Probabilistic properties of deterministic systems

Description

Title page Preface Chapter 1. Introduction 1.1 A simple system generating a density of states 1.2 The evolution of densities: an intuitive point of view 1.3 Trajectories versus densities Chapter 2. The toolbox 2.1 Measures and measure spaces 2.2 Lebesgue integration 2.3 Convergence of sequences of functions Chapter 3. Markov and Frobenius-Perron operators 3.1 Markov operators 3.2 The Frobenius-Perron operator 3.3 The Koopman operator Chapter 4. Studying chaos with densities 4.1 Invariant measures and measure-preserving transformations 4.2 Ergodic transformations 4.3 Mixing and exactness 4.4 Using the Frobenius-Perron and Koopman operators for c1assifying transformations 4.5 Kolmogorov automorphisms Chapter 5. The asymptotic properties of densities 5.1 Weak and strong precompactness 5.2 Properties of the averages A_nf 5.3 Asymptotic periodicity of {P^nf} 5.4 The existence of stationary densities 5.5 Ergodicity, mixing, and exactness 5.6 Asymptotic stability of {P^n} 5.7 Markov operators defined by a stochastic kernel 5.8 Conditions for the existence of lower-bound functions Chapter 6. The behavior of transformations on intervals and manifolds 6.1 Functions of bounded variation 6.2 Piecewise monotonic mappings 6.3 Piecewise convex transformations with a strong repellor 6.4 Asymptotically periodic transformations 6.5 Change of variables 6.6 Transformations on the real line 6.7 Manifolds 6.8 Expanding mappings on manifolds Chapter 7. Continuous time systems: an introduction 7.1 Two examples of continuous time systems 7.2 Dynamical and semidynamical systems 7.3 Invariance, ergodicity, mixing, and exactness in semidynamical systems 7.4 Semigroups of the Frobenius-Perron and Koopman operators 7.5 Infinitesimal operators 7.6 Infinitesimal operators for semigroups generated by systems of ordinary differential equations 7.7 Applications of the semigroups of the Frobenius-Perron and Koopman operators 7.8 The Hille-Yosida theorem and its consequences 7.9 Further applications of the Hille-Yosida theorem 7.10 The relation between the Frobenius-Perron and Koopman operators Chapter 8. Discrete time processes embedded in continuous time systems 8.1 The relation between discrete and continuous time processes 8.2 Probability theory and Poisson processes 8.3 Discrete time systems governed by Poisson processes 8.4 The linear Boltzmann equation: an intuitive point of view 8.5 Elementary properties of the solutions of the linear Boltzmann equation 8.6 Further properties of the linear Boltzmann equation 8.7 Effect of properties of the Markov operator on solutions of the linear Boltzmann equation 8.8 Linear Boltzmann equation with a stochastic kernel 8.9 The linear Tjon-Wu equation Chapter 9. Entropy 9.1 Basic definitions 9.2 Entropy of P^nf when P is a Markov operator 9.3 Entropy H(P^nf) when P is a Frobenius-Perron operator 9.4 Behavior of P^nf from H(P^nf) Chapter 10. Stochastic perturbation of discrete time systems 10.1 Independent random variables 10.2 Mathematical expectation and variance 10.3 Stochastic convergence 10.4 Discrete time systems with randomly applied stochastic perturbations 10.5 Discrete time systems with constantly applied stochastic perturbations 10.6 Small continuous stochastic perturbations of discrete time systems Chapter 11. Stochastic perturbation of continuous time systems 11.1 One-dimensional Wiener processes (Brownian motion) 11.2 d-Dimensional Wiener processes (Brownian motion) 11.3 The stochastic Itô integral: development 11.4 The stochastic Itô integral: special cases 11.5 Stochastic differential equations 11.6 The Fokker-Planck (Kolmogorov forward) equation 11.7 Properties of the solutions of the Fokker-Planck equation 11.8 Semigroups of Markov operators generated by parabolic equations 11.9 Asymptotic stability of solutions of the Fokker-Planck equation 11.10 An extension of the Liapunov function method References Notation and symbols Index