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I: splitting at heights
Grégory Miermont
To cite this version:
Grégory Miermont. Self-similar fragmentations derived from the stable tree I: splitting at heights.
Probability Theory and Related Fields, Springer Verlag, 2003, 127, pp.423-454. �10.1007/s00440-003- 0295-x�. �hal-00001550�
ccsd-00001550, version 1 - 10 May 2004
tree I : splitting at heights
Grégory Miermont
DMA , Éole Normale Supérieure,
and LPMA , Université Paris VI.
45, rue d'Ulm,
75230 Paris Cedex 05
∗
Résumé
Thebasiobjetweonsiderisaertainmodelofontinuumrandomtree,alled
thestabletree.Weonstruta fragmentation proess (F−(t), t≥0) outofthis tree
byremovingthevertiesloatedunderheightt.Thankstoaself-similarityproperty of the stable tree, we show that the fragmentation proess is also self-similar. The
semigroupand other features ofthe fragmentation aregiven expliitly. Asymptoti
resultsaregiven,aswellasaoupleofrelatedresultsonontinuous-state branhing
proesses.
Key Words. Self-similar fragmentation, stable tree, stable proesses, ontinuous-state
branhing proess.
A.M.S. Classiation. 60J25, 60G52, 60J80.
∗
ResearhsupportedinpartbyNSF GrantDMS-0071448
1 Introdution
Thereentadvanesinthestudyofoaleseneandfragmentationproessespointedat
thekeyroleplayedbytreestruturesinthistopi,bothatthedisreteandontinuouslevel
[15,3,4℄.Our goalhereis topushfurther the investigation,begunin[3,9℄,of aategory
offragmentationsobtained by utting aertain lass aontinuum randomtree. The tree
that was fragmented in the latter artiles is the Brownian Continuum Random Tree of
Aldous,andthe fragmentationisrelatedtotheso-alledstandardadditiveoalesent.The
family of trees we onsider is a natural but tehnially involved Lévy generalization of
the Brownian tree. It has been introdued in Duquesne and Le Gall [14℄, and impliitly
onsidered in the previous work of Kersting[18℄. Some of these trees, as their Brownian
ompanion,enjoy ertainself-similarproperties. Inthe present work the ruialproperty
is that when removing the verties of the stable tree loated under a xed height (or
distane to the root), the remaining objet is a forest of smaller trees that have the
same law as the originalone up to resaling. This is formalizedin Lemma 3 below. This
way of logging the stable tree indues a fragmentation proess whih by the property
explainedaboveturnsout tobeaself-similarfragmentation,the theoryof suhproesses
being extensively studied by Bertoin [8, 9, 10℄. The goal of this paper is to desribe the
harateristisandgivesomepropertiesofthisfragmentationproess.Wewillhavetouse
stohastiproessesandombinatorialapproahesinthesametime;inpartiular,wewill
enounter σ-nite generalizations of the (α, θ)-partitions of [26℄, whih are distributions on the set of partitions of N = {1,2, . . .}, as well as we will need the onstrution of
the stable tree out of Lévy proesses and its onnetion to ontinuous-state branhing
proesses(CSBP) explainedin [14℄.
In a ompanion paper [23℄ we will onsider another way of obtaining a self-similar
fragmentation by another uttingdevie on the stable tree, using the heuristi fat that
whenuttingatrandomonehub inthe the stabletree,the trunk andbranhes thathave
been separated are saled versions of the initial tree. Surprisingly, although this other
devie looks quite dierent from the rst (no mass is lost when utting a hub, whereas
there is a loss of mass when we throw everything that is loated under the height h), it
turnsoutthatthe onlydierenebetween thesetwofragmentationsisthespeed atwhih
fragments deay.
To state our main results, let us introdue quikly the already mentioned tree stru-
tures and fragmentation proesses, postponingthe detailsto afurther setion.
Let S = {s = (s1, s2, . . .) : s1 ≥ s2 ≥ . . . ≥ 0,P
i≥1si ≤ 1}. A ranked self-similar fragmentation proess (F(t), t ≥0) with index β ∈R is a S-valued Markov proess that
isontinuous inprobability,suh that F(0) = (1,0,0, . . .)and suh that onditionallyon
F(t) = (x1, x2, . . .), F(t+t′)has the law of the dereasing arrangement of the sequenes
xiF(i)(xβit′), where the F(i) are independent with the same law as F. That is, after time t, the dierent fragments evolve independently with a speed that depends on their size.
Ithas beenshown in[9℄ thatsuh fragmentationsare haraterized by a3-tuple(β, c, ν),
where β is the index, c ≥ 0 is an erosion real onstant saying that the fragments may
melt ontinuously at some rate depending on c, and ν is a σ-nite measure on S that
attributesmass 0to(1,0, . . .) andthat integrates s7→(1−s1). Thismeasure governs the
suddendisloationsinthefragmentationproess,andtheintegrabilityassumptionensures
thatthese disloationsdo not our too quikly,althoughthe fragmentation epohs may
form a dense subset of R+ as soon as ν(S) = +∞. When β < 0, a positive fration of
the mass an disappear within a nite time, even though there is noloss of mass due to
erosionnortosudden disloations.This phenomenon willberuialinthe fragmentation
F− below.
The treeswe are onsideringare ontinuumrandomtrees. Intuitively,they are metri
spaeswithaninnitelyramiedtreestruture,whihanbeonsideredasgenealogial
struturesombinedwithtwomeasures:aσ-nitelengthmeasuresupportedbytheskele-
ton of the tree and a nite mass measuresupported by its leaves, whihare everywhere
densein the tree. These trees an be dened inseveral equivalent ways :
asa weaklimitof Galton-Watson trees
through its height proess H, whih is a positive ontinuous proess on [0,1]. To a
point u ∈ [0,1] orresponds a vertex of the tree with height (distane to the root)
equaltoHu,andthemassmeasureonthetreeisrepresented byLebesgue'smeasure on[0,1]
through its expliit marginals, that is, the laws of subtrees spanned by a random
sampleof leaves.
We will have to use the seond (stohasti proess) and third (ombinatorial) points of
view. We know from the works of Duquesne and Le Gall [14℄ and Duquesne [13℄ that
one may denea partiular instane of tree, alled the stable tree with index α (for some α∈(1,2]).When α= 2, the stabletree is equalto the BrownianCRT of Aldous [2℄. We
willreall the rigorous onstrution of the height proess of the stable tree in Set. 2.2,
but let us state our results now. Fix α ∈ (1,2) and let (Hs,0 ≤ s ≤ 1) be the height
proess of the stable tree with index α.
The fragmentation proess, that we allF−, is dened as follows. Foreah t ≥ 0, let I−(t) be the open subset of (0,1)dened by
I−(t) ={s∈(0,1) : Hs > t}.
Withour intuitive interpretation of the height proess, I−(t) is the set of verties of the
tree with height > t. We denote by F−(t) the dereasing sequene of the lengths of the
onneted omponents of I−(t). Hene, F−(t) is the sequene of the masses of the tree
omponentsobtainedby uttingthestable treebelowheightt.Notie thatF− isadiret
generalization of the fragmentation F in [9, Setion 4℄. The boundedness of H implies
that F−(t) = (0,0, . . .) as soonas t≥max0≤s≤1Hs.
Proposition 1 The proess F− is a ranked self-similar fragmentation with index 1/α− 1∈(−1/2,0) and erosionoeient 0.
Notie that, as mentioned before, F− loses some mass, and eventually disappears
ompletelyin nite time even thoughthe erosion is 0. This is due, of ourse, to the fat
that the self-similarityindex is negative.
Our main result is a desription of the disloation measure ν−(ds) of F−. Let us
introdue some notation. For α ∈ (1,2), let (Tx, x ≥ 0) be a stable subordinator with Laplaeexponent λ1/α, thatis,Tx isthe sum ofthe magnitudesofthe atomsofaPoisson
point proess on (0,∞) with intensity cαxdr/r1+1/α, where cα = (αΓ(1−1/α))−1. We
denoteby ∆Tx =Tx−Tx− the jump atlevelx and by ∆T[0,x] the sequene of the jumps
of T before time x, and ranked in dereasing order.Dene the measure να onS by να(ds) =E
T1; ∆T[0,1]
T1
∈ds
(1)
wherethelastexpression meansthatforanypositivemeasurablefuntionG,the quantity να(G) isequal to E[T1G(T1−1∆T[0,1])].
Theorem 1 The disloation measure of F− isν− =Dανα, where
Dα = α(α−1)Γ 1−α1
Γ(2−α) = α2Γ 2− α1 Γ(2−α) .
Someomments aboutthis. First,the disloationmeasure harges onlythe sequenes
sforwhih
P
i≥1si = 1,thatis,nomass anbelostwithinasudden disloation.Seond, wereognize anexpressionloseto[27℄,ofaPoisson-Dirihlettypedistribution.However,
it has to be notied that this orresponds to a forbidden parametrization θ = −1, and
indeed,the measure that we obtain is innitesine E[T1] = ∞. This measure integrates 1−s1 though,justasithasto.Indeed,E[T1−∆1]is niteif∆1 denotes thelargestjump
of T beforetime 1. To see this, notie that ∆1 ≥∆∗1 where ∆∗1 is a size-biased pik from thejumps of T beforetime 1,and itfollowsfrom Lemma1inSet.2.1belowand saling
arguments that T −∆∗1 has niteexpetation.
Therestof thepaperisorganizedasfollows. InSet.2werst reallsomefatsabout
Lévy proesses, exursions, and onditioned subordinators. Then we give the rigorous
desription of the stable tree, and state some properties of the height proess that we
will need. Last we reall some fats about self-similar fragmentations. We then obtain
the harateristis of F− in Set. 3 and derive its semigroup. We insist on the fat that
knowing expliitly the semigroup of a fragmentation proess is in general a very om-
pliatedproblem, see [24℄ for somehow surprising negative results in this vein. However,
most of the fragmentation proesses that have been extensively studied in reent years
[3, 7, 22, 9℄ do have known, and sometimes strange-looking semigroups involving ondi-
tioned Poisson louds. And as a matter of fat, the fragmentation F+ onsidered in the
ompanion paper [23℄ has also anexpliit semigroup. We end the study of F− by giving
asymptotiresults for smalltimes in Set. 4. These results need some properties of on-
ditionedontinuous-time branhing proesses,whihareinthe veinofJeulin's resultsfor
the resaled Brownian exursion and its loal times. We prove these properties in Set.
5, where we give the rigorous denition of some proesses that are used heuristially in
Set.3 to onjeture the form of the disloationmeasure.
2 Preliminaries
2.1 Stable proesses, exursions, onditioned inverse subordina-
tor
Throughout the paper, we let(Xs, s ≥ 0) be the anonial proess in the Skorokhod
spae D([0,∞)) of àdlàg paths on [0,∞). Reall that a Lévy proess is a real-valued àdlàg proess with independent and stationary inrements. We x α ∈ (1,2). Let P be
thelawthatmakesX astableLévyproesswithnonegativejumpsandLaplaeexponent E[exp(−λXs)] = exp(λα)fors, λ≥0,whereE istheexpetationassoiatedwithP.Suh
a proess has innite variationand satises E[X1] = 0. When there is noambiguity, we
may sometimes speak of X as being itself the Lévy proess with law P. Writing this in
the formof the Lévy-Khinthine formula, wehave :
E[exp(−λXs)] = exp
s Z ∞
0
Cαdx
x1+α (e−λx−1 +λx)
, s, λ≥0, (2)
where Cα = α(α − 1)/Γ(2 − α). In partiular, the Lévy measure of X under P is Cαx−1−αdx1{x>0}. An importantproperty of X is then the salingproperty : under P,
1
λ1/αXλs, s≥0
= (Xd s, s≥0) for all λ >0.
It isknown [30℄ that under P, Xs has a density (ps(x), x∈R) for every s >0, suh that ps(x)is jointlyontinuous in xand s.
Exursions Let X bethe inmum proess of X,dened fors ≥0by Xs= inf{Xu,0≤u≤s}.
ByIt'sexursiontheoryforMarkovproesses,the exursionsawayfrom0ofthe proess X−X underP aredistributedaordingtoaPoissonpointproessthatanbedesribed by the It exursion measure, whih we all N. We now either onsider the proess X
underthelawP that makesitaLévy proessstartingat0,orunderthe σ-nitemeasure N under whih the sample paths are exursions with nite lifetime ζ (sine E[X1] = 0).
LetN(v)bearegularversionoftheprobabilitylawN(·|ζ =v),whihisweaklyontinuous
inv. That is,for any positiveontinuous funtional G, N(G) =
Z
(0,∞)
N(ζ ∈dv)N(v)(G)
andlimN(w)(G) = N(v)(G)asw→v.Suhaversionan beobtainedbysaling:forany
xed η >0,the proess
(v/ζ)1/αXζs/v,0≤s ≤v
under N(·|ζ > η) = N(·, ζ > η) N(ζ > η)
isN(v). See [12℄ for this and other interesting ways to obtain proesses with lawN(v) by
pathtransformations.Inpartiular,onehasthesalingpropertyatthelevelofonditioned
exursions : under N(v), v−1/αXvs,0≤s≤1
has law N(1).
First-passage subordinator Let T be the right-ontinuous inverse of the inreasing proess −X,that is,
Tx = inf{s≥0 :Xs<−x}.
Then itis known that underP, T isa subordinator, that is, aninreasing Lévy proess.
Aording to [6, Theorem VII.1.1℄, its Laplae exponent φ is the inverse funtion of the
restrition of the Laplae exponent of X to R+. Thus φ(λ) = λ1/α, and T is a stable
subordinator with index 1/α,as dened above. The Lévy-Khinthine formulagives,
E[exp(−λTx)] = exp(−xλ1/α) = exp
x Z ∞
0
cαdy
y1+1/α(1−e−λy)
for λ, x≥0.
where cα has been dened in the introdution. Reall our assumption that X has a
marginaldensity attimes under P,alledps(·). Thenunder P,the inverse subordinator
T has also biontinuous densities, given e.g. by [6,Corollary VII.1.3℄:
qx(s) = P(Tx ∈ds)
ds = x
sps(x). (3)
This equationan bederived fromthe ballottheorem of Takás [31℄.
Let usnow disuss the onditioned forms of distributions of the sequene ∆T[0,x]. An
easywaytoobtainnie regularversionsfortheseonditionallawsisdeveloped in[25,27℄,
and uses the notion of size-biased fragment. Preisely, the range of any subordinator,
with drift 0 say (whih we will assume in the sequel), between times 0 and x, indues
a partition of [0, Tx] into subintervals with sum Tx. Consider a sequene (Ui, i ≥ 1) of
independent uniform (0,1) variables, independent of T, and let ∆∗1(x),∆∗2(x), . . . be the
sequene ofthe lengthsof theseintervalsinthe orderinwhihthey are disovered by the
Ui's.That is, ∆∗1(x) is the length of the interval in whih TxU1 falls, ∆∗2(x) is the length
ofthe rst intervaldierent fromthe one ontaining TxU1 inwhihTxUi falls,and soon.
ThenPalmmeasureresults forPoissonloudsgivethefollowingresult(speializedtothe
ase of stablesubordinators).
Lemma 1 The joint law under P of (∆∗1(x), Tx) is
P(∆∗1(x)∈dy, Tx ∈ds) = cαxqx(s−y)
sy1/α dyds, (4)
and more generally for j ≥1, P ∆∗j(x)∈dy
Tx =s0,∆∗1(x) =s1, . . . ,∆∗j−1(x) =sj−1
= cαxqx(s−y) sy1/αqx(s) dy,
where s=s0−s1−. . .−sj−1.
This gives a nie regular onditional version for (∆∗i(x), i ≥ 1) given Tx, and thus
induesaonditionalversionfor∆T[0,x]givenTx,byranking,where∆T[0,x]isthesequene
of jumps of T beforex, ranked in dereasing order of magnitude.
2.2 The stable tree
Wenowintroduethemodels oftreeswe willonsider. Thissetionismainlyinspired
by [14, 13℄. With the notations of setion 2.1, for u ≥ 0, let R(u) be the time-reversed proess of X attime u :
R(u)s =Xu−X(u−s)− , 0≤s≤u.
Itis standard that this proess has the same lawas X killed attime uunder P.Letalso R(u)s = sup
0≤v≤s
R(u)v , 0≤s≤u
be its supremum proess. We let Hu be the loal time at 0 of the proess R(u) reeted
underits supremum R(u) up totime u. The normalizationan behosen sothat
Hu = lim
ε↓0
1 ε
Z u 0
1{R(u)s −R(u)s ≤ε}ds
It is known by [14, Theorem 1.4.3℄ that H admits a ontinuous version, with whih we
shall work in the sequel. It has to be notied that H is not a Markov proess (the only
exeptioninthe theory of Lévy trees isthe Brownian treeobtained when P isthe law of
Brownian motion with drift, whih has been exludedin our disussion). As a matter of
fat,it an beheked that H admits loalminima that are attained aninnite number
of times as soon as X has jumps, a property that sounds strange by ontrast with most
ofthe usually studied stohasti proesses.Tosee this, onsider ajumptime t of X, and
lett1, t2 > t so thatinft≤u≤tiXu =Xti and Xt− < Xti < Xt,i∈ {1,2}.Then itis easyto
see that Ht=Ht1 =Ht2 and that one may infat nd aninnite number of distint ti's
satisfyingthe propertiesof t1, t2. On the other hand,it is not diultto see that Ht isa
loalminimumof H.
Itisshownin[14℄thatthe denitionofH stillmakessense undertheσ-nitemeasure N rather than the probability law P. The proess H is then dened only on [0, ζ], and
weallit the exursion of the height proess. One an dene withoutdiulty, using the
salingproperty, the height proess underthe laws N(v) :this is simplythe lawof v
ζ
1−1/α
Hζt/v,0≤t≤v
!
underN(·, ζ > η)
Callitthelawoftheexursionoftheheightproesswithdurationv.Thefollowingsaling
property is the key for the self-similarityof F− : for every x >0,
(v1/α−1Hsv,0≤s≤1)under N(v) = (Hd s,0≤s≤1)under N(1). (5)
This property is inherited from the saling property of X, and it is easily obtained e.g.
by the above denition of H asan approximation.
An important tool for studying the height proess is its loal time proess, or width
proess, whih wewilldenote by (Lts, t≥0, s≥0). It an be obtained a.s. for every xed s, tby
Lts= lim
ε↓0
1 ε
Z s 0
1{t<Hu≤t+ε}du.
Ltsisthenthe densityofthe oupationmeasure ofH atlevelt andtimes.Fort= 0,one
hasthat(L0s, s≥0)isthe inverse ofthesubordinatorT,whihisareminisentofthefat
thattheexursions oftheheightproessareinone-to-oneorrespondenewithexursions
of X with the same lengths.Aording to the Ray-Knight theorem [14, Theorem 1.4.1℄,
for every x > 0, the proess (LtTx, t ≥ 0) is a ontinuous-time branhing proess with
