Markov Processes Volume 1
The modem theory of Markov processes has its origins in the studies of A. A. MARKOV (1906-1907) on sequences of experiments "connected in a chain" and in the attempts to describe mathematically the physical phenomenon known as Brownian motion (L. BACHELlE
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MATHEMATISCHEN WISSENSCHAFTEN IN EINZELDARSTELLUNGEN MIT BESONDERER BERUCKSICHTIGUNG DER ANWENDUNGSGEBIETE HERAUSGEGEBEN VON
J. L. DOOB· E. HEINZ· F. HIRZEBRUCH E.HOPF· H.HOPF· W.MAAK· W.MAGNUS F. K. SCHMIDT· K. STEIN GESCHAFTSFUHRENDE HERAUSGEBER
B.ECKMANN UND B. L.VAN DER WAERDEN ZURICH
BAND 121
SPRINGER-VERLAG BERLIN· GOTTINGEN . HEIDELBERG 1965
MARKOV PROCESSES BY
E. B. DYNKIN PROFESSOR OF MATHEMATICS UNIVERSITY OF MOSCOW
TRANSLATED WITH THE AUTHORIZATION AND ASSISTANCE OF THE AUTHOR BY
J. FABIUS, V. GREENBERG, A. MAITRA, G. MAJONE UNIVERSITY OF CALIFORNIA. BERKELEY
VOLUME I
1965 NEW YORK
ACADEMIC PRESS INC., PUBLISHERS BERLIN· GOTTINGEN . HEIDELBERG
SPRINGER-VERLAG BERLIN· GOTTINGEN . HEIDELBERG
Published in U. S. A. and Canada by ACADEMIC PRESS INC., PUBLISHERS
111 Fifth Avenue, New York, N. Y. 10003
ISBN 978-3-662-00033-5
ISBN 978-3-662-00031-1 (eBook)
DOI 10.1007/978-3-662-00031-1 Library of Congress Catalog Card Number 64-24812 All rights reserved No part of this book may be reproduced in any form, by microfilm or any other means, without written permission from the publishers
© BY SPRINGER- VERLAG
BERLIN· GOTTINGEN . HEIDELBERG 1965 SOFTCOVER REPRINT OF THE HARDCOVER lST EDITION 1965
Managing Editors: Prof. Dr. B. Eckmann, Eidgenossische Technische Hochschule Zurich Prof. Dr. B L f/I1n der Woe1'den, Mathemaluches In-stlllll der Umverstlaf Zurich
Preface The modem theory of Markov processes has its origins in the studies of A. A. MARKOV (1906-1907) on sequences of experiments "connected in a chain" and in the attempts to describe mathematically the physical phenomenon known as Brownian motion (L. BACHELlER 1900, A. EINSTEIN 1905). The first correct mathematical construction of a Markov process with continuous trajectories was given by N. WIENER in 1923. (This process is often called the Wiener process.) The general theory of Markov processes was developed in the 1930's and 1940's by A. N. KOLMOGOROV, W. FELLER, W. DOEBLlN, P. LEVY, J. L. DOOB, and others. During the past ten years the theory of Markov processes has entered a new period of intensive development. The methods of the theory of semigroups of linear operators made possible further progress in the classification of Markov processes by their infinitesimal characteristics. The broad classes of Markov processes with continuous trajectories became the main object of study. The connections between Markov processes and classical analysis were further developed. It has become possible not only to apply the results and methods of analysis to the problems of probability theory, but also to investigate analytic problems using probabilistic methods. Remarkable new connections between Markov processes and potential theory were revealed. The foundations of the theory were reviewed critically: the new concept of strong Markov process acquired for the whole theory of Markov processes great importance. This book attempts a systematic exposition of the modem theory of Markov processes. The newest directions, which have been barely treated
