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Approximation of Stochastic Invariant Manifolds

Author: Mickaël D. Chekroun

Publisher: Springer

Published: 2014-12-20

Total Pages: 136

ISBN-13: 331912496X

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

Author: Mickaël D. Chekroun

Publisher: Springer

Published: 2014-12-23

Total Pages: 141

ISBN-13: 3319125206

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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.

Invariant Manifolds for Stochastic PDE with Fractional Brownian Motion

Author: Alberto Ohashi

Publisher:

Published: 2007

Total Pages: 24

ISBN-13:

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Dynamics for a Random Differential Equation

Author: Junyilang Zhao

Publisher:

Published: 2018

Total Pages: 110

ISBN-13:

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In this dissertation, we first prove that for a random differential equation with the multiplicative driving noise constructed from a Q-Wiener process and the Wiener shift, which is an approximation to a stochastic evolution equation, there exists a unique solution that generates a local dynamical system. There also exist a local center, unstable, stable, centerunstable, center-stable manifold, and a local stable foliation, an unstable foliation on the center-unstable manifold, and a stable foliation on the center-stable manifold, the smoothness of which depend on the vector fields of the equation. In the second half of the dissertation, we show that any two arbitrary local center manifolds constructed as above are conjugate. We also show the same conjugacy result holds for a stochastic evolution equation with the multiplicative Stratonovich noise term as u ° dW.

Stochastic Pdes And Modelling Of Multiscale Complex System

Author: Wang Wei

Publisher: World Scientific

Published: 2019-05-07

Total Pages: 240

ISBN-13: 981120036X

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This volume is devoted to original research results and survey articles reviewing recent developments in reduction for stochastic PDEs with multiscale as well as application to science and technology, and to present some future research direction. This volume includes a dozen chapters by leading experts in the area, with a broad audience in mind. It should be accessible to graduate students, junior researchers and other professionals who are interested in the subject. We also take this opportunity to celebrate the contributions of Professor Anthony J Roberts, an internationally leading figure on the occasion of his 60th years birthday in 2017.

Infinite Dimensional And Finite Dimensional Stochastic Equations And Applications In Physics

Author: Wilfried Grecksch

Publisher: World Scientific

Published: 2020-04-22

Total Pages: 261

ISBN-13: 9811209804

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This volume contains survey articles on various aspects of stochastic partial differential equations (SPDEs) and their applications in stochastic control theory and in physics.The topics presented in this volume are:This book is intended not only for graduate students in mathematics or physics, but also for mathematicians, mathematical physicists, theoretical physicists, and science researchers interested in the physical applications of the theory of stochastic processes.

Stochastic Evolution Equations

Author: Wilfried Grecksch

Publisher: De Gruyter Akademie Forschung

Published: 1995

Total Pages: 188

ISBN-13:

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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.

Extremes and Recurrence in Dynamical Systems

Author: Valerio Lucarini

Publisher: John Wiley & Sons

Published: 2016-03-28

Total Pages: 367

ISBN-13: 1118632354

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Written by a team of international experts, Extremes and Recurrence in Dynamical Systems presents a unique point of view on the mathematical theory of extremes and on its applications in the natural and social sciences. Featuring an interdisciplinary approach to new concepts in pure and applied mathematical research, the book skillfully combines the areas of statistical mechanics, probability theory, measure theory, dynamical systems, statistical inference, geophysics, and software application. Emphasizing the statistical mechanical point of view, the book introduces robust theoretical embedding for the application of extreme value theory in dynamical systems. Extremes and Recurrence in Dynamical Systems also features: • A careful examination of how a dynamical system can serve as a generator of stochastic processes • Discussions on the applications of statistical inference in the theoretical and heuristic use of extremes • Several examples of analysis of extremes in a physical and geophysical context • A final summary of the main results presented along with a guide to future research projects • An appendix with software in Matlab® programming language to help readers to develop further understanding of the presented concepts Extremes and Recurrence in Dynamical Systems is ideal for academics and practitioners in pure and applied mathematics, probability theory, statistics, chaos, theoretical and applied dynamical systems, statistical mechanics, geophysical fluid dynamics, geosciences and complexity science. VALERIO LUCARINI, PhD, is Professor of Theoretical Meteorology at the University of Hamburg, Germany and Professor of Statistical Mechanics at the University of Reading, UK. DAVIDE FARANDA, PhD, is Researcher at the Laboratoire des science du climat et de l’environnement, IPSL, CEA Saclay, Université Paris-Saclay, Gif-sur-Yvette, France. ANA CRISTINA GOMES MONTEIRO MOREIRA DE FREITAS, PhD, is Assistant Professor in the Faculty of Economics at the University of Porto, Portugal. JORGE MIGUEL MILHAZES DE FREITAS, PhD, is Assistant Professor in the Department of Mathematics of the Faculty of Sciences at the University of Porto, Portugal. MARK HOLLAND, PhD, is Senior Lecturer in Applied Mathematics in the College of Engineering, Mathematics and Physical Sciences at the University of Exeter, UK. TOBIAS KUNA, PhD, is Associate Professor in the Department of Mathematics and Statistics at the University of Reading, UK. MATTHEW NICOL, PhD, is Professor of Mathematics at the University of Houston, USA. MIKE TODD, PhD, is Lecturer in the School of Mathematics and Statistics at the University of St. Andrews, Scotland. SANDRO VAIENTI, PhD, is Professor of Mathematics at the University of Toulon and Researcher at the Centre de Physique Théorique, France.

Advances in Nonlinear Geosciences

Author: Anastasios A. Tsonis

Publisher: Springer

Published: 2017-10-13

Total Pages: 707

ISBN-13: 3319588958

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Advances in Nonlinear Geosciences is a set of contributions from the participants of “30 Years of Nonlinear Dynamics” held July 3-8, 2016 in Rhodes, Greece as part of the Aegean Conferences, as well as from several other experts in the field who could not attend the meeting. The volume brings together up-to-date research from the atmospheric sciences, hydrology, geology, and other areas of geosciences and presents the new advances made in the last 10 years. Topics include chaos synchronization, topological data analysis, new insights on fractals, multifractals and stochasticity, climate dynamics, extreme events, complexity, and causality, among other topics.

Effective Dynamics of Stochastic Partial Differential Equations

Author: Jinqiao Duan

Publisher: Elsevier

Published: 2014-03-06

Total Pages: 283

ISBN-13: 0128012692

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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