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.

In engineering, physics and economics, many dynamical systems involving with stochastic components and random noise are often modeled by stochastic models. The stochastic effects of these models are often used to describe the uncertainty about the operating systems. Motivated by the development of analysis and theory of stochastic processes, as well as the studies of natural sciences, the theory of stochastic differential equations in infinite dimensional spaces evolves gradually into a branch of modern analysis. Many qualitative properties of such systems have been studied in the past few decades, among which, investigation of stability of such systems is often regarded as the first characteristic of the dynamical systems or models. In general, this thesis is mainly concerned with the studies of the stability property of stochastic differential equations in Hilbert spaces. Chapter 1 is an introduction to a brief history of stochastic differential equations in infinite dimensions, together with an overview of the studies. Chapter 2 is a presentation of preliminaries to some basic stochastic analysis. In Chapter 3, we study the stability in distribution of mild solutions to stochastic delay differential equations with Poisson jumps. Firstly, we use approximation of strong solutions to pass on the stability of strong solutions to the mild ones. Then, by constructing a suitable metric between the transition probability functions of mild solutions, we obtain the desired stability result under some suitable conditions. In Chapter 4, we investigate the stochastic partial delay differential equations with Markovian switching and Poisson jumps. By estimating the coefficients of energy equality, both the exponential stability and almost sure exponential stability of energy solutions to the equations are obtained. In Chapter 5, we study the relationship among strong, weak and mild solutions to the stochastic functional differential equations of neutral type. Finally, in Chapter 6, we study the asymptotic stability of two types of equations, impulsive stochastic delay differential equations with Poisson jumps and stochastic evolution equations with Poisson jumps. By employing the fixed point theorem, we derive the desired stability results under some criteria.

Since the publication of the first edition of the present volume in 1980, the stochastic stability of differential equations has become a very popular subject of research in mathematics and engineering. To date exact formulas for the Lyapunov exponent, the criteria for the moment and almost sure stability, and for the existence of stationary and periodic solutions of stochastic differential equations have been widely used in the literature. In this updated volume readers will find important new results on the moment Lyapunov exponent, stability index and some other fields, obtained after publication of the first edition, and a significantly expanded bibliography. This volume provides a solid foundation for students in graduate courses in mathematics and its applications. It is also useful for those researchers who would like to learn more about this subject, to start their research in this area or to study the properties of concrete mechanical systems subjected to random perturbations.

This advanced undergraduate and graduate text has now been revised and updated to cover the basic principles and applications of various types of stochastic systems, with much on theory and applications not previously available in book form. The text is also useful as a reference source for pure and applied mathematicians, statisticians and probabilists, engineers in control and communications, and information scientists, physicists and economists. Has been revised and updated to cover the basic principles and applications of various types of stochastic systems Useful as a reference source for pure and applied mathematicians, statisticians and probabilists, engineers in control and communications, and information scientists, physicists and economists

Based on the proceedings of the International Conference on Stochastic Partial Differential Equations and Applications-V held in Trento, Italy, this illuminating reference presents applications in filtering theory, stochastic quantization, quantum probability, and mathematical finance and identifies paths for future research in the field. Stochastic Partial Differential Equations and Applications analyzes recent developments in the study of quantum random fields, control theory, white noise, and fluid dynamics. It presents precise conditions for nontrivial and well-defined scattering, new Gaussian noise terms, models depicting the asymptotic behavior of evolution equations, and solutions to filtering dilemmas in signal processing. With contributions from more than 40 leading experts in the field, Stochastic Partial Differential Equations and Applications is an excellent resource for pure and applied mathematicians; numerical analysts; mathematical physicists; geometers; economists; probabilists; computer scientists; control, electrical, and electronics engineers; and upper-level undergraduate and graduate students in these disciplines.