Generators of Markov Chains: From a Walk in the Interior to a Dance on the Boundary

280 Pages·2020·3.71 MB·English

by  Adam Bobrowski

CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 190
EditorialBoard
J. BERTOIN, B. BOLLOBÁS, W. FULTON, B. KRA, I. MOERDIJK,
C. PRAEGER, P. SARNAK, B. SIMON, B. TOTARO
GENERATORSOFMARKOVCHAINS
ElementarytreatmentsofMarkovchains,especiallythosedevotedtodiscrete-timeand
finite state-space theory, leave the impression that everything is smooth and easy to
understand.ThisexpositionoftheworksofKolmogorov,Feller,Chung,Kato,andother
mathematical luminaries, which focuses on time-continuous chains but is not so far
frombeingelementaryitself,remindsusagainthattheimpressionisfalse:aninfinite,
butdenumerable,state-spaceiswherethefunbegins.
IfyouhavenotheardofBlackwell’sexample(inwhichallstatesareinstantaneous),
do not understand what the minimal process is, or do not know what happens after
explosion,diverightin.Butbewarelestyouareenchanted:‘therearemorespellsthan
yourcommonplacemagicianseverdreamedof.’
Adam Bobrowski is Professor and Chairman of the Department of Mathematics at
Lublin University of Technology, Poland. He is a pure mathematician who uses the
languageofoperatorsemigroupstodescribestochasticprocesses.Hehasauthoredand
coauthorednearly70papersonthesubjectandfivebooks,includingFunctionalAnaly-
sisforProbabilityandStochasticProcesses(2005)andConvergenceofOne-Parameter
OperatorSemigroups(2016).

CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS
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J.Bertoin,B.Bollobás,W.Fulton,B.Kra,I.Moerdijk,C.Praeger,P.Sarnak,B.Simon,B.Totaro
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‘FascinatingMarkovChains’byMarekBobrowski.Inadifferentversionofthis
cartoonbyW.Chojnacki,theroleoftheanonymousexaminerisplayedbyan
antagonistofA.A.Markov,thatis,PavelNiekrasov,whosays,‘Contrarytomy
beliefs,thesechainsarereallyfascinating!’See[50]fortheentirestory.
Generators of Markov Chains
From a Walk in the Interior to a Dance on the Boundary
ADAM BOBROWSKI
LublinUniversityofTechnology
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781108495790
DOI:10.1017/9781108863070
(cid:2)c AdamBobrowski2021
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TomydancingBeatka
Imetamanonce...towhomHeystexclaimed,innoconnectionwithanythingin
particular(itwasinthebilliard-roomoftheclub):‘Iamenchantedwiththese
islands!’Heshotitoutsuddenly,aproposdesbottes,astheFrenchsay,andwhile
chalkinghiscue.Andperhapsitwassomesortofenchantment.Therearemore
spellsthanyourcommonplacemagicianseverdreamedof.
J.Conrad(J.T.K.Korzeniowski),Victory
Niesprawiłes´mizawodu,synu.Przeciwnie,zadziwiłes´mnie.Zdołałes´dac´zsiebie
wie˛cej,niz˙emodciebieoczekiwał.
TeodorParnicki,Srebrneorły
Markovchainsmerelywalkintheirregularstatespace,butonthecliffsoftheir
boundaries,theydance.
JohannGottfriedvonSpacerniak
Contents
Preface pagexi
ANontechnicalIntroduction xv
1 AGuidedTourthroughtheLandofOperatorSemigroups 1
1.1 SemigroupsandGenerators 1
1.2 TheHille–YosidaTheorem 12
1.3 PerturbationTheorems 21
1.4 ApproximationandConvergenceTheorems 23
1.5 DualandSun-DualSemigroups 27
1.6 Appendix:OnConvergenceinl1 34
1.7 Notes 38
2 GeneratorsversusIntensityMatrices 39
2.1 TransitionMatricesandMarkovOperatorsinl1 39
2.2 TheMatrixofIntensities 44
2.3 TheUniformCase 49
2.4 TheGenerator 51
2.5 Intensities,Generators,andInfinitesimalDescription 64
2.6 BacktotheFirstKolmogorov–Kendall–ReuterExample 70
2.7 BacktotheSecondKolmogorov–Kendall–ReuterExample 76
2.8 Blackwell’sExample 82
2.9 Notes 87
3 BoundaryTheory:CoreResults 89
3.1 Kato’sTheorem 90
3.2 TheQuestionofExplosiveness 95
3.3 PureBirthProcessExample 105
ix