The fragmentation process of an infinite recursive tree and Ornstein-Uhlenbeck type processes

We consider a natural destruction process of an infinite recursive tree by removing each edge after an independent exponential time. The destruction up to time t is encoded by a partition $\Pi$(t) of N into blocks of connected vertices. Despite the lack of exchangeability, just like for an exchangeable fragmentation process, the process $\Pi$ is Markovian with transitions determined by a splitting rates measure r. However, somewhat surprisingly, r fails to fulfill the usual integrability condition for the dislocation measure of exchangeable fragmentations. We further observe that a time-dependent normalization enables us to define the weights of the blocks of $\Pi$(t). We study the process of these weights and point at connections with Ornstein-Uhlenbeck type processes.


Introduction
The purpose of this work is to investigate various aspects of a simple and natural fragmentation process on an infinite tree, which turns out to exhibit nonetheless some rather unexpected features.
Specifically, we first construct a tree T with set of vertices N = {1, . ..} by incorporating vertices one after the other and uniformly at random.That is, 1 is the root, and for each vertex i ≥ 2, we pick its parent u i according to the uniform distribution in {1, . . ., i − 1}, independently of the other vertices.We call T an infinite (random) recursive tree.Recursive trees are especially useful in computer science where they arise as data structures; see e.g. the survey by Mahmoud and Smythe [14] for background.
We next destroy T progressively by removing each edge e i connecting i to its parent u i at time ǫ i , where the sequence (ǫ i : i ≥ 2) consists of i.i.d.standard exponential variables, which are further independent of T. Panholzer [16] investigated costs related to this destruction process, whereas in a different direction, Goldschmidt and Martin [10] used it to provide a remarkable construction of the Bolthausen-Sznitman coalescent.We also refer to Kuba and Panholzer [13] for the study of a related algorithm for isolation of nodes, and to our survey [3] for further applications and many more references.
Roughly speaking, we are interested here in the fragmentation process that results from the destruction.We represent the destruction of T up to time t by a partition Π(t) of N into blocks of connected vertices.In other words, if we view the fragmentation of T up to time t as a Bernoulli bond-percolation with parameter e −t , then the blocks of Π(t) are the percolation clusters.Clearly Π(t) gets finer as t increases, and it is easily seen from the fundamental splitting property of random recursive trees that the process Π = (Π(t) : t ≥ 0) is Markovian.In this direction, we also recall that Aldous and Pitman [1] have considered a similar logging of the Continuum Random Tree that yields a notable fragmentation process, dual to the standard additive coalescent.We further point at the very recent work [11] in which the effects of repeated random removal of nodes (instead of edges) in a finite random recursive tree are analyzed.
It turns out that Π shares many features similar to homogeneous fragmentation processes as defined in [4,5].In particular, the transition kernels of Π are very similar to those of a homogeneous fragmentation; they are entirely determined by the so-called splitting rates r, which define an infinite measure on the space of partitions of N.However, there are also major differences: exchangeability, which is a key requirement for homogeneous fragmentation processes, fails for Π, and perhaps more notably, the splitting rates measure r does not fulfill the fundamental integral condition (6) which the splitting rates of homogeneous fragmentation processes have to satisfy.
It is known from the work of Kingman [12] that exchangeability plays a fundamental role in the study of random partitions, and more precisely, it lies at the heart of the connection between exchangeable random partitions (which are discrete random variables), and random mass-partitions (which are continuous random variables).In particular, the distribution of an exchangeable random partition is determined by the law of the asymptotic frequencies of its blocks B, |B| = lim Even though Π(t) is not exchangeable for t > 0, it is elementary to see that every block of Π(t), say B(t), has an asymptotic frequency.However this asymptotic frequency is degenerate, |B(t)| = 0 (note that if Π were exchangeable, this would imply that all the blocks of Π(t) would be singletons).We shall obtain a finer result and show that the limit lim exists in (0, ∞) almost surely.We shall refer to the latter as the weight of the block B(t) (we stress that this definition depends on the time t at which the block is taken), and another natural question about the destruction of T is thus to describe the process X of the weights of the blocks of the partition-valued process Π.
Because Π resembles homogeneous fragmentations, but with splitting rate measure r which does not fulfill the integral condition of the former, and because the notion (2) of the weight of a block depends on the time t, one might expect that X should be an example of a so-called compensated fragmentation which was recently introduced in [6].Although this is not exactly the case, we shall see that X fulfills closely related properties.Using well-known connections between random recursive trees, Yule processes, and Pólya urns, cf.[3], we shall derive a number of explicit results about its distribution.In particular, we shall show that upon a logarithmic transform, X can be viewed as a branching Ornstein-Uhlenbeck process.
The rest of this paper is organized as follows.In Section 2, we study the structure of the partition-valued process Π which stems from the destruction of T, stressing the resemblances and the differences with exchangeable fragmentations.In Section 3, we observe that after a suitable renormalization that depends on t, the blocks of the partition Π(t) possess a weight, and we relate the process of these weights to Ornstein-Uhlenbeck type processes.

Destruction of T and fragmentation of partitions
The purpose of this section is to show that, despite the lack of exchangeability, the partition valued process Π induced by the fragmentation of T can be analyzed much in the same way as a homogeneous fragmentation.We shall present the main features and merely sketch proofs, referring to Section 3.1 in [5] for details.
We start by recalling that a partition π of N is a sequence (π i : i ∈ N) of pairwise disjoint blocks, indexed in the increasing order of their smallest elements, and such that ⊔ i∈N π i = N.We write P for the space of partitions of N, which is a compact hypermetric space when endowed with the distance d(π, where π |B denotes the restriction of π to a subset B ⊆ N and [n] = {1, . . ., n} is the set of the n first integers.The space P B of partitions of B is defined similarly. We next introduce some spaces of functions on P. First, for every n ≥ 1, we write D n for the space of functions f : P → R which remain constant on balls with radius 1/n, that is such that f (π) = f (η) whenever the restrictions π |[n] and η |[n] of the partitions π and η to [n] coincide.Plainly, D n ⊂ D n+1 , and we set Observe that D ∞ is a dense subset of the space C(P) of continuous functions on P.
In order to describe a family of transition kernels which appear naturally in this study, we first need some notation.For every block B ⊆ N, write B(j) for the j-th smallest element of B (whenever it makes sense), and then, for every partition π ∈ P, B • π for the partition of B generated by the blocks B(π i ) = {B(j) : j ∈ π i } for i ∈ N. In other words, B • π is simply the partition of B induced by π when one enumerates the elements of B in their natural order.Of course, if the cardinality of B is finite, say equal to k ∈ N, then B • π does only depend on π through π |[k] , so that we may consider B • π also for π ∈ P [k] (or π ∈ P [ℓ] for any ℓ ≥ k).
In the same vein, for partitions η ∈ P B and every integer i ≥ 1, we write η • i π for the partition of B that results from fragmenting the i-th block of η by π, that is replacing the block η i in η by η i • π.Again, if k = #B < ∞, we may also take π ∈ P [ℓ] for ℓ ≥ k.
Finally, for every k ≥ 2, we consider a random partition of N that arises from the following Pólya urn.At the initial time, the urn contains k − 1 black balls labeled 1, . . ., k − 1 and a single red ball labeled k.Balls with labels k + 1, k + 2, . . .are colored black or red at random and then incorporated to the urn one after the other.More precisely, for n ≥ k, the color given to the n + 1-th ball is that of a ball picked uniformly at random when the urn contains n balls.This yields a random binary partition of N; we write p k for its law.We set which is thus an infinite measure on the set of binary partitions of N.
Recall that each edge of T is deleted at an exponentially distributed random time, independently of the other edges.This induces, for every t ≥ 0, a random partition Π(t) of N into blocks corresponding to the subsets of vertices which are still connected at time t.Observe that, by construction and the very definition of the distance on P, the process Π has càdlàg paths.
We are now able to state the main result of this section.
Theorem 1 (i) The process Π = (Π(t) : t ≥ 0) is Markovian and has the Feller property.We write G for its infinitesimal generator.
(ii) For every n ≥ 1, D n is invariant and therefore D ∞ is a core for G.
(iii) For every f ∈ D ∞ and η ∈ P, we have We stress that this characterization of the law of the process Π is very close to that of a homogeneous fragmentation.Indeed, one can rephrase well-known results (cf.Section 3.1.2in [5]) on the latter as follows.Every homogeneous fragmentation process Γ = (Γ t : t ≥ 0) is a Feller process on P, such that the sub-spaces D n are invariant (and hence D ∞ is a core).Further, its infinitesimal generator A is given in the form for every f ∈ D ∞ and η ∈ P, where s is some exchangeable measure on P.More precisely, s({1 N }) = 0, where for every block B ⊆ N, 1 B ∈ P B denotes the neutral partition which has a single non-empty block B, and Observe that the measure r fails to be exchangeable, but it fulfills (4); indeed, one has We shall now prepare the proof of Theorem 1.In this direction, it is convenient to introduce some further notation.Consider an arbitrary block B ⊆ N, a partition η ∈ P B and a sequence π (•) = (π (i) : i ∈ N) in P. We write η • π (•) for the partition of B whose family of blocks is given by those of η • i π (i) for i ∈ N. In words, for each i ∈ N, the i-th block of η is split according to the partition π (i) .Next, consider a probability measure q on P and a sequence (π (i) : i ∈ N) of i.i.d.random partitions with common law q.We associate to q a probability kernel Fr(•, q) on P B , by denoting the distribution of η • π (•) by Fr(η, q) for every η ∈ P B .We point out that if q is exchangeable, then η • π (•) has the same distribution as the random partition whose blocks are given by the restrictions π (i) |η i of π (i) to η i for i ∈ N, and Fr(•, q) thus coincides with the fragmentation kernel that occurs for homogeneous fragmentations, see Definition 3.2 on page 119 in [5].Of course, the assumption of exchangeability is crucial for this identification to hold.
Note that the restriction of partitions of N to [n] is compatible with the fragmentation operator Fr(•, •), in the sense that Proposition 1 For every n ∈ N, the process Its semigroup can be described as follows: for every s, t ≥ 0, the conditional distribution of , where q t denotes the distribution of Π(t).
Proof: The case n = 1 is clear since Π | [1] (t) = ({1}, ∅, . . . ) for all times t ≥ 0. Assume now n ≥ 2. The proof relies crucially on the so-called splitting property of random recursive trees that we now recall (see, e.g.Section 2.2 of [3]).Given a subset B ⊆ N, the image of T by the map j → B(j) which enumerates the elements of B in the increasing order, is called a random recursive tree on B and denoted by T B .In particular, for B = [n], the restriction of T to the first n vertices is a random recursive tree on [n].Imagine now that we remove k fixed edges (i.e.edges with given indices, say i 1 , . . ., i k , where 2 . Then, conditionally on the induced partition of [n], say η = (η 1 , . . ., η k+1 ), the resulting k + 1 subtrees are independent random recursive trees on their respective sets of vertices η j , j = 1, . . ., k + 1.

It follows easily from the lack of memory of the exponential distribution and the compatibility property (5) that the restricted process Π
where Fr(•, q t ) is here viewed as a probability kernel on P In order to describe the infinitesimal generator of the restricted processes Π |[n] for n ∈ N, we consider its rates of jumps, which are defined by where now π denotes a generic partition of [n] which has at least two (non-empty) blocks.The rates of jumps r π determine the infinitesimal generator G n of the restricted chain Π |[n] , specifically we have for f : π denotes the partition that results from fragmenting the i-th block of η according to π).This determines the distribution of the restricted chain Π |[n] , and hence, letting n vary in N, also characterizes the law of Π. Recall also that the measure r on P has been defined by (3).
Proposition 2 For every n ≥ 2 and every partition π of [n] with at least two (non-empty) blocks, there is the identity r π = r(P π ), where Proof: This should be intuitively straightforward from the connection between the construction of random recursive trees and the dynamics of Pólya urns.Specifically, fix n ≥ 2 and consider a partition π ∈ P [n] .If π consists in three or more non-empty blocks, then we clearly have lim since at least two edges have to be removed from T |[n] in order to yield a partition with three or more blocks.Assume now that π is binary with non-empty blocks π 1 and π 2 , and let k = min π 2 .
Then only the removal of the edge e k may possibly induce the partition π, and more precisely, if we write η for the random partition of [n] resulting from the removal of e k , then the probability that η = π is precisely the probability that in a Pólya urn containing initially k − 1 black balls labeled 1, . . ., k − 1 and a single red ball labeled k, after n − k steps, the red balls are exactly those with labels in π 2 .Since the edge e k is removed at unit rate, this gives lim in the notation of the statement.Note that the right-hand side can be also written as r(P π ), since p ℓ (P π ) = 0 for all ℓ = k.
Proposition 2 should be compared with Proposition 3.2 in [5]; we refer henceforth to r as the splitting rate of Π.
We have now all the ingredients necessary to establish Theorem 1.

Proof of Theorem 1:
From Proposition 1, we see that the transition semigroup of Π is given by (Fr(•, q t ) : t ≥ 0), and it is easily checked that the latter fulfills the Feller property; cf.Proposition 3.1(i) in [5].Point (ii) is immediate from the compatibility of restriction with the fragmentation operator, see (5).Concerning (iii), let f ∈ D n .Since f is constant on {η ∈ P : η |[n] = π}, it can naturally be restricted to a function f : P [n] → R. By the compatibility property (5), with r π ′ = r(P π ′ ) for a partition π ′ of [n] defined as in Proposition 2, we obtain π∈P r(dπ where G n is the infinitesimal generator of the restricted chain Π |[n] found above.This readily yields (iii).
Remark.It may be interesting to recall that the standard exponential law is invariant under the map t → − ln(1 − e −t ), and thus, if we set ǫi = − ln(1 − exp(−ǫ i )) (recall that ǫ i is the instant at which the edge connecting the vertex i to its parent is removed), then (ǫ i ) i≥2 is a sequence of i.i.d.exponential variables.The time-reversal t → − ln(1 − e −t ) transforms the destruction process of T into a construction process of T defined as follows.At each time ǫi , we create an edge between i ≥ 2 and its parent which is chosen uniformly at random in {1, . . ., i − 1}.It follows that the time-reversed process Π(t) = Π(− ln(1 − e −t )−), t ≥ 0, is a binary coalescent process such that the rate at which two blocks, say B and B ′ with min B < min B ′ , merge, is given by #{j ∈ B : j < min B ′ }.This can be viewed as a duality relation between fragmentation and coalescent processes; see Dong, Goldschmidt and Martin [9] and references therein.
Recall that for every k ≥ 2, a random binary partition with law p k resulting from the Pólya urn construction possesses asymptotic frequencies in the sense of (1).This readily entails the following result.
Proposition 3 For r-almost all binary partitions (B 1 , B 2 ) ∈ P, the blocks B 1 and B 2 have asymptotic frequencies, and more precisely, we have where f : [0, 1] 2 → R + denotes a generic measurable function.In particular, Proof: Indeed, it is a well-known fact of Pólya urns that for each k ≥ 2, p k -almost every partition (B 1 , B 2 ) has asymptotic frequencies with x ∈ (0, 1).
It is interesting to recall that the splitting rates s of a homogeneous fragmentation must fulfill the integrability condition which thus fails for r !
We next turn our attention to the Poissonian structure of the process Π, which can be rephrased in terms similar to those in Section 3.1 of [5].In this direction, we introduce a random point measure on R + × P × N as follows.Recall that ǫ i is the time at which the edge e i connecting the vertex i ∈ N to its parent in T is removed.Immediately before time ǫ i , the vertex i belongs to some block of the partition Π(ǫ i −), we denote the label of this block by k i (recall that Π is càdlàg and that blocks of a partition are labeled in the increasing order of their smallest element).Removing the edge e i yields a partition of that block B = Π k i (ǫ i −) into two sub-blocks, which can be expressed (uniquely) in the form B • ∆ i .This defines the binary partition ∆ i and hence the point measure M unambiguously.The process Π can be recovered from M, in a way similar to that explained on pages 97-98 in [5].Roughly speaking, for every atom of M, say (t, ∆, k), Π(t) results from partitioning the k-th block of Π(t−) using ∆, that is by replacing Π k (t−) by Π k (t−) • ∆.Adapting the arguments of Section 3.1.3in [5], we have the following result: The random measure M is Poisson with intensity λ ⊗ r ⊗ #, where λ denotes the Lebesgue measure on R + and # the counting measure on N.
Proof: Recall that we write 1 [n] = ([n], ∅, . ..) for the partition of [n] which consists of a single non-empty block.Consider a Poisson random measure M ′ with intensity λ ⊗ r ⊗ # as in the statement.Then M ′ has almost surely at most one atom in each fiber {t} ⊗ P ⊗ N, and the discussion below (4) shows that for each t ′ ≥ 0 and every n ∈ N, the number of atoms (t, π, k) and k ≤ n is finite.We may therefore define for fixed n ∈ N a P = for all integers n ≥ m.We deduce as in the proof of Lemma 3.3 in [5] that there exists a unique P-valued càdlàg function (Π ′ (t) : i (t), and it follows from the very construction of Π ′[n] (t) that the process Π ′ can be recovered from M ′ similarly to the description above the statement of the proposition.It remains to check that Π ′ and Π have the same law, which follows if we show that the restricted processes Π have the same law for each n ∈ N. Fix n ≥ 2, and denote by π a partition of [n] with at least two non-empty blocks.From the Poissonian construction of Π ′ [n] , with P π as in the statement of Proposition 2, we first see that , the jump rate of Π ′[n] from π ′ to π ′′ is non-zero only if π ′′ can be obtained from π ′ by replacing one single block of π ′ , say the k-th block π ′ k , by π ′ k • π, where π is some binary partition of [n].This observation and the last display readily show that Π ′ [n]  and Π |[n] have the same generator, and hence their laws agree.

The process of the weights
Even though the splitting rates measure r of the fragmentation process Π fails to fulfill the integral condition (6), we shall see that we can nonetheless define the weights of its blocks.The purpose of this section is to investigate the process of the weights as time passes.

The weight of the first block as an O.U. type process
In this section, we focus on the first block Π 1 (t), that is the cluster at time t which contains the root 1 of T. The next statement gathers its key properties, and in particular stresses the connection with an Ornstein-Uhlenbeck type process.
In the sequel, we shall refer to X 1 (t) as the weight of the first block (or the root-cluster) at time t.Before tackling the proof of Theorem 2, we make a couple of comments.Firstly, observe from (i) that lim t→∞ E(X 1 (t) q ) = Γ(q + 1), so that as t → ∞, Y (t) converges in distribution to the logarithm of a standard exponential variable.On the other hand, it is wellknown that the weak limit at ∞ of an Ornstein-Uhlenbeck type process is self-decomposable; cf.Section 17 in Sato [18].So (iii) enables us to recover the fact that the log-exponential distribution is self-decomposable; see Shanbhag and Sreehari [19].
Secondly, note that the Lévy-Khintchin formula for κ reads where γ = 0.57721 . . . is the Euler-Mascheroni constant.Indeed, this follows readily from the classical identity for the digamma function In turn, this enables us to identify the Lévy measure of L as Since the jumps of L and of Y coincide, the Lévy-Itō decomposition entails that the jump process of Y = ln X 1 is a Poisson point process with characteristic measure Λ.In this direction, recall from Proposition 3 that the distribution of the asymptotic frequency of the first block under the measure r of the splitting rates of Π is (1 − y) −2 dy, y ∈ (0, 1), and observe that the image of the latter by the map y → ln y is precisely Λ.This should of course not come as a surprise.
We shall present two proofs of Theorem 2(i); the first relies on the well-known connection between random recursive trees and Yule processes and is based on arguments due to Pitman.Indeed, #{j ≤ n : j ∈ Π 1 (t)} can be interpreted in terms of the two-type population system considered in Section 3.4 of [17], as the number of novel individuals at time t when the birth rate of novel offspring per novel individual is given by α = e −t , and conditioned that there are n individuals in total in the population system at time t.Part (i) of the theorem then readily follows from Proposition 3.14 in connection with Corollary 3.15 and Theorem 3.8 in [17].For the reader's convenience, let us nonetheless give a self-contained proof which is specialized to our situation.We further stress that variations of this argument will be used in the proofs of Proposition 5 and Corollary 2.
First proof of Theorem 2(i): Consider a population model started from a single ancestor, in which each individual gives birth a new child at rate one (in continuous time).If the ancestor receives the label 1 and the next individuals are labeled 2, 3, . . .according to the order of their birth times, then the genealogical tree of the entire population is a version of T. Further, if we write Z(s) for the number of individuals in the population at time s, then the process (Z(s) : s ≥ 0) is a Yule process, that is a pure birth process with birth rate n when the population has size n.Moreover, it is readily seen that the Yule process Z and the genealogical tree T are independent.

It is well-known that lim
Now we incorporate destruction of edges to this population model by killing merciless each new-born child with probability 1−p ∈ (0, 1), independently of the other children.The resulting population model is again a Yule process, say Z (p) = (Z (p) (s) : s ≥ 0), but now the rate of birth per individual is p.Therefore, we have also almost surely, where W (p) is another standard exponential variable.We stress that W (p) is of course correlated to W and not independent of T, in contrast to W .
In this framework, we identify for p = e −t almost surely.
Combining with (7), we arrive at which proves the first part of (i).
Now recall that the left-hand side above only depends on the genealogical tree T and the exponential random variables ǫ i attached to its edges.Therefore, it is independent of the Yule process Z and a fortiori of W . Since both W and W (p) are standard exponentials, the second part of (i) now follows from the moments of exponential random variables.
The second proof of Theorem 2(i) relies on more advanced features on the destruction of random recursive trees and Poisson-Dirichlet partitions.
Second proof of Theorem 2(i): It is known from the work of Goldschmidt and Martin [10] that the destruction of T bears deep connections to the Bolthausen-Sznitman coalescent.In this setting, the quantity #{j ≤ n : j ∈ Π 1 (t)} can be viewed as the number of blocks at time t in a Bolthausen-Sznitman coalescent on [n] = {1, . . ., n} started from the partition into singletons.On the other hand, it is known that the latter is a so-called (e −t , 0) partition; see Section 3.2 and Theorem 5.19 in Pitman [17].Our claims now follow from Theorem 3.8 in [17].
Proof of Theorem 2(ii): Let Π ′ 1 be an independent copy of the process Π 1 .Fix s, t ≥ 0 and put B = Π 1 (s), C = Π ′ 1 (t).Recall that B(j) denotes the j-th smallest element of B, and B(C) stands for the block {B(j) : j ∈ C}.By Proposition 1, there is the equality in distribution Π 1 (s + t) = B(C).From (i) we deduce that almost surely as n → ∞, and similarly C(n) ∼ (n/X ′ 1 (t)) e t as n → ∞, where X ′ 1 (t) has the same law as X 1 (t) and is further independent of (X 1 (r) : r ≥ 0).It follows that there are the identities Here, in the next to last equality we have used the fact that the n-th smallest element of B(C) is given by the C(n)-th smallest element of B, and for the last equality we have plugged in the asymptotic expressions for B(n) and C(n) that we found above.Our claim now follows easily.
We point out that, alternatively, the Markov property of X 1 can also be derived from the interpretation of #{j ≤ n : j ∈ Π 1 (t)} as the number of blocks at time t in a Bolthausen-Sznitman coalescent on [n]; see the second proof of Theorem 2(i) above.

Proof of Theorem 2(iii):
We first observe from (ii) that the process Y is Markovian with semigroup Q t (y, •) given by Next, recall from the last remark made after Theorem 2 that the function q → κ(q) = qψ(q + 1) is the cumulant-generating function of a spectrally negative Lévy process, say L = (L(t) : t ≥ 0).Consider then the Ornstein-Uhlenbeck type process U = (U(t) : t ≥ 0) that solves the stochastic differential equation that is, equivalently, U(t) = e −t t 0 e s dL(s).Then U is also Markovian with semigroup R t (u, •) given by R t (u, •) = P(e −t u + U(t) ∈ •).
So to check that the processes Y and U have the same law, it suffices to verify that they have the same one-dimensional distribution.
Remark.It would be interesting to understand whether with probability 1, the block Π 1 (t) has weights in the sense of Theorem 2(i) simultaneously for all t ≥ 0 (note that the asymptotic frequencies are equal to zero on (0, ∞) a.s.), and whether t → X 1 (t) is càdlàg.In this direction, we recall that each block of a standard homogeneous fragmentation process possesses asymptotic frequencies simultaneously for all t ≥ 0 a.s., see Proposition 3.6 in [5].Moreover, if B denotes a block of such a process, then the process t → |B(t)| is càdlàg.Here it should be observed that the first block Π 1 is the only block which is decreasing, in the sense that Π 1 (t ′ ) is contained in Π 1 (t) a.s.whenever t ′ ≥ t.This however does not imply monotonicity of X 1 (t) in t, which is crucial ingredient for the proof of the mentioned properties in the case of a homogeneous fragmentation.

Fragmentation of weights as a branching O.U. process
We next turn our interest to the other blocks of the partition Π(t); we shall see that they also have a weight, in the same sense as for the first block.In this direction, it is convenient to write first T i for the subtree of T rooted at i ≥ 1; in particular T 1 = T. Then for t ≥ 0, we write T i (t) the subtree of T i consisting of vertices j ∈ T i which are still connected to i after the edges e k with ǫ k ≤ t have been removed.Note that for i ≥ 2, T i (t) is a cluster at time t if and only if ǫ i ≤ t, an event which has always probability 1 − e −t and is further independent of T i (t).On that event, the vertex set of T i (t) is a block of the partition Π(t), and all the blocks of Π(t) arise in this form.
Lemma 1 For every t ≥ 0 and i ∈ N, the following limit exists in (0, ∞) a.s.Moreover, the process has the same law as (β e −t i X 1 (t) : t ≥ 0), where β i denotes a beta variable with parameter (1, i − 1) and is further independent of X 1 (t).
Proof: The recursive construction of T and T i has the same dynamics as a Pólya urn, and basic properties of the latter entail that the proportion β i of vertices in T i has the beta distribution with parameter (1, i−1).Further, enumerating the vertices of T i turns the latter into a random recursive tree.Our claim then follows readily from Theorem 2.
Lemma 1 entails that for every i ∈ N, the i-th block Π i (t) of Π(t) has a weight in the sense of (2), a.s.We write X i (t) for the latter and set X(t) = (X 1 (t), X 2 (t), . ..).We now investigate the process X = (X(t) : t ≥ 0).Firstly, using the functional equation of the gamma function, the integral representation of the beta function and the expression for the moments of X 1 (t) from Theorem 2(i), an easy calculation shows provided q > e t .In particular, for every t ≥ 0, the X i (t) can be sorted in the decreasing order.We write X ↓ (t) for the sequence obtained from X(t) by ranking the weights X i (t) decreasingly, where as usual elements are repeated according to their multiplicity.For q > 0, let x q i < ∞ , endowed with the ℓ q -distance.Similarly, denote by ℓ ∞↓ the space of ordered sequences of positive reals, endowed with the ℓ ∞ -distance.For the process X ↓ , we obtain the following characterization.
Corollary 1 Let T ∈ (0, ∞], and set q = e T (with the convention e ∞ = ∞).Then the process X ↓ = (X ↓ (t) : t < T ) takes its values in ℓ q↓ and is Markovian.More specifically, its semigroup can be described as follows.For s, t ≥ 0 with s + t < T , the law of X ↓ (s + t) conditioned on X ↓ (s) = (x 1 , . . . ) is given by the distribution of the decreasing rearrangement of the sequence (x e −t i x (i) j : i, j ∈ N), where ((x ) is a sequence of independent random elements in ℓ q↓ , each of them distributed as X ↓ (t).
Proof: The fact that X ↓ (t) ∈ ℓ q↓ for t < T follows from (8).The specific form of the semigroup can be deduced from Theorem 1 and a variation of the arguments leading to Theorem 2(ii).
We now draw our attention to the logarithms of the weights.In this direction, recall that for a homogeneous fragmentation, the process of the asymptotic frequencies of the blocks bears close connections with branching random walks.More precisely, the random point process with atoms at the logarithm of the asymptotic frequencies and observed, say at integer times, is a branching random walk; see [8] and references therein.This means that at each step, each atom, say y, is replaced by a random cloud of atoms located at y + z for z ∈ Z, independently of the other atoms, and where the random point process Z has a fixed distribution which does not depend on y nor on the step.In the same vein, we also point out that recently, a natural extension of homogeneous fragmentations, called compensated fragmentations, has been constructed in [7], and bears a similar connection with branching Lévy processes.Note that compensated fragmentations are ℓ 2↓ -valued processes, in contrast to X ↓ .

Our observations incite us to introduce
Theorem 3 The process with values in the space of point measure Y = (Y t : t ≥ 0) is an Ornstein-Uhlenbeck branching process, in the sense that it is Markovian and its transition probabilities can be described as follows: For every s, t ≥ 0, the conditional law of Y s+t given Y s = ∞ i=1 δ y i is given by the distribution of where the point measures are independent and each has the same law as Y t .
Furthermore, the mean intensity of Y t is determined by E e qy Y t (dy) = (q − 1) (e −t q − 1) Γ(q) Γ(e −t q) , q > e t .
Proof: The first claim follows from Theorem 1 and an adaption of the argument for Theorem 2(iii).For the second, we note that and the right hand side has already been evaluated in (8).
Next we give a description of the finite dimensional laws of X(t) = (X 1 (t), X 2 (t), . ..) for t > 0 fixed.In this direction, it is convenient to define two families of probability distributions.
The first family is indexed by j ∈ N and t > 0, and is defined as Note that the shifted distribution μj,t (k) = µ j,t (k + 1), k ≥ j, is sometimes called the negative binomial distribution parameters j and 1−e −t , that is the law of the number of independent trials for j successes when the success probability is given by 1 − e −t .
The second family is indexed by j ∈ N and k ≥ j, and can be described as follows.We denote by θ j,k the probability measure on the discrete simplex ∆ k,j = {(k 1 , . . ., k j ) ∈ N j : k 1 + • • • + k j = k}, such that θ j,k (k 1 , . . ., k j ) is the probability that on a random recursive tree of size k (that is on a random tree distributed as T |[k] ), after j − 1 of its edges chosen uniformly at random have been removed, the sequence of the sizes of the j subtrees, ordered according to the label of their root vertex, is given by (k 1 , . . ., k j ).
Remark.The distribution θ j,k is equal to δ k for j = 1.For j = 2, Meir and Moon [15] found the expression . Generalizing the proof of this formula given in [15] to higher j, we find (k ≥ j ≥ 3 and Proposition 5 Let j ∈ N, q 1 , . . ., q j+1 ≥ 0, and set k j+1 = 1.The Mellin transform of the vector (X 1 (t), . . .X j+1 (t)) for fixed t > 0 is given by Remark.By plugging in the definition of µ j,t (k), one checks that the right hand side is finite.
Proof: Fix t > 0, and set p = e −t .For ease of notation, we write Π i and X i instead of Π i (t) and X i (t).Furthermore, fix an integer j ∈ N and numbers k 1 , . . ., k j ∈ N.For convenience, set We first work conditionally on the event shall adapt the first proof of Theorem 2(i).Here, we consider a multi-type Yule process starting from k individuals in total such that k i of them are of type i, for each i = 1, . . ., j + 1.The individuals reproduce independently of each other at unit rate, and each child individual adopts the type of its parent.Then, if Z(s) stands for the total number of individuals at time s, we have that lim s→∞ e −s Z(s) = γ(k) almost surely, where γ(k) is distributed as the sum of k standard exponentials, i.e. follows the gamma law with parameters (k, 1).Now assume again that each new-born child is killed with probability 1 − p ∈ (0, 1), independently of each other.Writing Z (i,p) (s) for the size of the population of type i at time s (with killing), we obtain lim s→∞ e −ps Z (i,p) (s) = γ (i,p) (k i ), i = 1, . . ., j + 1, where the γ (i,p) (k i ) are independent gamma(k i , 1) random variables (they are however clearly correlated to the asymptotic total population size γ(k)).From the arguments given in the first proof of Theorem 2(i) it should be plain that conditionally on the event A k 1 ,...,k j , we have for the weights X i the representation and the X i are independent of γ(k).Now let q 1 , . . ., q j+1 ≥ 0 and put q = q 1 + • • • + q j+1 .Using the expression for the X i and independence, we calculate Therefore, again with k j+1 = 1, By induction on j we easily deduce that min Π j+1 −1 is distributed as the sum of j independent geometric random variables with success probability 1 − p, i.e. min Π j+1 − 1 counts the number of trials for j successes, so that P (min Π j+1 = k) = µ t,j (k).Moreover, it follows from the very definition of the blocks Π i and the fact that the exponentials attached to the edges of T |[k] are i.i.d. that This proves the proposition.

Asymptotic behaviors
We shall finally present some asymptotic properties of the process X of the weights.To start with, we consider the large time behavior.
Corollary 2 As t → ∞, there is the weak convergence where on the right-hand side, the W i are i.i.d.standard exponential variables.
Remark.This result is a little bit surprising, as obviously Π(∞) is the partition into singletons.That is Π i (∞) is reduced to {i} and hence has weight 1 if we apply (2) for t = ∞.In other words, the limits n → ∞ and t → ∞ may not be interchanged.

s→∞e
−s Z(s) = W almost surely, where W has the standard exponential distribution.As a consequence, if we write τ n = inf{s ≥ 0 : Z(s) = n} for the birth-time of the individual with label n, then lim n→∞ ne −τn = W almost surely.