This book has its roots in two different areas of mathematics: pure
mathematics, where structures are discovered in the context of other mathematical
structures and investigated, and applications of mathematics,
where mathematical structures are suggested by real–world problems arising
in science and engineering, investigated, and then used to address the
motivating problem. While there are philosophical differences between applied
and pure mathematical scientists, it is often difficult to sort them out.
The authors of this book reflect these different approaches.

Introduction 1
What Are Dynamical Systems? . . . . . . . . . . . . . . . . . . 1
What Is RandomNoise? . . . . . . . . . . . . . . . . . . . . . . 3
What Are Ergodic Theorems? . . . . . . . . . . . . . . . . . . . 8
What Happens for t Large? . . . . . . . . . . . . . . . . . . . . 18
What Is in This Book? . . . . . . . . . . . . . . . . . . . . . . . 21
1 Ergodic Theorems 49
1.1 Birkhoff’s Classical Ergodic Theorem . . . . . . . . . . . 49
1.1.1 Mixing Conditions . . . . . . . . . . . . . . . . . 53
1.1.2 Discrete–Time Stationary Processes . . . . . . . . 54
1.2 Discrete–TimeMarkov Processes . . . . . . . . . . . . . 56
1.3 Continuous–Time Stationary Processes . . . . . . . . . . 60
1.4 Continuous–TimeMarkov Processes . . . . . . . . . . . . 61
2 Convergence Properties of Stochastic Processes 64
2.1 Weak Convergence of Stochastic Processes . . . . . . . . 64
2.1.1 Weak Compactness in C . . . . . . . . . . . . . . 66
2.2 Convergence to a Diffusion Process . . . . . . . . . . . . 68
2.2.1 Diffusion Processes . . . . . . . . . . . . . . . . . 68
2.2.2 Weak Convergence to a Diffusion Process . . . . . 70
2.3 Central Limit Theorems for Stochastic Processes . . . . 73
viii Contents
2.3.1 Continuous–TimeMarkov Processes . . . . . . . . 73
2.3.2 Discrete–TimeMarkov Processes . . . . . . . . . 75
2.3.3 Discrete–Time Stationary Processes . . . . . . . . 76
2.3.4 Continuous–Time Stationary Processes . . . . . . 79
2.4 Large Deviation Theorems . . . . . . . . . . . . . . . . . 80
2.4.1 Continuous–TimeMarkov Processes . . . . . . . . 81
2.4.2 Discrete–TimeMarkov Processes . . . . . . . . . 86
3 Averaging 88
3.1 Volterra Integral Equations . . . . . . . . . . . . . . . . . 88
3.1.1 Linear Volterra Integral Equations . . . . . . . . 92
3.1.2 Some Nonlinear Equations . . . . . . . . . . . . . 97
3.2 Differential Equations . . . . . . . . . . . . . . . . . . . . 99
3.2.1 Linear Differential Equations . . . . . . . . . . . . 101
3.3 Difference Equations . . . . . . . . . . . . . . . . . . . . 102
3.3.1 Linear Difference Equations . . . . . . . . . . . . 105
3.4 Large Deviation for Differential Equations . . . . . . . . 105
3.4.1 Some Auxiliary Results . . . . . . . . . . . . . . . 106
3.4.2 Main Theorem. . . . . . . . . . . . . . . . . . . . 110
3.4.3 Systems with Additive Perturbations . . . . . . . 112
4 Normal Deviations 114
4.1 Volterra Integral Equations . . . . . . . . . . . . . . . . . 114
4.2 Differential Equations . . . . . . . . . . . . . . . . . . . . 120
4.2.1 Markov Perturbations . . . . . . . . . . . . . . . 127
4.3 Difference Equations . . . . . . . . . . . . . . . . . . . . 128
5 Diffusion Approximation 133
5.1 Differential Equations . . . . . . . . . . . . . . . . . . . . 133
5.1.1 Markov Jump Perturbations . . . . . . . . . . . . 134
5.1.2 Some Generalizations . . . . . . . . . . . . . . . . 140
5.1.3 GeneralMarkov Perturbations . . . . . . . . . . . 145
5.1.4 Stationary Perturbations . . . . . . . . . . . . . . 146
5.1.5 Diffusion Approximations to First Integrals . . . 150
5.2 Difference Equations . . . . . . . . . . . . . . . . . . . . 156
5.2.1 Markov Perturbations . . . . . . . . . . . . . . . 156
5.2.2 Diffusion Approximations to First Integrals . . . 161
5.2.3 Stationary Perturbations . . . . . . . . . . . . . . 166
6 Stability 172
6.1 Stability of Perturbed Differential Equations . . . . . . . 172
6.1.1 Jump Perturbations of Nonlinear Equations . . . 173
6.1.2 Stationary Perturbations . . . . . . . . . . . . . . 182
6.2 Stochastic Resonance for Gradient Systems . . . . . . . . 193
6.2.1 Large Deviations near a Stable Static State . . . 193
Contents ix
6.2.2 Transitions Between Stable Static States . . . . . 198
6.2.3 Stochastic Resonance . . . . . . . . . . . . . . . 199
6.3 Randomly Perturbed Difference Equations . . . . . . . . 200
6.3.1 Markov Perturbations: Linear Equations . . . . . 201
6.3.2 Stationary Perturbations . . . . . . . . . . . . . . 203
6.3.3 Markov Perturbations: Nonlinear Equations . . . 205
6.3.4 Stationary Perturbations . . . . . . . . . . . . . . 210
6.3.5 Small Perturbations of a Stable System . . . . . . 211
6.4 Convolution Integral Equations . . . . . . . . . . . . . . 216
6.4.1 Laplace Transforms and Their Inverses . . . . . . 218
6.4.2 Laplace Transforms of Noisy Kernels . . . . . . . 222
7 Markov Chains with Random Transition Probabilities 232
7.1 Stationary RandomEnvironment . . . . . . . . . . . . . 233
7.2 Weakly RandomEnvironments . . . . . . . . . . . . . . 243
7.3 Markov Processes with Randomly Perturbed
Transition Probabilities . . . . . . . . . . . . . . . . . . . 249
7.3.1 Stationary RandomEnvironments . . . . . . . . . 249
7.3.2 Ergodic Theorem for Markov Processes
in RandomEnvironments . . . . . . . . . . . . . 253
7.3.3 Markov Process in a Weakly Random
Environment . . . . . . . . . . . . . . . . . . . . . 254
8 Randomly Perturbed Mechanical Systems 257
8.1 Conservative Systems with Two Degrees of Freedom . . . 257
8.1.1 Conservative Systems . . . . . . . . . . . . . . . . 258
8.1.2 Randomly Perturbed Conservative Systems . . . 262
8.1.3 Behavior of the Perturbed System
near a Knot . . . . . . . . . . . . . . . . . . . . . 273
8.1.4 Diffusion Processes on Graphs . . . . . . . . . . . 285
8.1.5 Simulation of a Two-Well Potential Problem . . . 290
8.2 Linear Oscillating Conservative Systems . . . . . . . . . 290
8.2.1 Free Linear Oscillating Conservative Systems . . 290
8.2.2 Randomly Perturbed Linear
Oscillating Systems . . . . . . . . . . . . . . . . . 293
8.3 A Rigid Body with a Fixed Point . . . . . . . . . . . . . 297
8.3.1 Motion of a Rigid Body around
a Fixed Point . . . . . . . . . . . . . . . . . . . . 298
8.3.2 Analysis of Randomly PerturbedMotions . . . . 299
9 Dynamical Systems on a Torus 303
9.1 Theory of Rotation Numbers . . . . . . . . . . . . . . . . 303
9.1.1 Existence of the Rotation Number . . . . . . . . 305
9.1.2 Purely Periodic Systems . . . . . . . . . . . . . . 307
9.1.3 Ergodic Systems . . . . . . . . . . . . . . . . . . . 308
x Contents
9.1.4 Simulation of Rotation Numbers . . . . . . . . . . 310
9.2 Randomly Perturbed Torus Flows . . . . . . . . . . . . . 311
9.2.1 Rotation