ν(du) The generator is the fractional Laplace operator: Nonlocal generator: A = ( ) α 2 α = 2: Brownian motion B t, generator is local A = Fractional Fokker-Planck equation: Schertzer et al 2001 Goal: Exit time in the case of nonlocal generator?

How to compute mean exit time in Lévy case? Mean exit time T(x) := Eσ x (ω) AT(x) = 1, x D T(x) D c = 0. Generator A is non-local: Numerical simulation AT(x) = [T(x + u) T(x) I{ u < 1} ut (x)] ν(du) R\{0} Huiqin Chen, Jinqiao Duan, Xiaofan Li and Chengjian Zhang, 2009 Impact of non-gaussian noise on a system describing gene expression

Theorem: Exit time under α-stable Lévy motion Theorem ẋ = U (x) + ε L t, Mean exit time from a domain containing an unperturbed stable equilibrium: E x σ(ε) 1 ε α Imkeller and Pavlyukevich, 2006; Stoch. Proc. Appl.

Mean exit time under a family of symmetric Lévy noises Lévy jump measure: ν(du) = f(ln u ) f > 0 measurable Lemma du u 1+α, with 0 < α < 2 and ẋ = U (x) + ε L t, with Lévy jump measure ν(du) = f(ln u ) Main assumption: f(ln u/ε ) f(ε) du u 1+α. du u 1+α ν (du). Mean exit time from [ b, a] containing an unperturbed stable equilibrium: 1 1 E x σ(ε) ν. (R \ [ b, a]) f(ε)

Ideas of Proof: 1. Estimating R\[ K,K] ν(d(u ε )) 2. Weak convergence of measures f(ln u/ε ) f(ε) 3. Applying a result of Godovanchuk 1981 du u 1+α

Theorem: Mean exit time under a family of symmetric Lévy motion Theorem ẋ = U (x) + ε L t, 1 du with Lévy jump measure ν(du) = ln u +1. u 1+α Mean exit time from a domain containing an unperturbed stable equilibrium: ln ε E x σ(ε) O( ε α ) Yang and Duan: 2008; Stochastics and Dynamics

Mean exit time comparison Mean exit times are in the order: or symbolically: O( 1 ln ε ) < O( εα ε α ) < exp( C ε 2). Polynomial Combined polyn. and natural logarithm Exponential Brownian noise Our noise α-stable symmetric Lévy Noise

Dynamical systems issues: Exit time Bifurcation Invariant manifolds

What is bifurcation in random dynamical systems Bifurcation: Qualitative changes in dynamical behaviors when a parameter varies

Topological equivalence of random dynamical systems ϕ 1 (t,ω, x) & ϕ 2 (t,ω, x) topologically equivalent: If there exists a mapping α : Ω X 1 X 2 such that: (i) the mapping x α(ω, x) is a homeomorphism from X 1 onto X 2 for every ω Ω; (ii) the mappings ω α(ω, x 1 ) and ω α 1 (ω, x 2 ) are measurable for every x 1 X 1 and x 2 X 2 ; (iii) the cocycles ϕ 1 and ϕ 2 are cohomologous, i.e. ϕ 2 (t,ω,α(ω, x)) = α(θ t ω,ϕ 1 (t,ω, x)) for any x X 1.

Bifurcation in random dynamical systems Definition: ψ λ : A family of random dynamical systems, λ R Bifurcation point λ 0 : If the family is not structurally stable at λ 0, i.e., if in any neighborhood of λ 0 there are parameter values λ such that ψ λ and ψ λ0 are not topologically equivalent. L. Arnold: Random Dynamical Systems, 1998

Random isolated invariant set Random isolating neighborhood: A random compact set N(ω) that satisfies Inv N(ω) int N(ω), where int N(ω) denotes the interior of N(ω) and Inv N(ω) = {x N(ω) ϕ(n,ω, x) N(θ n ω), n Z}. Random isolated invariant set: S(ω) is a random isolated invariant set if there exists a random isolating neighborhood N(ω) such that S(ω) = Inv N(ω).

How to detect bifurcation? Approach by Conley index Definition of Conley index ϕ: the time-one map of a discrete random dynamical system S(ω): A random isolated invariant set for ϕ P = (N(ω), L(ω)) is a random filtration pair for S(ω). Let h P (S,ϕ) be the random homotopy class [ϕ P ] on the random pointed space N L (ω) with ϕ P a representative element. The random shift equivalent class of h P (S,ϕ), denoted by h(s,ϕ), is called the random Conley index for S(ω). Z. X. Liu: Journal of Differential Equations, 2008

Coney index for random dynamical systems Prime random isolated invariant set: S(ω) is an random isolated invariant set for ϕ, and for any proper subset S (ω) S(ω), S (ω), S (ω) is not a random isolated invariant set for ϕ. Extreme maximal random isolated invariant set: If ϕ has only finitely many prime random isolated invariant sets S 1 (ω), S 2 (ω),, S r (ω), then T ϕ (ω) = r S i (ω) is called an i=1 extreme maximal random isolated invariant set for ϕ.

M(T ϕλ )(ω) For an extreme maximal random isolated invariant set T ϕλ (ω): M(T ϕλ )(ω): The cardinal number of the set {N(ω) N(ω) is a prime random isolated invariant set for ϕ λ }. Note: If T ϕλ (ω) =, we set M(T ϕλ )(ω) = 0.

Bifurcation theorem ψ λ : A family of time-continuous random dynamical systems, λ R If for any neighborhood U containing λ 0 R there exists λ U such that or M(T ϕλ )(ω) M(T ϕλ0 )(ω) h(t ϕλ,ϕ λ ) h(t ϕλ0,ϕ λ0 ) for the corresponding discrete system ϕ λ (n,ω, x) for n Z, then λ 0 is a bifurcation point. Xiaopeng Chen, Jinqiao Duan, Xinchu Fu, 2009; Proc. of Ameri. Math. Soc.

Dynamical systems issues: Exit time Bifurcation Invariant manifolds

Stable and unstable manifolds Stable and unstable manifolds are geometric invariant structures Stable and unstable manifolds help us understand dynamics

Stable and unstable manifolds du = [Au + F(u)]dt + u(t ) dl t (ω) where stands for the Marcus canonical integral, which is a slight modification of the Stranotovich integrals. Duan, Liu, Lu and Schmafuss, 2009

Conclusions Dynamical Systems Driven by Non-Gaussian Noise Quantity, Phenomena, Invariant Structure: Exit time Bifurcation Invariant manifolds