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Lyapunov Exponents for Random Dynamical Systems

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Lyapunov Exponents for Random Dynamical Systems Lyapunov Exponents for Random Dynamical Systems DISSERTATION zur Erlangung des akademischen Grades Doctor rerum naturalium (Dr.rer.nat.) vorgelegt der Fakult¨at Mathematik und Naturwissenschaften der Technischen Universit¨at Dresden von Bachel.-Math. Doan Thai Son geboren am 05.10.1984 in Namdinh Gutachter: Prof. Stefan Siegmund Technische Universit¨at Dresden Prof. Nguyen Dinh Cong Hanoi Institute of Mathematics, Vietnam Eingereicht am: 15.1.2009 Tag der Disputation: 27.11.2009 Contents Introduction 1 1 Background on Random Dynamical Systems 9 1.1 Definition of Random Dynamical System . 9 1.2 Generation .................................. 16 1.2.1 Discrete Time: Products of Random Mappings . 16 1.2.2 Continuous Time 1: Random Differential Equations . 17 1.2.3 Continuous Time 2: Stochastic Differential Equations ...... 19 1.3 Multiplicative Ergodic Theorem in Rd ................... 20 1.3.1 SingularValues............................ 20 1.3.2 ExteriorPowers............................ 21 1.3.3 The Furstenberg-Kesten Theorem . 22 1.3.4 Multiplicative Ergodic Theorem . 24 1.4 Multiplicative Ergodic Theorem in Banach Spaces . ...... 25 2 Generic Properties of Lyapunov Exponents of Discrete Random Dy- namical Systems 29 2.1 TheSpaceofLinearCocycles . 29 2.2 Uniformly Hyperbolic Linear Cocycles . 31 2.2.1 Exponential Dichotomy . 31 2.2.2 Exponential Separation of Bounded Cocycles . 39 2.2.3 Exponential Dichotomy is Strictly Stronger than Exponential Sep- aration ................................ 41 2.2.4 Exponential Separation of Unbounded Cocycles . 44 2.3 An Open Set of Cocycles with Simple Lyapunov Spectrum but no Expo- nentially Separated Splitting . 50 3 Generic Properties of Lyapunov Exponents of Linear Random Differ- ential Equations 58 3.1 Spaces of Linear Random Differential Equations . ..... 59 3.2 Generic Properties of Lyapunov Exponents of Linear Random Differential Equations ................................... 60 i 4 Difference Equations with Random Delay 64 4.1 A Setting for Difference Equations with Random Delay . ..... 66 4.2 MET for Difference Equations with Random Delay . 70 4.3 SomeExamples................................ 78 4.3.1 BoundedRandomDelay. 79 4.3.2 Deterministic Delay . 82 5 Differential Equations with Random Delay 85 5.1 Differential Equations with Random Delay . 85 5.2 MET for differential equations with random delay . ..... 93 5.2.1 Integrability.............................. 93 5.2.2 KuratowskiMeasure . .. .. .. .. .. .. .. 95 5.2.3 Multiplicative Ergodic Theorem . 99 5.3 Differential equations with bounded delay . 104 6 Computational Ergodic Theorem 107 6.1 IteratedFunctionSystems . 109 6.1.1 Finite Iterated Function Systems . 109 6.1.2 Finite Iterated Function Systems with Place-dependent Probabilities113 6.1.3 Infinite Iterated Function Systems . 115 6.2 Computational Ergodic Theorem for Place-dependent IFS ........ 122 6.3 ComputationalErgodicTheoremforIIFS . 125 6.3.1 Approximating IIFS through a Sequence of IFS . 125 6.3.2 An Approximation and Convergence Result . 128 6.4 ProductsofRandomMatrices. 132 6.4.1 ProductsofRandomMatrices. 132 6.4.2 An Approximation and Convergence Result . 136 6.5 Examples ................................... 139 7 Outlook 146 7.1 One-SidedRDSonBanachSpace. 146 7.2 LyapunovnormforRDSonBanachSpace . 153 Appendices 155 A Birkhoff Ergodic Theorem 156 B Kingman Subadditive Ergodic Theorem 161 C Baire Category and Baire Class of Functions 165 Bibliography 167 Index 177 Introduction A dynamical system is a concept in mathematics where a specified rule describes the time dependence of a point in a state space. The mathematical models used to describe the motion of planets, the swinging of a clock pendulum in Newton’s mechanics, the flow of water in a pipe, chemical reactions, or even the number of fish each spring in a lake are examples of dynamical systems. A dynamical system is determined by a state space and a fixed evolution rule which describes how future states follow from the current state. The history of dynamical system began with the foundational work of Poincare [117] and Lyapunov [90] on the qualitative behavior of ordinary differential equations. The concept of dynamical systems was then introduced by Birkhoff [18], followed by impor- tant contributions by Markov [94], Nemytskii and Stefanov [107], Bhatia and Szegoe [16], Smale [127], among others. The main goal of this theory is to study the qualitative behavior of systems from geometrical and topological view points. The concept of random dynamical systems is a comparatively recent development combining ideas and methods from the well developed areas of probability theory and dynamical systems. Due to our inaccurate knowledge of the particular model or due to computational or theoretical limitations (lack of sufficient computational power, in- efficient algorithms or insufficiently developed mathematical and physical theory, for example), the mathematical models never correspond exactly to the phenomenon they are meant to model. Moreover, when considering practical systems we cannot avoid ei- ther external noise or inaccuracy errors in measurements, so every realistic mathematical model should allow for small errors along orbits. To be able to cope with unavoidable uncertainty about the ”correct” parameter values, observed initial states and even the specific mathematical formulation involved, we let randomness be embedded within the model. Therefore, random dynamical systems arise naturally in the modeling of many phenomena in physics, biology, economics, climatology, etc. The concept of random dynamical systems was mainly developed by Arnold [3] and his ”Bremen group”, based on the research of Baxendale [13], Bismut [19], Elworthy [52], Gihman and Skorohod [65], Ikeda and Watanabe [72] and Kunita [83] on two-parameter stochastic flows generated by stochastic differential equations. Three main classes of random dynamical systems are: Products of random maps: Let (Ω, , P) be a probability space and θ : Ω Ω • F → an ergodic transformation preserving the probability P. Let (X, ) be a measur- B 1 2 able space. For a given measurable function ψ : Ω X X we can define the × → corresponding random dynamical system ϕ : N Ω X X by × × → n 1 ψ(θ − ω) ψ(ω), if n 1, ϕ(n,ω)= ◦···◦ ≥ (1) ( idX , otherwise. The random dynamical system ϕ is said to be generated by the random mapping ψ. Conversely, every one-sided discrete time random dynamical system has the form (1), i.e. is a product of random mappings, or an iterated function system, or a system in a random environment (see Arnold [3, pp. 50]). Random differential equations: Let (Ω, , P) be a probability space and (θt)t R : • F ∈ Ω Ω an ergodic flow preserving the probability P. Let f : Ω Rd Rd be a → × → measurable function satisfying that for a fixed ω Ω the function (t,x) f(θ ω,x) ∈ 7→ t is locally Lipschitz in x, integrable in t and f(ω,x) α(ω) x + β(ω), k k≤ k k where t α(θ ω) and t β(θ ω) are locally integrable. Then the random differ- 7→ t 7→ t ential equation x˙ = f(θtω,x) uniquely generates a continuous random dynamical system ϕ : R Ω Rd Rd + × × → satisfying t ϕ(t,ω)x = x + f(θsω, ϕ(s,ω)x) ds, Z0 (we refer to Arnold [3, pp. 57–63] for more details). Stochastic differential equations: The classical Stratonovic stochastic differential • equation m dx = f (x )dt + f (x ) dW j, t R, t 0 t j t ◦ t ∈ Xj=1 where f0,...,fm are smooth vector fields, and W is standard Brownian motion of Rm generates a unique (up to indistinguishability) smooth random dynamical system ϕ over the filtered dynamical system describing Brownian motion (we refer to Arnold [3, pp. 68-107] for more details). For the gap between random dynamical systems and continuous skew products we refer to the paper by Berger and Siegmund [15]. Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. The concept was introduced by Lyapunov when studying the stability of non-stationary solutions of ordinary differential equations, and has been widely employed in studying dynamical systems since then. 3 The fundamental results on Lyapunov exponents for random dynamical systems on finite dimensional systems were first obtained by Oseledets [109] in 1968, which is now called the Oseledets Multiplicative Ergodic Theorem. Originally formulated for products of random matrices, it has been reformulated and reproved several times during the past thirty years. Basically, there are two classes of proofs. One makes use of the Kingman’s Subadditive Ergodic Theorem together with the polar decomposition of square matrices (see Arnold [3] and the references therein). The other one relies on the triangularization of a linear cocycle and the classical ergodic theorem for the triangular cocycles. This technique was also used in the contemporaneous paper of Millionˇsˇcikov [97] who inde- pendently derived a portion of the multiplicative ergodic theorem, and then taken up again by Johnson, Palmer and Sell [74] (assuming a topological setting for the metric dynamical system). Thanks to the multiplicative ergodic theorem of Oseledets [109], the Lyapunov spec- trum of products of random matrices is well defined (under some integrability conditions) and it is a generalization of the Lyapunov spectrum in the deterministic case and the Oseledets subspaces are generalizations of the eigenspaces. The study of the Lyapunov spectrum of linear cocycles is one of the central tasks of the theory of random dynamical systems (see Arnold [3]). In various situations it is of theoretical and practical importance to know when the Lyapunov spectrum is simple and the Oseledets splitting is exponentially separated. Recently, Arbieto and Bochi [2], Bochi [21], Bochi and Viana [22, 23], Bonatti and Viana [24] and Cong [36] have derived some new results on genericity of hyperbolicity of several classes of dynamical systems including smooth dynamical systems and linear cocycles. Let us mention here a result of Cong [36] stating that the set of cocycles with integral separation is open and dense in the space of all bounded Gl(d, R)-cocycles equipped with the uniform topology.
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