Invariant Markov Processes Under Lie Group Actions
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 probabili
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Invariant Markov Processes Under Lie Group Actions
Invariant Markov Processes Under Lie Group Actions
Ming Liao
Invariant Markov Processes Under Lie Group Actions
123
Ming Liao Department of Mathematics and Statistics Auburn University Auburn, AL, USA
ISBN 978-3-319-92323-9 ISBN 978-3-319-92324-6 (eBook) https://doi.org/10.1007/978-3-319-92324-6 Library of Congress Control Number: 2018946697 Mathematics Subject Classification (2010): 57S15, 60-02, 60B05, 60B15, 60G51, 60J25 © Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface
The invariance under a group action is a central theme in mathematics. In the probability theory, the invariance of probability distributions under translations or orthogonal transformations on Euclidean spaces has been well studied, and there is also a rich literature on the probability measures on group structures. However, there appears to be little systematic study of the invariance of probability distributions under a general group action. The purpose of this book is to provide a theory of the Markov processes that are invariant under Lie group actions in the sense of the distribution. The invariant Markov processes under transitive Lie group actions, including those in Lie groups invariant under left (or right) translations, may be identified with the processes that have independent and stationary increments. Such processes will be called Lévy processes because they are natural extensions of the classical Lévy processes in Euclidean spaces. The first half of the book (Chapters 1 through 5) is devoted to a theory of Lévy processe
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