Stochastic Dynamics Hans Crauel Matthias Gundlach, Editors Springer Contents ix Stochastic Flows with Random Hyperbolic Sets 139 References 144 Some Questions in Random Dynamical Systems Involving Real Noise Processes Russell Johnson Introduction Basic theory Random Orthogonal Polynomials A Random Bifurcation Problem: Introduction Robustness of Random Bifurcation An Example of Robust Random Bifurcation Other Applications References Topological, Smooth, and Control Techniques for Perturbed Systems Fritz Colonius and Wolfgang Kliemann Introduction Stochastic Systems, Control Flows, and Diffusion Processes: Basic Concepts Attractors, Invariant Measures, Control, and Chaos 3.1 Concepts from Topological Dynamics 3.2 Deterministic Perturbed Systems 3.3 Global Behavior of Markov Diffusion Systems 3.4 Invariant Measures 3.5 Attractors Global Behavior of Parameter Dependent Perturbed Systems References Perturbation Methods for Lyapunov Exponents Volker Wihstutz Introduction Basic Perturbation Schemes Asymptotics of Lyapunov Exponents Large Noise and Application to Stability Problems Open Problems References 147 147 150 154 160 162 169 171 174 181 181 183 187 187 188 192 195 199 202 205 209 209 216 219 229 234 235 10 The Lyapunov Exponent of the Euler Scheme for Stochastic Differential Equations Denis Talay 241 Introduction 241 An elementary example and objectives 244 Preface The conference on Random Dynamical Systems took place from April 28 to May 2, 1997, in Bremen and was organized by Matthias Gundlach and Wolfgang Kliemann with the help of Fritz Colonius and Hans Crauel It brought together mathematicians and scientists for whom mathematics, in particular the field of random dynamical systems, is of relevance The aim of the conference was to present the current state in the theory of random dynamical systems (RDS), its connections to other areas of mathematics, major fields of applications, and related numerical methods in a coherent way It was, however, not by accident that the conference was centered around the 60th birthday of Ludwig Arnold The theory of RDS owes much of its current state and status to Ludwig Arnold Many aspects of the theory, a large number of results, and several substantial contributions were accomplished by Ludwig Arnold An even larger number of contributions has been initiated by him The field benefited much from his enthusiasm, his openness for problems not completely aligned with his own research interests, his ability to explain mathematics to researchers from other sciences as well as his ability to get mathematicians interested in problems from applications not completely aligned with their research interests In particular, a considerable part of the impact stochastics had on physical chemistry as well as on engineering goes back to Ludwig Arnold He built up an active research group, known as “the Bremen group” While this volume was being prepared, a monograph on RDS authored by Ludwig Arnold appeared The purpose of the present volume is to document and, to some extent, summarize the current state of the field of RDS beyond this monograph The contributions of this volume emphasize stochastic aspects of dynamics They deal with stochastic differential equations, diffusion processes and statistical mechanics Further topics are large deviations, stochastic bifurcation, Lyapunov exponents and numerics Berlin, Germany Bremen, Germany Hans Crauel Volker Matthias Gundlach August 1998 Contents Preface Contributors and Speakers Lectures Stochastic Dynamics: Building upon Two Cultures Stability Along Trajectories at a Stochastic Bifurcation Point Peter Baxendale Introduction Stability for {xt : t ≥ 0} Stability for {(xt , θt ) : t ≥ 0} ˜ Formula for λ Ratio of the two Lyapunov exponents Rotational symmetry Translational symmetry Homogeneous stochastic flows References v xiii xix xxi 1 10 12 15 18 20 24 Bifurcations of One-Dimensional Stochastic Differential Equations Hans Crauel, Peter Imkeller, and Marcus Steinkamp Introduction Invariant measures of one-dimensional systems Bifurcation Sufficient criteria for the finiteness of the speed measure References 27 27 30 39 41 46 P-Bifurcations in the Noisy Duffing–van der Pol Equation Yan Liang and N Sri Namachchivaya Introduction Statement of the Problem Deterministic Global Analysis Stochastic Analysis 49 49 52 53 56 viii Contents The Phenomenological Approach Mean First Passage Time Conclusions References 59 65 67 68 The Stochastic Brusselator: Parametric Noise Destroys Hopf Bifurcation Ludwig Arnold, Gabriele Bleckert, and ´ Klaus Reiner Schenk-Hoppe Introduction The Deterministic Brusselator The Stochastic Brusselator Bifurcation and Long-Term Behavior 4.1 Additive versus multiplicative noise 4.2 Invariant measures 4.3 What is stochastic bifurcation? 4.4 P-bifurcation 4.5 Lyapunov exponents 4.6 Additive noise destroys pitchfork bifurcation 4.7 No D-Bifurcation for the stochastic Brusselator References 71 71 73 73 78 78 78 81 81 82 84 86 90 Numerical Approximation of Random Attractors Hannes Keller and Gunter Ochs Introduction Definitions 2.1 Random dynamical systems 2.2 Random attractors 2.3 Invariant measures and invariant manifolds A numerical algorithm 3.1 The concept of the algorithm 3.2 Implementation 3.3 Continuation of unstable manifolds The Duffing–van der Pol equation 4.1 The deterministic system 4.2 The stochastic system Discussion References 93 93 94 94 95 97 98 98 100 103 104 104 105 109 114 Random Hyperbolic Systems Volker Matthias Gundlach and Yuri Kifer Introduction Random Hyperbolic Transformations Discrete Dynamics on Random Hyperbolic Sets Ergodic Theory on Random Hyperbolic Sets 117 117 119 124 133 x Contents The linear case The nonlinear case Expansion of the discretization error Comments on numerical issues References 246 252 254 255 256 11 Towards a Theory of Random Numerical Dynamics Peter E Kloeden, Hannes Keller, and ă Bjorn Schmalfuò Introduction Deterministic Numerical Dynamics Random and Nonautonomous Dynamical Systems Pullback Attractors of NDS and RDS Pullback Attractors under Discretization Discretization of a Random Hyperbolic Point Open questions References 259 259 260 264 266 270 275 279 280 12 Canonical Stochastic Differential Equations based on L´vy Processes and Their Supports e Hiroshi Kunita Introduction Stochastic flows determined by a canonical SDE with jumps driven by a L´vy process e Supports of L´vy processes and stochastic flows e driven by them Applications of the support theorem Proofs of Theorems 3.2 and 3.3 References 13 On the Link Between Fractional and Stochastic Calculus ă Martina Zahle Introduction Notions and results from fractional calculus An extension of Stieltjes integrals An integral operator, continuity and contraction properties Integral transformation formulae An extension of the integral and its stochastic version Processes with generalized quadratic variation and Itˆ formula o Differential equations driven by fractal functions of order greater than one half 283 283 285 287 290 292 303 305 305 306 309 312 314 315 316 320 Contents 10 xi Stochastic differential equations driven by processes with absolutely continuous generalized covariations 321 References 324 14 Asymptotic Curvature for Stochastic Dynamical Systems Michael Cranston and Yves Le Jan Introduction Isotropic Brownian flows Random walks on diffeomorphisms of Rd The convergence of the second fundamental forms References 327 327 330 331 333 337 15 Stochastic Analysis on (Infinite-Dimensional) Product Manifolds Sergio Albeverio, Alexei Daletskii, and Yuri Kondratiev Introduction Main geometrical structures and stochastic calculus on product manifolds 2.1 Main notations 2.2 Differentiable and metric structures Tangent bundle 2.3 Classes of vector and operator fields 2.4 Stochastic integrals 2.5 Stochastic differential equations 2.6 Stochastic differential equations on product groups Quasi-invariance of the distributions Stochastic dynamics for lattice models associated with Gibbs measures on product manifolds 3.1 Gibbs measures on product manifolds 3.2 Stochastic dynamics 3.3 Ergodicity of the dynamics and extremality of Gibbs measures Stochastic dynamics in fluctuation space 4.1 Mixing properties and space of fluctuations 4.2 Dynamics in fluctuation spaces References 357 359 359 362 364 16 Evolutionary Dynamics in Random Environments Lloyd Demetrius and Volker Matthias Gundlach Introduction Population Dynamics Models The Thermodynamic Formalism Perturbations of Equilibrium States 371 371 373 377 381 339 339 343 343 344 348 349 350 351 353 353 355 426 P Kotelenez (iv) Let Mi (v) be the martingale part of B(ψi (v)), i = 1, Since M1 (v) − M2 (v) = B(ψ1 (v)) − B(ψ2 )) we obtain (using, e.g., Ikeda and Watanabe [20, Ch II, §2] and an elementary calculation) c[ u ≤ 2(B(ψ1 (v)) − B(ψ2 (v)))d(M1 (v) − M2 (v))] ≤ (77) sup(B(ψ1 (v)) − B(ψ2 (v)))2 + 8c4 B(ψ1 (u)) − B(ψ2 (u)) , v≤u where c > is an arbitrary positive constant (v) Since |B (x)x| is bounded we obtain from (72)-(77), the BurkholderDavis-Gundy inequality and Gronwall’s lemma E sup(B(ψ1 (v))−B(ψ2 (v)))2 ≤ cF,J ,B,T E v≤u + B(Z1 (t − v)) − B(Z2 (t−v)) u ˜ {ρ2 (ˇ(v, Z1 , r), r(v, Z2 , r))+ r ˇ ˜ 0,λ + ( ˆ ΛL (ˇ(v, Z2 , r)−p) r L∈{F,J } ˆ ˆ ·|B(Z1 (t−v, p))−B(Z2 (t−v, p))|dp)2 }dv Now, using (35) we obtain (71) In order to derive a solution of (25) we will now assume: ˆ Hypothesis Z ∈ W0,2,λ,[0,∞) is a fixed random input Since we want to be conceptual rather than computational we make an assumption on the initial condition Hypothesis X0 is F0 -measurable, and it does not depend on r ∈ Rd Moreover, there are constants < c < c < ∞ such that c ≤ |X0 (ω)| ≤ c a.s Lemma 6.7 Suppose Hypothesis 3, (with m > d + 2), 5, and For ˆ Zi ∈ W0,2,λ,[0,∞) set Yi (t, r) := Y (t, Zi , X0 , r) := Y (t, Z, Zi , X0 , r), i = 1, Then, for any t > and c := cF,J ,λ,T,X0 ˜ sup E B(Y1 (t))−B(Y2 (t)) 0≤t≤T ˜ 0,λ ≤ c T E B(Z1 (s))−B(Z2 (s)) 0,λ ds (78) Proof Fix t ∈ [0, T ] Then, by (61) and the properties of B (B(Y1 (t, r)) − B(Y2 (t, r)))2 ≤ c(B(ψ1 (t, r)) − B(ψ2 (t, r)))2 Application of (71) with change of variables t − v → s and integration against λ(r)dr implies (78) Now we can solve (32) on H0,λ under the previous assumptions Microscopic and Mezoscopic Models for Mass Distributions 427 Theorem 6.8 Suppose Hypothesis 3, (with m > d + 2), 5, and Then (32) has a unique weak (Itˆ) solution X(·, X0 ) ∈ H0,λ,[0,∞) o Proof (i) By Theorem 6.2 the solution of the bilinear SPDE (31) satisˆ ˆ fies the relation Y (·, Z, X0 ) := Y (·, Z, Z, X0 ) ∈ H0,λ,[0,∞) for any Z, Z ∈ ˆ in our notation, we W0,2,λ,[0,∞) Hence, suppressing the dependence on Z iteratively define Yn+1 (·, Yn , X0 ), n ≥ 1, Y1 ≡ X0 (78) implies that this ˜ sequence has a unique fixed point X := limn→∞ B(Yn ) on W0,2,λ,[0,∞) Set ˜ X(t, r) := B −1 (X(t, r)) (ii) Let f : R → R be bounded and continuous Then we easily verify E f (Yn (t)) − f (X(t)) 0,λ → 0, as n → ∞ (79) In particular, choosing f with |f |(x) ≤ |x| for all x ∈ R, we obtain by (58) E f (X(t)) ≤ lim inf n→∞ E Yn (t) ≤ cF,J ,λ,T E|X0 |2 Choos0,λ 0,λ ing fm ≥ such that fm (x) ≤ |x| and fm (x) ↑ |x| for all x ∈ R, monotone convergence implies sup E X(t) 0≤t≤T 0,λ ≤ cF,J ,λ,T E|X0 |2 (80) In particular, X ∈ W0,2,λ,[0,∞) Next, set µ(dr) := λ(r)dr and let dT be the Lebesgue measure on [0, T ] Then µ and dT are finite Borel measures and by Chebyshev’s inequality and (58) dT ⊗ µ ⊗ P {(t, r, ω) : |Yn (t, r, ω)| ≥ N } ≤ cF,J ,λ,T E|X0 |2 N2 Hence, |Yn |2 is uniformly integrable on L1 ([0, T ] × Rd × Ω, dT ⊗ µ ⊗ P ) By T (79) and (80) this implies E Yn (t) − X(t) dt → 0, as n → ∞ 0,λ (iii) Repeating the arguments of the proof of Theorem 6.2, it follows that X(·, X, X0 ) is a solution of (31) in H0,λ with Z ≡ X and that it is an element of Wm,2,λ,[0,∞) Hence, X(·, X, X0 ) is a strong solution of (32) on H0,λ (cf Da Prato and Zabczyk [9]) and, therefore, X(·, X, X0 ) ∈ H0,λ,[0,∞) The uniqueness follows directly from Theorem 6.2 Mezoscopic Models with Creation/Annihilation The assumption of mass conservation was used to derive SPDE’s for interacting and diffusing particle systems by essentially considering the motion of N particles in such a way that >0 = a+ and −