Stability of Infinite Dimensional Stochastic Differential Equations with Applications

by Liu; Kai

• ISBN13: 9781584885986
• ISBN10: 158488598X
• Format: Hardcover
• Copyright: 2005-08-23
• Publisher: Chapman & Hall/

Summary

Stochastic differential equations in infinite dimensional spaces are motivated by the theory and analysis of stochastic processes and by applications such as stochastic control, population biology, and turbulence, where the analysis and control of such systems involves investigating their stability. While the theory of such equations is well established, the study of their stability properties has grown rapidly only in the past 20 years, and most results have remained scattered in journals and conference proceedings.This book offers a systematic presentation of the modern theory of the stability of stochastic differential equations in infinite dimensional spaces - particularly Hilbert spaces. The treatment includes a review of basic concepts and investigation of the stability theory of linear and nonlinear stochastic differential equations and stochastic functional differential equations in infinite dimensions. The final chapter explores topics and applications such as stochastic optimal control and feedback stabilization, stochastic reaction-diffusion, Navier-Stokes equations, and stochastic population dynamics.In recent years, this area of study has become the focus of increasing attention, and the relevant literature has expanded greatly. Stability of Infinite Dimensional Stochastic Differential Equations with Applications makes up-to-date material in this important field accessible even to newcomers and lays the foundation for future advances.

Table of Contents

Preface	ix
1 Stochastic Differential Equations in Infinite Dimensions	1	(38)
1.1 Notations, Definitions and Preliminaries	1	(10)
1.2 Wiener Processes and Stochastic Integration	11	(5)
1.3 Stochastic Evolution Equations	16	(13)
1.3.1 Variational Approach and Strong Solutions	18	(4)
1.3.2 Semigroup Approach and Mild Solutions	22	(7)
1.4 Definitions and Methods of Stability	29	(8)
1.5 Notes and Comments	37	(2)
2 Stability of Linear Stochastic Differential Equations	39	(42)
2.1 Stable Semigroups	39	(15)
2.1.1 Lyapunov Functions	42	(9)
2.1.2 A Useful Stability Criterion	51	(3)
2.2 Lyapunov Equations and Stability	54	(16)
2.2.1 Characterization of Mean Square Stability	55	(8)
2.2.2 Almost Sure Pathwise Stability	63	(7)
2.3 Uniformly Asymptotic Stability	70	(3)
2.4 Some Examples	73	(5)
2.5 Notes and Comments	78	(3)
3 Stability of Nonlinear Stochastic Differential Equations	81	(94)
3.1 Equivalence of Lρ-Stability and Exponential Stability	81	(13)
3.1.1 An Extension of Linear Stability Criteria	83	(5)
3.1.2 Comparison Approaches	88	(6)
3.2 A Coercive Decay Condition	94	(8)
3.3 Stability of Semilinear Stochastic Evolution Equations	102	(8)
3.4 Lyapunov Functions for Strong Solutions	110	(11)
3.5 Two Applications	121	(10)
3.5.1 Stability in Probability	121	(3)
3.5.2 Ultimate Boundedness and Invariant Measures	124	(7)
3.6 Further Results on Invariant Measures	131	(6)
3.7 Stability, Ultimate Boundedness of Mild Solutions and Invariant Measures	137	(13)
3.7.1 Lyapunov Functions for Mild Solutions	137	(6)
3.7.2 Ultimate Boundedness and Invariant Measures	143	(7)
3.8 Decay Rates of Systems	150	(9)
3.8.1 Decay in the ρ-th Moment	151	(3)
3.8.2 Almost Sure Pathwise Decay	154	(5)
3.9 Stabilization of Systems by Noise	159	(7)
3.9.1 Nonlinear Deterministic Equations	160	(4)
3.9.2 Nonlinear Stochastic Equations	164	(2)
3.10 Lyapunov Exponents and Stabilization	166	(7)
3.11 Notes and Comments	173	(2)
4 Stability of Stochastic Functional Differential Equations	175	(54)
4.1 Linear Deterministic Equations	175	(17)
4.1.1 Stable Semigroups (Finite Delays)	176	(10)
4.1.2 Stable Semigroups (Infinite Delays)	186	(6)
4.2 Stability Equivalence and Reduction of Neutral Equations	192	(20)
4.2.1 Stability of Retarded Stochastic Systems	192	(9)
4.2.2 Stability of Neutral Stochastic Systems	201	(11)
4.3 Decay Criteria of Stochastic Delay Differential Equations	212	(8)
4.3.1 Nonlinear Coercive Conditions for Decay	213	(4)
4.3.2 Linear Stability Conditions	217	(3)
4.4 Razumikhin Type Stability Theorems	220	(7)
4.5 Notes and Comments	227	(2)
5 Some Related Topics of Stability and Applications	229	(38)
5.1 Parabolic Equations with Boundary and Pointwise Noise	229	(7)
5.2 Stochastic Stability and Quadratic Control	236	(10)
5.2.1 Optimal Control on a Finite Interval	237	(4)
5.2.2 Optimal Control on an Infinite Interval	241	(5)
5.3 Feedback Stabilization of Stochastic Differential Equations	246	(5)
5.4 Stochastic Models in Mathematical Physics	251	(6)
5.4.1 Stochastic Reaction-Diffusion Equations	251	(3)
5.4.2 Stochastic Navier-Stokes Equations	254	(3)
5.5 Stochastic Systems Related to Multi-Species Population Dynamics	257	(7)
5.6 Notes and Comments	264	(3)
Appendix	267	(6)
A The Proof of Proposition 4.2.4	267	(1)
B Existence and Uniqueness of Strong Solutions of Stochastic Delay Differential Equations	268	(5)
References	273	(24)
Index	297