Excessive Measures
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R.K. Getoor
Excessive Measures
Birkhauser Boston· Basel • Berlin
R.K. Getoor Department of Mathematics University of California, San Diego La Jolla, CA 92093, USA
Library of Congress Cataloging-in-Publication Data Getoor, R.K. (Ronald Kay), 1929Excessive measureS/R.K. Getoor. p. cm.-(Probabilityand its applications Includes bibliographical references. ISBN-13:978-1-4612-8036·1 1. Excessive measures (Mathematics) 2. Markov processes. 3. Measure theory. 4. Potential theory (Mathematics) I. Title. II. Series. QA274.7.G47 1990 89-18654 519.2'33---dc20 Printed on acid-free paper.
© Birkhiiuser Boston, 1990 Softcover reprint of the hardcover 1st edition 1990 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system. or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the copyright owner. ISBN-13:978-1-4612-8036·1 001: 10.1007/978-1-4612-3470·8
e-ISBN-13:978-1-4612-3470·8
Camera-ready copy supplied by author using TEX.
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Preface The study of the cone of excessive measures associated with a Markov process goes back to Hunt's fundamental memoir [H57]. However until quite recently it received much less attention than the cone of excessive functions. The fact that an excessive function can be composed with the underlying Markov process to give a supermartingale, subject to secondary finiteness hypotheses, is crucial in the study of excessive functions. The lack of an analogous construct for excessive measures seemed to make them much less tractable to a probabilistic analysis. This point of view changed radically with the appearance of the pioneering paper by Fitzsimmons and Maisonneuve [FM86] who showed that a certain stationary process associated with an excessive measure could be used to study excessive measures probabilistically. These stationary processes or measures had been constructed by Kuznetsov [Ku74] extending earlier work of Dynkin. It is now common to call them Kuznetsov measures. Following the FitzsimmonsMaisonneuve paper there was renewed interest and remarkable progress in the study of excessive measures. The purpose of this monograph is to organize under one cover and prove under standard hypotheses many of these recent results in the theory of excessive measures. The two basic tools in this recent development are Kuznetsov measures mentioned above and the energy functional. The energy functional has a long history that may be traced back to Hunt, but its systematic use in the study of excessive measures seems to be more recent. However, see [CL75] for its definition and use in an abstract setting. Also it was used for other purposes by Meyer in [Me68] and [Me73]. A third ingredient in this development is the use of two Riesz type
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decompositions of an excessive measure: the first into dissipative and conservative parts is due to Dynkin [Dy80]-see also Blumenthal [BI]; the second into a potential and a harmonic part is due originally to Getoor and G
