TWO COALESCENTS DERIVED FROM THE RANGES OF STABLE SUBORDINATORS

Let M (cid:11) be the closure of the range of a stable subordinator of index (cid:11) 2 ]0 ; 1[. There are two natural constructions of the M (cid:11) ’s simultaneously for all (cid:11) 2 ]0 ; 1[, so that M (cid:11) (cid:18) M (cid:12) for 0 < (cid:11) < (cid:12) < 1: one based on the intersection of independent regenerative sets and one based on Bochner’s subordination. We compare the corresponding two coalescent processes de(cid:12)ned by the lengths of complementary intervals of [0 ; 1] n M 1 − (cid:26) for 0 < (cid:26) < 1. In particular, we identify the coalescent based on the subordination scheme with the coalescent recently introduced by Bolthausen and Sznitman [6].


Introduction
A ranked coalescent process describes the evolution of a system of masses which aggregate randomly as time passes. More formally, write S ↓ for the state space of decreasing positive sequences v := (v(n), n ∈ N) with n v(n) = 1, so each term v(n) can be viewed as the mass of some fragment of a unit mass. In a ranked coalescent process (V ρ , ρ ∈ I) parameterized by some interval I ⊆ R, for each ρ < ρ V ρ (n) = k∈Π(ρ,ρ ,n) where Π(ρ, ρ , n) indicates which of the masses present at time ρ have coalesced by time ρ to form the nth largest mass present at time ρ . The Π(ρ, ρ , n) for n ∈ N are the blocks of some random partition of N, and these partitions are subject to a consistency requirement as ρ and ρ vary. We call the time reversal of a ranked coalescent process a ranked fragmentation process. See [8] for a general framework for the analysis of such processes, and [1,3,6,8,17] for some specific examples.
We recall in Section 2.3 the precise definition of the semigroup of this Bolthausen-Sznitman coalescent (V BS e −t ; t > 0). The equality in distribution V BS α d = V (M α ), for each fixed α ∈]0, 1[, suggests the possibility of constructing a family of nested stable regenerative sets (M α , 0 < α < 1) such that (V (M α ), 0 < α < 1) is a realization of (V BS α , 0 < α < 1). To this end we consider Bochner's subordination [5]. Recall that if σ α and σ α are independent stable subordinators, then the subordinate process σ α • σ α has the same law as σ αα . By application of Kolmogorov's extension theorem we can justify the following construction: where (σ * α , 0 < α < 1) is a family of stable subordinators such that for every 0 < α n < . . . < α 1 < 1, the joint distribution of σ * α 1 , . . . , σ * αn is the same as that of σ α 1 , . . . , σ αn defined as follows. Consider a family of n independent stable subordinators, τ β 1 , . . . , τ βn with indices β 1 , . . . , β n ∈ ]0, 1] such that α i = β 1 . . . β i for i = 1, . . . , n, and set Our main result, which we prove in Section 2.3 using excursion theory and special properties of Poisson-Dirichlet laws, is the following identity: Theorem 3 There is the equality of finite-dimensional distributions of ranked fragmentation processes This result is closely related to another recent construction of the Bolthausen-Sznitman coalescent in [4], based on the genealogy of Neveu's continuous-state branching process. We explain the connection between the two constructions in Section 2.4. See also [17] for quite a different construction, which yields generalizations of the Bolthausen-Sznitman coalescent.
There is another natural construction of nested stable regenerative sets M α which is implicit in the literature. Recall from [15] that M α can be constructed for each fixed α ∈]0, 1[ as the zero set of a Bessel process of dimension 2 − 2α started at 0. The additivity property of squares of Bessel processes then justifies the following construction [24,19]: Construct on a common probability space the squares of Bessel processes with dimension δ, X δ,• , in such a way that the path-valued process (X δ, process with independent increments and càdlàg paths, and set M ∩ α = {t ≥ 0 : X 2−2α,t = 0} . The Poisson structure of jumps of (X δ,• , δ ≥ 0), described in [19] (see also [16,Proposition 14]), shows that the process created by Construction 4 can be represented by a simpler random covering scheme [10]: That M ∩ α is the closure of the range of a stable subordinator of index α can be read from [10, p. 180]. Implicit in Constructions 4 and 5 is the well known fact that if M α and M α are independent, then the intersection M α ∩ M α has the same law as Plainly, the set-valued processes (M ∩ α , 0 < α < 1) and (M * α , 0 < α < 1) have the same onedimensional distributions, meaning that M ∩ α and M * α have the same law as M α in (1) for each fixed α. But these families do not have the same finite-dimensional distributions. Indeed, we will show in Section 3 that the ranked fragmentation processes (V (M ∩ α ), 0 < α < 1) and (V (M * α ), 0 < α < 1) do not have the same laws. So Theorem 3 is false with M ∩ α instead of M * α . Nonetheless, there are some striking resemblances between the two families, and it is interesting to investigate the similarities and differences.
One similarity involves the so-called age A α of M α at time 1. That is, A α is the length of the component interval of [0, 1]\M α that contains 1 (or equivalently, 1 − A α is the largest point in To be more precise [20], A α is a size-biased pick from the components of V (M α ), meaning that where V (M α )(n) is the length of the nth longest component interval of [0, 1] ∩ M α . Here and in the sequel, we develop notation for M α , and modify by a superscript ∩ or * to replace M α by M ∩ α or M * α . It is known from [7] that there is a gamma subordinator (Γ ρ , ρ ≥ 0) such that for each fixed ρ ∈]0, 1[ As an extension of this, we obtain the following proposition: There is the equality of finitedimensional distributions In other words, each of these process has a right-continuous version which is the cumulative distribution function of a Dirichlet random measure on ]0, 1[ governed by Lebesgue measure.
This Proposition is a consequence of [2, Proposition 8], as it is easily checked that for every 0 < α 1 < . . . < α n < 1, the embeddings are compatible with the regenerative property in the sense [2]. We also give another proof of Proposition 6 in Section 4. Finally, we mention some open problems about the coalescent V (M ∩ 1−ρ ), 0 < ρ < 1 in section 5.

Preliminaries
exists with probability 1. Moreover, if V α = V (M α ) then L 1,α coincides with the local time of the random set [0, 1] ∩ M α . Alternatively, L 1,α can be constructed as the first passage time above level 1 for the stable subordinator σ α , provided that the latter has been suitably normalized (which induces no loss of generality by the scaling property). See equations (2.c-h) in [20] for details. The rest of this section is organized as follows. In the next subsection, we develop material on the 'excursions' of the random set M * α away from M * γ for arbitrary 0 < γ < α < 1 . This is used in the third subsection to prove Theorem 3. In the final subsection, we relate Theorem 3 to a different construction of the Bolthausen-Sznitman coalescent process based on Bochner's subordination that has been recently obtained in [4], using a multidimensional extension of an identity due to Pitman and Yor [20].

Normalized excursion and meander
In this subsection, we work in the canonical space Ω of closed subsets ω ⊆ [0, ∞[ with 0 ∈ ω. We give Ω the topology of Matheron [14] and the corresponding Borel field B(Ω). We write n for the operator of normalization of compact sets. That is, if max ω < ∞, then and k r for the killing operator at r ≥ 0 For ω ∈ Ω let V (ω) be the sequence of ranked lengths of component intervals of [0, 1]\ω. Note that V (ω) = V (k 1 (ω)), and that V (ω) ∈ S ↓ provided the Lebesgue measure of ω equals 0, as it is for almost all ω with respect to the distribution of M α for each 0 < α < 1. Here as before, M α := {σ α (t), t ≥ 0} cl for σ α a stable subordinator with index α, and we now regard M α as a random variable with values in Ω. Our analysis relies on the following identity in distribution [20, Theorem 1.1]: where the common distribution of both sides is the Poisson-Dirichlet law P D(α, 0).
We focus now on the joint law of (M * γ , M * α ) for arbitrary 0 < γ < α < 1. Let β := γ/α, so 0 < β < 1. Suppose that σ α and τ β are independent stable subordinators of indices α, β ∈]0, 1[, and that M * α and M * γ are the closed ranges of σ α and σ α • τ β respectively. Consider the random countable subset of [0, ∞[×Ω defined by the points As both σ α and τ β have independent and stationary increments, and τ β is independent of σ α , it follows by the argument of Itô [11] that the points (6) are points of a Poisson point process [0, ∞[×Ω with intensity measure dt ν α,γ (dω), for some measure ν α,γ on Ω, call it the law of excursions of M * α away from M * γ . By construction of the point process there is the formula Here ct −β−1 dt is the Lévy measure of τ β , with c > 0 a constant whose value is not relevant. Note that c may change from line to line in the sequel. We define the meander M * me α,γ to be a random set distributed according to an excursion conditioned to have length at least 1 and restricted to the unit interval. That is, We also define a normalized excursion M * ex α,γ to be a random set distributed as the excursion conditioned to have length at least 1 and then normalized in order to have unit length. That is Consider now V (M * ex α,γ ) and V (M * me α,γ ), the ranked lengths of intervals that result from the partition of [0, 1] induced by M * me α,γ and M * ex α,γ respectively.
Proof: Recall γ = αβ. First, observe that the scaling property combined with Fubini's theorem yields that the distribution of the normalized excursion is given for every B ∈ B(Ω) by In particular, the distribution of V (M * ex α,γ ) is absolutely continuous with respect to that of V (n • k σα(1) (M * α )) with density proportional to (σ α (1)) γ . As (σ α (1)) −α coincides with the local time of the normalized set n•k σα(1) (M * α ), we deduce from (5) . On the other hand, {σ α (t) > 1} = {L 1,α < t} up to a null set, and it then follows that the law of the meander is given for every B ∈ B(Ω) by In particular, the law of Next, following [21,17], we can associate to any parameters α ∈]0, 1[ and θ > −α an (α, θ)fragmentation kernel on S ↓ as follows. We introduce first a sequence

Lemma 9
For each choice of α and γ with 0 Proof: Let I 0 be the right-most interval component of [0, 1]\M * γ , and write (I k , k ∈ N), for the sequence of the remaining open intervals of this decomposition, ranked according to the decreasing order of lengths. For every integer k ≥ 0, let k ∈]0, 1[ be the left-end point of the interval I k and denote by Z k the Ω-valued random variable such that Note that the sequence of the lengths |I 0 |, |I 1 |, . . . is measurable with respect to k 1 (M * γ ) and that its increasing rearrangement is V (M * γ ). Standard arguments of excursion theory using the scaling property imply that Z 0 , Z 1 , . . . is a sequence of independent variables which is also independent of k 1 (M * γ ). Moreover, Z 0 has the law of the meander M * me α,γ , and for k ≥ 1, each Z k has the law of the normalized excursion M * ex α,γ . By construction, V (M * α ) is the decreasing rearrangement of the elements of the sequences |I 0 |V (Z 0 ), |I 1 |V (Z 1 ), . . . As we know from Lemma We recall next a basic duality relation between Poisson-Dirichlet fragmentation and coagulation kernels. Associated with each probability distribution Q on S ↓ there is a Markov kernel on S ↓ , the Q-coagulation kernel, denoted Q-COAG, which is defined as follows [6,17]. Let V be a random element of S ↓ with distribution Q, and given V let

Proof of Theorem 3
As shown in [6], the semigroup of the Bolthausen-Sznitman coalescent (V BS e −t ; t > 0) introduced in Theorem 1 is provided by the family of coagulation kernels (e −t , 0)-COAG, t > 0 . It follows using (5) and Lemma 11 that there is equality of twodimensional distributions in (2). To complete the proof, it remains only to establish the Markov property of ranked fragmentation process (V * α , 0 < α < 1), where we abbreviate V * α := V (M * α ). Denote by G α the sigma field generated by the family of random closed sets Because of the way that σ * α can be recovered from M * α , the sigma field G α coincides with the sigma field generated by the family of processes Combined with Bochner's subordination, this shows that for every 0 < α < α < 1, the conditional distribution of k 1 (M * α ) given G α only depends on k 1 (M * α ). In other words, the set-valued process (k 1 (M * α ), 0 < α < 1) is Markovian with respect to (G α ).

Another construction of the Bolthausen-Sznitman coalescent
Recently, another simple connection linking the nested family (M * α , 0 < α < 1) to the Bolthausen-Sznitman coalescent has been obtained in [4]. Specifically, fix α 1 ∈]0, 1[, and for α ∈ [α 1 , 1[ define T α by the identity Then consider the normalized sets It is immediately checked that (N * α , α 1 ≤ α < 1) is a nested family of closed subsets of the unit interval with zero Lebesgue measure. Theorem 8 in [4] states the following identity of finite dimensional distributions of ranked fragmentation processes: It follows from (9) and Theorem 3 that Observe that identity of one-dimensional distributions in (10) just rephrases (5). The purpose of this subsection is to point out that our approach yields a refinement of (10), and hence enables us to recover (9) via Theorem 3.
We define W * (n 1 , . . . , n k ) analogously by replacing M * α i by N * α i for i = 1, . . . , k. We can now state the following result which obviously encompasses (10):
On the one hand, it follows readily from Itô's excursion theory and the scaling property that the S ↓ -valued random variables W (1, ·), W (2, ·), . . . are i.i.d. and are jointly independent of V (N * α 1 ) = (W * (n), n ∈ N). Moreover, using the notation of Section 3.2, each has the same law as V (M * ex α 1 ,α 2 ). On the other hand, recall that A * α 1 denotes the age for M * α 1 , and work conditionally on A * α 1 = V * (p) for an arbitrary p ∈ N. In other words, we work conditionally on the event that the right-most interval component of [0, 1]\M * α 1 has rank p when the interval components are ordered by decreasing lengths. It follows again from Itô's excursion theory and the scaling property that the S ↓ -valued random variables V (1, ·), V (2, ·), . . . are independent and are jointly independent of V * α 1 = (V * (n), n ∈ N). Moreover, in the notation of Section 3.2, the law of V (p, ·) is that of V (M * me α 1 ,α 2 ), and for r = p the law of V (r, ·) is that of V (M * ex α 1 ,α 2 ). But we know from Lemma 8 that V (M * ex α 1 ,α 2 ) and V (M * me α 1 ,α 2 ) are identical in distribution, and the conclusion follows.

The ranked coalescent
In this section, we show that the ranked coalescent based on the intersection scheme is different from the one based on the subordination scheme. First recall Constructions 2 and 5, the definition (4) of the local time, and that we are using superscripts in the obvious notation. For every 0 < α < β < 1 the joint laws of (V * α , V * β ) and (V ∩ α , V ∩ β ) are distinct because where the generic sequence v belongs to some subset of S ↓ with positive P D(β, 0)-measure. The proof of (11) relies on the following lemma.
Next, recall Construction 5. For every ε > 0, write N (ε) 1+α−β for the set left uncovered by intervals ]x, x + z[ corresponding to points (a, x, z) of the Poisson process with 1 − β < a ≤ 1 − α and z ≥ ε. It is easily checked that We then introduce and point out the straightforward estimate Combining this with the fact that All that we need now is to check that there is some constant number c > 0 such that To see this, observe that for η ∈]0, ε[, one has 1+α−β denotes the set left uncovered by intervals ]x, x + z[ corresponding to points (a, x, z) of the Poisson process with 1 − β < a ≤ 1 − α and η ≤ z < ε. Using the elementary identity P(s ∈ N (η,ε) we deduce that for every fixed 0 < s < t, the process L (ε) Standard arguments involving the martingale convergence theorem and (14) show that L (ε) α (t) converges in L 1 (P) as ε → 0+ to, say, L α (t), and that the increasing process (L α (t), t ≥ 0) is continuous. It is plain from the construction that L α (·) is an additive functional that only increases on M ∩ α , and thus is must be proportional to the local time process L ∩ ·,α on M ∩ α . Thus (15) is established and the proof of (13) is complete.

Now if (11) failed, then the conditional expectation
would be given by some functional f α/β (L ∩ 1,β ) of the local time of V ∩ β (by Lemma 13, because L ∩ 1,β is measurable with respect to V ∩ β ). To see that this is absurd, observe that for every η > 0, we would have for a.e. > 0 According to [20,Prop. 6.3], given V ∩ β , A ∩ β is a size-biased choice from V ∩ β . It follows readily that the random variable greater than 1 − η, with positive probability (because the conditional probability given L ∩ 1,β = that the first element of V ∩ β being larger than 1 − η is strictly positive). We conclude that we would have and since η > 0 can be chosen arbitrarily small and α − β < 0, that f α/β ≡ ∞. Hence cannot be a functional of L ∩ 1,β , and by (12), this establishes (11).

The partition-valued process
In this section, we shall investigate the partition-valued process for the intersection scheme.
For n ∈ N, let P n be the finite set of all partitions of the set [n] := {1, . . . , n}. Following [12,13,8,6] a ranked coalescent (V 1−ρ ; 0 < ρ < 1), defined by V α := V (M α ) for a family of nested closed sets with zero Lebesgue measure, (M α , 0 < α < 1), is conveniently encoded as a family of P n -valued processes with step-function paths (Π n,α ; 0 < α < 1), n = 1, 2, . . . as follows. Let U 1 , U 2 , . . . be i.i.d. uniform [0, 1] variables, independent of (M α , 0 < α < 1). Let Π n,α be the partition of [n] generated by the random equivalence relation ∼ α where i ∼ α j if and only if U i and U j fall in the same component interval of the complement of M α . It is plain that Π n,α a refinement of Π n,α for α > α , the distribution of each Π n,α is exchangeable, that is invariant under the natural action of permutations of [n] on P n , and the Π n,α are consistent as n varies, meaning that Π m,α is the restriction to [m] of Π n,α for n < m. For each α the sequence Π n,α , n = 1, 2, . . . then induces a random partition of the set of all positive integers, each of whose classes has an almost sure limiting frequency; the ranked values of these frequencies define the random vector V α ∈ S ↓ .
Each of the partition-valued processes (Π BS n,e −t ; t > 0) corresponding to the Bolthausen-Sznitman coalescent is a Markov chain with stationary transition probabilities such that whenever that the state of the process is a partition with b blocks, each k-tuple of these blocks is merging to form a single block at rate As observed in [17], collision rates λ b,k specified by the above integral, with Λ(dx) instead of dx, serve to define a consistent family of coalescent Markov chains and hence an S ↓ -valued coalescent process for an arbitrary positive and finite measure Λ on [0, 1]. The case Λ = δ 0 , a unit mass at 0, is Kingman's coalescent in which every pair of blocks coalesces at rate 1.

The age
In this section, we shall give a direct proof of Proposition 6. Recall that the age A α is defined by . Consider first the case of M ∩ α , and suppose that M ∩ α is constructed as in Construction 4 as the zero set of a Bessel process X 2−2α,• . If we pick parameters 0 < α 1 < . . . < α n < 1, set 2 − 2α i = β i + . . . + β n for i = 1, . . . , n, and introduce independent squares of Bessel processes X (1) , . . . , X (n) started at 0 with respective dimensions β 1 , . . . , β n , then X (i) + · · · + X (n) is the square of a Bessel process of dimension (2 − 2α i ), and its zero set is a version of M ∩ α i . Consider an exponential time T which is independent of the family of Bessel processes, and write for the last passage time at the origin of X (i) , . . . , X (n) before T .
On the one hand, the decomposition at a last passage time tells us that for each i = 1, . . . , n, the processes (X are independent. It follows that g (1) , g (2) − g (1) , . . . , g (n) − g (n−1) , T − g (n) are independent random variables.
On the other hand, we know from [7] that there is some gamma process (Γ t , 0 ≤ t ≤ 1) such that for each i, g (i) has the same distribution as Γ α i . We deduce from above that in fact the (n + 1)-tuples g (1) , . . . , g (n) , T and Γ α 1 , . . . , Γ α n+1 have the same law, where α n+1 = 1. The scaling property now implies that which is equivalent to our statement for A ∩ .
A similar argument applies for A * . More precisely, if we write for the age process related to M * α i , then it is easy to check that for each i = 1, . . . , n, the (n − i + 1)-tuple A * α i (t), . . . , A * αn (t) , t ≥ 0 is a Markov process, and the set of its passage times at the origin coincides with M * α i . This enables us to follow the same argument as above.
We do not believe the same identity should hold for V ∩ α , 0 < α < 1 , but we do not have a rigorous argument. The problem of specifying the distribution of the process V ∩ α , 0 < α < 1 is open.
More generally, it is a natural problem to describe more explicitly such features of the coalescent process (V ∩ 1−ρ ; 0 < ρ < 1) as the laws of associated partition-valued processes, as considered in [6,17] for the Bolthausen-Sznitman coalescent. The study of such problems is complicated by the fact in contrast to the Bolthausen-Sznitman coalescent, neither (V ∩ 1−ρ ; 0 < ρ < 1) nor its associated partition-valued processes appear to have the Markov property. While it is easily seen from the Poisson construction that the set-valued process (M ∩ α ; 0 < α < 1) is Markov, this Markov property does not propagate to (V ∩ α ; 0 < α < 1) by Dynkin's criterion for a function of a Markov process to be Markov, because the way that the restriction of M ∩ α to [0, 1] evolves depends on the ordering of the intervals, not just V ∩ α . It might be that (V ∩ α ) is Markov by the criterion of Rogers-Pitman [23]. According to [20,Prop. 6.3] there is a conditional law for [0, 1] ∩ M ∩ α given V ∩ α which is the same for all α: to reconstruct [0, 1] ∩ M ∩ α from V ∩ α , place an interval whose length is a size-biased choice from V ∩ α at the right end, then precede it by intervals with lengths from the rest of V ∩ α put in exchangeable random order. But to apply the result of [23] there is an intertwining identity of kernels to be verified, and it does not seem easy to decide if this identity holds.