Lecture Notes in Mathematics 2001Editors:J.-M. Morel, CachanF. Takens, GroningenB. Teissier, ParisL evy in Stanford (Permission granted by G.L. Alexanderson)L evy Matters is a subseries of the Springer Lecture Notes in Mathematics, devoted to the dissem-inationofimportantdevelopmentsintheareaofStochasticsthatarerootedinthetheoryofL evyprocesses. Eachvolumewill containstate-of-the-art theoretical resultsaswell asapplicationsofthis rapidly evolvingeld, with special emphasis on the caseof discontinuouspaths. Contributionsto this series byleadingexperts will presentor surveynew andexciting areasof recenttheoreticaldevelopments, or will focus on some of the more promising applications in related elds. In this wayeach volume will constitute a reference text that will serve PhD students, postdoctoral researchers andseasoned researchers alike.EditorsOle E. Barndorff-NielsenThiele Centre for Applied Mathematicsin Natural ScienceDepartment of Mathematical SciencesArhus University, Ny Munkegade 118DK-8000Arhus C, Denmarkoebn@imf.au.dkJean BertoinLaboratoire de Probabilit eset Mod` eles Al eatoiresUniversit e Paris 6 Pierre et Marie CurieCase courrier 188, 4 Place Jussieu75252 Paris Cedex 05, Francejean.bertoin@upmc.frJean JacodInstitut de Math ematiques de JussieuCNRS-UMR 7586Universit e Paris 6 Pierre et Marie CurieCase courrier 188, 4 Place Jussieu75252 Paris Cedex 05, Francejean.jacod@upmc.frClaudia Kl uppelbergTUM Institute for Advanced Study& Zentrum MathematikTechnische Universit at M unchenBoltzmannstrae 385747 Garching bei M unchen, Germanycklu@ma.tum.deManaging EditorsVicky FasenZentrum MathematikTechnische Universit at M unchenBoltzmannstrae 385747 Garching bei M unchen, Germanyfasen@ma.tum.deRobert StelzerTUM Institute for Advanced Study& Zentrum MathematikTechnische Universit at M unchenBoltzmannstrae 385747 Garching bei M unchen, Germanyrstelzer@ma.tum.deThe volumes in this subseries are published under the auspices of the Bernoulli Society.Thomas DuquesneOleg ReichmannKen-iti SatoChristoph SchwabL evy Matters IRecent Progress in Theoryand Applications: Foundations, Treesand Numerical Issues in FinanceWith a Short Biography of Paul L evyby Jean JacodEditors:Ole E. Barndorff-NielsenJean BertoinJean JacodClaudia Kl uppelberg1 3Thomas DuquesneUniversit e Paris 6 Pierre et Marie CurieLaboratoire de Probabilit es et Mod` elesAl eatoiresCase courrier 188, 4 Place Jussieu75252 Paris CX 05Francethomas.duquesne@upmc.frKen-iti SatoHachiman-yama 1101-5-103Tenpaku-KuNagoya 468-0074Japanken-iti.sato@nifty.ne.jpOleg ReichmannETH Z urichSeminar f ur Angewandte MathematikR amistrasse 1018092 Z urichSwitzerlandoleg.reichmann@math.ethz.chChristoph SchwabETH Z urichSeminar f ur Angewandte MathematikR amistrasse 1018092 Z urichSwitzerlandchristoph.schwab@math.ethz.chISBN: 978-3-642-14006-8 e-ISBN: 978-3-642-14007-5DOI: 10.1007/978-3-642-14007-5Springer Heidelberg Dordrecht London New YorkLecture Notes in Mathematics ISSN print edition: 0075-8434ISSN electronic edition: 1617-9692Library of Congress Control Number: 2010933508Mathematics Subject Classication (2010): 60G51, 60E07, 60J80, 45K05, 65N30, 28A78, 60H05,60G57, 60J75cSpringer-Verlag Berlin Heidelberg 2010This work is subject to copyright.All rightsare reserved,whether the whole or part of the materialisconcerned, specically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproduction on microlm or in any other way, and storage in data banks. Duplication of this publicationor parts thereof is permittedonly under the provisionsof the German CopyrightLaw of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violationsare liable to prosecution under the German Copyright Law.Theuseofgeneral descriptivenames,registerednames,trademarks, etc. inthispublicationdoesnotimply, even in the absence of a specic statement, that such names are exempt from the relevant protectivelaws and regulations and therefore free for general use.Cover photograph: cBildarchiv des Mathematischen Forschungsinstituts OberwolfachCover design: WMXDesign GmbH, Heidelberg, GermanyPrinted on acid-free paperspringer.comPrefaceOver thepast 10-15years, wehaveseenarevival of general L evyprocessestheory as well as a burst of new applications. In the past, Brownian motion or thePoisson process have been considered as appropriate models for most applications.Nowadays, the need for more realistic modelling of irregular behaviour of phenom-ena in nature and society like jumps, bursts, and extremes has led to a renaissance ofthe theory of general L evy processes. Theoretical and applied researchers in eldsas diverse as quantumtheory, statistical physics, meteorology, seismology, statistics,insurance, nance,andtelecommunication haverealisedtheenormous exibilityof L evy models in modelling jumps, tails, dependence and sample path behaviour.L evyprocesses orL evydriven processesfeature sloworrapid structural breaks,extremal behaviour, clustering, and clumping of points.Tools and techniques fromrelated but disctinct mathematical elds, such as pointprocesses, stochastic integration, probability theory in abstract spaces, and differen-tial geometry, have contributed to a better understanding of L evy jump processes.Asinmanyotherelds, theenormouspowerofmoderncomputershasalsochanged theview ofL evy processes. Simulation methods for paths of L evy pro-cesses andrealisations of their functionalshavebeendeveloped. MonteCarlosimulation makes it possible to determine the distribution of functionals of samplepaths of L evy processes to a high level of accuracy.This development of L evy processes was accompanied and triggered by a seriesof Conferences on L evy Processes: Theory and Applications. The First and SecondConferences were held in Aarhus (1999, 2002), the Third in Paris (2003), the Fourthin Manchester (2005), and the Fifth in Copenhagen (2007).To show the broad spectrum of these conferences, the following topics are takenfrom the announcement of the Copenhagen conference: Structural results for L evy processes: distribution and path properties L evy trees, superprocesses and branching theory Fractal processes and fractal phenomena Stable and innitely divisible processes and distributions Applications in nance, physics, biosciences and telecommunications L evy processes on abstract structures Statistical, numerical and simulation aspects of L evy processes L evy and stable random elds.vvi PrefaceAt the Conference on L evy Processes: Theory and Applications in Copenhagenthe idea was born to start a series of Lecture Notes on L evy processes to bear witnessof the exciting recent advances in the area of L evy processes and their applications.Its goal is the dissemination of important developments in theory and applications.Each volume will describe stateof the art results of this rapidly evolving subjectwithspecialemphasisonthenon-Brownianworld. Leadingexpertswillpresentnewexcitingelds, orsurveysofrecentdevelopments, orfocusonsomeofthemostpromisingapplications.Despiteitsspecialcharacter,eacharticleiswrittenin an expository style, normally with an extensive bibliography at the end. In thisway each article makes an invaluable comprehensive reference text. The intendedaudience are PhD and postdoctoral students, or researchers, who want to learn aboutrecentadvancesinthetheoryofL evyprocessesandtogetanoverviewofnewapplications in different elds.Now, with the eld in full ourish and with future interest denitely increasingitseemedreasonabletostartaseriesofLectureNotesinthisarea. Thepresentvolume is the rst in the series, and future volumes will appear over time under thecommon name L evy Matters, in tune with the developments in the eld. L evyMatters will appear as a subseries of the Springer Lecture Notes in Mathematics,thus ensuring wide dissemination of the scientic material. The expository articlesin this rst volume have been chosen to reect the broadness of the area of L evyprocesses.The rst article by Ken-iti Sato characterises extensions of the class of selfde-composabledistributionson Rd. Theyaregivenastwofamilieseachwithtwocontinuous parameters ofclassesofdistributions ofimproper stochasticintegralslimt_t0f (s)dXsfor appropriate non-random functionsf and L evy processes X.Many known classes appear as limiting cases in some parameters: the Thorin class,the Goldie-Steutel-Bondesson class, and the class of completely selfdecomposabledistributions. Moreover, the theory of fractional integrals of measures is built.The second article by Thomas Duquesne discusses Hausdorff and packing mea-sures of stabletrees.Stabletrees areaspecial classof L evy trees,which form aclass of random compact metric spaces, and were introduced by Le Gall and Le Jan(1998) as the genealogy of continuous state branching processes. It is shown thatlevel sets of stable trees are the sets of points situated at a given distance from theroot. In contrast to Brownian trees, for non-Brownian stable trees there is no exactpacking measure for level sets,i.e.thesetsofpoints situatedatagiven distancefrom the root.The third (and last) article by Oleg Reichmann and Christoph Schwab presentsnumerical solutions to Kolmogorov equations, which arise for instance in nancialengineering, when L evy or additive processes model the dynamics of the risky as-sets. Solution algorithms based on wavelet representations for the Dirichlet and freeboundary problems connected to barrier and American style contracts are presented.L evycopulasareusedfor asystematicconstructionof parametricmultivariateFeller-L evy processes. Numerical aspects of the implementation and Monte Carlopath simulation techniques are addressed.Preface viiWe take the possibility to acknowledge the very positive collaboration with therelevant Springer staff and the Editors of the LN Series, and the (anonymous) refer-ees of the three articles.We hope that the readers of this and subsequent volumes enjoy learning aboutthe high potential of L evy processes in theory and applications. Researchers withideas for contributions to further volumes in the L evy Matters series are invited tocontact any of the Editors with proposals or suggestions.June 2010 Ole E. Barndorff-Nielsen (Aarhus)Jean Bertoin (Paris)Jean Jacod (Paris)Claudia Kl uppelberg (Munich)ContentsFractional Integrals and Extensions of Selfdecomposability . . . . . . . . . . . . . . . . . . 1Ken-iti Sato1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 Characterizations of Selfdecomposable Distributions. . . . . . . . . . . . . . . . . . . 21.2 Nested Classes of Multiply Selfdecomposable Distributions . . . . . . . . . . . 41.3 Continuous-Parameter Extension of Multiple Selfdecomposability . . . . 41.4 Stable Distributions and the Class L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Fractional Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6 The Classes Kp,and Lp,Generated by StochasticIntegral Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.7 Remarkable Subclasses of ID. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Fractional Integrals and Monotonicity of Order p>0 . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 One-to-One Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 More Properties and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Preliminaries in Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1 L evyKhintchine Representation of Innitely DivisibleDistributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Radial and Spherical Decompositions of -Finite Measures on Rd. . . . 273.3 Weak Mean of Innitely Divisible Distributions . . . . . . . . . . . . . . . . . . . . . . . . 293.4 Stochastic Integral Mappings of Innitely Divisible Distributions . . . . . 313.5 Transformation of L evy Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 FirstTwo-Parameter Extension Kp,oftheClassLof Selfdecomposable Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1 f and Lffor f =. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2p,andLp,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3 Range ofLp,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.4 Classes Kp,, K0p,, and Kep,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 One-Parameter Subfamilies of Kp, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.1 Kp,, K0p,, and Kep,for p(0, ) with Fixed . . . . . . . . . . . . . . . . . . . . . . . . . 575.2 Kp,, K0p,, and Kep,for (, 2) with Fixed p. . . . . . . . . . . . . . . . . . . . . . . 65ixx Contents6 Second Two-Parameter Extension Lp,of the ClassLof Selfdecomposable Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.1 p,and Lp, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.2 Range of Lp,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.3 Classes Lp,, L0p,, and Lep,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.4 Relation Between Kp,and Lp,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 One-Parameter Subfamilies of Lp,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.1 Lp,, L0p,, and Lep,for p(0, ) with xed . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.2 Lp,, L0p,, and Lep,for (, 2) with Fixed p . . . . . . . . . . . . . . . . . . . . . . . 87References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Packing and Hausdorff Measures of Stable Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Thomas Duquesne1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932 Notation, Denitions and Preliminary Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1022.1 Hausdorff and Packing Measures on Metric Spaces. . . . . . . . . . . . . . . . . . . .1022.2 Height Processes and L evy Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1032.3 Estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1133 Proofs of the Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1173.1 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1173.2 Proof of Proposition 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1213.3 Proof of Proposition 1.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1223.4 Proof of Theorems 1.6 and 1.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135Numerical Analysis of Additive, L evy and Feller Processeswith Applications to Option Pricing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .137Oleg Reichmann and Christoph Schwab1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1382 Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1402.1 Time-Homogeneous Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1402.2 Time-Inhomogeneous Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1443 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1454 Multivariate Model Setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1484.1 Copula Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1484.2 Sector Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1514.3 A Class of Admissible Market Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1545 Variational PIDE Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1565.1 European Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1575.2 American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1605.3 Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1646 Wavelets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1686.1 Spline Wavelets on an Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1696.2 Tensor Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1736.3 Space Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1746.4 Wavelet Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .177Contents xi7 Computational Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1787.1 Time Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1787.2 Numerical Quadratures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1798 Alternative Pricing Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1818.1 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1818.2 Fourier Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1839 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1859.1 Univariate Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1869.2 Multidimensional Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .190List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .192References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .193Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .197A Short Biography of Paul L evyThe rst volume of the series L evy Matters would not be complete without a shortsketch about the life and mathematical achievements of the mathematician whosename has been borrowed and used here. This is more a form of tribute to Paul L evy,who not only invented what we call now L evy processes, but also is in a sense thefounder of the way we are now looking at stochastic processes, with emphasis onthe path properties.Paul L evy was born in 1886, and lived until 1971. He studied at the Ecole Poly-technique in Paris, and was soon appointed as professor of mathematics in the sameinstitution, a position that he held from 1920 to 1959. He started his career as an an-alyst, with 20 published papers between 1905 (he was then 19 years old) and 1914,and he became interested in probability by chance, so to speak, when asked to give aseries of lectures on this topic in 1919 in that same school: this was the starting pointof an astounding series of contributions in this eld, in parallel with a continuingactivity in functional analysis.Verybriey, onecanmentionthat heis themathematicianwhointroducedcharacteristic functions in full generality, proving in particular the characterisationtheoremand the rst L evys theorem about convergence. This naturally led him tostudy more deeply the convergence in law with its metric, and also to consider sumsof independent variables, a hot topic at the time: Paul L evy proved a form of the0-1 law, as well as many other results, for series of independent variables. He alsointroduced stable and quasi-stable distributions, and unravelled their weak and/orstrong domains of attractions, simultaneously with Feller.Then we arrive at the book Th eorie de laddition des variables al eatoires, pub-lishedin1937, andinwhichhesummarizeshisndingsabout what hecalledadditive processes (the homogeneous additive processes are now called L evy pro-cesses,buthedidnot restricthisattention tothehomogeneous case).Thisbookcontains a host of new ideas and new concepts: the decomposition into the sum ofjumps at xed times and the rest of the process; the Poissonian structure of the jumpsfor an additive process without xed times of discontinuities; the compensationof those jumps so that one is able to sum up all of them; the fact that the remainingcontinuous part is Gaussian. As a consequence, he implicitly gave the formula pro-viding the formof all additive processes without xed discontinuities, nowcalled theL evy-It o Formula, and he proved the L evy-Khintchine formula for the characteristicxiiixiv A Short Biography of Paul L evyfunctions ofallinnitelydivisibledistributions. But,asfundamental asallthoseresults are, this book contains more: new methods, like martingales which, althoughnot given a name, are used in a fundamental way; and also a new way of looking atprocesses, which is the pathwise way: he was certainly the rst to understand theimportance of looking at and describing the paths of a stochastic process, instead ofconsidering that everything is encapsulated into the distribution of the processes.Thisisofcoursenot theendofthestory. Paul L evyundertookaverydeepanalysis of Brownian motion, culminating in his book Processus stochastiques etmouvement brownien in 1948, completed by a second edition in 1965. This is aremarkable achievement, inthespirit ofpathproperties, andagain itcontains somany deep results: the L evy modulus of continuity, the Hausdorff dimension of thepath, themultiple points, theL evy characterisation theorem. Heintroduced localtime, proved the arc-sine law. He was also the rst to consider genuine stochasticintegrals, with the area formula. In this topic again, his ideas have been the originof a huge amount of subsequent work, which is still going on. It also laid some ofthe basis for the ne study of Markov processes, like the local time again, or thenew concept of instantaneous state. He also initiated the topic of multi-parameterstochastic processes, introducing in particular the multi-parameter Brownianmotion.Asshouldbequiteclear,theaccountgivenheredoesnotdescribethewholeofPaulL evysmathematicalachievements,andonecanconsultformanymoredetails the rst paper (by Michel Lo` eve) published in the rst issue of the Annalsof Probability (1973). It also does not account for the humanity and gentleness ofthe person Paul L evy. But I would like to end this short exposition of Paul L evyswork by hoping that this new series will contribute to fullling the program, whichhe initiated.Jean Jacod (Paris)Fractional Integrals and Extensionsof SelfdecomposabilityKen-iti SatoAbstract After characterizations of the class L of selfdecomposable distributionsonRdarerecalled, theclasses Kp,andLp,withtwocontinuousparameters0 0andn Rdfor n = 1, 2, . . . suchthat the law of bnYn +nconverges weakly to a distribution as n andthat bnZk:k = 1, . . . , n; n = 1, 2, . . . is a null array (that is, for any > 0,max1knP([bnZk[ >) 0 as n ). Then L. Conversely, any L isobtained in this way.(c) Given ID, let X()t: t 0 be a L evy process on Rd(that is, a stochasticprocesscontinuous inprobability, startingat0, withtime-homogeneous in-dependent increments, with cadlag paths) having distribution at time 1. If_[x[>1log[x[(dx) < , then the improper stochastic integral _0esdX()sisdenable and its distribution =L__0esdX()s_(1.4)is selfdecomposable. Here L(Y) denotes the distribution (law) of a randomelement Y. Conversely, any L is obtained in this way. On the other hand,if_[x[>1log[x[(dx) =, then_0esdX()sis not denable. (See Section 3.4for improper stochastic integrals.)To see that of (1.4) is selfdecomposable, notice that_0esdX()s=_logb0esdX()s+_logbesdX()s= I1 +I2,I1 and I2 are independent, andI2 =_0elogbsdX()logb+s = b1_0esdY()s,where Y()s is identical in law with X()s, and hence satises (1.2).(d) Let Yt : t 0 be an additive process on Rd, that is, a stochastic process con-tinuousinprobabilitywithindependentincrements, withcadlagpaths, andwith Y0 = 0. If, for some H > 0, it is H-selfsimilar (that is, for any a > 0, thetwo processes Yat : t 0 and aHYt : t 0 have an identical law), then thedistribution of Y1is in L. Conversely, for any L and H> 0, there is aprocess Yt : t 0 satisfying these conditions and L(Y1) = .Historically, selfdecomposabledistributionswereintroducedbyL evy[18]in1936 and written in his 1937 book [19] under the name lois-limites, to charac-terize the limit distributions in (b). L evy wrote in [18, 19] that this characterizationproblem had been posed by Khintchine, and Khintchines book [16] in 1938 calledthese distributions of class L. The book [9] of Gnedenko and Kolmogorov usesthe same naming. Lo` eves book [20] uses the name selfdecomposable.Theproperty(c) givesacharacterizationof thestationarydistributionof anOrnsteinUhlenbeck type process (sometimes called an OrnsteinUhlenbeck pro-cess driven by a L evy process) Vt : t 0 dened byVt = etV0 +_t0estdX()s,4 K. Satowhere V0and X()t:t 0 are independent. The stationary OrnsteinUhlenbecktype process and the selfsimilar process in the property (d) are connected via theso-calledLamperti transformation(see[11, 26]). Forhistorical factsconcerning(c) see [33], pp. 5455.The proofs of (a)(d) and many examples of selfdecomposable distributions arefound in Satos book [39].The main purpose of the present article is to give two families of subclasses ofID, with two continuous parameters, related to L, using improper stochastic integralsand extending the characterization (c) of L.1.2 Nested Classes of Multiply Selfdecomposable DistributionsIf L, then, for any b > 1, the distribution b in (1.2) is innitely divisible anduniquely determined by and b. If L and b L for all b > 1, then is calledtwice selfdecomposable. Let n be a positive integer 3. A distribution is called ntimesselfdecomposable, if Landif bisn 1timesselfdecomposable. LetL1,0 = L1,0(Rd) = L(Rd) and let Ln,0 = Ln,0(Rd) be the class of n times selfdecom-posable distributions on Rd. Then we haveID L = L1,0 L2,0 L3,0 . (1.5)TheseclassesandtheclassL(Rd)inSection1.4wereintroducedbyUrbanik[52, 53] and studied by Sato [37] and others. (In [37, 52, 53] the class Ln,0 is writtenas Ln1, but this notation is inconvenient in this article.)An n times selfdecomposable distribution is characterized by the property that ID with L evy measure having radial decomposition (1.3) in (a) with k(r) =h(logr) for some function h(y) monotone of order n for each (see Section 1.5andProposition2.11forthemonotonicity ofordern). Inproperty(b), Ln,0is characterized by theproperty that L(Zk) Ln1,0for k = 1, 2, . . ..In (c), Ln,0is characterized by Ln1,0in (1.4). A direct generalization of (1.4) usingexp(s1/n) or, equivalently, exp((n! s)1/n) in place of esis also possible. In (d), Ln,0if and only if, for any H, the corresponding process Yt : t 0 satisesL(YtYs) Ln1,0 for 0 < s 0. He introducedfractional times multiple Selfdecomposability and used fractional integrals and frac-tional difference quotients. On the one hand he extended the denition of n timesFractional Integrals and Extensions of Selfdecomposability 5Selfdecomposability based on (1.2) to fractional times Selfdecomposability in theformofinniteproducts. Ontheotherhand heextended essentiallytheformula(1.4) in the characterization (c), considering its L evy measure.Directly using improper stochastic integrals with respect to L evy processes, wewill dene and study thedecreasing classesLp,0forp > 0, which generalize thenestedclassesLn,0forn = 1, 2, . . ..ThustheresultsofThuwillbereformulatedas a special case in a family Lp,with two continuous parameters 0 < p < and 0, (z)t= (t1/z), z Rd. We say that is -stable if ID and, for any t > 0, there is t Rdsuch that (z)t= (t1/z)exp(it, z)),z Rd. (When is a -distribution, this terminology is not the same as in Sato [39].)Let S0 =S0(Rd) and S =S(Rd) be the class of strictly -stable distributionson Rdand the class of -stable distributions on Rd, respectively. Let S = S(Rd)be the class of stable distributions on Rd. That is, S=

0 0), respectively.The properties of fractional integrals of functions are studied in M. Riesz [32],Ross(ed.)[35], Samko,Kilbas, andMarichev[36], Kamimura[15], andothers.Williamson [56] studied fractional integrals of measures on R+ for p 1 and intro-duced the concept ofp-times monotonicity. But we do not assume any knowledgeof them.In Sections 2.12.3 we build the theory of the fractional integral mappings IpandIp+forp (0, ) from the point of view that they are mappings from measures tomeasures. A basic relation is the semigroup property IqIp= Ip+qand Iq+Ip+ = Ip+q+.An important property that both Ipand Ip+are one-to-one is proved. The relationbetween the theories on R and R+is not extension and restriction. We need boththeories, as will be mentioned at the end of Section 6.2.8 K. Sato1.6 The Classes Kp, and Lp, Generated by StochasticIntegral MappingsThe formula (1.4) gives a mapping from ID(Rd) to ID(Rd). Thus =L__0esdX()s_. (1.11)Thedomainof istheclassof forwhichtheimproper stochasticintegral in(1.11) is denable.For functionsf (s) in a suitable class, we are interested in the mapping ffrom ID to ID dened by =f =L__0f (s)dX()s_. (1.12)The domain D(f ) is the class of for which the improper stochastic integral in(1.12) is denable. The range is dened by R(f ) =f : D(f ).Let us consider three families of functions. For 0 < p b,where c+ is the same as in (c) andc = cp_0(u +1)p1udu = cpB(1 , p) =1p/(p1p).Note that fp(b) = cp_b (s b)p1ds = . Thisfp is a (0, ]-valued contin-uous function on R, strictly increasing on (, b), equal to at b, and strictlydecreasing on (b, ). For any p/ > p, thisfp is not monotone of order p/ by thesame reason as in (b).(e) Let 0 < p 0. Then f iscompletelymonotone on R+, because, for any p > 0, we can choose = p +and applyExample 2.17 (c). Alternatively, use Proposition 2.11.(b) Let f (r)=erfor r R. Then f is completely monotone on R. UseProposition 2.11 or cp_r(s r)p1esds = cp_0up1eurdu = er.(c) Letf (r) =_arcsin(1 r), 0 < r < 1,0, r 1.Thenf is monotone of order 2 on R+, since it is decreasing and convex. Foranyp > 2, f is not monotone of orderp on R+. To prove this, supposef ismonotone of order p >2 on R+. Thenf (r)dr =Ip+ for some Mp1(R+).Hencef (r)dr = I1+ with = Ip1+. On the other handf (r) =_rg(s)ds with g(s) = (1 (1 s)2)1/21(0,1)(s).Hence (ds) =g(s)ds by Theorem2.10. Hence g(s) is equal almost everywhereon R+to a function monotone of orderp 1. Sincep 1 > 1, it follows thatg(s)isequal almost everywhere on R+toan absolutely continuous function(Proposition 2.14). This is absurd.(d) Letf (r) =_logr, 0 < r < 1,0, r 1.Then, similarly to the previous example,f is monotone of order 2 on R+ but isnot monotone of order p on R+ for any p > 2.Example 2.19. Let g(r) =r2+1r, r R, and h(r) = g(r), r R, with (0, ). The function g is monotone of order 2 on R, since g(r) > 0, g/(r) = 1 r(r2+1)1/2> 0, andFractional Integrals and Extensions of Selfdecomposability 25g//(r) = (r2+1)1/2r2(r2+1)3/2= (r2+1)3/2> 0,g(r) =[r[_1 +[r[2r =[r[(1 +O([r[2)) r = O(r1), r .Let us show the following.(a) For every > 0, his not monotone of order p on R for any p > +1.(b) For every > 0, his monotone of order 1 on R.(c) The following statement is true for n = 1, 2, 3. For any n, his monotoneof order n +1 on R.We have g(r) =2[r[ + O([r[1), r . Hence we see (a) byvirtue ofProposition 2.13 (iii), because h(r)/[r[p1 2/[r[p1as r .Wehave(b), sinceh/ = (r2+1r)1r2+1(r _r2+1) =hr2+1, (2.17)which is negative on R. We have (c) for n = 1, sinceh// =_rh(r2+1)3/2 h/r2+1_=h(r2+1)3/2(r +_r2+1), (2.18)which is positive on R for 1.The following recursion formula is known for the derivatives of h([30] p. 41):(r2+1)h( j+2)+(2 j +1)rh( j+1)+( j22)h( j)= 0. (2.19)Indeed, this istrue for j = 0from (2.17) and (2.18); if(2.19) istrue for agivenj 0, then its differentiation shows that it is true withj +1 in place ofj.Now let us prove (c) for n = 2. It follows from (2.17), (2.18), and (2.19) that(r2+1)h///=3rh//(1 2)h/ =3rh(r2+1)3/2(r +_r2+1) + (1 2)hr2+1=h(r2+1)3/2(3r_r2+1+(2+2)r2+(21))=h(r2+1)3/2[32(_r2+1+r)2+( 2)( 1)r2+( 2)( +12)],which is negative on R for 2.Let us prove (c) for n = 3. We have(r2+1)h(4)=5rh/// (4 2)h//=5rh(r2+1)5/2(3r_r2+1+(2+2)r2+(21))(4 2)h(r2+1)3/2(r +_r2+1)26 K. Sato=h(r2+1)5/2[(2+11)r2_r2+1+(24)_r2+1+6(2+1)r3+3(223)r]=h(r2+1)5/2[32(2+1)(_r2+1+r)3+32(29)r+(362+11 6)r2_r2+1+(33224 32)_r2+1]=h(r2+1)5/2[32(2+1)(_r2+1+r)3+32(29)(_r2+1+r)+( 3)( 2)( 1)r2_r2+1+( 3)(24)_r2+1],which is positive on R for 3. This shows (c) for n = 3.Remark 2.20. Open question: In the notation of Example 2.19, is hmonotone oforder +1 for every > 0?Some transformations more general than the fractional integral Ip+are studiedby Maejima, P erez-Abreu, and Sato [24], which is related to [23].3 Preliminaries in Probability Theory3.1 L evyKhintchine Representation of InnitelyDivisible DistributionsWe also use a representation of the cumulant function C(z) of ID other than(1.1) in the formC(z) =12z, Az) +_Rd_eiz,x)1 iz, x)1 +[x[2_(dx) +i

, z). (3.1)Here

is an element of Rd; A and are common to (1.1) and (3.1). Throughoutthis article

is used in this sense. It follows from (1.1) and (3.1) that

=_[x[1x[x[21 +[x[2(dx) +_[x[>1x1 +[x[2(dx). (3.2)The triplets (A, , )and (A, ,

)are both calledtheL evyKhintchinetriplet of . Each has its own advantage and disadvantage. Weak convergence of asequence of innitely divisible distributions can be expressed by the correspondingtriplets of the type (A, ,

), but cannot by the triplets of the type (A, , ).This is because theintegrand inthe integral term iscontinuous with respect toxin (3.1), but not continuous in (1.1). On the other hand the formulas derived fromFractional Integrals and Extensions of Selfdecomposability 27(A, , ) areoften simpler than those derived from (A, ,

). Seethebook[39] for details. In [39] the author uses the symbol in the sense of , but in thepapers [40][44] in the sense of

.The and

are both called the location parameter of . They depend on thechoice of the integrand in the L evyKhintchine representation. Many other choicesoftheintegrandarefoundintheliterature. Kwapie nandWoyczy nski [17]andRajput and Rosinski [31] use some form other than in (1.1) and (3.1). Maruyama[29] uses still another form.3.2 Radial and Spherical Decompositions of -FiniteMeasures on RdAmeasure (B), B B(Rd), iscalled -niteifthereisaBorel partitionBn,n = 1, 2, . . ., of Rdsuch that (Bn) < . The following propositions give two de-compositions of -nite measures on Rd.Proposition 3.1. Let be a -nite measure on Rdsatisfying (0) = 0. Thenthere are a -nite measure on S = : [[ =1 with (S) 0 and a measurablefamily : S of -nite measures on R+ with (R+) > 0 such that(B) =_S(d)_R+1B(r)(dr), B B(Rd). (3.3)Here and are uniquely determined in the following sense: if ((d), ) and(/(d), /) both satisfy these conditions, then there is a measurable function c()on S such that0 < c() 0, then let f (x) = 2n/an for x Bn. If an = 0, thenlet f (x) = 1forx Bn. Letb = _Rd0 f (x)(dx). Wehave0 < b n=12n.Let (dx) = b1f (x)(dx), which is a probability measure. Using the conditionaldistribution theorem, we nd a probability measureon S and a measurable family : S of probability measures on R+ such that(B) =_S(d)_R+1B(r)(dr), B B(Rd).28 K. SatoThus(B) =_Bb f (x)1(dx) =_S(d)_R+1B(r)b f (r)1(dr).Let =and (dr) = b f (r)1(dr).Then is a -nitemeasure on R+for each and (3.3) holds. To see the uniqueness, let ((d), ) be the pair justconstructed, and let (/(d), /) be another decomposition of . Then, for everyE B(S),(E) =(E) =((0, )E) =_(0,)Eb1f (x)(dx)=_E/(d)_R+b1f (r)/(dr).Let c() = _R+ b1f (r)/(dr). Thenc()is positivefor all andnitefor/-a. e. .Modifyc()ona /-nullsetsothat(3.4)holds.Nowwehave(3.5).Then (3.6) also follows. It follows that (3.4)(3.6) hold for two arbitrary decompo-sitions with an appropriate c(). .Remark 3.2. If ,= 0, then we can choose the measure in Proposition 3.1 to be aprobability measure. Indeed,in the proof is a probability measure.Proposition 3.3. Let be a -nite measure on Rdsatisfying (0) = 0. Thentherearea -nitemeasure on R+with (R+) 0andameasurable familyr: r R+ of -nite measures on S = : [[ = 1 with r(S) > 0 such that(B) =_R+ (dr)_S1B(r)r(d), B B(Rd). (3.7)Here and rare uniquely determined in the following sense: if ( (dr), r) and( /(dr), /r) both satisfy these conditions, then there is a measurable function c(r)on R+ such that0 < c(r) 0. We have gp,1(t) = cpt1(1 p)cplogt +O(1), t 0,since gp,1(t) = cp_t1+_1t((1 u)p11)u2du 1_= cp_t1+(1 p)_1tudu +_1t((1 u)p11 (1 p)u)u2du 1_.Hences = cp fp,1(s)1(1 p)cplogfp,1(s) +O(1), s ,that is,fp,1(s) = cps1(1 p)cps1 fp,1(s)logfp,1(s) +O(s1 fp,1(s)). (4.25)Fractional Integrals and Extensions of Selfdecomposability 43On the other hand we have gp,1(t) = cpt1+o(t1), t 0,s = cp fp,1(s)1+o( fp,1(s)1), s .fp,1(s) = cps1(1 +o(1)),successively. The last formula and (4.25) yield (4.23) with o(s2logs) in place ofO(s2). Then this and (4.25) give (4.23). .If < 0, thenD0( p,) = D( p,) = De( p,) = ID(Rd). If 0, thenD0( p,), D( p,), and De( p,) are described by Theorems 4.2 and 4.4 by virtueof Proposition 4.6. As a consequence, they do not depend on p. We notice thatp,is trivial if 2.For < < and p>0 we deneLp,= Lfwith f =fp,as inDenition 3.25. Again by Proposition 4.6, Theorem4.1 is applied to the descriptionof D( Lp,), which does not depend onp. If 2, thenLp,is trivial. If < 0,then D( Lp,) =ML.If D( Lp,), thenLp,(B) =_0ds_Rd1B( fp,(s)x)(dx)=_10d gp,(t)_Rd1B(tx)(dx)= cp_10(1 t)p1t1dt_Rd1B(tx)(dx)(4.26)forB B(Rd 0). ThisshowsthatLp,isanexampleof -transformationsstudied by Barndorff-Nielsen, Rosi nski, and Thorbjrnsen [2].The familyLp,satises the following identity.Theorem 4.7. Let 0, and q > 0. ThenLp+q, =Lq,pLp, =Lp,Lq,p. (4.27)Proof. Let ML(Rd). Let ( j), j =1, 2, 3, 4, be measures on Rdwith( j)(0) = 0 satisfying(1)(B) =_0ds_Rd1B( fp,(s)x)(dx),(2)(B) =_0ds_Rd1B( fq,p(s)x)(1)(dx),(3)(B) =_0ds_Rd1B( fq,p(s)x)(dx),(4)(B) =_0ds_Rd1B( fp,(s)x)(3)(dx)44 K. Satofor B B(Rd 0). Then(2)(B) = cq_10(1 t)q1t+p1dt_Rd1B(tx)(1)(dx)= cqcp_10(1 t)q1t+p1dt_10(1 u)p1u1du_Rd1B(tux)(dx)= cqcp_Rd(dx)_10(1 t)q1t+p1dt_101B(tux)(1 u)p1u1du= cqcp_Rd(dx)_10(1 t)q1dt_t01B(wx)(t w)p1w1dw= cqcp_Rd(dx)_101B(wx)w1dw_1w(1 t)q1(t w)p1dt= cqcp_Rd(dx)_101B(wx)(1 w)p+q1w1dw_10(1 y)q1yp1dy(by change of variables y = (t w)/(1 w))= cp+q_10(1 w)p+q1w1dw_Rd1B(wx)(dx).Hence it follows from Denition 3.25 that(2) ML(Rd) D( Lp+q,).On the other hand,(2) ML(Rd) (1) D( Lq,p)and D( Lp+q,) D( Lp,)by Proposition 4.6. Hence D( Lp+q,) D( Lp,),Lp, D( Lq,p)andLp+q, =Lq,pLp,. Similarly, in order to seeLp+q, =Lp,Lq,p, observethat(4)(B) = cp_10(1 u)p1u1du_Rd1B(ux)(3)(dx)= cpcq_10(1 u)p1u1du_10(1 t)q1t+p1dt_Rd1B(utx)(dx),which equals (2)(B) in the preceding calculus. .Fractional Integrals and Extensions of Selfdecomposability 45Corollary 4.8. We haveR( Lp,) R( Lp/,) for 0 < p < p/ 0. For < 0 we havef(s) = 0 for s .Asymptotic behaviors off(s) are as follows.Proposition 5.1. As s 0,f(s) logs for R. (5.4)As s ,f0(s) exp(c s), (5.5)f(s) (s)1/for > 0, (5.6)f1(s) = s1s2logs +O(s2), (5.7)wherec =_1u1eudu _10u1(1 eu)du. (5.8)Proof. Since g(t) t1et, t , we havelims0f(s)log(1/s) = limttlog(1/g(t)) = limt1t1et/g(t) = 1,that is, (5.4) holds. To see (5.5), note thatg0(t) =_1tu1du +_1tu1(eu1)du +_1u1eudu =logt +c +o(1)58 K. Satoas t 0 and hence s = log f0(s) +c +o(1), s . To see (5.6), see that g(t) =1t(1 +o(1)), t 0, equivalently, s =1f(s)(1 +o(1)), s .Assertion (5.7): We haveg1(t) =_tu2du +_1tu2(eu1 +u)du _1tu1du +_1u2(eu1)du=t1+logt +O(1), t 0and hence s =f1(s)1+log f1(s) +O(1), s , which is written tof1(s) = s1+s1f1(s)log f1(s) +O(s1f1(s)), s . (5.9)Sincef1(s) = s1(1 +o(1)) from (5.6), we obtain from (5.9)f1(s) = s1s2logs +o(s2logs), s .Putting this again in (5.9), we arrive at (5.7). .We denef(0) = for convenience. Thenf(s) is locally square-integrable onR+. We have =L__0f(s)dX()s_,that is, = f with f =fin(3.24) whenever theimproper stochasticintegral is denable. If 0. Then Lp,if andonlyifhasaradialdecomposition ((d), u1h(logu)du) satisfying (6.20).(ii) Let 1 0. Then Lp,if and only if has a radial decom-position ((d), u1h(logu)du) satisfying (6.20) and has mean 0.Proof. Assertion (i) is from Proposition 6.8 and Theorem 6.9. Let us prove asser-tion (ii). Let Lp,. Then Lep,from (6.26), and Theorem 6.9 says that has((d), u1h(logu)du)satisfying(6.20). Wehave= p,forsome D(p,). Thus, byTheorems6.2and6.3, _[x[>2(log[x[)p1[x[(dx)< and _Rd x(dx) = 0. Hence= _[x[>1x(dx). Let l = lp,. It followsfromProposition 5.9 that_0ds_[l(s)x[>1[l(s)x[(dx) 1[x[(dx). It follows that =_0ds_[l(s)x[>1l(s)x(dx), (6.29)and hence =_[x[>1x(dx), that is, has mean 0. Conversely, assume that has the property stated and that has mean 0. Then by Theorem 6.9, Lep,and R(Lp,). Choose such that Lp, =. Then (6.28) and (6.29) hold with in place of . Let A = (_0lp,(s)2ds)1Aand = _[x[>1x(dx). Then IDwith triplet (A, , ) belongs to D(p,) from Theorem6.3 and we have p, =..Remark 6.13. Open problem: Describe the classes Lp,1(Rd) and L0p,1(Rd) for p >0.Theorem 6.14. Let 0. Then is a nite measure if and onlyifis a nite measure. In particular, for any < 0 and p > 0, Lp,containssome compound Poisson distribution.Proof. This is proved by the same idea as Theorem 4.24. The key formulas are, for0and similarly dened,_0u1h(logu)du = cp_(0,)r0(dr)_r0u1(log(r/u))p1du =for 0 and(Rd) = ()p(Rd)for < 0. .6.4 Relation Between Kp, and Lp,We have K1, = L1,, K01, = L01,, and Ke1, = Le1,for 0. ThenLq,Lp, =Lp+q,. (7.1)Proof. First note that a special case of (2.4) with =0 givescpcq_0u(r)q1(r u)p1dr = cp+q(u)p+q1, u < 0,that is, for 0 < w < 1,cpcq_1w(logu)q1(log(w/u))p1u1du = cp+q(logw)p+q1. (7.2)Given ML(Rd), let ( j)(0) = 0, j = 1, 2, and(1)(B) =_0ds_Rd1B(lp,(s)x)(dx),(2)(B) =_0ds_Rd1B(lq,(s)x)(1)(dx)Fractional Integrals and Extensions of Selfdecomposability 79for B B(Rd 0). Then(2)(B) = cq_10(logu)q1u1du_Rd1B(ux)(1)(dx)= cqcp_10(logu)q1u1du_10(logt)p1t1dt_Rd1B(utx)(dx)= cqcp_10(logu)q1u1du_u0(log(w/u))p1w1dw_Rd1B(wx)(dx)= cqcp_Rd(dx)_101B(wx)w1dw_1w(logu)q1(log(w/u))p1u1du= cp+q_10(logw)p+q1w1dw_Rd1B(wx)(dx),using (7.2). Hence(2) ML D(Lp+q,).On the other hand,(2) ML (1) D(Lq,).Hence D(Lp+q,) (1) D(Lq,), D(Lp,), Lp, =(1).It follows that D(Lp+q,)=D(Lq,Lp,) and that, if D(Lp+q,), thenLp+q, =Lq,Lp,. .Corollary 7.2. We haveR(Lp,) R(Lp/,) for 0.(i) If D0(p+q,), then D0(p,), p, D0(q,), andp+q, =q,p, (7.4)(ii) If ,= 1, thenp+q, =q,p,(7.5)Proof. Let us prove (i). Let D0(p+q,). As in the proof of Theorem 7.1,cpcq_10(logu)q1u1du_10[C(tuz)[(logt)p1t1dt= cp+q_10[C(wz)[(logw)p+q1w1dw,80 K. Satowhich is nite since D0(p+q,). Then, we can use Fubinis theoremand obtaincpcq_10(logu)q1u1du_10C(tuz)(logt)p1t1dt= cp+q_10C(wz)(logw)p+q1w1dw.We have D0(p,) from (6.4), andcq_10[Cp,(uz)[(logu)q1u1ducpcq_10(logu)q1u1du_10[C(tuz)[(logt)p1t1dt 0byTheorem 6.3. If D(p+q,), then D(q,p,) and q,p, =p+q,by (i). It remains to show that D(q,p,) D(p+q,). Let D(q,p,).Thismeansthat D(p,)and p, D(q,). Hence D(Lp,)andLp, D(Lq,). Hence we have D(Lp+q,) from Theorem 7.1. Hence De(p+q,). Now, if < 1, then D(p+q,) since De(p+q,) = D(p+q,).If > 1, then _Rd x(dx) = 0 from D(p,), using Theorem 6.3, and hence D(p+q,). .Corollary 7.4. For any positive integer n and (, 1) (1, 2), we haven,0 = . .nand n, =1,

1,. .n,where is dened by (1.11).Proof. Combine (7.5) with 1,0 = and 1, =1,. .Remark 7.5. Open question: Is (7.5) also true for = 1?Corollary 7.6. For pR(Lp/,), since h(y) is monotone oforder p but not of order p/ (Example 2.17 (a)). It follows that Lep,

p/>pLep/, ,= / 0.If < 1, thenthisalsosaysthat L0p,

p/>pL0p/, ,=/ 0. If1 < < 2, thenlet IDbe suchthat = and, recalling that _[x[>1[x[(dx) < ,choose =_[x[>1x(dx) to see that L0p,

p/>pL0p/,by Theorem 6.12. Assuming that = 1,let satisfy _S(d) = 0andlet IDbesuchthat Lp,1= and = 0. We consider the proof of Theorem 6.7 and see that can be chosen to beof polar product type with the same . Hence_Rdx(1[lp,1(s)x[11[x[1)(dx) = 0,which shows that D0(p,1) by Proposition 3.18. Thus = p,1has = and = 0 and belongs to L0p,1

p/>pL0p/,1. .Remark 7.8. SinceLp,= L0p,for (, 1) (1, 2), Lp,hasthepropertiessimilar to Corollary 7.6 and Theorem 7.7 if ,= 1. Open question: Is it true thatLp,1 Lp/,1 for 0 < p < p/ and Lp,1

p/>pLp/,1 for p > 0?If 0, then the class L0p, is continuous for decreasing p in the following sense.Theorem 7.9. Let 0. Then

q(0,p)L0q, = L0p,. (7.8)Proof. Let

q(0,p)L0q,. It is enough to prove that L0p,. Let ((d), u1h(logu)du) be a radial decomposition of . For any q (0, p) there is q D(Iq)such thath(y) = cq_(y,)(s y)q1q(ds), y R.Fix for the moment and omit the subscript . For < a < b _[x[1[x[2(dx) =_S(d)_10u1h(logu)du=_S(d)_10u1du_(0,)u0 (d)=_S(d)_10udu_(,)u(d)=_S(d)_(,)(d)_10u1du,where we dene(E) =_(0,)1E( +)0 (d), E B((, )).Since_10 u1du = for 2, we obtain ([2, )) = 0 for -a. e. . We have_[x[1[x[2(dx) =_S(d)_(,2)(2 )1(d).We also have >_[x[>1(dx) =_S(d)_1u1h(logu)du=_S(d)_1u1du_(0,)u0 (d)=_S(d)_1u1du_(,2)u(d)=_S(d)_(,2)(d)_1u1du,and_1u1du = for 0. Hence, if < 0, then ((, 0]) = 0 for -a. e. .For any < 2 we have_[x[>1(dx) =_S(d)_(0,2)1(d).Similarly, it follows from(B) =_S(d)_01B(u)u1h(logu)du (7.18)86 K. Satothat(B) =_S(d)_(0,2)(d)_01B(u)u1du. (7.19)Themeasure (d)(d)onS ( 0, 2)iswrittento (d)(d), where(d) is a measure on ( 0, 2) satisfying_(0,2)(1+(2 )1)(d) 1ln,1(s)x0(dx).Choose ID such that = 0, A =__0ln,1(s)2ds_1A, and = _[x[>1x0(dx). Thenit followsfromProposition3.18that D0(n,1)and n,1=. Hence L0n,1 Ln,1. Similarly, if > 1andif Ln,/ = L0n,/, then Ln, = L0n,.Step 3. To show the strictness of the inclusion, let be a non-zero nite measureon S and let h(y) = (y)n11(,0)(y), which is monotone of order n on R(Example2.17 (a)). Then ((d), u1h(logu)du) is a radial decomposition of a L evy mea-sure , since _10 u1h(logu)du < . Let ID with = . Then Len,but , Len,/, as is seen by an argument similar to the proof of Theorem 5.19. Indeed,we haveu1h(logu) = u/1h

(logu)forh

(y) = e(/)y(y)n11(,0)(y),which is not monotone of any order on R from Proposition 2.13 (iii). Strictness ofthe rst and second inclusions in (7.22) is obtained from that of the third. .Remark 7.20. Open question: Is (7.22) true for p R+ in place of n?Acknowledgments The author thanks Makoto Maejima and Vctor P erez-Abreu for their constantencouragement byproposingandexploringmanyproblemsrelatedtothesubject andfortheirvaluable comments during preparationof this work, and Yohei Ueda for his helpful remarks forimprovement of Section 6.1.Fractional Integrals and Extensions of Selfdecomposability 89References1. Barndorff-Nielsen, O.E., Maejima, M., Sato, K.: Some classes of innitely divisible distribu-tions admitting stochastic integral representations. Bernoulli 12, 133 (2006)2. Barndorff-Nielsen, O.E., Rosi nski, J., Thorbjrnsen, S.: General transformations. ALEALat. Am. J. Probab. Math. Statist. 4, 131165 (2008)3. 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Verojatnost. i Primenen. 10, 672692 (1965)Packing and Hausdorff Measures of Stable TreesThomas DuquesneAbstract InthispaperwediscussHausdorffandpackingmeasuresofrandomcontinuous trees called stable trees. Stable trees form a specic class of L evy trees(introduced by Le Gall and Le Jan in [33]) that contains Aldouss continuum ran-dom tree which corresponds to the Brownian case. We provide results for the wholestable trees and for their level sets that are the sets of points situated at a given dis-tance from the root. We rst show that there is no exact packing measure for levelsets. We also prove that non-Brownian stable trees and their level sets have no exactHausdorff measure with regularly varying gauge function, which continues previousresults from [14].AMS Subject Classication 2000:Primary: 60G57, 60J80Secondary: 28A78Keywords Hausdorff measureL evy treesLocal time measureMass measure Packing measureStable trees1 IntroductionStable trees are particular instances of L evy trees that form a class of random com-pact metric spaces introduced by Le Gall and Le Jan in [33]as the genealogy ofContinuous State Branching Processes (CSBP for short). The class of stable treescontains Aldous continuumrandomtree that corresponds to the Brownian case (see[2, 3]). Stable trees (and more generally L evy trees) are the scaling limit of Galton-Watsontrees(see[12]Chapter2and[10]).Various geometric anddistributionalT. Duquesne (

)Universit e Paris 6 Pierre et Marie Curie, Laboratoire de Probabilit es et Mod` eles Al eatoires,Case courrier 188, 4 Place Jussieu, 75252 Paris CX 05, Francee-mail: thomas.duquesne@upmc.frT. Duquesne et al., L evy Matters I: Recent Progress in Theory and Applications:Foundations, Trees and Numerical Issues in Finance, Lecture Notes in Mathematics 2001,DOI 10.1007/978-3-642-14007-5 2, cSpringer-Verlag Berlin Heidelberg 20109394 T. Duquesneproperties ofL evytrees(andofstabletrees,consequently) havebeenstudiedin[13] and in Weill [40]. An alternative construction of L evy trees is discussed in [16].Stable trees have been also studied in connection with fragmentation processes: seeMiermont [35, 36],Haasand Miermont [23],Goldschmidt and Haas [21]for thestable cases and see Abraham and Delmas [1] for related models concerning moregeneral L evy trees.Fractal properties of stable trees have been discussed in [13] and [14]: Hausdorffand packing dimensions of stable trees are computed in [13] and the exact Hausdorffmeasure of Aldous continuum random tree is given in [14]. The same paper con-tains partial results for the non-Brownian stable trees that suggest there is no exactHausdorff measure inthese cases.In this paper weprove there is no exact pack-ing measure for the level sets of stable trees (including the Brownian case) and wealso prove that there is no exact Hausdorff measure with regularly varying gaugefunction for the non-Brownian stable trees and their level sets.Beforestatingthe mainresults of thepaper, let us recall thedenitionofstable CSBPs and the denition of stable trees that represent the genealogy of sta-bleCSBPs. CSBPsaretime-andspace-continuous analoguesofGalton-WatsonMarkov chains. They have been introduced by Jirina [26] and Lamperti [30] as the[0, ]-valued Feller processes that are absorbed in states 0 and and whosekernel semi-group (pt(x, dy); x [0, ], t [0, )) enjoys the branching property:pt(x, ) pt(x/, ) =pt(x + x/, ), for everyx, x/ [0, ] andeveryt [0, ). Aspointed out in Lamperti [30], CSBPs are time-changed spectrally positive L evy pro-cesses. Namely, let Y = (Yt, t 0) be a L evy process starting at 0 that is dened onaprobabilityspace (, F, P)andthathasnopositivejump.Letx (0, ). SetAt = infs 0 : _s0 du/(Yu +x) > t for any t 0, and Tx = infs 0 : Ys = x,with the convention that inf / 0 = . Next set Zt = XAtTxif AtTxis nite and setZt = if not. Then, Z = (Zt, t 0) is a CSBP with initial state x (see Helland [25]for a proof in the conservative cases). Recall that the distribution of Yis character-ized by its Laplace exponent given by E[exp(Yt)] = exp(t()), t, 0 (seeBertoin [5], Chapter 7). Consequently, the law of the CSBP Z is also characterisedby and it is called its branching mechanism.We shall restrict to -stable CSBPs for which () =, 0, where (1, 2].Thecase= 2shall bereferredtoastheBrowniancase(andthecorrespond-ing CSBPis theFeller diffusion) and the cases 1 < < 2 shall be referred toasthenon-Brownian stablecases.LetZbea -stableCSBPdenedon (, F, P).As aconsequenceof aresult duetoSilverstein[38], thekernel semigroupofZischaracterisedasfollows: forany, s, t 0, onehasE[exp(Zt+s)[Zs] =exp(Zsu(t, )), where u(t, ) is the unique nonnegative solution of u(t, )/t =u(t, )andu(0, ) = . Thisordinarydifferentialequationcanbeexplicitlysolved as follows.u(t, ) =_(1)t +11_11, t, 0. (1.1)Packing and Hausdorff Measures of Stable Trees 95It is easy to deduce from this formula that -stable CSBPs get almost surely extinctin nite time with probability one: P(t 0 : Zt = 0) = 1. We refer to Bingham [6]for more details on CSBPs.L evytreeshavebeenintroducedbyLeGallandLeJanin[33]viaacodingfunction called the height process whose denition is recalled in Section 2.2. Let usbriey recall the formalismdiscussed in [13] where L evy trees are viewed as randomvariables taking values in the space of all compact rooted R-trees. Informally, anR-tree is a metric space (T , d) such that for any two points and / in Tthere isa unique arc with endpoints and / and this arc is isometric to a compact intervalof the real line. A rooted R-tree is a R-tree with a distinguished point that we denoteby and that we call the root. We say that two rooted R-trees are equivalent if thereis a root-preserving isometry that maps one onto the other. Instead of considering allcompact rooted R-trees, we introduce the set T of equivalence classes of compactrooted R-trees. Evans, Pitman and Winter in [19] noticed that T equipped with theGromov-Hausdorff distance [22], is a Polish space (see Section 2.2 for more details).With any stable exponent (1, 2] one can associate a sigma-nite measure on T called the law of the -stable tree. Although is an innite measure, onecanprove thefollowing: Dene (T ) = supTd(, )that isthetotalheightof T . Then, for any a (0, ), one has((T ) > a) = ((1)a)11.Stable trees enjoy the so-called branching property, that obviously holds true forGalton-Watson trees. More precisely, for every a >0, under the probability measure( [(T ) > a) and conditionally given the part of Tbelow level a, the subtreesabove level a are distributed as the atoms of a Poisson point measure whose intensityis a random multiple of , and the random factor is the total mass of the a-localtime measure that is dened below (see Section 2.2 for a precise denition). It isimportant to mention that Weill in [40] proves that the branching property charac-terizes L evy trees, and therefore stable trees.We now dene by an approximation with Galton-Watson trees as follows. Letbe a probability distribution on the set of nonnegative integers N. We rst assumethat k0 k(k) = 1 and that is in the domain of attraction of a -stable distribu-tion. More precisely, let Y1 be a random variable such that logE[exp(Y1)] = ,for any [0, ). Let (Jk, k 0) be an i.i.d. sequence of r.v. with law . We as-sume there exists an increasing sequence (ap, p 0) of positive integers such that(ap)1(J1 + +Jp p)converges indistributionto Y1. Denoteby aGalton-Watsontreewithoffspring distribution that canbeviewed as arandom rootedR-tree (, , )byaffecting length 1toeachedge. Thus, (,1p, )isthetree whose edges are rescaled by a factor 1/p and we simply denote it by1p. Then, forany a (0, ), the law of1punder P( [1p() > a) converges weakly in T to theprobability distribution ( [(T ) >a), when p goes to . This result is Theorem4.1 [13].Let us introduce two important kinds of measures dened on -stable trees. Let(T , d, ) be a -stable tree. For every a > 0, we dene the a-level set T (a) of T96 T. Duquesneas the set of points that are at distance a from the root. Namely,T (a) :=_ T: d(, ) = a_. (1.2)Wethen dene the random measureaon T (a) in the following way. For every >0, write T(a) for the nite subset of T (a) consisting of those vertices that havedescendants at level a +. Then, -a.e. for every bounded continuous functionfon T , we havea, f ) = lim0((1))11T(a)f (). (1.3)The measure ais a nite measure on T (a) that is called the a-local time measureof T . We refer to [13] Section 4.2 for the construction and the main properties ofthe local time measures (a, a 0) (see also Section 2.2 for more details). Theorem4.3 [13] ensures we can choose a modication of the local time measures (a, a 0)in such a way that a ais -a.e. cadlag for the weak topology on the space ofnite measures on T .We next dene the mass measure m on the tree Tbym =_0daa. (1.4)The topological support of m is T . Note that the denitions of the local time mea-sures and of the mass measure only involve the metric properties of T .Let us mention that -stable trees enjoy the following scaling property: For anyc (0, ), the law of (T , cd, ) under is c1/(1). Then, it is easy to showthat for any a, c (0, ) the law of c1/(1)a/c) under is the law of a) underc1/(1)(here, b) stands for the total mass of the b-local time measure). Sim-ilarly, thelawofc/(1)m)under isthelawof m)underc1/(1). Since

aand mareinsomesensethemost spreadout measures onrespectively T (a)and T , these scaling properties give a heuristic explanation for the following re-sults that concern the fractal dimensions of stable trees (see [13] for a proof): Forany a (0, ), -a.e. on T (a) ,= / 0 the Hausdorff and the packing dimensions ofT (a) are equal to 1/( 1) and -a.e. the Hausdorff and the packing dimensionsof Tare equal to /( 1).In this paper wediscuss ner results concerning possible exact Hausdorff andpacking measures for stable trees and their level sets. We rst state a result concern-ing the exact packing measure for level sets.To that end, let us briey recall thedenition of packing measures. Packing measures have been introduced by Taylorand Tricot in [39]. Though their construction is done in Euclidian spaces, it easilyextends to metric spaces and more specically to -stable trees. More precisely, forany Tand any r [0, ), let us denote byB(, r) (resp. B(, r)) the closed(resp. open) ball of Twith center and radius r. Let A Tand (0, ). An-packing of A is a countable collection of pairwise disjoint closed ballsB(xn, rn),n 0, suchthat xn Aandrn . Werestrict ourattentiontopackingmea-sures associated with a regular gauge function in the following sense: A functionPacking and Hausdorff Measures of Stable Trees 97g : (0, r0) (0, ) is a regular gauge function if it is continuous, non decreasing, iflim0+g = 0 and if there exists a constant C (1, ) such thatC > 1: g(2r) Cg(r) , r (0, r0/2). (1.5)Such a property shall be referred to as a C-doubling condition. We then setPg(A) = lim0sup_n0g(rn); ( B(xn, rn), n 0) packing of A_(1.6)that is the g-packing pre-measure of A and we dene the g-packing outer measureof A asPg(A) = inf_n0Pg(En); A _n0En_. (1.7)As in Euclidian spaces, Pg is a Borel regular metric outer measure (see Section 2.1for more details). The original denition of packing measures [39] makes use, as setfunction, of the diameter of open ball packing instead of the radius of closed ballpacking. As pointed out by H. Haase [24], diameter-type packing measures may benot Borel regular: H. Joyce [28] provides an explicit example where this problemoccurs. In our setting, we do not face such a problem and our results hold true fordiameter-type and radius-type packing measures as well thanks to specic propertiesof compact real trees (see Section 2.1 for a brief discussion). The following theoremshows that the level sets of stable trees have no exact packing measure, even in theBrownian case.Theorem 1.1. Let (1, 2] and let us consider a -stable tree (T , d, ) under itsexcursion measure . Let g : (0, 1) (0, ) be any function such thatlimr0r11g(r) = 0. (1.8)(i) Ifn1_2n1g(2n)_< , then for any a (0, ), -a.e. on T (a) ,=/ 0and for a-almost all , we haveliminfn

a(B(, 2n))g(2n)=. (1.9)Moreover, if g is a regular gauge function, then Pg(T (a)) = 0, -a.e.(ii) Ifn1_2n1g(2n)_=, then for any a (0, ), -a.e. and for a-almostall , we haveliminfn

a(B(, 2n))g(2n)= 0 . (1.10)Moreover, if g is a regular gauge function, then Pg(T (a)) =, -a.e. on theevent T (a) ,= / 0.98 T. DuquesneThis result is not surprising, even in the Brownian case, for it has been proved in[34] that super-Brownian motion with quadratic branching mechanism has no exactpacking measure in the super-critical dimension d 3 and [34] provides a test thatis close in some sense to the test given in the previous theorem.Remark 1.2. For any p 1, dene recursively the functions logp by log1 = log andlogp+1 = logplog. The previous theorem provides the following family of criticalgauge functions for packing measures of level sets of a -stable tree: For any Rand any p 1, setgp,(r) =r11(log(1/r). . . logp(1/r))1(logp+1(1/r)).If >1, then for any a (0, ), one has Pgp, (T (a)) = 0, -a.e. and if 1,then for any a (0, ), one has Pgp, (T (a)) =, -a.e. on the event T (a) ,= / 0.Remark 1.3. Although the level sets of stable trees have no exact packing measure,the whole -stable tree has an exact packing measure as shown in the preprint [11].More precisely, for any r (0, 1/e), we setg(r) =r1(loglog1/r)11.Then, there exists c0 (0, ) such that Pg = c0m, -a.e.Remark 1.4. Note that no other condition than (1.8) is imposed on g for the test onthe lower density ofato hold true. A similartest holds true when the sequenceofradii (2n, n 0)isreplacedby (sRn, n 0), withs ]0, 1[,andR > 1.Aspointed out by Berlinkov (Lemma 1 [4]), the result can rephrased under the formof an integral test: indeed, if we assume that g : (0, 1) (0, ) is non-decreasingand that it satises (1.8) and the doubling condition (1.5), then, Theorem 1.1 easilyimplies the following integral test.(i) If _0+g(r)rdr< , thenforanya (0, ), -a.e. on T (a) ,=/ 0andfor

a-almost all , we haveliminfr0

a(B(, r))g(r)=.(ii) If_0+g(r)rdr =, then for any a (0, ), -a.e. for a-almost all , we haveliminfr0

a(B(, r))g(r)= 0.Let usbrieyrecall thedenitionof Hausdorff measuresona-stabletree(T , d). Let us xaregular gaugefunctiong. For anysubset E T , wesetPacking and Hausdorff Measures of Stable Trees 99diam(E) =supx,yE d(x, y) that is the diameter of E. For any AT , the g-Hausdorffmeasure of A is then given byHg(A) = lim0inf_n0g(diam(En)); diam(En) 1. If we assume that g : (0, 1) (0, ) is non-decreasing and that it satises(1.12)andthedoublingcondition(1.5), then, Proposition1.5easilyimpliesthefollowing integral test.(i) If_0+drg(r)11GCthat is the set of the gauge functions we shall consider. Let us mention that, insteadof regular gauge functions, someauthors speakof blanketed Hausdorff functionsafter Larman [31].Let (T , d) be an uncountable complete and separable metric space. Let us xg GC. Recallfrom(1.7)thedenitionoftheg-packing measurePgandfrom(1.11) thedenition oftheg-Hausdorff measure Hg.Weshallusethefollowingcomparison results.Lemma 2.1. (Taylor and Tricot [39], Edgar [17]). Let g GC. Then, for any niteBorel measure on Tand for any Borel subset A of T , the following holds true.(i) If liminfr0(B(,r))g(r)1, for any A, then Pg(A) C2(A).(ii) If liminfr0(B(,r))g(r)1, for any A, then Pg(A) (A).(iii) If limsupr0(B(,r))g(r)1, for any A, then Hg(A) C1(A).(iv) If limsupr0(B(,r))g(r)1, for any A, then Hg(A) C(A).Points (iii)and (iv) inEuclidian spaces are stated inLemmas 2 and 3 inRogersand Taylor [37]. Points (i) and (ii) in Euclidian spaces can be found in Theorem5.4 in Taylor and Tricot [39]. We refer to Edgar [17] for a proof of Lemma 2.1 forgeneral metric spaces: For (i) and (ii), see Theorem 4.15 [17] in combination withProposition 4.24 [17]. For (iii) and (iv), see Theorem 5.9 [17].Remark 2.2. As already mentioned, the original denition of packing measures [39]uses the diameter of open ball packing instead of the radius of closed ball packing.As pointed out by H. Haase [24], diameter-type packing measures may be not Borelregular: H. Joyce [28] provides an explicit example where this problem occurs (seealso H. Joyce [27] and H. Joyce and D. Preiss [29]). Let us briey explain why therePacking and Hausdorff Measures of Stable Trees 103is no such problem when dealing with stable trees and more generally with compactreal trees: we now assume that (T , d) is a compact real tree with a distinguishedpoint that is called the root. In real trees, any two points and /are joined bya unique arc with endpoints and / and this arc is isometric to a compact intervalof the real line (see Denition 2.3). Let A Tand (0, ). Let us say that an-open ball packing of A is a countable collection of pairwise disjoint open ballsB(n, rn), n 0, such that n A and diam(B(n, rn)) . Let g be a regular gaugefunction that satises a C-doubling condition. For any , we set

P()g(A) = sup_n0g_diam(B(n, rn))_; (B(n, rn))n0, -op. ball pack. of A_

Pg(A) = lim0

P()g(A) and

Pg(A) = inf_n0

Pg(En); A _n0En_.Following exactly the proof of Edgar Proposition 5.2 [17], one sees that Pgis ametric outer measure. Let us now briey explain why it is equivalent to the radius-type measure using closed ball packing, as dened previously by (1.6) and (1.7).Let us recall notation (T ) = supd(, ); T that is the total height of T .Next, let Tand r (0, ). We denote the actual radius of the open ball B(, r)by R(B(, r)) = supd(, /); / B(, r). Note thatR(B(, r)) = r = r diam(B(, r)) 2r. (2.1)If there exists 0Tsuch that d(0, ) r, then R(B(, r)) =r, since we can ndpoints on the unique geodesic joining to 0 that are at distance r/ from , for anyr/ < r. If R(B(, r)) < r, we therefore see that (T ) < 2r. By (2.1), we then get T, 0 < r 0 for any t (0, ).Weeasily deduce from (2.3) the following scalingproperty for Hunder N: For any c (0, ) and for any measurable function F:D([0, ), R) [0, ), one hasc1N_F(c1Hct, t 0)_= N_F(Ht, t 0)_. (2.5)Local Times of the Height ProcessWe recall here from [12] Chapter 1 Section 1.3 the following result: There existsa jointly measurable process (Las , a, s 0) such that P-a.s. for any a 0, s Lasiscontinuous and non-decreasing and such thatt, a 0, lim0E_ sup0st1_s0dr1a 0, let us set v(b) = N(supt[0,] Ht> b). The continuity of Hand the Poisson decomposition (2.4) obviously imply that v(b) < , for any b > 0.It is moreover clear that v is non-increasing and limv = 0. For every a (0, ),we then dene a continuous increasing process (Lat , t [0, ]), such that for everyb (0, ) and for any t 0, one haslim0N_1supH>bsup0st1_s0dr1a 0 , v(a) = N_supHt a_= N_La ,= 0_=_(1)a_11. (2.12)L evy TreesWe rst dene R-trees (or real trees) that are metric spaces that generalisegraph-trees.Denition 2.3. Let (T, ) be a metric space. It is a real tree iff the following holdstrue for any 1, 2 T.(a) There is a unique isometry f1,2 from[0, (1, 2)] into T such that f1,2(0) =1 andf1,2((1, 2)) = 2. We denote by [[1, 2]] the geodesic joining 1to 2. Namely, [[1, 2]] :=f1,2([0, (1, 2)]).(b) Ifj isacontinuous injective map from [0, 1]into T,suchthat j(0) = 1andj(1) =2, then we have j([0, 1]) = [[1, 2]].A rooted R-tree is an R-tree (T, ) with a distinguished point r called the root. .Packing and Hausdorff Measures of Stable Trees 107Amongmetricspaces, R-treesarecharacterizedbytheso-calledfourpointsinequality that is expressed as follows. Let (T, ) be a connected metric space. Then,(T, ) is a R-tree iff for any 1, 2, 3, 4 T, we have(1, 2)+(3, 4) _(1, 3)+(2, 4)__(1, 4)+(2, 3)_. (2.13)WerefertoEvans[18]ortoDress, MoultonandTerhalle[9]foradetailedac-count on this property. The set of all compact rooted R-trees can be equipped withthepointed Gromov-Hausdorff distancein thefollowing way. Let (T1, 1, r1)and(T2, 2, r1) be two compact pointed metric spaces. They can be compared one witheach other thanks to the pointed Gromov-Hausdorff distance dened bydGH(T1, T2) = inf H_j1(T1), j2(T2)__j1(r1), j2(r2)_.Here the inmum is taken over all (j1, j2, (E, )),where (E, ) is a metric space,where j1 : T1 E and j2 : T2 E are isometrical embeddings and where H standsfor the usual Hausdorff metric on compact subsets of (E, ). Obviously dGH(T1, T2)only depends on the isometry classes of T1 and T2 that map r1 to r2. In [22], Gromovproves that dGH is a metric on the set of the equivalence classes of pointed compactmetric spaces that makes it a complete and separable metric space. Let us denoteby T, the set of all equivalence classes of rooted compact real-trees. Evans, Pitmanand Winter observed in [19] that T is dGH-closed. Therefore, (T, dGH) is a completeseparable metric space (see Theorem 2 of [19]).Let us briey recall how R-trees can be obtained via continuous functions. Weconsider a continuous function h : [0, ) R such that there exists a [0, ) suchthat h is constant on [a, ). We denote by hthe least of such real numbers a andwe view has the lifetime of h. Such a continuous function is said to be a codingfunction. To avoid trivialities, we also assume that h is not constant. Then, for everys, t 0, we setbh(s, t) = infr[st,st]h(r) and dh(s, t) = h(s) +h(t) 2bh(s, t). (2.14)Clearlydh(s, t) =dh(t, s). It is easytocheckthat dhsatises thefour pointsinequality, which implies that dhis a pseudo-metric. We then introduce theequivalence relation s h t iff dh(s, t) = 0 (or equivalently iff h(s) = h(t) = bh(s, t))and we denote by Th the quotient set [0, h]/ h. Standard arguments imply that dhinduces a metric on Th that is also denoted by dh to simplify notation. We denote byph : [0, h] Th the canonical projection. Since h is continuous, ph is a continuousfunctionfrom[0, h] equippedwiththeusual metriconto(Th, dh). Thisimpliesthat (Th, dh) is a compact and connected metric space that satises the four pointsinequality. It is therefore a compact R-tree. Next observe that for any t0, t1 [0, h]such that h(t0) = h(t1) =minh, we have ph(t0) = ph(t1); so it makes sense to denetherootof (Th, dh)by h =ph(t0). Weshallrefertotherootedcompact R-tree(Th, dh, h) as to the tree coded by h.108 T. DuquesneWe next dene the -stable tree as the tree coded by the -stable height process(Ht, 0 t ) under the excursion measure Nand to simplify notation we set(TH, dH, H) = (T , d, ).We also set p = pH : [0, ] T . Note that = p(0). Since H =0 and since Ht >0,for any t (0, ), is the only time t [0, ] distinct from 0 such that p(t) =.Let us denote byTthe root-preserving isometry class of (T , d, ). It is provedin [13] thatTis measurable in (T, dGH). We then dene as the distribution ofTunder N.Remark 2.4. Wehave stated themain results of thepaper under because it ismore natural and because has anintrinsic characterization as shownby Weillin [40]. However, each time we make explicit computations with stable trees, wehavetoworkwithrandomisometryclassesofcompactrealtrees, whichcausestechnicalproblems (mostlymeasurabilityproblems). Toavoidtheseunnecessarycomplications during the intermediate steps of the proofs, we prefer to work withthe specic compact rooted real tree (T , d, ) coded by the -stable height processHunder Nrather than to work directly under . So, we prove the results of thepaper for (T , d, ) under N, which easily implies the same results under . .The Local Time Measures and the Mass Measure on -Stable TreesAsabovementioned, wenowworkwiththe -stabletree (T , d, )codedbyHunder the excursion measure N. A certain number of denitions and ideas can beextendedfromgraph-treestorealtreessuchasthedegreeofavertex.Namely,forany T , wedenotebyn()the(possiblyinnite)numberofconnectedcomponents of the open set T . We say that n() is the degree of . Let bea vertex distinct from the root. If n() = 1, then we say that is a leaf of T ; ifn() = 2, then we say that is a simple point; if n() 3, then we say that isa branching point of T . If n() = , we then speak of as an innite branchingpoint. Wedenote by Lf(T ) the set leaves of T , we denote by Br(T ) the set ofbranching points of Tand we denote by Sk(T ) = T Lf(T ) the skeleton of T .Note that the closure of the skeleton is the whole tree Sk(T ) =T . Let us mentionthat His not constant on every non-empty open subinterval of [0, ], N-a.e. Thiseasily entails the following characterisation of leaves in terms of the height process:For any t (0, ),p(t) Lf(T ) > 0 , infs[t,t]Hsand infs[t,t+]Hs 0, v(a) = N_T (a) ,= / 0_= N (a,= 0) =_(1)a_11. (2.17)As already mentioned, the a-local time measure acan be dened in a purely metricway by (1.3) and there exists a modication of local time measures (a, a 0) suchthat a ais N-a.e. cadlag for the weak topology on the space of nite measureson T . Except in the Brownian case, a ais not continuous and Theorem 4.7 [13]asserts that there is a one-to-one correspondence between the times of discontinuityof a a, the innite branching points of Tand the jumps of the excursion Xofthe underlying -stable L evy process. More precisely, a is a time-discontinuity ofa aiff there exists a (unique) innite branching point aT (a) such that a =

a+aa. Moreover, a point Tis an innite branching point iff there existst [0, ] such thatp(t) = and Xt> 0; if furthermore = a, then a = Xt.Now, observe that if T (a) is an atom of a, the denition (1.3) of aentails thatis an innite branching point and that a is a time-discontinuity of a a. Thus, =a. Recall that the Ray-Knight theoremfor H asserts that a a) is distributedasaCSBP(under itsexcursion measure), whichhas noxed time-discontinuity.This (roughly) explains the following.a > 0, Na.e. ais diffuse. (2.18)We refer to [13] for more details.The Branching Property for HWenow describe the distribution of excursions oftheheight process above levelb(orequivalently ofthecorresponding stabletreeabove levelb). Letusxb (0, ), and denote by (gbj, dbj ), j Ib, the connected components of the open sett 0 : Ht> b. For anyj Ib, denote by Hb, jthe corresponding excursion of Hthat dened by Hb, js= H(gbj+s)dbjb, s 0.Thisdecompositionisinterpretedintermsofthetreeasfollows. Recall thatB(, b) stands for the closed ball with center and radius b. Observe that the con-nected components of the open set T B(, b) are the subtreesTb,oj:= p((gbj, dbj )),j Jb. The closure Tbjof Tb,ojis simply bjTb,oj, where bj= p(gbj) = p(dbj ),that is the points on the b-level set T (b) at which Tb,ojis grafted. Observe that therooted compact R-tree (Tbj, d, bj ) is isometric to the tree coded by Hb, j.110 T. DuquesneWe then deneHbs= Hbs , where for every s 0, we have setbs= inf_t 0 :_t0ds1Hsb > s_.The processHb= ( Hbs , s 0) is the height process below b and the rooted compactR-tree ( B(, b), d, )isisometrictothetreecodedbyHb.LetGbbethesigma-eld generated byHbaugmented by the N-negligible sets. From the approximation(2.9), it follows that Lbis measurable with respect to Gb. We next use the followingnotationN(b)= N([ supH > b) (2.19)that is a probability measure and we dene the following point measure on [0, ) D([0, ), R):Mb = jIb(Lbgbj,Hb, j). (2.20)The branching property at level b then asserts that under N(b), conditionally givenGb, Mbis distributedas aPoissonpoint measurewithintensity1[0,Lb](x)dx N(dH). We refer to Proposition 1.3.1 in [12] or to the proof of Proposition 4.2.3[12]. Let us mention that it is possible to rewrite intrinsically the branching propertyunder : we refer to Theorem 4.2 [13] for more details.Spinal Decomposition at a Random TimeWe recall another decomposition of the height process (and therefore of the corre-sponding tree) that is proved in [12] Chapter 2 and in [13] under a more explicitform (see also [15] for further applications). This decomposition is used in a crucialway in the proof of the upper- and lower-density results for the local times measuresand the massmeasure. Let us introduce an auxiliary probability space (, F, P)that is assumed to be rich enough to carry the various independent randomvariableswe shall need.Let (Ut, t 0) be a subordinator dened on (, F, P) with initial value U0 = 0and with Laplace exponent /() =1, 0. LetN =jI(rj, Hj)(2.21)be a random point measure on [0, ) D([0, ), R) dened on (, F, P) such thata regular version of the law of N conditionally given Uis that of a Poisson pointmeasure with intensity dUrN(dH). Here dUrstands for the (random) Stieltjesmeasure associatedwith thenon-decreasing path r Ur.For any a (0, ),wealso setN a=jI1[0,a](rj)(rj , Hj). (2.22)Packing and Hausdorff Measures of Stable Trees 111We next consider the -height process Hunder its excursion measure N. For anyt 0, we setHt:= (H(ts)+, s 0) (here, ()+ stands for the positive part function)andHt:= (H(t+s), s 0). We also dene the randompoint measure Nt on [0, )D([0, ), R) byNt =N ( Ht) +N ( Ht) :=jJt(rtj,Ht, j), (2.23)whereforanycontinuous functionh : [0, ) [0, )withcompactsupport,thepoint measureN (h) is denedas follows: Set h(t) =inf[0,t]handdenoteby(gi, di), i I(h)theexcursion intervalsofh hawayfrom0thatarethecon-nected component of the open set t 0 : h(t) h(t) > 0. For any i I(h), sethi(s) = ((h h)((gi +s) di), s 0). We then dene N (h) as the point measureon [0, ) D([0, ), R) given byN (h) =iI(h)(h(gi),hi).Lemma 3.4 in [13] asserts the following. For any a and for any nonnegative measur-able function Fon the set of positive measures on [0, ) D([0, ), R) (equippedwith the topology of vague convergence), one hasN__0dLatF_Nt__= E[F(N a)] . (2.24)We shall refer to this identity as to the spinal decomposition of H at a random time.Let us briey interpret this decomposition in terms of the -stable tree TcodedbyH. Choose t (0, )andset =p(t) T . Thenthegeodesic [[, ]]isin-terpreted as the ancestral line of . Let us denote by Toj, j J, the connectedcomponents of the open set T [[, ]] and denote by Tj the closure of Toj. Then,there exists a point j[[, ]] such that Tj=j Toj. Recall notation (rtj, Ht, j),j Jt from (2.23). The specic coding of Tby H entails that for anyj Jthereexists a uniquej/ Jtsuch that d(, j ) = rtj/and such that the rooted compactR-tree (Tj , d, j ) is isometric to the tree coded by Ht, j/Wenowcomputem( B(p(t), r))intermsof Ntasfollows. First, recall from(2.14) the denition of b(s, t) and d(s, t). Note that if Hs = b(s, t) with s ,= t, thenp(s) Sk(T ) by (2.15). Let us x a radius r in (0, Ht). Then, (2.16) entailsm_B(p(t), r)_=_01d(s,t)rds =_010 0. Recall the denition of the a-local time measure a(whose totalmass a) is equal to La) and recall thatN_1 ea)_= N_1 eLa_= u(a, ) =_(1)a +11_11. (2.31)Nextrecall from(2.19) thedenitionofN(a).Weeasilydeducefrom(2.12) and(2.31) thatN(a)_exp(a))_= 1 _(1)a11 +(1)a1_ 11. (2.32)Consequently,a11a) under N(a)(law)= 1) under N(1). (2.33)Lemma 2.5. For any (1, 2], we haveN(1)_1) x_ x0+x1(1)2().Proof. From (2.32), we getN(1)_exp(1))_ (1)( 1)2.The desired result is then a direct consequence of a Tauberian theoremdue to Feller:see [20] Chapter XIII 5 (see also [8] Theorem 1.7.1, p. 38). .Recall the notation N and the denition of Lr(a) from (2.29). For any 0 r/r 2a, we setr/,r(a) = jI1[ar2 , ar/2)(rj) Larjj. (2.34)Observe that0 r/ r 2a , Lr(a) =r/,r(a) +Lr/(a). (2.35)Lemma 2.6. Let (rn, n 0) beasequencesuchthat 0< rn+1 rn 2aandlimnrn = 0. Then, therandom variables (rn+1,rn(a), n 0)areindependent andLr0(a) = n0rn+1,rn(a). (2.36)114 T. DuquesneProof. First, note that (2.36) is a direct consequence of the denitions of r/,r(a) andof Lr(a). Let us prove the independence property. Recall that conditionally givenU,N is a Poisson point process with intensity dUtN(dH). Elementary propertiesof Poisson point processes and the denition of the rn+1,rn(a)s entail that the ran-domvariables (rn+1,rn(a), n 0) are independent conditionally givenU. Moreover,the conditional distribution of rn+1,rn(a) given Uonly involves the increments ofU on [rn+1, rn], which easily implies the desired result since U is a subordinator. .Remark 2.7. The previous lemma and (2.35) imply that for any 0 r/r 2a, onehas Lr(a) r/,r(a) and that r/,r(a) is independent of Lr/(a). Observe also that theprocess r Lr(a) has independent increments.Lemma 2.8. For any 0 r/ r 2a, we haveE_exp(r/,r(a))=_1