Stochastic Population Models: Measure-Valued and Partition-Valued Formulations

Transcription

1 Stochastic Population Models: Measure-Valued and Partition-Valued Formulations Diplomarbeit Humboldt-Universität zu Berlin Mathematisch-aturwissenschaftliche Fakultät II Institut für Mathematik eingereicht von icolas Simon Perkowski Betreuer: Prof. Dr. Peter Imkeller Berlin, den 4. ovember 2009

2 Acknowledgements I would like to thank my advisor Peter Imkeller for his support and his mentoring. Also I would like to thank Jochen Blath for giving me great further suggestions and ideas and for fruitful discussions. And last but not least I would like to thank Jean Bertoin for introducing me to the fascinating subjects treated in this work. 1

3 Contents 1 Introduction 4 2 Preliminaries Exchangeable Random Partitions Partitions of [n] Mass Partitions Exchangeable Random Partitions Exchangeable Coalescents Definition and Classification Examples Some Properties of Coalescents Exchangeable Coalescents and Martingale Problems Exchangeable Coalescents in Discrete Time Exchangeable Coalescents and Flows of Bridges Bridges and Exchangeable Partitions Flows of Bridges Fleming-Viot Process Weak Convergence Results Convergence of Rescaled Markov Chains Convergence of Markov Processes An Application Ξ-Fleming-Viot Processes Definition and Construction of the Ξ-Fleming-Viot Process Some Properties of the Ξ-Fleming-Viot Process Discrete Time Ξ-Fleming-Viot Processes Cannings Population Model The Model Convergence Results Convergence Criteria Convergence Results for Schweinsberg s Model Preliminary Results Proof of Theorem 6.1, Proof of Theorem 6.1, Proof of Theorem 6.1, Proof of Theorem 6.1, Proof of Theorem 6.1, A Poisson Point Processes 83 B Subordinators 84 C Martingale Problems 84 2

4 D Regular Variation 86 3

5 1 Introduction Consider the easiest model in population genetics: the Wright-Fisher model. That is, we consider a population that develops over time. The population is supposed to be haploid, i.e. each individual has exactly one ancestor. The generations are non-overlapping and of constant size. Further suppose that there is an infinite number of generations both in the future and in the past. Each individual in generation n chooses its ancestor uniformly among the individuals of generation n 1, independently of the choices of the other individuals. n=-2 n=-1 n=0 n=1 n=2 Figure 1: An example of the genealogical tree for a population of size seven for the generations 2 to 2. If we model the development of the distribution of genetic types forward in time, we obtain a measure-valued process in the limit for large populations: the so called Fleming-Viot process Kurtz, If we model the genealogical tree backward in time, we obtain a partition-valued process in the limit for large populations: Kingman s coalescent Kingman, 1982b. Those two processes are dual to each other. This was shown by Dawson and Hochberg They proved the duality of the Fleming-Viot process to a function-valued process, but their formulation can be easily adapted to prove the duality of Fleming-Viot process and Kingman s coalescent. The Wright-Fisher model is a special case of a class of population models that was introduced by Cannings 1974, Möhle and Sagitov 2001 studied the partition-valued formulation of Cannings model and obtained a general class of coalescents in the limit for large populations, so called exchangeable coalescents. Schweinsberg 2000a classified those exchangeable coalescents and proved that they are in one-to-one correspondance with finite 4

6 measures Ξ on the infinite simplex { := x 1, x 2,... R : x 1 x 2 0 and } x i 1 This is why exchangeable coalescents are also called Ξ-coalescents. If we consider only measures on that are concentrated on sequences of the form x 1, 0, 0,... and can thus be interpreted as measures on [0, 1], we also speak of Λ-coalescents. Bertoin and Le Gall 2003 introduced a generalisation of the Fleming-Viot process, so called Λ-Fleming-Viot processes, for which they gave an explicit Poisson construction. Also they showed that Λ-Fleming-Viot processes and Λ-coalescents are dual to each other. Ξ-Fleming-Viot processes that are a generalisation of Λ-Fleming-Viot processes were introduced explicitly by Birkner et al who gave a fundamentally different construction of these processes than Bertoin and Le Gall 2003 gave for their Λ-coalescents. In this work we want to generalize the result of Bertoin and Le Gall We will construct Ξ-Fleming-Viot processes and we will show the duality of Ξ-Fleming-Viot processes and Ξ-coalescents. Bertoin and Le Gall 2003 point out the possibility of such a generalisation and they state that details are left to the interested reader. Having obtained the duality between Ξ-coalescents and Ξ-Fleming-Viot processes, it is not surprising that we will be able to show convergence of the measure-valued formulation of Cannings model towards Ξ-Fleming- Viot processes. Finally, we slightly generalize a realistic population model introduced by Schweinsberg This population model is in the class of Cannings models and we can use the before obtained convergence results to show the convergence towards coalescents or Fleming-Viot processes, depending on the considered formulation. In the entire text, we will always consider the Borel σ-algebra, unless it is noted otherwise. We will denote the Borel σ-algebra of a topological space E by BE. 2 Preliminaries Unless it is noted otherwise, everything in this section is a translation of the corresponding sections from Perkowski Exchangeable Random Partitions In this chapter we introduce the important correspondance between exchangeable random partitions and mass partitions Partitions of [n] Definition Let B, B, be a subset of := {1, 2,... }. A partition π of B is a family of disjoint blocks π i : i such that i π i = B. We suppose that the π i are always enumerated by increasing order of their least element. 2. For a partition π of B, #π := is the number of non-empty blocks of π, i.e. #π := sup{i : π i }. 3. For i B, πi is the number of the block of π that contains i. 5

7 4. P n is the space of partitions of [n] := {1,..., n}, equipped with the discrete topology. 0 n is the partition of [n] in singletons. 5. P is the space of partitions of [ ] :=. 0 is the partition of in singletons. 6. For n, m n and π P n let R m π be the restriction of π to [m]: R m π is the unique partition in P m such that for i, j m, i and j are in the same block of R m π if and only if they are in the same block of π. 7. For n and π, π P n, we write π π if π is coarser than π, i.e. if π is obtained by coagulating blocks of π. We write π π if π is obtained by coagulating exactly two blocks of π. We introduce the notation i π j to express that i and j are in the same block of π. We define a distance ρ on P : ρπ, π := 2 inf{n:rnπ Rnπ } We would like P to be a Polish space. In fact it is even a compact metric space: Proposition 2.2. P equipped with the distance ρ is a compact metric space. Proof. We will show that P, ρ is complete and that each sequence in P admits a Cauchy subsequence. Let π n be a Cauchy sequence in P, and let m. So there is m such that for each n, n m we have ρπ n, π n < 2 m. So the sequence R m π n n is constant for large enough n. We define a partition π P such that i π j if and only if i πn j for each n that is large enough. The definition of ρ immediately implies the convergence of π n towards π. Let π n be a sequence in P. We consider R 2 π n. Since P 2 is finite, there is an infinite constant subsequence R 2 π nk. Then we consider R 3 π nk and select another infinite constant subsequence R 3 π nkl, etc. We obtain a Cauchy subsequence by choosing a diagonal sequence of this collection of subsequences of π n Mass Partitions Definition 2.3. A mass partition is a real-valued sequence x 1, x 2,... such that x 1 x 2 0 and x i 1 We define x 0 := 1 x i We denote by the infinite simplex of mass partitions. is a compact metric space: 6

8 Proposition 2.4. equipped with the uniform distance dx, x := max{ x i x i : i }, x, x is a compact space. Uniform convergence is equivalent to simple convergence. Proof. The equivalence of uniform convergence and simple convergence is a direct consequence of the fact that for each x = x i in we have x i 1/i for all i. Let x n be a sequence in. We want to show that x n admits a convergent subsequence. Since x n 1 n is a bounded sequence in R, we can choose a convergent subsequence x n k 1 x 1. ow we can choose a subsequence x n k l 2 of x n k 2 that converges to a x 2 R. We repeat this for each i. Then we choose a diagonal subsequence of all those subsequences. Denote that subsequence of x n by x m. So for each i, x m i converges to x i when m. Of course the limit x i is still monotone, i.e. x 1 x Fatou s lemma yields x i 1 Thus x = x i is in. Since uniform convergence is equivalent to simple convergence, x m converges uniformly to x. Example 2.5. Let ξ t, 0 t 1 be a pure jump subordinator with jumps a 1 a 2... in decreasing order. In the Appendix B there is an overview of subordinators. So a 1 /ξ 1, a 2 /ξ 1,... is a random point in, and the distribution of ξ t corresponds to a distribution on. Let α 0, 1 and let ξ t, t [0, 1] be a subordinator with Laplace exponent Φq = cq α = cα Γ1 α 0 1 e qx x α 1 dx. for some c > 0. Here, Γ is the gamma function, Γα = x α 1 e x dx. Such a ξ 0 t is called stable subordinator of index α. The Lévy measure of ξ is given by cα Λ α dx = Γ1 α x α 1 dx It satisfies Λ α x, = c Γ1 α x α The corresponding distribution on is called Poisson-Dirichlet distribution of index α,0, PDα,0. ote that the parameter c has no influence on the PDα, 0-distribution, since k α c corresponds to kξ t, t [0, 1] this can be immediately seen by calculating the Laplace exponent Exchangeable Random Partitions To define exchangeable random partitions, we first need to define permutations: A permutation of [n] for n is a bijective map from [n] to [n]. A permutation of is a bijective map σ from to such that there exists an with σn = n for each n. For each permutation σ of [n], n and for each partition π P n we define the partition ˆσπ as follows: for i, j [n], σi ˆσπ σj if and only if i π j. 7 k

9 Definition 2.6. A random partition π of [n] with n is called exchangeable if the law of π is invariant under permutations of [n], i.e. if for each permutation σ of [n], ˆσπ has the same distribution as π. Definition 2.7. A partition π of is said to have asymptotic frequencies if for each block B of π: 1 lim n n n 1 {i B} exists With the paintbox construction of Kingman 1982b we can associate an exchangeable random partition to each mass partition: Definition Let x. Let ξ n n be a sequence of independent and identically distributed i.i.d. random variables, such that Pξ 1 = i = x i, i, Pξ 1 = 0 = 1 Given the values of the ξ n, we define a partition π P such that i j are in the same block of π if and only if ξ i = ξ j > 0 So all i with ξ i = 0 are singletons of π. We denote the distribution of π by P x. P x is called a paint box distribution. To motivate this name, imagine that each number i corresponds to a color. 0 corresponds to a magic paint that has a different color each time it is used. Each element j is painted with the colour ξ j. Then all the elements with the same color are put in the same block of π. 2. For a distribution ν on we define a mixture of paint boxes: P ν dπ := P x dπνdx It is easily verified that those paint boxes correspond to exchangeable partitions that almost surely a.s. possess asymptotic frequencies. The second statement is obtained with the law of large numbers. Indeed, every exchangeable random partition is given by a mixture of paint boxes. To prove this, we will need de Finetti s theorem. The following version is Theorem 3.1 of Aldous 1985: Theorem 2.9 de Finetti. Let Z i i be an exchangeable sequence of real-valued random variables. That is, for each permutation σ of, Z σi i has the same distribution as Z i i. Then there exists a random probability measure µ on R cf. Definition A.1, such that Z i is i.i.d. conditionally on the σ-algebra created by µ PZ i A µω = µω, A ow we are ready to state the main result of this section. This theorem was established by Kingman The following proof is taken from Aldous 1985, Proposition 11.9, and we use details from the more elaborate version of Bertoin 2006, Theorem x i

10 Theorem 2.10 Kingman. Let π be an exchangeable random partition of. Then π a.s. possesses asymptotic frequencies. Let X 1 X 2... be the ordered sequence of the asymptotic frequencies of the different blocks of π where X n := 0 if π has less than n non-empty blocks. Then X = X 1, X 2,... is a.s. in, and conditionally on X, π has the distribution P X. In particular Pπ A = P x AGdx where G is the distribution of X. Proof. 1. b : is called selection map for the partition η if for all i, j in the same block of η we have bi = bj = k where k is an element of the same block of η. So let b be a selection map for π. Let ξ i i be an i.i.d. sequence that is uniformly distributed on [0, 1] notation: ξ i U[0, 1], independent of π and of b. We define Z i := ξ bi. Since b and π are independent of ξ i, the distribution of Z i i does not depend on the selection map b. 2. The sequence Z i is exchangeable: Let σ be a permutation of. We have where Z σi = ξ bσi = ξ b i ξ i := ξ σi and b i := σ 1 b σi b is a selection map for σ 1 π: Let i and j be in the same block of σ 1 π: Then σi and σj are in the same block of π, and thus bσi = bσj = σk for a certain k such that σk and σi are in the same block of π. But that means that k and i are in the same block of σ 1 π. Further we have b i = b j = σ 1 σk = k and therefore b is a selection map for σ 1 π. ξ i is an i.i.d. sequence that is uniformly distributed on [0, 1] and that is independent of σ 1 π and of b. Since π is exchangeable, σ 1 π has the same distribution as π, and thus Z σi has the same distribution as Z i. 3. We use de Finetti s theorem Theorem 2.9. Let µ be a random probability measure for Z i as in the theorem. We can choose it such that for each ω, the mass of µω is concentrated on [0, 1]. Let fµω be the ordered sequence µ 1 ω µω 2... of atoms of µω. That is, µ 1 ω is the size of the largest atom of µω, etc. We define µ n ω := 0 if µω has less than n atoms. Conditionally on µ, the distribution of π is given by P fµ : Let qx := inf{y : µ[0, y] x} 9

11 be the random quantile function of µ. We define θ := {x 0, 1 : ɛ > 0 such that qx = qy if y x < ɛ} The intervall lengths of θ correspond to the atom sizes of µ. Let V i i be an i.i.d. sequence, V i U[0, 1], independent of π, of Z i, and of µ. Then PqV 1 x µ = Pµ[0, x] V 1 µ = µ[0, x] so conditionally on µ, qv i has the same distribution as Z i. We define a partition π such that i and j are in the same block of π if and only if qv i = qv j. Conditionally on µ, π has the same distribution as π. But i and j are in the same block of π if and only if V i and V j are in the same intervall of θ. So conditionally on µ, π and therefore also π has the paint box distribution P fµ. We could define W i := k if V i is in the k-th largest intervall of θ and W i := 0 if V i is in no intervall of θ to see that we are really in the paint box setting. 4. Conclusion: We have Pπ A µ = P fµ A and conditionally on µ, π has asymptotic frequencies fµ. In particular, π a.s. possesses asymptotic frequencies. By taking expectations on both sides we get Pπ A = P x AGdx where G is the distribution of fµ, i.e. the distribution of the asymptotic frequencies of π. 2.2 Exchangeable Coalescents Definition and Classification We introduce coalescents with simultaneous multiple collisions and we show a correspondance between such coalescents and finite measures on the infinite simplex. Definition A coalescent is a stochastic process Πt t 0 with values in P n for n that is a.s. right-continuous and possesses left limits càdlàg and such that for all s > t 0: Πt is a refinement of Πs, i.e. Πt Πs. Definition Let B be a subset of. Let π be a partition of B. Let m #π and let π P m. We define the partition Coagπ, π as follows: Coagπ, π j := π i, j #π i π j where Coagπ, π j is the jth block of Coagπ, π. The coagulation operator has two elementary properties that will be very useful: 10

12 1. For π, π P, n, we have R n Coagπ, π = CoagR n π, R n π 2. If all the terms in the following equation are well-defined we have Coagπ, Coagπ, π = CoagCoagπ, π, π Definition Let b, r, k 1,..., k r 2, s 0 := {0, 1,... }, and b = k k r + s. π P b is called a b; k 1,..., k r ; s-partition if π has non-ordered blocks B 1,..., B r of respective sizes k 1,..., k r, and s singletons. Definition Let π P n, n and > b = #π. π is a b; k 1,..., k r ; s-collision of π if π = Coagπ, π where π is any b; k 1,..., k r ; s-partition. Here we will only consider coalescents that are Markov processes and for which the rate of each b; k 1,..., k r ; s-collision is the same. Definition Let m. A coalescent Πt t 0 with values in P m is called coalescent with simultaneous multiple collisions c.s.m.c. or exchangeable coalescent if for all n, m, n m: and R n Πt t 0 is a Markov chain with values in P n when R n Πt has b blocks, each b; k 1,..., k r ; s-collision happens with rate λ b;k1,...,k r;s If Π0 = 0 m, then Π is called standard. An important example of such coalescents is given by Kingman s coalescent. For this coalescent, the collision rates are λ b;2;b 2 = 1 for each b, and every other rate is 0. This means that the jump rate from π to π is 1 if π is formed from π by coagulating exactly 2 of its blocks, and otherwise the rate is 0. This process was introduced by Kingman 1982b to study the genealogy of large populations. The new idea that proved to be very successful was to consider a process with values in P n. Kingman proved that this coalescent arises in the limit for large populations in a number of models: The Wright-Fisher model, the Moran model which we will not study here, but also the general Cannings model if we assume the family sizes to be sufficiently bounded this will be expressed more precisely later in this text. The mathematical properties of Kingman s coalescent are described in Kingman 1982a. In 1998, Bolthausen and Sznitman 1998 introduced another exchangeable coalescent. This paved the way for the general classification of those processes: In 1999, Pitman 1999 and Sagitov 1999 introduced independently of each other coalescents with multiple collisions. Those are exchangeable coalescents with λ b;k1,...,k r;s = 0 for r > 1, i.e. each λ that is not of the form λ b;k;b k is 0. This evidently means that for such coalescents we can have a collision of several blocks not just of two blocks as for Kingman s coalescent, but a.s. no two such collisions happen at the same time. Coalescents with simultaneous multiple collisions were obtained the first time by Möhle and Sagitov 2001 as limits of Cannings population models. A classification of c.s.m.c. s was given by Schweinsberg 2000a. In this article Schweinsberg proved that c.s.m.c. s are in one-to-one correspondance with finite measures on the space of mass partitions : 11

13 Theorem Let {λ b;k1,...,k r;s : r, b, k 1,..., k r 2, s 0, b = r j=1 k j + s} be a family of positive i.e. 0 numbers. Then there exists a standard coalescent with simultaneous multiple collisions with values in P with collision rates λ b;k1,...,k r;s, if and only if there is a finite measure Ξ on, Ξ = Ξ 0 + cδ 0 where Ξ 0 has no atom in 0 := 0, 0,..., δ 0 is the Dirac mass in 0 and c 0, such that Q k1,...,k λ b;k1,...,k r;s = r;sx Ξ 0 dx + c1 {r=1,k=2} with 1 j=1 x2 j s s l s Q k1,...,k r;sx := x k 1 i l 1... x kr i r x ir+1... x ir+l 1 x j 2 l=0 i 1 =i r+l For each c.s.m.c., the associated measure Ξ is uniquely determined. Remark. 1. ote that the integral in 1 is well-defined, as Ξ 0 has no atom in The formula 1 is the formula that was originally established by Schweinsberg 2000a. There is another formula given by Bertoin Bertoin considers the measure / νdx := 1 Ξ 0 dx + cδ 0 that is not necessarily finite on. j=1 Definition A c.s.m.c. Πt t 0 with rates λ b;k1,...,k r;s given by 1 is called Ξ-coalescent. Poissonian Construction To show that condition 1 is sufficient, we construct a Ξ- coalescent with a Poisson point process construction cf. Appendix A for an overview of Poisson point processes. This construction was originally given by Schweinsberg 2000a, but we present the slightly adapted version of Bertoin 2006, Chapter onetheless some details in the proof are taken from Schweinsberg 2000a. Let ν be a σ-finite measure on such that ν{0} = 0 and x 2 jνdx < 3 j=1 Let c 0. We associate a σ-finite measure µ on P to ν and c: For i, j let κi, j be the unique partition of that consists of one block of size two, {i, j}, and otherwise only of singletons. We define µdπ := P x dπνdx + c x 2 j j=i+1 j=1 1 κi,j dπ 4 Since ν is σ-finite, µ is σ-finite as well. Let et t 0 be a Poisson point process of intensity µ. We will use et to construct processes Π n t t 0 with values in P n. Then we will see that all the Π n are compatible: a.s. 12

14 R m Π n t = Π m t for each t. Therefore we can define a process Πt t 0 with values in P such that R n Πt = Π n t for t 0, n. For n we define and for k, l : A n := {π P : R n π 0 n } A k,l := {π P : k and l are in the same block of π} 5 We have µa n = n 2 n n k=1 l=k+1 j=1 µa k,l = n n k=1 l=k+1 x 2 jνdx + 1 < P x A k,l νdx + 1 The last inequality comes from 3. We define T 0,n := 0 and for k 1: T k,n := inf{t > T k 1,n : et A n }. Since µa n <, the T k,n correspond to jump times of a Poisson process. Thus they are without cluster point and we have et k,n A n for k 1. Given a partition π P, we define Π π n0 := R n π and Π π nt k,n := CoagΠ π nt k 1,n, et k,n ow let m < n. Since A m A n, Π π m and Π π n are constant on the intervall [T k,n, T k+1,n for each k 0. Thus it suffices to verify the equality Π π mt = R m Π π nt a.s. for t {T k,n : k 0 }. For k = 0 this is trivial. Let k 1. Recall that for a partition η, ηi is the number of the block containing i. Let i, j [m]. Then i and j are in the same block of Π π mt k,n if and only if Π π mt k 1,n i and Π π mt k 1,n j are in the same block of et k,n. On the other side i and j are in the same block of Π π nt k,n and thus of R m Π π nt k,n if and only if Π π nt k 1,n i and Π π nt k 1,n j are in the same block of et k,n. But since the blocks of partitions are enumerated by increasing order of their least element, and since by induction hypothesis Π π mt k 1,n = R m Π π nt k 1,n, we have Π π nt k 1,n i = Π π mt k 1,n i for each i [m]. We obtain Π π mt k,n = R m Π π nt k,n. The construction of Π π is now evident: Let i, j, then i and j are in the same block of Π π t if they are in the same block of Π π max{i,j} t. Using the definition of the topology on P it is evident that Π π is càdlàg and that for each t < s, Π π t is a refinement of Π π s. Therefore we constructed a coalescent. Given a finite measure Ξ = Ξ 0 + cδ 0 on, we define νdx := Ξ 0dx and we construct j=1 x2 j Π π exactly like we just did. It remains to show that Π π is a Ξ-coalescent. Proposition 2.18 Sufficient Condition of Theorem The process Π π t t 0 constructed as above is a Ξ-coalescent. Proof. 1. R n Π π is a Markov chain: R n Π π = Π π n where Π π n is the process of the construction. By using the construction and the independent increments 55 of Poisson point processes, it is easily verified that Π π n is a Markov chain. 13

15 2. Each b; k 1,..., k r ; s-collision has the rate λ b;k1,...,k r;s: Let n. Let π P such that R n π has b blocks. Let π be a b; k 1,..., k r ; s- partition. We denote its non-ordered blocks of size 2 by B 1,..., B r. The jump rate of R n Π π 0 = Π π n0 to CoagR n π, π is given by µa,π with A,π := {η P : R b η = π }. We calculate P x A,π : Recall that P x was constructed by i.i.d. variables ξ i. If R b η = π, there exist necessarily 0 l s, i 1 i r+l all 0 and 1 m 1 < < m l b such that ξ m = i j for m B j, 1 j r ξ mj = i r+j, 1 j l ξ m = 0 for m b, m / j=1,...,r B j {m1,..., m l } By summing up all the possible combinations we obtain P x A,π = s l=0 s x k1 i l 1... x kr i r x ir+1... x ir+l 1 i 1 =i r+l s l x j = Q k1,...,k r;sx 6 j=1 This implies λ b;k1,...,k r;s = µa,π = Q k1,...,k r;sxνdx + c1 {r=1,k=2} = Q k1,...,k r;sx/ Ξ 0 dx + c1 {r=1,k=2} j=1 x 2 j and this is the desired formula 1. ecessary Condition of Theorem 2.16 Given the λ b;k1,...,k r;s, we will construct a σ-finite measure µ on P. Then we will associate a σ-finite measure ν on and a c 0 to µ. We will see that ν satisfies 3, and we will be able to define a finite measure on by setting Ξdx := j=1 x2 jνdx + cδ 0. Then we will see that the rates λ b;k1,...,k r;s are given by 1. We choose this complicated way to obtain the results of Schweinsberg 2000a that we want to use with the methods of Bertoin 2006 that reveal more about the structure of coalescents with simultaneous multiple collisions. Definition Given π P n, n, we define for m, m > n: A m,π := {π P m : R n π = π} Proposition There is an unique measure µ on P such that µa,π = λ b;k1,...,k r;s for each b; k 1,..., k r ; s-partition π. This measure satisfies 1. µ is invariant under permutations of then µ is called exchangeable, 2. µ{0 } = 0, 3. µ{π P : R n π 0 n } < for each n 14

16 Proof. For each b; k 1,..., k r ; s-partition π with r > 0 let q π := λ b;k1,...,k r;s We define A n := σ{a,π : π P n \{0 n }} and A := n A n It is easily verified that A is an algebra. We define a measure µ 0 on A by µ 0 A,π := q π To verify that µ 0 is σ-additive, we consider π P n and m > n. Since R n R m Π 0 = R n Π 0, we have q π = q π 7 π A m,π which is the same as µ 0 A,π = µ 0 π A m,π A,π = µ 0 A,π π A m,π µ 0 is evidently additive on A n, thus we have a σ-additive measure on an algebra A. We can use Caratheodory s extension theorem to extend µ 0 to an unique measure µ on BP \{0 } = σa if we consider each A n as sub-set of P \{0 } rather than P. To obtain a measure on σp, we define µ{0 } := 0. µ satisfies condition 2 by definition. Condition 3 is satisfied since µ{π P : R n π 0 n } = π P n\{0 n} Condition 1 is satisfied since q π = qˆσπ for each permutation σ of [n]. Proposition Let µ be the measure of Proposition There are a unique measure ν on and a unique c 0 such that µdπ = P x dπνdx + c δ κi,j dπ. ν satisfies We even have a stronger result: ν0 = 0 and j=1 j=i+1 x 2 jνdx <. 1. µ-almost every a.e. π has asymptotic frequencies 2. ν is given by νdx = 1 {x 0} µ π dx where π denotes the asymptotic frequency of π, and 1 { π 0}µdπ = P x dπνdx 15 q π

17 3. 1 { π =0}µdπ = c j=i+1 δ κi,jdπ Proof. 1. For n we introduce µ n dπ := 1 {Rnπ 0 n}µdπ. Since µ{π : R n π 0 n } < cf. Proposition 2.20, µ n is a finite measure on P. Let µn be the image measure of µ n under π π where i π j n + i π n + j Since µ is exchangeable, µ n is a finite exchangeable measure on P. From Kingman s theorem applied to µ n./ µ n P we obtain that µ n -a.e. π possesses asymptotic frequencies and that µ n is given by µn dπ = P x dπ µ n π dx 8 Let A := {π : π possesses asymptotic frequencies}. We have µ{0 } = 0 and the asymptotic frequencies of a partition π do not depend on R n π for n <. Thus µa = lim n µ n A = lim n µ n {π : π possesses asymptotic frequencies} = lim n µn A = 1 which yields the first statement of the theorem. 2. By using the same measure extension argument as in the proof of Proposition 2.20, we see that it suffices to show µr k π = π k, π 0 = P x R k π = π k 1 {x 0} µ π dx 9 for k and π k P k. So let k and π k be given. By monotone convergence we obtain µr k π = π k, π 0 = lim n µr k π = π k, π 0, π {k+1,...,k+n} 0 {k+1,...,k+n} where 0 {k+1,...,k+n} is the partition of {k + 1,..., k + n} into singletons. Since µ is exchangeable, this expression equals = lim n µn R k π = π k, π 0 8 = lim P n x R k π = π k 1 {x 0} µn π dx With the same argument that we used in the proof of 1., we see that π does not change under π π; hence we obtain = lim P n x R k π = π k 1 {x 0} µ n π dx = lim P n x R k π = π k 1 {x 0} µ π dx, R n π 0 n 16

18 By monotone convergence and using µ{0 } = 0, we obtain the desired equation: µr k π = π k, π 0 = P x R k π = π k 1 {x 0} µ π dx Hence it suffices to define νdx := 1 {x 0} µ π dx. It remains to show that But this is easy now: j=1 x 2 jνdx = x 2 jνdx < j=1 P x 1 π 2νdx 9 = µr 2 π = {1, 2}, π 0 µr 2 π 0 2 < The last inequality is condition 3 of Proposition Let µdπ := 1 {1 π 2, π =0} µdπ and let µ be the image measure of µ under π π where i π j 2 + i π 2 + j µ is a finite exchangeable measure on P and under µ, a.e. π has the asymptotic frequency 0. Hence µ is a Dirac mass in 0. Since µ is exchangeable and µp <, µ j 3 : 1 π j = µ1 π j = 0. j=1 Therefore µ = cδ κ1,2 for some c 0. Since µ is exchangeable, we deduce 1 { π =0}µdπ = c j=i+1 δ κi,j To obtain the rates λ b;k1,...,k r;s, we first calculate λ b;k1,...,k r;s as a function of ν: Let π be a b; k 1,..., k r ; s-partition. Then λ b;k1,...,k r;s = µa,π = P x A,π νdx + c 6 = Q k1,...,k r;sxνdx + c1 {r=1,k1 =2} j=i+1 δ κi,j A,π By defining Ξdx := j=1 x2 jνdx, we obtain a finite measure on such that the λ b;k1,...,k r;s are given by 1. 17

19 2.2.2 Examples Without doubt the most prominent example of an exchangeable coalescent is Kingman s coalescent. It corresponds to Ξ = δ 0. This coalescent has some interesting properties: Proposition Let Πt, t 0 be a standard Kingman coalescent with values in P. 1. Π comes down from infinity. This means that for each t > 0, a.s. #Πt <. Further, a.e. block of Πt is of infinite size. 2. D t := #Πt, t > 0 is a pure death process with death rate k 2, k. More precisely,d t is a Markov process with values in and with jump rates for all k. 1 lim h h PD t+h = l D t = k = 3. Each trajectory of Πt passes by a sequence { k 2, l = k 1 0, otherwise... R k R k 1 R 2 R 1 where R k is the state of Π when #Π = k. The sequence R k is independent of D t, it is Markovian, and for each k, conditionally on R k+1 = π, R k is distributed uniformly on the k+1 2 partitions that are obtained by coagulating exactly two blocks of π. 4. As a consequence of 2. and 3. we obtain: For all S BP PR t S = PD t = kpr k S k=1 The proof of this proposition can be found in Kingman 1982a, Theorem 4. An entire class of Ξ-coalescents that are particularly easy to describe are coalescents with multiple asynchronous collisions that were introduced independently by Pitman 1999 and Sagitov Definition A coalescent with multiple asynchronous collisions or simple coalescent is an exchangeable coalescent that corresponds to a finite measure Λ on which satisfies Λ{x = x 1, x 2,... : x 2 > 0} = 0 In this case we could rather consider the image measure of Λ under the projection x 1, x 2,... x 1. Thus we can view Λ as a finite measure on [0, 1]. In this setting the rates λ b;k1,...,k r;s are given by λ b;k := λ b;k;b k = x k 2 1 x b k Λdx [0,1] and all other rates are 0. In words, a simple coalescent is an exchangeable coalescent without simultaneous collisions. At each collision time, several blocks are selected and united to form a single new block. Pitman showed in Proposition 23 of Pitman 1999 that each simple coalescent comes down from infinity or stays infinite, which means that the coalescent a.s. has an infinite number of blocks at each time t. 18

20 Example For r, s > 0, we can consider Λ = Betar, s. Betar, s is the distribution on [0, 1] with density x r 1 1 x s 1 where B is the beta-function, Br, s Br, s = ΓrΓs Γr + s = In this case the jump rates are given by 1 0 x r 1 1 x s 1 dx λ b;k = Bk + r 2, b + s k. Br, s Schweinsberg showed in Schweinsberg 2000b, Example 15, that a standard Betar, s-coalescent comes down from infinity if and only if r < 1. In the case r = s = 1, Beta1, 1 is the uniform distribution on [0, 1]. We denote it by U. The U-coalescent has jump rates λ b;k = k 2!b k! b 1! and was introduced by Bolthausen and Sznitman The standard U-coalescent does not come down from infinity Some Properties of Coalescents Elementary Properties Remark. 1. With the Poisson-construction one can easily see that a Ξ-coalescent Π π t t 0 with Π π 0 = π is obtained by defining Π π t := Coagπ, Πt, t 0 where Πt t 0 is a standard Ξ-coalescent. We even have a stronger result: Conditionally on Π π t, Π π t + s s 0 has the same distribution as CoagΠ π t, Πs s If Πt t 0 is a standard exchangeable coalescent, then for each t 0, Πt is a random exchangeable partition. This is equally verified with the Poissonian construction since the measure µ that we had constructed on P was exchangeable and the coagulation of two independent exchangeable partitions is still exchangeable cf. Bertoin 2006, Lemma Let Ξ be a finite measure on with Ξ 0. The case Ξ = 0 is trivial, since in that case all jump rates are 0. We define G := Ξ/Ξ. Then G is a probability on, and with the definition of the jump rates 1, we see that the rates of the G-coalescent are given by dividing the rates of the Ξ-coalescent by Ξ. Modulo a change of the time scale we can therefore suppose Ξ = 1. 19

21 Lemma The jump rates {λ b;k1,...,k r;s : r, b, k 1,..., k r 2, s 0, b = k k r + s} of an exchangeable coalescent satisfy the following consistency relation: r λ b;k1,...,k r;s = λ b+1;k1,...,k m 1,k m+1,k m+1,...,k r;s + sλ b+1;k1,...,k r,2;s 1 m=1 + λ b+1;k1,...,k r;s+1 10 where we define λ b;k1,...,k r; 1 := 0. This equation can be rewritten as r λ b;k1,...,k r;s+1 =λ b;k1,...,k r;s m=1 λ b+1;k1,...,k m 1,k m+1,k m+1,...,k r;s sλ b+1;k1,...,k r,2;s 1 11 This is a recurrence equation that allows us the calculate all the rates when we are only given the λ b;k1,...,k r;0, b, r, k 1,..., k r 2, b = k k r We do not give the proof here. This is Lemma 18 in Schweinsberg 2000a. The proof is elementary and it is based on the fact that R n R n+1 Π = R n Π for an exchangeable coalescent Π. oting this, one distinguishes the different possibilities for the behaviour of n+1 in R n+1 Π, and one gets the desired equation. Behaviour at Collision Times Lemma Let Ξ be a probability on and let Πt t 0 be a standard Ξ-coalescent. Let for i j τ i,j := inf{t 0 : i Πt j}. Let B 1, B 2,... be the blocks of Πτ i,j that are possibly empty for large enough k. Let π P #Πτi,j be the unique partition with k π l if and only if B k and B l are in the same block of Πτ i,j. Then π is the restriction of a partition π P to {1,..., #Πτ i,j }. π is invariant under permutations of that do not change Πτ i,j i and Πτ i,j j, and π a.s. possesses asymptotic frequencies that have distribution Ξ. Sketch of the proof. 1. Without loss of generality we suppose that Π is given by the Poissonian construction. Since Πt is exchangeable for each t, it suffices to show the statement for i, j = 1, We have τ 1,2 = inf{t 0 : et A 1,2 } where A 1,2 is defined as in 5. It suffices to show eτ 1,2 Ξ. Let S B. We define A S 1,2 := {ε A 1,2 : ε S}. The formula 57 of the Appendix A yields: P eτ 1,2 S = Peτ 1,2 A S 1,2 57 = µas 1,2 µa 1,2 [ / 1 = x 2 µa 1,2 j1 {x S} x 2 jξ 0 dx + c = 1 µa 1,2 j=1 j=1 [ Ξ0 S + c1 {0 S} ] = 1 µa 1,2 ΞS 20 j=i+1 δ {κi,j A S 1,2 } ]

22 Since µa 1,2 = 1, the proof is complete. Coming from Infinity and Proper Frequencies Let Π t : t 0 be a simple standard coalescent with rates λ b;k. We denote by γ b the rate with which the number of blocks of R b Π decreases, i.e. γ b := b k=2 k 1 b k λb;k Schweinsberg 2000b showed that Π comes down from infinity if and only if b=2 γ 1 b < For general exchangeable coalescents we do not know an equally simple condition that is equivalent to the coming down from infinity. But there is a nice result on the asymptotic frequencies: Definition Let π P be a partition that possesses asymptotic frequencies and let x 1, x 2,... be the ordered sequence of its frequencies. We say that π has proper frequencies if x j = 1 Otherwise we say that π has dust. j=1 Proposition Let Ξ = Ξ 0 + cδ 0 be a finite measure on with Ξ0 = 0 and c 0. Let Π t : t 0 be a standard Ξ-coalescent and let t > 0. Then Π t a.s. has proper frequencies if and only if c > 0 or if / x j x 2 jξ 0 dx = j=1 j=1 Proof. Let ν be the distribution of the asymptotic frequencies of Π t. Then the distribution of Π t is given by P x dπνdx With the definition of the paint box P x we see that Π t a.s. has proper frequencies if and only if {1} a.s. is not a block of Π t. Without loss of generality we suppose that Π is given by the Poisson construction with Poisson point process et t 0 of intensity µ. We define A := {π P : {1} is no block of π} and T A := inf{t 0 : et A}. Then {1} is a block of Π t if and only if T A > t. We have PT A > t = 0 if and only if µa =. But / µa = P x AΞ 0 dx x 2 j + c 1 {κi,j} A = j=1 j=1 j=1 j=i+1 / x j x 2 jξ 0 dx + c 1 and this is infinite if and only if c > 0 or j=1 x j j=2 / j=1 x2 jξ 0 dx =. 21

23 Feller Property We recall the definition of a Feller process cf. Revuz and Yor 1999, Definition 2.1 and Definition 2.5 of Chapter III.. Since P is compact, C 0 P = CP. Definition A Feller semi-group on CP is a family of linear positive i.e. f 0 P t f 0 operators T t t 0 on CP such that 1. T 0 = Id and T t 1 for t 0, 2. T t+s = T t T s for t, s 0 and 3. lim t 0 T t f = f in CP for each f CP. A Feller process is a Markov process with a Feller semi-group. Proposition 2.30 Feller Property. Let Ξ be a finite measure on. Each Ξ-coalescent is a Feller process in its canonic filtration. Its semi-group is given by where Π is a standard Ξ-coalescent. P t fπ = EfCoagπ, Π t Proof. Let Π t : t 0 be a Ξ-coalescent. Without loss of generality we can suppose that Π is given by the Poissonian construction with Poisson point process e. We already remarked that conditionally on Π t, Π t+s : s 0 has the same distribution as CoagΠt, Πs : s 0 where Π is a standard Ξ-coalescent that is independent of Π. Indeed this remains true if we condition on Π r : 0 r t since et + s s 0 is independent of er 0 r<t. So Π t is a Markov process. It remains to show that its semi-group P t : t 0 is Feller. It suffices to show that P t CP CP and that for each f CP and for each π P, we have lim t 0 P t fπ = fπ cf. Proposition 2.4 of Chapter III of Revuz and Yor Let π P, let Π π t : t 0 be a Ξ-coalescent with Π π 0 = π and let Π t : t 0 be a standard Ξ-coalescent that is independent of Π π. Let f CP. We have P t fπ = EfΠ π t = EfCoagπ, Π t But it is easily verified that the Coag operator is continuous from P P to P. With dominated convergence we obtain the continuity of P t f. It remains to show that lim t 0 P t fπ = fπ. But this follows immediately since Π is right-continuous, Π 0 = 0 and Coagπ, 0 = π. Then we use once again dominated convergence and we obtain the desired result. We remark that as a consequence each Ξ-coalescent admits the strong Markov property cf. Theorem 3.1 in Chapter III. of Revuz and Yor Exchangeable Coalescents and Martingale Problems We want to show that the Ξ-coalescent is the unique solution to an easily described martingale problem. Let λ b;k1,...,k r;s be the rates of a Ξ-coalescent. We write λ π := λ b;k1,...,k r;s for every b; k 1,..., k r ; s-partition π. Let D := {F CP : n, F CP n, F π = F R n π π}. We define an operator Q : D CP, F η P n λ η F CoagR n, η F R n 22

24 Because of the consistency relation 10 this operator is well-defined, and of course it is just the restriction of the infinitesimal generator of the Ξ-coalescent to D. So we know that the Ξ-coalescent with starting distribution ν is a solution to the Q, ν-martingale problem cf. Appendix C for an overview of martingale problems. Proposition For Q and ν as above, every solution to the Q, ν-martingale problem has the same finite-dimensional distributions as the Ξ-coalescents starting with distribution ν. Any solution with càdlàg paths is a Ξ-coalescent. Proof. Let Π be a solution. Then for any n, R n Π is a solution to the Q n, ν n -martingale problem with Q n : BP n BP n, Q n F = η P n λ η F Coag, η F where BP n is the space of bounded measurable functions on P n and ν n := ν Rn 1. But for a finite state space there is uniqueness for any martingale problem cf. example in Appendix C. That means that for every solution Π of the Q, ν-martingale problem the finite-dimensional distributions of R n Π are uniquely determined. The functions depending only on R n π form an algebra in CP that separates points and contains constants. So it is dense in the uniform topology by the Stone-Weierstrass theorem. Thus we obtain the uniqueness of the finitedimensional distributions for solutions to the martingale problem. Since the Ξ-coalescents is a solution, this means that any solution has the same finite-dimensional distributions as the Ξ-coalescent. We immediately obtain from Proposition C.3 in Appendix C that a solution with càdlàg paths is a Ξ-coalescent Exchangeable Coalescents in Discrete Time In this section we introduce a discrete time version of the Ξ-coalescent. Under certain assumptions we will obtain such processes as limits of Cannings population models. For a Ξ-coalescent to exist it is necessary that Ξ satisfies an additional condition. Proposition Let {p b;k1,...,k r;s : b, r, k 1,..., k r 2, s 0, b = r j=1 k j + s} be a family of non-negative numbers. Then there exists a discrete time process Y m : m 0 with values in P with Y 0 = 0 and such that for n, R n Y m is a Markov chain that satisfies for all π with #π = b, for each b; k 1,..., k r ; s-collision ε of π and for all m 0 : PR n Y m + 1 = ε R n Y m = π = p b;k1,...,k r;s if and only if p b;k1,...,k r;s = Q k1,...,k r;sx Ξdx 12 j=1 x2 j for a finite measure Ξ on, without atom in 0, which satisfies / 1 x 2 j Ξdx 1 13 j=1 In this case, the measure Ξ is uniquely determined. 23

25 In the demonstration we simply reduce the discrete time case to the continuous time case. Proof. 1. ecessary condition and uniqueness: Let Y be such a discrete time process. Let t, t 0 be a Poisson process with parameter 1, independent of Y. Define Πt := Y t, t 0. Then Π is a standard exchangeable coalescent with jump rates {p b;k1,...,k r;s}. Hence there exists a unique finite measure Ξ on such that the p b;k1,...,k r;s are given by 12. Let λ b be the total collision rate of a Ξ-coalescent with b blocks, i.e. λ b = = b/2 r=1 {k 1,...,k r} b/2 r=1 {k 1,...,k r} b; k 1,..., k r ; sλ b;k1,...,k r;s b; k 1,..., k r ; sp b;k1,...,k r;s 1 where b; k 1,..., k r ; s is the number of b; k 1,..., k r ; s-partitions in P b, and x is the largest integer that is smaller than x. We necessarily have λ b 1 for all b. Let µ and c 0 be associated to Ξ like in the Poissonian construction. We have λ b = µ{η : R b η 0 b } b = P x P \{0 } Ξ 0 dx + c j=1 x2 j 1 Ξ 0 dx + c j=1 x2 j j=i+1 j=i+1 δ κi,j P \{0 } δ κi,j P \{0 } For this expression to be 1, it is necessary that Ξ has no atom in 0 and satisfies Sufficient condition: Let Ξ be a finite measure on that has no atom in 0 and that satisfies 13. Let Πt be a standard Ξ-coalescent, given by the Poissonian construction. Let et and µ be as in the Poissonian construction. We have µp \{0 } = P x P \{0 } Ξdx 1 j=1 x2 j If we define T 0 := 0, T k := inf{t > T k 1 : et P \{0 }}, k 1, we obtain a sequence 0 = T 0 < T 1 <.... Let I m : m 0 be an i.i.d. sequence of Bernoulli variables, independent of e, such that PI m = 1 = µp \{0 }. Let S m := I I m. We define a discrete time Markov process Y by setting Y m := ΠT Sm. Let n, let π P n with b blocks, and let ε be a b; k 1,..., k r ; s-collision of π, ε = Coagπ, η with η P b. Using the strong Markov property of the Poisson point process e and the property 57 from Appendix A, we obtain PR n Y m + 1 = ε R n Y m = π = PI m+1 = 1PR n ΠT Sm+1 = ε R n ΠT Sm = π p b;k1,...,k r;s = µp \{0 }PR b et Sm+1 = η = µp \{0 } µp \{0 } = p b;k1,...,k r;s 24

26 Since we saw that for every discrete time exchangeable coalescent with transition probabilities p b;k1,...,k r;s there exists a continuous time exchangeable coalescent with jump rates p b;k1,...,k r;s, we know that the p b;k1,...,k r;s must also satisfy the recursion Exchangeable Coalescents and Flows of Bridges Bridges and Exchangeable Partitions In this chapter we present an interesting correspondance between exchangeable coalescents and flows of bridges that was established by Bertoin and Le Gall Definition A bridge is a stochastic process Br : r [0, 1] such that 1. B0 = 0, B1 = 1, B has increasing càdlàg paths. 2. For all n : B1/n B0, B2/n B1/n,..., B1 B1 1/n is an exchangeable vector. The general classification of processes with exchangeable increments was given by Kallenberg 1973, Theorem 2.1. In our setting this result can be expressed as follows: Proposition 2.34 Kallenberg. Br : r [0, 1] is a bridge if and only if there is a random variable X with values in and an i.i.d. sequence U i i of uniform variables on [0, 1], independent of X, such that Br : r [0, 1] has the same distribution as 1 X j r + X j 1 {Uj r} : r [0, 1] j=1 j=1 In the following we will always assume that B is of this form. We can associate an exchangeable partition to each flow of bridges. We define the càdlàg inverse of B: B 1 s := inf{r [0, 1] : Br > s}, s [0, 1 et B 1 1 := 1. The lengths of the constant intervalls of B 1 correspond exactly to the jump sizes of B. Let V i i be an i.i.d. sequence of uniform random variables on [0, 1]. We define a partition πb such that i πb j if and only if B 1 V i = B 1 V j In what follows we suppose that the sequence V i to define πb is always the same, for each choice of B. By combining Theorem 36 of Pitman 1999 with Theorem 2.3 of Kallenberg 1973 we obtain: Proposition Let B n be a sequence of bridges with respective jump sizes X n i i, and let B be a bridge with jump sizes X. Then the following conditions are equivalent: 1. πb n n πb in distribution on P 2. X nn X in distribution on 3. B nn B in distribution on the space D[0, 1], [0, 1] of càdlàg functions on [0, 1] with values in [0, 1], equipped with the Skorohod topology. 25

27 Remark. 1. If B and B are independent bridges, then B B is a bridge as well: The only property that is not obvious is the exchangeability of the increments. Let n and let f : R n R be a bounded measurable function. By conditioning on B and by using the independence of B and B we obtain EfB B 1/n B B 0/n,..., B B n/n B B n 1/n = EφB 0, B 1/n,..., B n with φt 0,..., t n = EfBt 1 Bt 0,..., Bt n Bt 1. But B has exchangeable increments, so φ only depends on t 1 t 0,..., t n t n 1. Let ψ be such that ψt 1 t 0,..., t n t n 1 = φt 0,..., t n. Then we have EfB B 1/n B B 0/n,..., B B n/n B B n 1/n = EψB 1/n B 0,..., B 1 B 1 1/n Since B has exchangeable increments and since ψ is a bounded measurable function, we obtain the exchangeability of the increments of B B. 2. B B 1 = B 1 B 1 The following result is Corollary 1 of Bertoin and Le Gall We do not give the proof here, but it is not at all trivial. Proposition Let k 2, and let B 1,..., B k be independent bridges. We define C l := B 1 B l, l = 1,..., k Then conditionally on πc 1,..., πc l 1, πc l has the same distribution as the coagulation of πc l 1 by an independent partition that is distributed like πb l Flows of Bridges Definition A family B s,t : < s t < of bridges is a flow of bridges if 1. For each s t u: B s,u = B s,t B t,u. 2. The distribution of B s,t does not depend on t s. 3. For < t 1 < < t n <, the bridges B t1,t 2,..., B tn 1,t n are independent. 4. B s,s = Id for all s and B 0,t t 0 Id in probability in the Skorohod topology. We can associate an exchangeable coalescent to each flow of bridges: Proposition Let B be a flow of bridges. We define for each t 0 Π t := πb 0,t. Then Π t : t 0 has a càdlàg modification that is a standard exchangeable coalescent. Proof. Let 0 t 0 < < t n. By Proposition 2.36, conditionally on Π t0,..., Π tn 1, Π tn has the same distribution as the coagulation of Π tn 1 by an independent partition that is distributed like πb 0,tn tn 1 = Π tn tn 1. So Π is a Markov process with semi-group P t fη = EfCoagη, πb 0,t 26

28 This is a Feller semi-group. This is shown exactly as in the proof of Proposition 2.30: We use the continuity of Coag from P P to P and the fact that B 0,t converges in probability to the identity when t tends to 0. Then we obtain the convergence of P t fη to fη when t 0 from Proposition Since Π is a Feller process, it has a càdlàg modification cf. Theorem 2.7 in Chapter III of Revuz and Yor It remains to show that for each n R n Π is a Markov process such that each b; k 1,..., k r ; s- collision has the same rate λ b;k1,...,k r;s. The Markov property is easily obtained with the property R n Coagη, ε = CoagR n η, R n ε of the coagulation operator. Like this we see that R n Π has the semi-group Pt n fη = EfCoagη, R n Π t Since Π t is an exchangeable partition for each t, each b; k 1,..., k r ; s-collision has the same rate. Therefore the càdlàg modification of Π is an exchangeable coalescent. We would like to establish a correspondance between flows of bridges and exchangeable coalescents. It remains to show the injectivity and the surjectivity of the map B s,t πb 0,t. More precisely we would like to show: 1. Let B and B be two flows of bridges with the same finite-dimensional distributions. Then πb 0,t t 0 has the same finite-dimensional distributions as πb 0,t t Let Π be a standard exchangeable coalescent. Then there exists a flow of bridges B such that Π and πb 0,t t 0 have the same finite-dimensional distributions. The first statement is more or less obvious: This is just Proposition 2.35 and an application of the stationarity and independence properties of flows of bridges. We will show the second statement with a Poissonian construction. Let u i i [0, 1] and let x i i. We define b ui,x i r := 1 x i r + x i 1 {r ui } ote that if u i is an i.i.d. sequence of uniform variables on [0, 1], then b ui,x i is a bridge. Let ν be a finite measure on with ν{0} = 0. Let U := U U... on [0, 1] where U is the uniform distribution on [0, 1]. Let et : t R be a Poisson point process of intensity U ν on [0, 1]. A Poisson point process with real-valued index t instead of positive t is defined exactly as an usual Poisson point process, just that in this case we consider a Poisson random measure on R E rather than R + E. Since ν and U are finite measures, e a.s. only has a finite number of points on s, t] for all finite s t. Let t 1, u 1 i, x 1 i,..., t k, u k i, x k i be those points with s < t 1 < < t k t. We define If e has no points on s, t], we define B s,t := Id. B s,t := b u 1 i,x 1 i b u k i,x k i 14 Proposition B s,t : < s t < is a flow of bridges 27