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Presents a unified treatment of stochastic differential equations in abstract, mainly Hilbert, spaces.
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 establ
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.
With this hands-on introduction readers will learn what SDEs are all about and how they should use them in practice.
This textbook provides the first systematic presentation of the theory of stochastic differential equations with Markovian switching. It presents the basic principles at an introductory level but emphasizes current advanced level research trends. The material takes into account all the features of Ito equations, Markovian switching, interval systems and time-lag. The theory developed is applicable in different and complicated situations in many branches of science and industry.
Random Differential Inequalities
This book develops a unified approach to qualitative analysis of large scale systems described by many diversified types of equations.
Where do solutions go, and how do they behave en route? These are two of the major questions addressed by the qualita tive theory of differential equations. The purpose of this book is to answer these questions for certain classes of equa tions by recourse to the framework of semidynamical systems (or topological dynamics as it is sometimes called). This approach makes it possible to treat a seemingly broad range of equations from nonautonomous ordinary differential equa tions and partial differential equations to stochastic differ ential equations. The methods are not limited to the examples presented here, though. The basic idea is this: Embed some representation of the solutions of the equation (and perhaps the equation itself) in an appropriate function space. This space serves as the phase space for the semidynamical system. The phase map must be chosen so as to generate solutions to the equation from an initial value. In most instances it is necessary to provide a "weak" topology on the phase space. Typically the space is infinite dimensional. These considerations motivate the requirement to study semidynamical systems in non locally compact spaces. Our objective here is to present only those results needed for the kinds of applications one is likely to encounter in differen tial equations. Additional properties and extensions of ab stract semidynamical systems are left as exercises. The power of the semidynamical framework makes it possible to character- Preface ize the asymptotic behavior of the solutions of such a wide class of equations.
The first paper in the volume, Stochastic Evolution Equations by N V Krylov and B L Rozovskii, was originally published in Russian in 1979. After more than a quarter-century, this paper remains a standard reference in the field of stochastic partial differential equations (SPDEs) and continues to attract attention of mathematicians of all generations, because, together with a short but thorough introduction to SPDEs, it presents a number of optimal and essentially non-improvable results about solvability for a large class of both linear and non-linear equations.