The stability of stochastic differential equations in abstract, mainly Hilbert, spaces receives a unified treatment in this self-contained book. It covers basic theory as well as computational techniques for handling the stochastic stability of systems from mathematical, physical and biological problems. Its core material is divided into three parts devoted respectively to the stochastic stability of linear systems, non-linear systems, and time-delay systems. The focus is on stability of stochastic dynamical processes affected by white noise, which are described by partial differential equations such as the Navier–Stokes equations. A range of mathematicians and scientists, including those involved in numerical computation, will find this book useful. It is also ideal for engineers working on stochastic systems and their control, and researchers in mathematical physics or biology.
Author(s): Kai Liu
Series: London Mathematical Society Lecture Note Series (453)
Publisher: Cambridge University Press
Year: 2019
Language: English
Pages: 277
Frontmatter......Page 2
Contents......Page 6
Preface......Page 8
1.1 Linear Operators, Semigroups, and Examples......Page 12
1.2 Stochastic Processes and Martingales......Page 27
1.3 Wiener Processes and Stochastic Integration......Page 33
1.4 Stochastic Differential Equations......Page 37
1.5 Definitions and Methods of Stochastic Stability......Page 46
1.6 Notes and Comments......Page 55
2.1 Deterministic Linear Systems......Page 57
2.2 Lyapunov Equations and Stochastic Stability......Page 72
2.3 Systems with Boundary Noise......Page 93
2.4 Exponentially Stable Stationary Solutions......Page 99
2.5 Some Examples......Page 103
2.6 Notes and Comments......Page 106
3.1 An Extension of Linear Stability Criteria......Page 109
3.2 Comparison Approach......Page 116
3.3 Nonautonomous Stochastic Systems......Page 119
3.4 Stability in Probability and Sample Path......Page 133
3.5 Lyapunov Function Characterization......Page 142
3.6 Two Applications......Page 154
3.7 Invariant Measures and Ultimate Boundedness......Page 161
3.8 Decay Rate......Page 170
3.9 Stabilization of Systems by Noise......Page 179
4.1 Deterministic Systems......Page 189
4 Stability of Stochastic Functional Differential Equations......Page 188
4.2 Linear Systems with Additive Noise......Page 203
4.3 Linear Systems with Multiplicative Noise......Page 207
4.4 Stability of Nonlinear Systems......Page 217
4.5 Notes and Comments......Page 234
5.1 Applications in Mathematical Biology......Page 237
5.2 Applications in Mathematical Physics......Page 243
5.3 Applications in Stochastic Control......Page 250
5.4 Notes and Comments......Page 254
A Proof of Theorem 4.1.5......Page 256
B Proof of Proposition 4.1.7......Page 258
C Proof of Proposition 4.1.10......Page 259
D Proof of Proposition 4.1.14......Page 261
References......Page 263
Index......Page 276
