Exchangeable Fragmentation-Coalescence processes and their equilibrium measures

We define and study a family of Markov processes with state space the compact set of all partitions of N that we call exchangeable fragmentation-coalescence processes. They can be viewed as a combination of exchangeable fragmentation as defined by Bertoin and of homogenous coalescence as defined by Pitman and Schweinsberg or Mohle and Sagitov. We show that they admit a unique invariant probability measure and we study some properties of their paths and of their equilibrium measure.

References as to the fields of application of coalescence and fragmentation models (physical chemistry, astronomy, biology, computer sciences...) may be found in Aldous [2] -mainly for coalescence-and in the proceedings [9] for fragmentation (some further references can be found in the introduction of [4]). Clearly, many fragmentation or coalescence phenomena are not "pure" in the sense that both are present at the same time. For instance, in the case of polymer formation there is a regime near the critical temperature where molecules break up and recombine simultaneously. Another example is given by Aldous [2], when, in his one specific application section, he discusses how certain liquids (e.g., olive oil and alcohol) mix at high temperature but separate below some critical level. When one lowers very slowly the temperature through this threshold, droplets of one liquid begin to form, merge and dissolve back very quickly.
It appears that coalescence-fragmentation processes are somewhat less tractable mathematically than pure fragmentation or pure coalescence. One of the reasons is that by combining these processes we lose some of the nice properties they exhibit when they stand alone, as for instance their genealogic or branching structure. Nevertheless, it is natural to investigate such processes, and particularly to look for their equilibrium measures.
In this direction Diaconis, Mayer-Wolf, Zeitouni and Zerner [11] considered a coagulation-fragmentation transformation of partitions of the interval (0, 1) in which the merging procedure corresponds to the multiplicative coalescent while the splittings are driven by a quadratic fragmentation. By relating it to the random transposition random walk on the group of permutations, they were able to prove a conjecture of Vershik stating that the unique invariant measure of this Markov process is the Poisson-Dirichlet law. We would also like to mention the work of Pitman [21] on a closely related split and merge transformation of partitions of (0, 1) as well as Durrett and Limic [12] on another fragmentation-coalescence process of (0, 1) and its equilibrium behavior. However, a common characteristic of all these models is that they only allow for binary splittings (a fragment that splits creates exactly two new fragments) and pairwise coalescences. Furthermore the rate at which a fragment splits or merges depends on its size and on the size of the other fragments.
Here, we will focus on a rather different class of coagulation-fragmentations that can be deemed exchangeable or homogeneous. More precisely, this paper deals with processes which describe the evolution of a countable collection of masses which results from the splitting of an initial object of unit mass. Each fragment can split into a countable, possibly finite, collection of sub-fragments and each collection of fragments can merge. One can have simultaneously infinitely many clusters that merge, each of them containing infinitely many masses.
We will require some homogeneity property in the sense that the rate at which fragments split or clusters merge does not depend on the fragment sizes or any other characteristic and is not time dependent.
Loosely speaking, such processes are obtained by combining the semigroups of a homogenous fragmentation and of an exchangeable coalescent. Exchangeable coalescents, or rather Ξ-coalescents, were introduced independently by Schweinsberg [23] 2 and by Möhle and Sagitov [19] who obtained them by taking the limits of scaled ancestral processes in a population model with exchangeable family sizes. Homogeneous fragmentations were introduced and studied by Bertoin [6,7,8].
The paper is organized as follows. Precise definitions and first properties are given in Section 3. Next, we prove that there is always a unique stationary probability measure for these processes and we study some of their properties. Section 5 is dedicated to the study of the paths of exchangeable fragmentation-coalescence processes.
The formalism used here and part of the following material owe much to a work in preparation by Bertoin based on a series of lectures given at the IHP in 2003, [5].

Preliminaries
Although the most natural state space for processes such as fragmentation or coalescence might be the space of all possible ordered sequence of masses of fragments as in the case of pure fragmentation or pure coalescence, we prefer to work with the space P of partitions of N. An element π of P can be identified with an infinite collection of blocks (where a block is just a subset of N and can be the empty set) π = (B 1 , B 2 , ...) where ∪ i B i = N, B i ∩ B j = ø when i = j and the labelling corresponds to the order of the least element, i.e., if w i is the least element of B i (with the convention min ø = ∞) then i ≤ j ⇒ w i ≤ w j . The reason for such a choice is that we can discretize the processes by looking at their restrictions to [n] := {1, ..., n}.
As usual, an element π ∈ P can be identified with an equivalence relation by setting i π ∼ j ⇔ i and j are in the same block of π.
Thus we may define the exchangeable fragmentation-coalescence processes by their restrictions to [n].
Let us now define deterministic notions which will play a crucial role in the forthcoming constructions. We define two operators on P, a coagulation operator, π, π ∈ P → Coag(π, π ) (the coagulation of π by π ) and a fragmentation operator π, π ∈ P, k ∈ N → Frag(π, π , k) (the fragmentation of the k-th block of π by π ).
• Take π = (B 1 , B 2 , ...) and π = (B 1 , B 2 , ...). Then, for k ≤ #π, where #π is the number of non-empty blocks of π, the partition Frag(π, π , k) is the relabelled collection of blocks formed by all the B i for i = k, plus the sub-blocks of B k given by π |B k .
Similarly, when π ∈ P n and π ∈ P or π ∈ P k for k ≥ #π one can define Coag(π, π ) as above and when π ∈ P or π ∈ P m for m ≥ Card(B k ) (and k ≤ #π) one can define Frag(π, π , k) as above. Then 0 is the neutral element for Coag, i.e., for each π ∈ P Coag(π, 0) = Coag(0, π) = π, (for π ∈ ∪ n≥2 P n , as Coag(0, π) is not defined, one only has Coag(π, 0) = π) and 1 is the neutral element for Frag, i.e., for each π ∈ P one has Similarly, when π ∈ ∪ n≥2 P n , for each k ≤ #π one only has Note also that the coagulation and fragmentation operators are not really reciprocal because Frag can only split one block at a time.
Much of the power of working in P instead of S ↓ comes from Kingman's theory of exchangeable partitions. For the time being, let us just recall the basic definition. Define the action of a permutation σ : N → N on P by A random element Π of P or a P valued process Π(·) is said to be exchangeable if for any permutation σ such that σ(n) = n for all large enough n one 3 Definition, characterization and construction of EFC processes

Definition and characterization
We can now define precisely the exchangeable fragmentation-coalescence processes and state some of their properties. Most of the following material is very close to the analogous definitions and arguments for pure fragmentations (see [6]) and coalescences (see [20,23]).
• Its restrictions Π |[n] are càdlàg finite state Markov chains which can only evolve by fragmentation of one block or by coagulation.
Observe that this definition implies that Π(0) should be exchangeable. Hence the only possible deterministic starting points are 1 and 0 because the measures δ 1 (·) and δ 0 (·) (where δ • (·) is the Dirac mass in •) are the only exchangeable measures of the form δ π (·). If Π(0) = 0 we say that the process is started from dust, and if Π(0) = 1 we say it is started from unit mass.
Note that the condition that the restrictions Π |[n] are càdlàg implies that Π itself is also càdlàg.
Fix n and π ∈ P n . For convenience we will also use the following notations for the transition rates: For π ∈ P m \{0 m } where m = #π the number of non-empty blocks of π, call C n (π, π ) := q n (π, Coag(π, π )) the rate of coagulation by π . For k ≤ #π and π ∈ P |B k | \{1 |B k | } where |B k | is the cardinality of the k-th block, call F n (π, π , k) := q n (π, Frag(π, π , k)) the rate of fragmentation of the kth block by π .
We will say that an EFC process is non-degenerated if it has both a fragmentation and coalescence component, i.e., for each n there are some π 1 = 1 n and π 2 = 0 n such that F n (1 n , π 1 , 1) > 0 and C n (0 n , π 2 ) > 0.
Of course the compatibility of the Π |[m] and the exchangeability requirement entail that not every family of transition rates is admissible. In fact, it is enough to know how Π |[m] leaves 1 m and 0 m for every m ≤ n to know all the rates q n (π, π ). Proposition 2. There exist two families ((C n (π)) π∈P n \{0 n } ) n∈N and ((F n (π)) π∈P n \{1 n } ) n∈N such that for every m ≤ n and for every π ∈ P n with m blocks (#π = m) one has 1. For each π ∈ P m \{0 m } q n (π, Coag(π, π )) = C n (π, π ) = C m (π ).
3. All other transition rates are zero.
As the proof of this result is close to the arguments used for pure fragmentation or pure coalescence and is rather technical, we postpone it until section 6.
Observe that, for n fixed, the finite families (C n (π)) π∈Pn\{0n} and (F n (π)) π∈P n \{1 n } may be seen as measures on P n . The compatibility of the Π |[n] (·) implies the same property for the (C n , F n ), i.e., as measures, the image of C n+1 (resp. F n+1 ) by the projection P n+1 → P n is C n (resp. F n ), see Lemma 1 in [6] for a precise demonstration in the case where there is only fragmentation (C ≡ 0), the general case being a simple extension. Hence, by Kolmogorov's extension Theorem, there exists a unique measure C and a unique measure F on P such that for each n and for each π ∈ P n such that π = 1 n (resp. π = 0 n ) C n (π) = C({π ∈ P : π |[n] = π}) resp. F n (π) = F ({π ∈ P : π |[n] = π}).
Furthermore, as we have observed, the measures C n and F n are exchangeable. Hence, C and F are exchangeable measures. They must also satisfy some integrability conditions because the Π |[n] (·) are Markov chains and have thus a finite jump rate at any state. For π ∈ P define Q(π, n) := {π ∈ P : π |[n] = π |[n] }. Then for each n ∈ N we must have and It is clear that we can suppose without loss of generality that C and F assign no mass to the respective neutral elements for Coag and Frag, i.e., C(0) = 0 and F (1) = 0.
Here are three simple examples of exchangeable measures.
1. Let n the partition that has only two non empty blocks: N\{n} and {n}. Then the (infinite) measure e(·) = n∈N δ n (·) (where δ is the Dirac mass) is exchangeable. We call it the erosion measure .
2. For each i = j ∈ N, call i,j the partition that has only one block which is not a singleton: {i, j}. Then the (infinite) measure κ(·) = i<j∈N δ i,j (·) is exchangeable. We call it the Kingman measure.
be a sequence of independent variables with respective law given by a random variable π with value in P by letting i π ∼ j ⇔ X i = X j . Following Kingman, we call π the x-paintbox process and denote by µ x its distribution. Let ν be a measure on S ↓ , then the mixture µ ν of paintbox processes directed by ν, i.e., for A ⊆ P is an exchangeable measure. We call it the ν-paintbox measure.
Extending seminal results of Kingman [17], Bertoin has shown in [6] and in [5] that any exchangeable measure on P that satisfies (1) (respectively (2)) is a combination of κ and a ν-paintbox measure (resp. e and a ν-paintbox process). Hence the following proposition merely restates these results.
Proposition 3. For each exchangeable measure C on P such that C({0}) = 0, and C(P\Q(0, n)) < ∞, ∀n ∈ N there exists a unique c k ≥ 0 and a unique measure ν Coag on S ↓ such that and For each exchangeable measure F on P such that F ({1}) = 0 and F (P\Q(1, n)) < ∞, ∀n ∈ N there exists a unique c e ≥ 0 and a unique measure ν Disl on S ↓ such that and F = c e e + µ ν Disl .
The two integrability conditions on ν Disl and ν Coag (4) and (3) ensure that C(P\Q(0, n)) < ∞ and F (P\Q(1, n)) < ∞. See [6] for the demonstration concerning F . The part that concerns C can be shown by the same arguments.
The condition on ν Disl (4) may seem at first sight different from the condition that Bertoin imposes in [6] and which reads and on the other hand Thus the above proposition implies that for each EFC process Π there is a unique (in law) exchangeable fragmentation Π (F ) (t) -the fragmentation whose law is characterized by the measure F -and a unique (in law) exchangeable coalescence Π (C) (t) -the coalescent whose law is characterized by the measure C-such that Π is a combination of Π (F ) and Π (C) in the sense that its transition rates respectively in the coalescence and the fragmentation sense are the same as those of Π (F ) and Π (C) . This was not obvious a priori because some kind of compensation phenomenon could have allowed weaker integrability conditions. One can sum up the preceding analysis in the following characterization of exchangeable fragmentation-coalescence processes.
Proposition 4. The distribution of an EFC process Π(·) is completely characterized by the initial condition (i.e., the law of Π(0)), the measures ν Disl and ν Coag as above and the parameters c e , c k ∈ R+.
Remark : The above results are well known for pure fragmentation or pure coalescence. If, for instance, we impose F (P) = 0 (i.e., there is only coalescence and no fragmentation, the EFC process is degenerated), the above proposition shows that our definition agrees with Definition 3 in Schweinsberg [23]. On the other hand if there is only fragmentation and no coalescence, our definition is equivalent to that given by Bertoin in [6], which relies on some fundamental properties of the semi-group. There, the Markov chain property of the restrictions is deduced from the definition as well as the characterization of the distribution by c and ν Disl .
Nevertheless, the formulation of Definition 1 is new. More precisely, for pure fragmentations, Definition 1 only requires that the process Π and its restrictions should be Markov and exchangeable and furthermore that only one block can fragmentate at a time. Point 2 of Proposition 2 then implies that Π has the fragmentation and homogeneity properties. We say that Π has the fragmentation property if each fragment evolves independently of the past and of the other fragments. This is obvious from the observation that the splitting rates in Proposition 2 of the blocks of Π |[n] only depend on their size. The fact that all transition rates can be expressed in terms of F n (π) -the rates at which 1 n splits-implies the homogeneity property, i.e., each fragment splits according to a fragmentation which has the same law as the original one, up to the scale-factor. In [6] homogeneous fragmentations are rather defined as exchangeable Markov processes whose semi-group have the fragmentation and homogeneity properties.

Poissonian construction
As for exchangeable fragmentation or coalescence, one can construct EFC processes by using Poisson point processes (PPP in the following). More precisely let P C = ((t, π (C) (t)), t ≥ 0) and P F = ((t, π (F ) (t), k(t)), t ≥ 0) be two independent PPP in the same filtration. The atoms of the PPP P C are points in R + × P and its intensity measure is given by dt ⊗ (µ ν Coag + c k κ). The atoms of P F are points in R + × P × N and its intensity measure is dt ⊗ (c e e + µ ν Disl ) ⊗ # where # is the counting measure on N and dt is the Lebesgue measure.
Take π ∈ P an exchangeable random variable and define a family of P n -valued processes Π n (·) as follows: for each n fix Π n (0) = π |[n] and • if t is not an atom time neither for P C or P F then Π n (t) = Π n (t−), • if t is an atom time for P C such that (π (C) (t)) |[n] = 0 n then • if t is an atom time for P F such that k(t) < n and (π (F ) (t)) |[n] = 1 n then Π n (t) = Frag(Π n (t−), π (F ) (t), k(t)).
Note that the Π n are well defined because on any finite time interval, for each n, one only needs to consider a finite number of atoms. Furthermore P C and P F being independent in the same filtration, almost surely there is no t which is an atom time for both PPP's. This family is constructed to be compatible and thus defines uniquely a process Π such that Π |[n] = Π n for each n. By analogy with homogeneous fragmentations ( [6]) and exchangeable coalescence ( [20,23]) the following should be clear. Proof. It is straightforward to check that the restrictions Π |[n] (t) are Markov chains whose only jumps are either coagulations or fragmentations. The transition rates are constructed to correspond to the characteristics c k , ν Coag , c e and ν Disl . The only thing left to check is thus exchangeability. Fix n ∈ N and σ a permutation of [n], then (σ(Π n (t))) t≥0 is a jump-hold Markov process. Its transition rates are given by q (σ) n (π, π ) = q n (σ −1 (π), σ −1 (π )). Suppose first that π = Frag(π, π , k) for some π . Note that there exists a unique l ≤ #π and a permutation σ of [m] (where m = |B k | is the cardinality of the k-th block of π we want to split) such that Using Proposition 2 we then obtain that q (σ) n (π, π ) = q n (σ −1 (π), σ −1 (π )) = q n (σ −1 (π), Frag(σ −1 (π), σ (π ), l)) The same type of arguments show that when π = Coag(π, π ) for some π we also have q (σ) n (π, π ) = q n (π, π ). Thus, Π n and σ(Π n ) have the same transition rates and hence the same law.
As this is true for all n, it entails that Π and σ(Π) also have the same law.
Proof. Call C f the set of functions which is dense in the space of continuous functions of P → R. The first point is clear for a function Φ ∈ C f (because the first jump-time of Φ(Π(·)) is distributed as an exponential variable with finite mean). We conclude by density. For the second point, consider π, π ∈ P such that d(π, π ) < 1/n (i.e., π |[n] = π |[n] ) then use the same PPP P C and P F to construct two EFC processes, Π(·) and Π (·), with respective starting points Π(0) = π and Π (0) = π . By construction Π |[n] (·) = Π |[n] (·) in the sense of the identity of the paths. Hence ∀t ≥ 0, d(Π(t), Π (t)) < 1/n.
Hence, when considering an EFC process, one can always suppose that one works in the usual augmentation of the natural filtration F t which is then right continuous.
As a direct consequence, one also has the following characterization of EFC's in terms of the infinitesimal generator : Let (Π(t), t ≥ 0) be an EFC process, then the infinitesimal generator of Π, denoted by A, acts on the functions f ∈ C f as follows: where F = c e e + µ ν Disl and C = c k κ + µ ν Coag . Indeed, take f ∈ C f and n such that π |[n] = π |[n] ⇒ f (π) = f (π ), then as Π |[n] (·) is a Markov chain the above formula is just the usual generator for Markov chains. Transition rates have thus the required properties and hence this property characterizes EFC processes.

Asymptotic frequencies
When A is a subset of N we will writē When the equalityλ A = λ A holds we call A , the asymptotic frequency of A, the common value which is also the limit If all the blocks of π = (B 1 , B 2 , ...) ∈ P have an asymptotic frequency we define Λ(π) = ( B 1 , B 2 , ...) ↓ the decreasing rearrangement of the B i 's.

Theorem 7. Let Π(t) be an EFC process. Then
exists almost surely simultaneously for all t ≥ 0, and (X(t), t ≥ 0) is a Feller process.
The proof (see section 6), which is rather technical, uses the regularity properties of EFC processes and the existence of asymptotic frequencies simultaneously for all rational times t ∈ Q. We call the process X(t) the associated ranked-mass EFC process.
Remark The state space of a ranked mass EFC process X is S ↓ . Thus, our construction of EFC processes Π in P started from 0 gives us an entrance law Q (0,0,...) for X. More precisely, call Q (0,0,...) (t) the law of X(t) conditionally on X(0) = (0, 0, ...). Then, for all t ≥ 0, there is the identity Q (0,0,...) (t) = Λ(P 0 (t)) where Λ(P 0 (t)) is the image of P 0 (t), the distribution of Π(t) conditionally on Π(0) = 0, by the map Λ. The ranked frequencies of an EFC process started from 0 defines a process with this entrance law that comes from dust at time 0+, i.e., the largest mass vanishes almost surely as t 0. The construction of this entrance law is well known for pure coalescence processes, see Pitman [20] for a general treatment, but also Kingman [16] and Bolthausen-Sznitman [10, Corollary 2.3] for particular cases.

Equilibrium measures
Consider an EFC process Π which is not trivial, i.e., ν Coag , ν Disl , c e and c k are not zero simultaneously.
Theorem 8. There exists a unique (exchangeable) stationary probability measure ρ on P and one has where δ π is the Dirac mass at π. Furthermore, Π(·) converges in distribution to ρ as t → ∞.
Proof. If the process Π is a pure coalescence process (i.e., c e = 0 and ν Disl (·) ≡ 0) it is clear that 1 is an absorbing state towards which the process converges almost surely. In the pure fragmentation case it is 0 that is absorbing and attracting.
In the non-degenerated case, for each n ∈ N, the process Π |[n] (·) is a finite state Markov chain. Let us now check the irreducibility in the nondegenerated case. Suppose first that ν Disl (S ↓ ) > 0. For every state π ∈ P n , if Π |[n] (t) = π there is a positive probability that the next jump of Π |[n] (t) is a coalescence. Hence, for every starting point Π |[n] (0) = π ∈ P n there is a positive probability that Π |[n] (·) reaches 1 n in finite time T before any fragmentation has occurred. Now take x ∈ S ↓ such that x 2 > 0 and recall that µ x is the x-paintbox distribution. Then for every π ∈ P n with #π = 2 (recall that #π is the number of non-empty blocks of π) one has That is the n-restriction of the x-paintbox partition can be any partition of [n] into two blocks with positive probability. More precisely if π ∈ P n is Hence, for any π ∈ P with #π = 2, the first transition after T is 1 n → π with positive probability. As any π ∈ P n can be obtained from 1 n by a finite series of binary fragmentations we can iterate the above idea to see that with positive probability the jumps that follow T are exactly the sequence of binary splitting needed to get to π and the chain is hence irreducible. Suppose now that ν Disl ≡ 0, there is only erosion c e > 0, and that at least one of the following two conditions holds then almost the same proof applies. We first show that the state 0 n can be reached from any starting point by a series of splittings corresponding to erosion, and that from there any π ∈ P n is reachable through binary coagulations.
In the remaining case (i.e., c k = 0, ν Disl ≡ 0 and there exists k > 0 such that the situation is slightly different in that P n is not the irreducible class. It is easily seen that the only partitions reachable from 0 n are those with at most k non-singletons blocks. But for every starting point π one reaches this class in finite time almost surely. Hence there is no issues with the existence of an invariant measure for this type of Π |[n] , it just does not charge partitions outside this class. Thus there exists a unique stationary probability measure ρ (n) on P n for the process Π | [n] . Clearly by compatibility of the Π |[n] (·) one must have where Proj P n (ρ (n+1) ) is the image of ρ (n+1) by the projection on P n . This implies that there exists a unique probability measure ρ on P such that for each n one has ρ (n) (·) = Proj P n (ρ)(·). The exchangeability of ρ is a simple consequence of the exchangeability of Π. Finally, the chain Π | [2] (·) is specified by two transition rates {1}{2} → {1, 2} and {1, 2} → {1}{2}, which are both non-zero as soon as the EFC is non-degenerated. Hence, Hence, when we have both coalescence and fragmentation ρ ∈ {δ 1 , δ 0 }.

The Π |[n]
(·) being finite state Markov chains, it is well known that they converge in distribution to ρ (n) , independently of the initial state. By definition of the distribution of Π this implies that Π(·) converges in distribution to ρ.
Although we cannot give an explicit expression for ρ in terms of c k , ν Coag , c e and ν Disl , we now relate certain properties of ρ to these parameters. In particular we will ask ourselves the following two natural questions: • under what conditions does ρ charge only partitions with an infinite number of blocks, resp. a finite number of blocks, resp. both ?
• under what conditions does ρ charge partitions with dust (i.e., parti- The proofs of the results in the remainder of this section are placed in section 6.

Number of blocks
We will say that an EFC process fragmentates quickly if c e > 0 or ν Disl (S ↓ ) = ∞. If this is not the case (i.e., c e = 0 and ν Disl (S ↓ ) < ∞) we say that it fragmentates slowly.
We first examine whether of not ρ charges partitions with a finite number of blocks.

Remarks :
1. The proof of the second point uses the connection with a work of Lambert [18] on which some details are given in section 5. In the case where we drop the hypothesis ν Coag (S ↓ ) = 0, although adding coalescence should reinforce the conclusion, we are only able to prove that it holds under the stronger condition 2. This implies that for an EFC process with a binary fragmentation component, a Kingman coalescence component and no erosion (i.e., 3. Finally, an interesting question is the case of an EFC process for which the measure ν Coag satisfies the condition given in [23] for a coalescent to come down from infinity. It is not self-evident that the condition of slow fragmentation is enough to ensure that ρ only charges partitions with finitely many blocks. The reason is that even though each fragment then splits at a finite rate, as we start with an infinite number of fragments the fragmentation could fix the process at an infinite level.

Dust
For any fixed time t the partition Π(t) is exchangeable. Hence, by Kingman's theory of exchangeable partition (see [1] for a simple proof of this result), its law is a mixture of paintbox processes. A direct consequence is that every block B i (t) of Π(t) is either a singleton or an infinite block with strictly positive asymptotic frequency. Recall that the asymptotic frequency of a block B i (t) is given by so part of Kingman's result is that this limit exists almost surely for all i simultaneously. The asymptotic frequency of a block corresponds to its mass, thus singletons have zero mass, they form what we call dust. More precisely, for π ∈ P define the set When π is exchangeable we have almost surely For fragmentation or EFC processes, dust can be created via two mechanisms: either from erosion (that's the atoms that correspond to the erosion measure c e e when c e > 0), or from sudden splitting which corresponds to atoms associated to the measure µ ν Disl where ν Disl is simply ν Disl restricted to {x ∈ S ↓ : i x i < 1}. Conversely, in the coalescence context mass can condensate out of dust, thus giving an entrance law in S ↓ , see [20].
The following theorem states that when the coalescence is strong enough in an EFC process, the equilibrium measure does not charge partitions with dust. We say that an EFC process coalesces quickly (resp. slowly) Theorem 10. Let (Π(t), t ≥ 0) be an EFC process that coalesces quickly and ρ its invariant probability measure. Then ρ({π ∈ P : dust(π) = ø}) = 0.
In case of no fragmentation, this follows from Proposition 30 in [23].

Equilibrium measure for the ranked mass EFC process
For ρ the equilibrium measure of some EFC process with characteristics ν Disl , ν Coag , c e and c k , let θ be the image of ρ by the map P → S ↓ : π → Λ(π).
Proposition 11. Let X be a ranked-mass EFC process with characteristics ν Disl , ν Coag , c e and c k . Then θ is its unique invariant probability measure.
Proof. As for each fixed t one has it is clear that θ is an invariant probability measure.
Suppose that θ is an invariant measure for X and fix t ≥ 0. Hence if X(0) has distribution θ so does X(t) = Λ(Π(t)). As Π(t) is exchangeable it is known by Kingman's theory of exchangeable partitions (see [1]) that Π(t) has law µ θ (·) the mixture of paintbox processes directed by θ. This implies that µ θ (·) is invariant for Π and hence µ θ (·) = ρ(·) and thus θ is the unique invariant measure for X. One of the problems tackled by Pitman [20] and Schweinsberg [24,23] about coalescent processes is whether or not they come down from infinity. Let us first recall some of their results. By definition if Π C (·) is a standard coalescent Π C (0) = 0 and thus #Π C (0) = ∞. We say that Π C comes down We know by Lemma 31 in [24], which is a generalization of Proposition 23 in [20], that if ν Coag (∆ f ) = 0 the coalescent either stays infinite or comes down from infinity.
For b ≥ 2 let λ b denote the total rate of all collisions when the coalescent has b blocks Let γ b be the total rate at which the number of blocks is decreasing when the coalescent has b blocks, < ∞, then the coalescent comes down from infinity. The converse is not always true but holds for instance for the important case of the Λ-coalescents (i.e., those for which many fragments can merge into a single block, but only one such merger can occur simultaneously).
These kinds of properties concerns the paths of the processes, and it seems that they bear no simple relation with properties of the equilibrium measure. For instance the equilibrium measure of a coalescent that stays infinite is δ 1 (·) and it therefore only charges partitions with one block, but its path lies entirely in the subspace of P of partitions with an infinite number of blocks.
Let Π(·) be an EFC process. Define the sets Clearly every arrival time t of an atom of P C such that In the same way an arrival time t of an atom then both G and G c are everywhere dense, and this independently of the starting point which may be 1 or 0.
The following proposition shows that when the fragmentation rate is infinite, G is everywhere dense. Recall the notation Theorem 12. Let Π be an EFC process that fragmentates quickly. Then, a.s. G is everywhere dense. More precisely and for all n ≥ 2 We begin with the proof that G is a.s. everywhere dense.
As G = ∩G k we only need to show that a.s. for each k ∈ N the set G k is everywhere dense and open to conclude with Baire's Theorem. The proof relies on two lemmas. Lemma 13. Let Π be an EFC process that fragmentates quickly started from , of which at least one is not a singleton and is thus in fact infinite when seen in N. The time of coalescence of i 1 and i 2 -the first time at which they are in the same block again-is exponentially distributed with parameter Hence if we define then almost surely we can find n 2 > n 1 large enough such that the first time ..} the collection of atom times of P C such that a coalescence occurs on the k + 1 first blocks if there are more than k + 1 blocks, i.e., π (C) (t) ∈ Q(0, k + 1) (recall that Q(0, k + 1) = {π ∈ P : π |[k+1] = 0 k+1 }). Suppose t ∈ G k , then by construction inf{s > t : s ∈ G c k } ∈ Γ k (because one must at least coalesce the first k + 1 distinct blocks present at time t before having less than k blocks). As the t (k) i are stopping times, the strong Markov property and the first lemma imply that G c k ⊆ Γ k . Hence G k is a dense open subset of R+.
We can apply Baire's theorem to conclude that ∩ k G k = G is almost surely everywhere dense in R+.
We now turn to the proof of (5) and (6). As G c = ∪G c n , it suffices to show (6) for some n ∈ N.
Recall from the proof of Lemma 14 that G c .} the set of coalescence times at which π (C) (t) ∈ Q(0, k + 1). Now fix n ∈ N and consider simultaneously the sequence (t have no other accumulation points than ∞, thus, by Theorem 12, there exists r is a multiple collision time, and more precisely t (k) 1 must be a collision time such that #π As this is true for each n almost surely, the conclusion follows. This finishes the proof of Theorem 12.
As recently noted by Lambert [18], there is an interpretation of some EFC processes in terms of population dynamics. More precisely if we consider an EFC process (Π(t), t ≥ 0) such that ν Disl (S ↓ ) < ∞ and (H) and there is a negative density-dependence due to competition pressure. This is reflected by the Kingman coalescence phenomenon which results in a quadratic death rate term. The natural death rate is set to 0, i.e., there is no linear component in the death rate. In this context, an EFC process that comes down from infinity corresponds to a population started with a very large size. Lambert has shown that a sufficient condition to be able to define what he terms a logistic branching process started from infinity is More precisely, this means that if P n is the law of the N-valued Markov chain (Y (t), t ≥ 0) started from Y (0) = n with transition rates then P n converge weakly to a law P ∞ which is the law of a N ∪ ∞-valued Markov process (Z(t), t ≥ 0) started from ∞, with the same transition semigroup on N as Y and whose entrance law can be characterized. Moreover, if we call τ = inf{t ≥ 0 : As #Π(·) has the same transition rates as Y (·) and the entrance law from ∞ is unique, these processes have the same law. Hence the following is a simple corollary of Lambert's result. Proof. If T = inf{t : #Π(t) < ∞}, Lambert's result implies that E(T ) < ∞ and hence T is almost surely finite. A simple application of Proposition 23 in [20] and Lemma 31 in [23] shows that if there exists t < ∞ such that #Π(t) < ∞ then inf{t : #Π(t) < ∞} = 0. To conclude, we can use Theorem 3.1 in [18] to see that as #Π(t) is positive recurrent in N, if Π(0) = 1 (or any partition with a finite number of blocks) then inf{t ≥ 0 : #Π(t) = ∞} = ∞. This entails that when an EFC process satisfying (H) and (L) reaches a finite level it cannot go back to infinity. As inf{t : #Π(t) < ∞} = 0, this means that ∀t > 0, #Π(t) < ∞.
If we drop the hypothesis ν Disl (S ↓ ) < ∞ (i.e., we drop (L) and we suppose ν Disl (S ↓ ) = ∞), the process Π stays infinite (Theorem 12). We now show that nevertheless, for a fixed t, almost surely B 1 (t) > 0. We define by induction a sequence of integers (n i ) i∈N as follows: we fix n 1 = 1, t 1 = 0 and for each i > 1 we chose n i such that there exists a time t i < t such that t i is a coalescence time at which the block 1 coalesces with the block n i and such that n i > w n i−1 (t i−1 ) where w k (t) is the least element of the kth block at time t. This last condition ensures that (w n i (t i )) is a strictly increasing sequence because one always has w n (t) ≥ n. The existence of such a construction is assured by the condition c k > 0. Hence at time t one knows that for each i there has been a coalescence between 1 and w n i (t i ). Consider (Π The fragmentations are given by the same PPP P F used to construct Π (and hence the processes are coupled). It should be clear that if w n i (t i ) is in the same block with 1 for Π (F ) (t) the same is true for Π(t) because it means that no dislocation separates 1 from w n i (t i ) during [0, t] for Π (F ) and hence also for Π, thus As by construction B Hence for all t > 0 one has P ({1} ⊂ dust(Π(t))) = 0 and thus P (dust(Π(t)) = ø) = 0.
Otherwise said, when ν Disl (S ↓ ) = ∞ the fragmentation part does not let a "(H)" EFC process come down from infinity, but it let the dust condensates into mass. Note that "binary-binary" 3 EFC processes are a particular case. The question of the case ν Disl (S ↓ ) < ∞ but (L) is not true remains open.

Missing mass trajectory
This last remark prompts us to study in more generality the behavior of the process of the missing mass In [20] it was shown (Proposition 26) that for a pure Λ-coalescence started from 0 (i.e., such that ν Coag ({x ∈ S ↓ : has the following behavior: and then D(t) almost surely jumps from 1 to 0 immediately (i.e., D(t) = 0 for all t > 0,) • or the coalescence is slow (c k = 0 and S ↓ ( i x i )ν Coag (dx) < ∞) and one has that ξ(t) is a drift-free subordinator whose Lévy measure is the image of ν Coag (dx) via the map x → − log(1 − x 1 ).
In the following we make the following hypothesis about the EFC process we consider The last assumption means that sudden dislocations do not create dust.
Before going any further we should also note that without loss of generality we can slightly modify the PPP construction given in Proposition 5 : We now suppose that P F is the sum of two point processes P F = P Disl + P e where P Disl has measure intensity µ ν Disl ⊗ # and P e has measure intensity c e e. If t is an atom time for P Disl one obtains Π(t) from Π(t−) as before, if P e has an atom at time t, say (t, k(t) ), then Π(t−) is left unchanged except for k(t) which becomes a singleton if this was not already the case. Furthermore, if t is an atom time for P C we will coalesce B i (t−) and B j (t−) at time t if and only if w i (t−) and w j (t−) (i.e., the least elements of B i (t−) and B j (t−) respectively) are in the same block of π (C) (t). This is equivalent to saying that from the point of view of coalescence the labelling of the block is the following: if i is not the least element of its block B i is empty, and if it is the least element of its block then B i is this block. To check this, one can for instance satisfy that the transition rates of the restrictions Π |[n] (·) are left unchanged.

Proposition 16. Let Π be an EFC process satisfying (H'). Then ξ is a solution of the SDE
where σ is a drift-free subordinator whose Lévy measure is the image of The case when c e = 0 is essentially a simple extension of Proposition 26 in [20] which can be shown with the same arguments. More precisely, we use a coupling argument. If we call (Π (C) (t), t ≥ 0) the coalescence process started from Π(0) and constructed with the PPP P C , we claim that for all t dust(Π(t)) = dust(Π (C) (t)).

This is clear by observing that for a given i if we define
is necessarily a collision time which involves {i} and the new labelling convention implies that Furthermore, given a time t, if i ∈ dust(Π(t)) then ∀s ≥ 0 : i ∈ dust(Π(t + s)). Hence for all t ≥ 0 one has dust(Π(t)) = dust(Π (C) (t)) and thus Proposition 26 of [20] applies.

Note that dust(Π(t))∩[n] can be strictly included in the set of the singletons of the partition Π |[n] (t).
Observe that the process D n is a Markov chain with state-space {0, 1/n, ..., (n − 1)/n, 1}. We already know that D is a càdlàg process and that almost surely, for all t ≥ 0 one has D n (t) → D(t).

First we show that
hence we will work on the right hand term. Fix S ∈ T n N . First we wish to control P (sup t≤θ (D n (S + t) − D n (S)) ≥ ). Observe that the times t at which ∆(D n (t)) = D n (t) − D n (t−) > 0 all are atom times of P F such that π (F ) (t) = i for some i ≤ n (recall that i is the partition of N that consists of two blocks: {i} and N\{i}) because under (H'), the only way in which dust can be created is erosion. Hence, clearly, The process is a sum of n independent standard Poisson processes with intensity c e , hence for each η > 0 and > 0 there exists θ 0 ≤ /c e and n 0 such that for each θ ≤ θ 0 and n ≥ n 0 one has where the first equality is just the strong Markov property in S and the last inequality is a consequence of the law of large numbers (observe that ). Hence, the bound is uniform in S and one has that for each θ ≤ θ 0 and n ≥ n 0 Let us now take care of P (sup t≤θ (D n (S) − D n (S + t)) ≥ ). We begin by defining a coupled coalescence process as follows: we let Π S (·) corresponds to P C . More precisely, if P C has an atom at time S + t, say π (C) (S + t), we coalesce Π We claim that for each t ≥ 0 Indeed suppose j ∈ {i ≤ n : ∀s ∈ [S, S + t] i ∈ dust n (s)}, then for r ≤ t one has j ∈ dust(π (C) (S + r)) (recall that the π (C) (·) are the atoms of P C ) and hence j has not yet coalesced for the process Π and thus (for the second inequality observe that t → {i ≤ n : ∀s ∈ [S, S + t] i ∈ dust n (s)} is decreasing). We can now apply the strong Markov property for the PPP P C at time S and we see that Define ξ n (t) := − log(D coag n (0, t)).
We know that almost surely, for all t ≥ 0 one has ξ n (t) → ξ(t) where ξ(t) is a subordinator whose Lévy measure is given by the image of ν Coag by the Thus, for any η > 0 there exists a θ 1 such that if θ < θ 1 one has lim sup n P (ξ n (θ) > − log(1 − )) < η. This bound being uniform in S, the conditions for applying Aldous' criterion are fulfilled.
It is not hard to see that D n (·), which takes its values in {0, 1/n, ..., n/n}, is a Markov chain with the following transition rates: • if k < n it jumps from k/n to (k + 1)/n with rate c e n(1 − k/n), • if k > 0 it jumps from k/n to r/n for any r in 0, ...., k with rate Hence, if A n is the generator of the semi-group of D n one necessarily has for any non-negative continuous f We wish to define the A n so they will have a common domain, hence we will let A n be the set of pairs of functions f, g such that f : Hence A n is multivalued because for each function f , any function g such that g(k/n) is given by (7)will work. In the following we focus on the only such g n which is linear on each [k/n, (k + 1)/n].
We know that D n ⇒ D in the space of càdlàg functions and that D n is solution of the martingale problem associated to A n . Define In the following f = sup x∈ [0,1] |f (x)|.
Proof. We decompose g n into g n = g (1) n + g (2) n where both g (1) n and g (2) n are linear on each [k/n, (k + 1)/n] and g (1) n (k/n) = One has that f ((k+1)/n)−f (k/n) Hence, as f is continuous on [0, 1], one has that Let us now turn to the convergence of g n (2). For a fixed x and a fixed θ one has that and [nx] when n → ∞. We need this convergence to be uniform in x. We proceed in two steps: first it is clear that For the second part fix > 0. There exists η > 0 such that ∀x, y Next it is clear that for each η > 0 there is a n 0 ∈ N such that ∀n ≥ n 0 and ∀x ∈ [η, 1] one has Hence, for n ≥ n 0 and x > θ and [nx] Hence, for any > 0, by choosing and η small enough, one can ensure that there exists a n 1 such that for all n ≥ n 1 one has We can bound θ) [nx]−r as follows: Hence one has that Finally we conclude that We can then apply the dominated convergence theorem and we get Hence, one has g (2) n − g (2) → 0 where g (2) One can now use Lemma IV. 5.1 (p. 196) in [14] to see that D must be solution of the Martingale Problem associated to A. Hence one can use Theorem III.2.26 in [15] to see that D is solution of where N (t) is a PPP with values in [0, 1] whose measure intensity is the image of ν Coag by the map Recall that ξ(t) = − log(D(t)) and observe that D is a bounded variation process. Some straightforward calculus then shows Hence, we conclude that where σ is a drift-free subordinator whose Lévy measure is the image of ν Coag by x → − log(1 − i x i ).

Proof of Proposition 2
The compatibility of the chains Π |[n] can be expressed in terms of transition rates as follows: For m < n ∈ N and π, π ∈ P n one has q n (π, π ).

Proof of Theorem 7
We first have to introduce a bit of notation: let B(i, t) denote the block that contains i at time t and define In the following we will use repeatedly a coupling technique that can be described as follows: Suppose Π is an EFC process constructed with the PPP P F and P C , we choose T a stopping time for Π, at time T we create a fragmentation process (Π (F ) (T + s), s ≥ 0) started from Π (F ) (T ) = Π(T ) and constructed with the PPP (P F (T + s), s ≥ 0). We call (B Consider an exchangeable variable Π = (B 0 , B 1 , B 2 , ...) where as before B 0 is used to contain all the singletons, and a fixed subset A = (a 1 , a 2 This implies that almost surely This result can easily be extended to the case of a random set A which is independent of Π by conditioning on A. Hence, if we start a homogeneous fragmentation (Π (F ) (T + s), s ≥ 0) from a partition that does not necessarily admit asymptotic frequencies, say Π (F ) (T ) = (..., A, ....) (i.e., A is one of the block in Π (F ) (0)), we still have that if a designates the least element of A then almost surelȳ and λ (F ) (a, T + s) → λ A when s 0. To prove Theorem 7, it suffices to prove the existence of the asymptotic frequency of B 1 (t) simultaneously for all t, the same proof then applies to the B(i, t) for each i. As Π(t) is an exchangeable process we already know that B 1 (q) exists simultaneously for all q ∈ Q. For such q we thus have thatλ 1 (q) = λ 1 (q). Hence, it suffices to show thatλ 1 (t) and λ 1 (t) are both càdlàg processes. In the following we write q t or q t to mean q converges to t in Q from below (resp. from above).
The first step is to show that: Lemma 19. Almost surely, the process (L(t), t ≥ 0) defined by exists and is càdlàg.
Proof. Using standard results (see for instance [22,Theorem 62.13]), and recalling thatλ 1 and λ 1 coincide on Q, one only need to show that q → λ 1 (q) = λ 1 (q) is a regularisable process, that is lim q tλ Using [22,Theorem 62.7], one only has to check that whenever N ∈ N and a, b ∈ Q with a < b, almost surely we have is the number of upcrossings ofλ 1 from a to b during [0, N ]. By definition sup{λ 1 (q) : q ∈ Q + ∩ [0, N ]} ≤ 1 and sup{λ 1 (q) : q ∈ Q + ∩ [0, N ]} ≤ 1. Suppose that q ∈ Q is such thatλ 1 (q) > b. Then if we define s = inf{r ≥ 0 :λ 1 (q + r) ≤ a} one can use the Markov property and the coupling with a fragmentation (Π (F ) (q + r), r ≥ 0) started from Π(q), constructed with the PPP (P F (q + r), r ≥ 0) to see that s ≥ θ where θ is given by θ := inf{r ≥ 0 :λ (F ) 1 (t + r) ≤ a}. If one has a sequence L 1 < R 1 < L 2 < R 2 , .... in Q such thatλ 1 (L i ) < a < b <λ 1 (R i ), then one has that for each i, Proof. For the first point, suppose that iλ i (T ) = 1 + γ > 1. Then there exists N ∈ N such that i≤Nλ i (T ) > 1 + γ/2. Call w i (t) the least element of B i (t). Let S be the random stopping time defined as the first time after T such that S is a coalescence involving at least two of the w N (T ) first blocks Hence, between T and S, for each i ≤ N one has that w i (T ) is the least element of its block. Applying the Markov property in T we have that S − T is exponential with a finite parameter and is thus almost surely positive.
Define (Π (F ) (T + s), s ≥ 0) as the fragmentation process started from Π(T ) and constructed from the PPP (P F (T +s), s ≥ 0). On the time interval [T, S] one has that for each i, the block of Π (F ) that contains w i is included in the block of Π that contains w i (because the last might have coalesced with blocks whose least element is larger than w N (T )).
Fix > 0, using (11) and the above remark, one has that for each i ≤ N there exists a.s. a θ i > 0 such that for all t ∈ [T, T + θ i ] one has As the random set Q ∩ [T, T + θ] is almost surely not empty, choosing small enough yields a contradiction with the fact that almost surely for all t ∈ Q one has λ i (t) ≤ 1.
Fix > 0, the first part of the lemma implies that there exists N ∈ N such that i≥N λ i (T ) ≤ .
To conclude the proof of the first point of Theorem 7, observe that as the map Π(t) →λ 1 (t) is measurable in F t , the right-continuous usual augmentation of the filtration, one has that for any > 0 inf{t : | lim sup s 0λ are stopping times for Π in F. The above lemma applies and hence this stopping times are almost surely infinite. The same argument works for λ 1 . This shows thatλ 1 and λ 1 are almost surely right-continuous processes. As they coincide almost surely with L on the set of rationals, they coincide everywhere and hence their paths are almost surely càdlàg. Before we can prove rigourously that X(t) is a Feller process, as stated in Theorem 7, we have to pause for a moment to define a few notions related to the laws of EFC processes conditioned on their starting point. By our definition, an EFC process Π is exchangeable. Nevertheless, if P is the law of Π and P π is the law of Π conditionally on Π(0) = π, one has that as soon as π = 0 or 1, the process Π is not exchangeable under P π (because for instance Π(0) is not exchangeable). The process Π conditioned by Π(0) = π (i.e., under the law P π ) is called an EFC evolution. Clearly one can construct every EFC evolution exactly as the EFC processes, or more precisely, given the PPP's P F and P C one can then choose any initial state π and construct the EFC evolution Π, Π(0) = π with P F and P C as usually. Let us first check quickly that under P π we still have the existence of X(t) simultaneously for all t.
In the following we will say that a partition π ∈ P is good if Λ(π) exists, there are no finite blocks of cardinality greater than 1 and either dust(π) = ø or dust(π) > 0.
Lemma 21. For each π ∈ P such that π is good, then P π -a.s. the process X(t) = Λ(Π(t)) exists for all t simultaneously and we call Q π its law.
Proof. Consider π = (B 1 , B 2 , ...) a good partition. For each i ∈ N such that #B i = ∞, let f i : N → N be the only increasing map that send B i on N. Let B 0 = ∪ i:#B i <∞ B i and if B 0 is infinite(which is the case whenever it is not empty) set g : N → N the unique increasing map that send B 0 onto N.
Using the exchangeability properties attached to the PPP's P F and P C one can easily see that for each i ∈ N such that #B i = ∞, are EFC processes with initial state 1 for the first ones and 0 for the later. Hence for each i one has that f i (Π(t) |B i ) has asymptotic frequencies X (i) (t) := Λ(f i (Π(t) |B i )) simultaneously for all t. Thus it is not hard to see from this that Π(t) |B i has asymptotic frequencies simultaneously for all t, namely B i X (i) (t). Fix > 0, there exists N such that If we call Π(t) = (B 1 (t), B 2 (t), ...) the blocks of Π(t), we thus have that for j ∈ N fixedλ As is arbitrary this shows that almost surely sup t≥0 sup i∈N (λ i (t) − λ i (t)) = 0. We call Q π the law of X(t) under P π .
If t is an atom time for P C , say π (C) (t), then Π 2 also coalesces at t, and if the blocks i and j merge at t in Π 1 then the blocks φ(t, i) and φ(t, j) merge at t for Π 2 , hence the coalescence is made with the usual rule by the partition φ −1 (t, π (C) (t)). CallP C the point process image of P C by the transformation (t, π (C) (t)) → (t, φ −1 (t, π (C) (t))).
We now show thatP C andP F are PPP with the same measure intensity as P C and P F respectively. The idea is very close to the proof of Lemma 3.4 in [3]. Let us begin withP F . Let A ⊂ P such that (µ ν Disl + c e e)(A) < ∞ and define The process N A is increasing, càdlàg and has jumps of size 1 because by construction the N Define .
A are orthogonal (because they do not share any jump-time) and hence the oblique bracket of M A is given by Hence M A is a L 2 martingale. This shows that N A (t) is increasing càdlàg with jumps of size 1 and has (µ ν Disl + c e e)(A)t as compensator. We conclude that N A (t) is a Poisson process of intensity (µ ν Disl +c e e)(A). Now take B ⊂ P such that A∩B = ø and consider N A (t) and N B (t), clearly they do not share any jump time because the N is a PPP with measure-intensity (µ ν Disl + c e e). Now, by the same arguments P (2) is also a PPP with measure-intensity (µ ν Disl + c e e) independent of P (1) F . By iteration we see thatP F is a PPP with measure intensity (µ ν Disl + c e e) ⊗ #.
Let us now treat the case ofP C . The main idea is very similar since the first step is to show that for n ∈ N fixed and π ∈ P such that π |[n] = 0 n one has that the counting process is a Poisson process with intensity (µ ν Coag + c k κ) (Q(π, n)).
For each unordered collection of n distinct elements in N, say a = a 1 , a 2 , ..., a n , let σ a be a permutation such that for each i ≤ n, σ a (i) = a i .
By exchangeability N a,π (t) is a Poisson process with measure intensity (µ ν Coag +c k κ)(Q(π, n)). By construction We see that we are in a very similar situation as before: the N a,π (t) are not independent but at all time t there is exactly one a such that and hence one can define orthogonal martingales M a (t) as we did for the M (i) A (t) above and conclude in the same way that N π,n (t) is a Poisson process with measure intensity (µ ν Coag + c k κ)(Q(π, n)). If we now take π ∈ P such that π |[n] = π |[n] we have that N π,n (t) and N π ,n (t) are independent because for each fixed a the processes given by the equation and dM a,π (t) = n i=1 1 {φ(t,a i )=i} dN a,π (t) respectively do not have any common jumps. Hence N π,n (t) and N π ,n (t) are independent and thus we conclude thatP C is a PPP with measure intensity µ ν Coag + c k κ.
Putting the pieces back together we see that Π 2 is an EFC evolution with law P π 2 and same characteristics as Π 1 .
For each π ∈ P such that Λ(π) = x exists, and for each k ∈ N we define n π (k) the label of the block of π which corresponds to x k , i.e., B nπ(k) = x k . In the case where two B k 's have the same asymptotic frequency we use the order of the least element, i.e., if there is an i such that x i = x i+1 one has n π (i) < n π (i + 1). The map i → n π (i) being bijective, call m π its inverse. Furthermore we define B 0 = ∪ i: B i =0 B i and x 0 = B 0 = 1 − i∈N x i . In the following we will sometimes write π = (B 0 , B 1 , ...) and x = (x 0 , x 1 , ...).
A simple application of Dynkin's criteria for functionals of Markov processes to be Markov (see Theorem 10.13 (page 325) in [13]) concludes the proof of the "Markov" part of Proposition 7. For the "Fellerian" part, for x ∈ S ↓ , call (Q x (t), t ≥ 0) the law of X(t) conditionally on X(0) = x. As X is rightcontinuous we must only show that for t fixed x → Q x (t) is continuous.
Let x (n) → x when n → ∞. The idea is to construct a sequence of random variables X (n) (t) each one with law Q xn (t) and such that X (n) (t) → X(t) almost surely and where X(t) has law Q x (t).
Using the Lemma 23 one has that This entails that for each j one has a.s.
Hence, almost surely, X (n) (t) → X(t) as n → ∞ and thus Q x (n) (t) → Q x (t) in the sense of convergence of finite dimensional marginals.
Let us begin with (18). Define Φ(q) := c e (q + 1) + This function was introduced by Bertoin in [6], where it plays a crucial role as the Laplace exponent of a subordinator; in particular, Φ is a concave increasing function. When k is an integer greater or equal than 2, Φ(k − 1) is the rate at which {[k]} splits, i.e., it is the arrival rate of atoms (π (F ) (t), k(t), t) of P F such that π (F ) |[k] (t) = 1 k and k(t) = 1. More precisely c e k is the rate of arrival of atoms that correspond to erosion and S ↓ (1 − i x k i )ν Disl (dx) is the rate of arrival of dislocations. Hence, for π ∈ P n such that #π = K, say π = (B 1 , B 2 , ...., B K , ø, ø, ...), one has q n (π, D n,K ) = because it only takes a fragmentation that creates at least one new block to enter D n,K . First, observe that i:|B i |>1 c e |B i | ≥ c e (n − K + 1), next, note thatΦ is also concave and increasing for the same reason that Φ is and furthermorẽ This ensure that for any n 1 , n 2 , ..., n K ∈ (N * ) K such that K i=1 n i = n one has On the other hand it is clear that for π ∈ D n,K the rate q n (π, A n,K ) only depends on #π and K (by definition the precise state π and n play no role in this rate). By compatibility it is easy to see that if π, π are such that #π > #π = K + 1 then q n (π, A n,K ) ≥ q n (π , A n,K ).

Proof of Theorem 9, part 2
Proof. We use the connection explained before Proposition 15. The set of conditions in Theorem 9, part 2 is just (L) and (H). Hence we can apply Theorem 3.1 in [18] to see that the process #Π(t) started from ∞ is positive recurrent in N and converges in distribution to some probability distribution on N.
Recall that A n,b designates the set of partitions π ∈ P n such that #π ≤ b and D n,b = P n \A n,b . For each b observe that min π∈D n,b ∩I n {q n (π, I c n )} = q n (π , I c n ) where π can be any partition in P n such that π ∈ I n and #π = b + 1. We can thus define f (b) := min π∈D n,b ∩In {q n (π, I c n )}.
If c k > 0 and π ∈ D n,b ∩ I n one can exit from I n by a coalescence of the Kingman type. This happens with rate greater than c k b. If ν Coag (S ↓ ) > 0 one can also exit via a coalescence with multiple collision, and this happens with rate greater than This ζ(b) is the rate of arrival of atoms π (C) (t) of P C such that π (C) (t) ∈ I b and which do not correspond to a Kingman coalescence. Thus sup b∈N ζ(b) is the rate of arrival of "non-Kingman" atoms π (C) (t) of P C such that π (C) (t) ∈ I := ∩ n I n . This rate being S ↓ ( i x i ) ν Coag (dx) and ζ(b) being an increasing sequence one has Thus it is clear that, under the conditions of the proposition, f (b) → ∞ when b → ∞. On the other hand, when π ∈ I c n , the rate q n (π, I n ) is the speed at which 1 is isolated from all the other points, thus by compatibility it is not hard to see that where q 2 is the rate at which 1 is isolated from its first neighbor (the inequality comes from the inclusion of events). Hence, ρ (n) (π)q n (π, I c n ) ≤ π ∈I c n ρ (n) (π )q n (π , I n ) ≤ π ∈I c n ρ (n) (π )q 2 ≤ q 2 which yields ρ (n) (I n ∩ D n,b ) ≤ q 2 /f (b). Now as ρ is exchangeable one has ρ(I ∩A b ) = 0 where I = ∩ n I n and A b = ∩ n A n,b (exchangeable partitions who have dust have an infinite number of singletons, and thus cannot have a finite number of blocks). Hence ρ (n) (I n ∩ A n,b ) → 0.