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Author: Ming Liao
Publisher: Springer
ISBN: 3319923242
Category : Mathematics
Languages : en
Pages : 363
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Book Description
The purpose of this monograph is to provide a theory of Markov processes that are invariant under the actions of Lie groups, focusing on ways to represent such processes in the spirit of the classical Lévy-Khinchin representation. It interweaves probability theory, topology, and global analysis on manifolds to present the most recent results in a developing area of stochastic analysis. The author’s discussion is structured with three different levels of generality:— A Markov process in a Lie group G that is invariant under the left (or right) translations— A Markov process xt in a manifold X that is invariant under the transitive action of a Lie group G on X— A Markov process xt invariant under the non-transitive action of a Lie group GA large portion of the text is devoted to the representation of inhomogeneous Lévy processes in Lie groups and homogeneous spaces by a time dependent triple through a martingale property. Preliminary definitions and results in both stochastics and Lie groups are provided in a series of appendices, making the book accessible to those who may be non-specialists in either of these areas. Invariant Markov Processes Under Lie Group Actions will be of interest to researchers in stochastic analysis and probability theory, and will also appeal to experts in Lie groups, differential geometry, and related topics interested in applications of their own subjects.
Author: Ming Liao
Publisher: Springer
ISBN: 3319923242
Category : Mathematics
Languages : en
Pages : 363
Get Book
Book Description
The purpose of this monograph is to provide a theory of Markov processes that are invariant under the actions of Lie groups, focusing on ways to represent such processes in the spirit of the classical Lévy-Khinchin representation. It interweaves probability theory, topology, and global analysis on manifolds to present the most recent results in a developing area of stochastic analysis. The author’s discussion is structured with three different levels of generality:— A Markov process in a Lie group G that is invariant under the left (or right) translations— A Markov process xt in a manifold X that is invariant under the transitive action of a Lie group G on X— A Markov process xt invariant under the non-transitive action of a Lie group GA large portion of the text is devoted to the representation of inhomogeneous Lévy processes in Lie groups and homogeneous spaces by a time dependent triple through a martingale property. Preliminary definitions and results in both stochastics and Lie groups are provided in a series of appendices, making the book accessible to those who may be non-specialists in either of these areas. Invariant Markov Processes Under Lie Group Actions will be of interest to researchers in stochastic analysis and probability theory, and will also appeal to experts in Lie groups, differential geometry, and related topics interested in applications of their own subjects.
Author: Stefania Ugolini
Publisher: Springer Nature
ISBN: 303087432X
Category : Mathematics
Languages : en
Pages : 273
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Book Description
This book grew out of the Random Transformations and Invariance in Stochastic Dynamics conference held in Verona from the 25th to the 28th of March 2019 in honour of Sergio Albeverio. It presents the new area of studies concerning invariance and symmetry properties of finite and infinite dimensional stochastic differential equations.This area constitutes a natural, much needed, extension of the theory of classical ordinary and partial differential equations, where the reduction theory based on symmetry and invariance of such classical equations has historically proved to be very important both for theoretical and numerical studies and has given rise to important applications. The purpose of the present book is to present the state of the art of the studies on stochastic systems from this point of view, present some of the underlying fundamental ideas and methods involved, and to outline the main lines for future developments. The main focus is on bridging the gap between deterministic and stochastic approaches, with the goal of contributing to the elaboration of a unified theory that will have a great impact both from the theoretical point of view and the point of view of applications. The reader is a mathematician or a theoretical physicist. The main discipline is stochastic analysis with profound ideas coming from Mathematical Physics and Lie’s Group Geometry. While the audience consists essentially of academicians, the reader can also be a practitioner with Ph.D., who is interested in efficient stochastic modelling.
Author: Ming Liao
Publisher: Cambridge University Press
ISBN: 9780521836531
Category : Mathematics
Languages : en
Pages : 292
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Book Description
Up-to-the minute research on important stochastic processes.
Author: Olav Kallenberg
Publisher: Springer Nature
ISBN: 3030618714
Category : Mathematics
Languages : en
Pages : 946
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Book Description
The first edition of this single volume on the theory of probability has become a highly-praised standard reference for many areas of probability theory. Chapters from the first edition have been revised and corrected, and this edition contains four new chapters. New material covered includes multivariate and ratio ergodic theorems, shift coupling, Palm distributions, Harris recurrence, invariant measures, and strong and weak ergodicity.
Author: Mark Pollicott
Publisher: Cambridge University Press
ISBN: 0521576881
Category : Mathematics
Languages : en
Pages : 496
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Book Description
A mixture of surveys and original articles that span the theory of Zd actions.
Author: Mark Iosifovich Freĭdlin
Publisher: Springer Science & Business Media
ISBN: 9780817636968
Category : Mathematics
Languages : en
Pages : 460
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Book Description
Eugene B. Dynkin published his first paper, on Markov chain theory, whilst still an undergraduate student at Moscow State University. He went on to make fundamental contributions to the theory of Markov processes and to Lie groups, generating entire schools in these areas.
Author: Robert J. Zimmer
Publisher: University of Chicago Press
ISBN: 022656827X
Category : Mathematics
Languages : en
Pages : 724
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Book Description
Robert J. Zimmer is best known in mathematics for the highly influential conjectures and program that bear his name. Group Actions in Ergodic Theory, Geometry, and Topology: Selected Papers brings together some of the most significant writings by Zimmer, which lay out his program and contextualize his work over the course of his career. Zimmer’s body of work is remarkable in that it involves methods from a variety of mathematical disciplines, such as Lie theory, differential geometry, ergodic theory and dynamical systems, arithmetic groups, and topology, and at the same time offers a unifying perspective. After arriving at the University of Chicago in 1977, Zimmer extended his earlier research on ergodic group actions to prove his cocycle superrigidity theorem which proved to be a pivotal point in articulating and developing his program. Zimmer’s ideas opened the door to many others, and they continue to be actively employed in many domains related to group actions in ergodic theory, geometry, and topology. In addition to the selected papers themselves, this volume opens with a foreword by David Fisher, Alexander Lubotzky, and Gregory Margulis, as well as a substantial introductory essay by Zimmer recounting the course of his career in mathematics. The volume closes with an afterword by Fisher on the most recent developments around the Zimmer program.
Author: Radu Zaharopol
Publisher: Springer
ISBN: 3319057235
Category : Mathematics
Languages : en
Pages : 389
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Book Description
The structure of the set of all the invariant probabilities and the structure of various types of individual invariant probabilities of a transition function are two topics of significant interest in the theory of transition functions, and are studied in this book. The results obtained are useful in ergodic theory and the theory of dynamical systems, which, in turn, can be applied in various other areas (like number theory). They are illustrated using transition functions defined by flows, semiflows, and one-parameter convolution semigroups of probability measures. In this book, all results on transition probabilities that have been published by the author between 2004 and 2008 are extended to transition functions. The proofs of the results obtained are new. For transition functions that satisfy very general conditions the book describes an ergodic decomposition that provides relevant information on the structure of the corresponding set of invariant probabilities. Ergodic decomposition means a splitting of the state space, where the invariant ergodic probability measures play a significant role. Other topics covered include: characterizations of the supports of various types of invariant probability measures and the use of these to obtain criteria for unique ergodicity, and the proofs of two mean ergodic theorems for a certain type of transition functions. The book will be of interest to mathematicians working in ergodic theory, dynamical systems, or the theory of Markov processes. Biologists, physicists and economists interested in interacting particle systems and rigorous mathematics will also find this book a valuable resource. Parts of it are suitable for advanced graduate courses. Prerequisites are basic notions and results on functional analysis, general topology, measure theory, the Bochner integral and some of its applications.
Author: L.A. Bunimovich
Publisher: Springer Science & Business Media
ISBN: 9783540663164
Category : Mathematics
Languages : en
Pages : 476
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Book Description
This EMS volume, the first edition of which was published as Dynamical Systems II, EMS 2, familiarizes the reader with the fundamental ideas and results of modern ergodic theory and its applications to dynamical systems and statistical mechanics. The enlarged and revised second edition adds two new contributions on ergodic theory of flows on homogeneous manifolds and on methods of algebraic geometry in the theory of interval exchange transformations.
Author: Isabelle Chalendar
Publisher: Cambridge University Press
ISBN: 1139503294
Category : Mathematics
Languages : en
Pages : 298
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Book Description
One of the major unsolved problems in operator theory is the fifty-year-old invariant subspace problem, which asks whether every bounded linear operator on a Hilbert space has a nontrivial closed invariant subspace. This book presents some of the major results in the area, including many that were derived within the past few years and cannot be found in other books. Beginning with a preliminary chapter containing the necessary pure mathematical background, the authors present a variety of powerful techniques, including the use of the operator-valued Poisson kernel, various forms of the functional calculus, Hardy spaces, fixed point theorems, minimal vectors, universal operators and moment sequences. The subject is presented at a level accessible to postgraduate students, as well as established researchers. It will be of particular interest to those who study linear operators and also to those who work in other areas of pure mathematics.
