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Author: Boris L. Rozovsky
Publisher: Springer
ISBN: 3319948938
Category : Mathematics
Languages : en
Pages : 340
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Book Description
This monograph, now in a thoroughly revised second edition, develops the theory of stochastic calculus in Hilbert spaces and applies the results to the study of generalized solutions of stochastic parabolic equations. The emphasis lies on second-order stochastic parabolic equations and their connection to random dynamical systems. The authors further explore applications to the theory of optimal non-linear filtering, prediction, and smoothing of partially observed diffusion processes. The new edition now also includes a chapter on chaos expansion for linear stochastic evolution systems. This book will appeal to anyone working in disciplines that require tools from stochastic analysis and PDEs, including pure mathematics, financial mathematics, engineering and physics.
Author: Boris L. Rozovsky
Publisher: Springer
ISBN: 3319948938
Category : Mathematics
Languages : en
Pages : 340
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Book Description
This monograph, now in a thoroughly revised second edition, develops the theory of stochastic calculus in Hilbert spaces and applies the results to the study of generalized solutions of stochastic parabolic equations. The emphasis lies on second-order stochastic parabolic equations and their connection to random dynamical systems. The authors further explore applications to the theory of optimal non-linear filtering, prediction, and smoothing of partially observed diffusion processes. The new edition now also includes a chapter on chaos expansion for linear stochastic evolution systems. This book will appeal to anyone working in disciplines that require tools from stochastic analysis and PDEs, including pure mathematics, financial mathematics, engineering and physics.
Author: Vladimir S. Korolyuk
Publisher: Springer Science & Business Media
ISBN: 940114625X
Category : Mathematics
Languages : en
Pages : 195
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Book Description
In this monograph stochastic models of systems analysis are discussed. It covers many aspects and different stages from the construction of mathematical models of real systems, through mathematical analysis of models based on simplification methods, to the interpretation of real stochastic systems. The stochastic models described here share the property that their evolutionary aspects develop under the influence of random factors. It has been assumed that the evolution takes place in a random medium, i.e. unilateral interaction between the system and the medium. As only Markovian models of random medium are considered in this book, the stochastic models described here are determined by two processes, a switching process describing the evolution of the systems and a switching process describing the changes of the random medium. Audience: This book will be of interest to postgraduate students and researchers whose work involves probability theory, stochastic processes, mathematical systems theory, ordinary differential equations, operator theory, or mathematical modelling and industrial mathematics.
Author: Peter H. Baxendale
Publisher: World Scientific
ISBN: 9812706623
Category : Science
Languages : en
Pages : 416
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Book Description
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.
Author: Valery I. Klyatskin
Publisher: Elsevier
ISBN: 008050485X
Category : Science
Languages : en
Pages : 211
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Book Description
Fluctuating parameters appear in a variety of physical systems and phenomena. They typically come either as random forces/sources, or advecting velocities, or media (material) parameters, like refraction index, conductivity, diffusivity, etc. The well known example of Brownian particle suspended in fluid and subjected to random molecular bombardment laid the foundation for modern stochastic calculus and statistical physics. Other important examples include turbulent transport and diffusion of particle-tracers (pollutants), or continuous densities (''oil slicks''), wave propagation and scattering in randomly inhomogeneous media, for instance light or sound propagating in the turbulent atmosphere.Such models naturally render to statistical description, where the input parameters and solutions are expressed by random processes and fields.The fundamental problem of stochastic dynamics is to identify the essential characteristics of system (its state and evolution), and relate those to the input parameters of the system and initial data.This raises a host of challenging mathematical issues. One could rarely solve such systems exactly (or approximately) in a closed analytic form, and their solutions depend in a complicated implicit manner on the initial-boundary data, forcing and system's (media) parameters . In mathematical terms such solution becomes a complicated "nonlinear functional" of random fields and processes.Part I gives mathematical formulation for the basic physical models of transport, diffusion, propagation and develops some analytic tools.Part II sets up and applies the techniques of variational calculus and stochastic analysis, like Fokker-Plank equation to those models, to produce exact or approximate solutions, or in worst case numeric procedures. The exposition is motivated and demonstrated with numerous examples.Part III takes up issues for the coherent phenomena in stochastic dynamical systems, described by ordinary and partial differential equations, like wave propagation in randomly layered media (localization), turbulent advection of passive tracers (clustering).Each chapter is appended with problems the reader to solve by himself (herself), which will be a good training for independent investigations.·This book is translation from Russian and is completed with new principal results of recent research.·The book develops mathematical tools of stochastic analysis, and applies them to a wide range of physical models of particles, fluids, and waves.·Accessible to a broad audience with general background in mathematical physics, but no special expertise in stochastic analysis, wave propagation or turbulence
Author: Henry P. McKean
Publisher: American Mathematical Society
ISBN: 1470477874
Category : Mathematics
Languages : en
Pages : 159
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Book Description
This little book is a brilliant introduction to an important boundary field between the theory of probability and differential equations. —E. B. Dynkin, Mathematical Reviews This well-written book has been used for many years to learn about stochastic integrals. The book starts with the presentation of Brownian motion, then deals with stochastic integrals and differentials, including the famous Itô lemma. The rest of the book is devoted to various topics of stochastic integral equations, including those on smooth manifolds. Originally published in 1969, this classic book is ideal for supplementary reading or independent study. It is suitable for graduate students and researchers interested in probability, stochastic processes, and their applications.
Author: Henrik Jeldtoft Jensen
Publisher: World Scientific Publishing Company
ISBN: 1848169957
Category : Science
Languages : en
Pages : 300
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Book Description
Dynamical evolution over long time scales is a prominent feature of all the systems we intuitively think of as complex — for example, ecosystems, the brain or the economy. In physics, the term ageing is used for this type of slow change, occurring over time scales much longer than the patience, or indeed the lifetime, of the observer. The main focus of this book is on the stochastic processes which cause ageing, and the surprising fact that the ageing dynamics of systems which are very different at the microscopic level can be treated in similar ways.The first part of this book provides the necessary mathematical and computational tools and the second part describes the intuition needed to deal with these systems. Some of the first few chapters have been covered in several other books, but the emphasis and selection of the topics reflect both the authors' interests and the overall theme of the book. The second part contains an introduction to the scientific literature and deals in some detail with the description of complex phenomena of a physical and biological nature, for example, disordered magnetic materials, superconductors and glasses, models of co-evolution in ecosystems and even of ant behaviour. These heterogeneous topics are all dealt with in detail using similar analytical techniques.This book emphasizes the unity of complex dynamics and provides the tools needed to treat a large number of complex systems of current interest. The ideas and the approach to complex dynamics it presents have not appeared in book form until now./a
Author: Matthias Birkner
Publisher: World Scientific
ISBN: 9811206104
Category : Mathematics
Languages : en
Pages : 363
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Book Description
Interacting particle systems are Markov processes involving infinitely many interacting components. Since their introduction in the 1970s, researchers have found many applications in statistical physics and population biology. Genealogies, which follow the origin of the state of a site backwards in time, play an important role in their studies, especially for the biologically motivated systems.The program Genealogies of Interacting Particle Systems held at the Institute for Mathematical Sciences, National University of Singapore, from 17 July to 18 Aug 2017, brought together experts and young researchers interested in this modern topic. Central to the program were learning sessions where lecturers presented work outside of their own research, as well as a normal workshop. This is reflected in the present volume which contains two types of articles:Written by respected researchers, including experts in the field such as Steve Evans, member of the US National Academy of Sciences, as well as Anton Wakolbinger, Andreas Greven, and many others, this volume will no doubt be a valuable contribution to the probability community.
Author: Dmitri Koroliouk
Publisher: John Wiley & Sons
ISBN: 1119851246
Category : Mathematics
Languages : en
Pages : 345
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Book Description
Within the field of modeling complex objects in natural sciences, which considers systems that consist of a large number of interacting parts, a good tool for analyzing and fitting models is the theory of random evolutionary systems, considering their asymptotic properties and large deviations. In Random Evolutionary Systems we consider these systems in terms of the operators that appear in the schemes of their diffusion and the Poisson approximation. Such an approach allows us to obtain a number of limit theorems and asymptotic expansions of processes that model complex stochastic systems, both those that are autonomous and those dependent on an external random environment. In this case, various possibilities of scaling processes and their time parameters are used to obtain different limit results.
Author: Wilfried Grecksch
Publisher: De Gruyter Akademie Forschung
ISBN:
Category : Mathematics
Languages : en
Pages : 188
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Book Description
The authors give a self-contained exposition of the theory of stochastic evolution equations. Elements of infinite dimensional analysis, martingale theory in Hilbert spaces, stochastic integrals, stochastic convolutions are applied. Existence and uniqueness theorems for stochastic evolution equations in Hilbert spaces in the sense of the semigroup theory, the theory of evolution operators, and monotonous operators in rigged Hilbert spaces are discussed. Relationships between the different concepts are demonstrated. The results are used to concrete stochastic partial differential equations like parabolic and hyperbolic Ito equations and random constitutive equations of elastic viscoplastic materials. Furthermore, stochastic evolution equations in rigged Hilbert spaces are approximated by time discretization methods.
Author: Mark A. Pinsky
Publisher: World Scientific
ISBN: 9789810205591
Category : Science
Languages : en
Pages : 158
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Book Description
Random evolution denotes a class of stochastic processes which evolve according to a rule which varies in time according to jumps. This is in contrast to diffusion processes, which assume that the rule changes continuously with time. Random evolutions provide a very flexible language, having the advantage that they permit direct numerical simulation-which is not possible for a diffusion process. Furthermore, they allow connections with hyperbolic partial differential equations and the kinetic theory of gases, which is impossible within the domain of diffusion proceses. They also posses great geometric invariance, allowing formulation on an arbitrary Riemannian manifold. In the field of stochastic stability, random evolutions furnish some easily computable models in which to study the Lyapunov exponent and rotation numbers of oscillators under the influence of noise. This monograph presents the various aspects of random evolution in an accessible and interesting format which will appeal to a large scientific audience.
