The Stable Manifold Theorem For Semilinear Stochastic Evolution Equations And Stochastic Partial Differential Equations
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The Stable Manifold Theorem For Semilinear Stochastic Evolution Equations And Stochastic Partial Differential Equations
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Author : Salah-Eldin A. Mohammed
language : en
Publisher: American Mathematical Soc.
Release Date : 2008
The Stable Manifold Theorem For Semilinear Stochastic Evolution Equations And Stochastic Partial Differential Equations written by Salah-Eldin A. Mohammed and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Evolution equations categories.
The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations and stochastic partial differential equations near stationary solutions.
Memoirs Of The American Mathematical Society
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Author :
language : en
Publisher:
Release Date : 1950
Memoirs Of The American Mathematical Society written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1950 with Evolution equations categories.
Effective Dynamics Of Stochastic Partial Differential Equations
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Author : Jinqiao Duan
language : en
Publisher: Elsevier
Release Date : 2014-03-06
Effective Dynamics Of Stochastic Partial Differential Equations written by Jinqiao Duan and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-03-06 with Mathematics categories.
Effective Dynamics of Stochastic Partial Differential Equations focuses on stochastic partial differential equations with slow and fast time scales, or large and small spatial scales. The authors have developed basic techniques, such as averaging, slow manifolds, and homogenization, to extract effective dynamics from these stochastic partial differential equations. The authors’ experience both as researchers and teachers enable them to convert current research on extracting effective dynamics of stochastic partial differential equations into concise and comprehensive chapters. The book helps readers by providing an accessible introduction to probability tools in Hilbert space and basics of stochastic partial differential equations. Each chapter also includes exercises and problems to enhance comprehension. New techniques for extracting effective dynamics of infinite dimensional dynamical systems under uncertainty Accessible introduction to probability tools in Hilbert space and basics of stochastic partial differential equations Solutions or hints to all Exercises
Discovering Evolution Equations With Applications
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Author : Mark McKibben
language : en
Publisher: CRC Press
Release Date : 2011-06-03
Discovering Evolution Equations With Applications written by Mark McKibben and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-06-03 with Mathematics categories.
Most existing books on evolution equations tend either to cover a particular class of equations in too much depth for beginners or focus on a very specific research direction. Thus, the field can be daunting for newcomers to the field who need access to preliminary material and behind-the-scenes detail. Taking an applications-oriented, conversational approach, Discovering Evolution Equations with Applications: Volume 2-Stochastic Equations provides an introductory understanding of stochastic evolution equations. The text begins with hands-on introductions to the essentials of real and stochastic analysis. It then develops the theory for homogenous one-dimensional stochastic ordinary differential equations (ODEs) and extends the theory to systems of homogenous linear stochastic ODEs. The next several chapters focus on abstract homogenous linear, nonhomogenous linear, and semi-linear stochastic evolution equations. The author also addresses the case in which the forcing term is a functional before explaining Sobolev-type stochastic evolution equations. The last chapter discusses several topics of active research. Each chapter starts with examples of various models. The author points out the similarities of the models, develops the theory involved, and then revisits the examples to reinforce the theoretical ideas in a concrete setting. He incorporates a substantial collection of questions and exercises throughout the text and provides two layers of hints for selected exercises at the end of each chapter. Suitable for readers unfamiliar with analysis even at the undergraduate level, this book offers an engaging and accessible account of core theoretical results of stochastic evolution equations in a way that gradually builds readers’ intuition.
Amplitude Equations For Stochastic Partial Differential Equations
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Author : Dirk Blomker
language : en
Publisher: World Scientific
Release Date : 2007
Amplitude Equations For Stochastic Partial Differential Equations written by Dirk Blomker and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007 with Mathematics categories.
Rigorous error estimates for amplitude equations are well known for deterministic PDEs, and there is a large body of literature over the past two decades. However, there seems to be a lack of literature for stochastic equations, although the theory is being successfully used in the applied community, such as for convective instabilities, without reliable error estimates at hand. This book is the first step in closing this gap. The author provides details about the reduction of dynamics to more simpler equations via amplitude or modulation equations, which relies on the natural separation of time-scales present near a change of stability. For students, the book provides a lucid introduction to the subject highlighting the new tools necessary for stochastic equations, while serving as an excellent guide to recent research.
Probability And Partial Differential Equations In Modern Applied Mathematics
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Author : Edward C. Waymire
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-06-14
Probability And Partial Differential Equations In Modern Applied Mathematics written by Edward C. Waymire and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-06-14 with Mathematics categories.
"Probability and Partial Differential Equations in Modern Applied Mathematics" is devoted to the role of probabilistic methods in modern applied mathematics from the perspectives of both a tool for analysis and as a tool in modeling. There is a recognition in the applied mathematics research community that stochastic methods are playing an increasingly prominent role in the formulation and analysis of diverse problems of contemporary interest in the sciences and engineering. A probabilistic representation of solutions to partial differential equations that arise as deterministic models allows one to exploit the power of stochastic calculus and probabilistic limit theory in the analysis of deterministic problems, as well as to offer new perspectives on the phenomena for modeling purposes. There is also a growing appreciation of the role for the inclusion of stochastic effects in the modeling of complex systems. This has led to interesting new mathematical problems at the interface of probability, dynamical systems, numerical analysis, and partial differential equations. This volume will be useful to researchers and graduate students interested in probabilistic methods, dynamical systems approaches and numerical analysis for mathematical modeling in the sciences and engineering.
Approximation Of Stochastic Invariant Manifolds
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Author : Mickaël D. Chekroun
language : en
Publisher: Springer
Release Date : 2014-12-20
Approximation Of Stochastic Invariant Manifolds written by Mickaël D. Chekroun and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-12-20 with Mathematics categories.
This first volume is concerned with the analytic derivation of explicit formulas for the leading-order Taylor approximations of (local) stochastic invariant manifolds associated with a broad class of nonlinear stochastic partial differential equations. These approximations take the form of Lyapunov-Perron integrals, which are further characterized in Volume II as pullback limits associated with some partially coupled backward-forward systems. This pullback characterization provides a useful interpretation of the corresponding approximating manifolds and leads to a simple framework that unifies some other approximation approaches in the literature. A self-contained survey is also included on the existence and attraction of one-parameter families of stochastic invariant manifolds, from the point of view of the theory of random dynamical systems.
Stochastic Parameterizing Manifolds And Non Markovian Reduced Equations
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Author : Mickaël D. Chekroun
language : en
Publisher: Springer
Release Date : 2014-12-23
Stochastic Parameterizing Manifolds And Non Markovian Reduced Equations written by Mickaël D. Chekroun and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-12-23 with Mathematics categories.
In this second volume, a general approach is developed to provide approximate parameterizations of the "small" scales by the "large" ones for a broad class of stochastic partial differential equations (SPDEs). This is accomplished via the concept of parameterizing manifolds (PMs), which are stochastic manifolds that improve, for a given realization of the noise, in mean square error the partial knowledge of the full SPDE solution when compared to its projection onto some resolved modes. Backward-forward systems are designed to give access to such PMs in practice. The key idea consists of representing the modes with high wave numbers as a pullback limit depending on the time-history of the modes with low wave numbers. Non-Markovian stochastic reduced systems are then derived based on such a PM approach. The reduced systems take the form of stochastic differential equations involving random coefficients that convey memory effects. The theory is illustrated on a stochastic Burgers-type equation.
Center Manifolds For Semilinear Equations With Non Dense Domain And Applications To Hopf Bifurcation In Age Structured Models
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Author : Pierre Magal
language : en
Publisher: American Mathematical Soc.
Release Date : 2009
Center Manifolds For Semilinear Equations With Non Dense Domain And Applications To Hopf Bifurcation In Age Structured Models written by Pierre Magal and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with Bifurcation theory categories.
Several types of differential equations, such as delay differential equations, age-structure models in population dynamics, evolution equations with boundary conditions, can be written as semilinear Cauchy problems with an operator which is not densely defined in its domain. The goal of this paper is to develop a center manifold theory for semilinear Cauchy problems with non-dense domain. Using Liapunov-Perron method and following the techniques of Vanderbauwhede et al. in treating infinite dimensional systems, the authors study the existence and smoothness of center manifolds for semilinear Cauchy problems with non-dense domain. As an application, they use the center manifold theorem to establish a Hopf bifurcation theorem for age structured models.
New Trends In Stochastic Analysis And Related Topics
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Author : Huaizhong Zhao
language : en
Publisher: World Scientific
Release Date : 2012
New Trends In Stochastic Analysis And Related Topics written by Huaizhong Zhao and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.
The volume is dedicated to Professor David Elworthy to celebrate his fundamental contribution and exceptional influence on stochastic analysis and related fields. Stochastic analysis has been profoundly developed as a vital fundamental research area in mathematics in recent decades. It has been discovered to have intrinsic connections with many other areas of mathematics such as partial differential equations, functional analysis, topology, differential geometry, dynamical systems, etc. Mathematicians developed many mathematical tools in stochastic analysis to understand and model random phenomena in physics, biology, finance, fluid, environment science, etc. This volume contains 12 comprehensive review/new articles written by world leading researchers (by invitation) and their collaborators. It covers stochastic analysis on manifolds, rough paths, Dirichlet forms, stochastic partial differential equations, stochastic dynamical systems, infinite dimensional analysis, stochastic flows, quantum stochastic analysis and stochastic Hamilton Jacobi theory. Articles contain cutting edge research methodology, results and ideas in relevant fields. They are of interest to research mathematicians and postgraduate students in stochastic analysis, probability, partial differential equations, dynamical systems, mathematical physics, as well as to physicists, financial mathematicians, engineers, etc.