Lookdown representation for tree-valued Fleming-Viot processes

Transcription

1 Lookdown representation for tree-valued Fleming-Viot processes Stephan Gufler April 15, 2014 arxiv: v1 [math.pr] 14 Apr 2014 Abstract We construct the tree-valued Fleming-Viot process from the Lookdown model. The tree-valued Fleming-Viot process was introduced by Greven, Pfaffelhuber, and Winter [14]. We generalize this process to include the cases with multiple reproduction events and simultaneous multiple reproduction events. We obtain a process which jumps at the times of large reproduction events. We state well-posed martingale problems. In case that the associated coalescent comes down from infinity, the construction from the Lookdown model allows to read off a process with values in the space of measure-preserving isometry classes of compact metric measure spaces, endowed with the Gromov-Hausdorff-Prohorov metric. This process has a. s. càdlàg paths with additional jumps at the times when old families become extinct. Keywords: Tree-valued Fleming-Viot process, Lookdown model, Λ-coalescent, Ξ-coalescent, marked metric measure space, marked Gromov-weak topology, jointly exchangeable array, martingale problem, evolving coalescent, Gromov-Hausdorff-Prohorov topology. AMS MSC 2010: Primary 60J25, Secondary 60K35, 60G09, 92D10. 1 Introduction The tree-valued Fleming-Viot process was introduced by Greven, Pfaffelhuber, and Winter [14] as a stochastic process with values in the space of equivalence classes of ultrametric measure spaces which describes the evolving genealogical trees in a neutral haploid population in the limit of infinite population size. The theory of metric measure spaces is developed in the work of Gromov [15] and Greven, Pfaffelhuber, and Winter [13]. The space of suitable equivalence classes of metric measure spaces can be endowed with the Gromov-weak topology which is defined in terms of the distribution of the random matrix of distances between individuals drawn independently from the metric space according to a sampling measure. The measure-valued Fleming-Viot process describes the evolution of type distributions. Donnelly and Kurtz [8] give a representation for the measure-valued Fleming-Viot Goethe-Universität, Institut für Mathematik, Postfach111932, Fach187, 60054Frankfurtam Main, Germany, gufler@math.uni-frankfurt.de 1

2 process and more general measure-valued processes in terms of a particle system, the Lookdown model. From this model, genealogical trees which occur in the tree-valued Fleming-Viot process can be read off directly. This connection is considered already in [14]. However, the tree-valued Fleming-Viot process is constructed there via a different method: Tree-valued processes are read off from finite Moran models and shown to converge weakly to the solution of a well-posed martingale problem as population size tends to infinity. The semi-metric space of all individuals in the Lookdown model, endowed with the genealogical distances, was introduced by Pfaffelhuber and Wakolbinger [21] as the Lookdowngraph. Inthepresent article, wecallthemetriccompletionoftheunionofthisspace with the ancestors of the individuals at time zero the Lookdown graph. The completion of the metric space obtained from the individuals at a fixed time, which corresponds to the Kingman coalescent, is studied by Evans [10]. In this article, we construct the treevalued Fleming-Viot process from the Lookdown model. More precisely, we construct a family of probability measures (m t,t R + ) on the Lookdown graph (Z,ρ) such that the process (Z,ρ,m t,t R + ) is a tree-valued Fleming-Viot process, where Z,ρ,m t denotes an equivalence class of the metric measure space (Z,ρ,m t ). This is a main result in this article and part of Proposition 4. In Theorem 1 and Corollary 2, we give a deterministic function which, on an event of probability 1, maps a realization of a Poisson random measure and a realization of the random distance matrix obtained from the initial state to a realization of the tree-valued Fleming-Viot process. The construction from the Lookdown model is extended to the cases with multiple and with simultaneous multiple reproduction events. We apply this construction to study path properties, convergence to equilibrium (Theorem 2), well-posed martingale problems, and a process in a stronger topology, where additional jumps can appear. Coalescents with multiple collisions were introduced by Pitman [23] and Sagitov [25]. Coalescents with simultaneous multiple collisions were introduced by Schweinsberg [26] and Möhle and Sagitov [20]. In [26] they are constructed from a Poisson random measure and characterized by a measure Ξ on the infinite simplex; and in [20] they are obtained as limiting models for the genealogy in Cannings models. Birkner, Blath, Möhle, Steinrücken, and Tams [3] study the Lookdown model in the setting with simultaneous multiple reproduction events to give a construction of the measure-valued Ξ-Fleming-Viot process. The tree-valued Fleming-Viot process introduced in [14] in the setting with binary reproduction events has continuous paths. We generalize the tree-valued Fleming-Viot process to the case with multiple and with simultaneous multiple reproduction events. In the Lookdown model, the number of reproduction events which change the genealogical distances among the first finitely many levels is finite in compact time intervals. In the cases of multiple or simultaneous multiple birth events, the phenomenon of dust may appear, namely that every individual takes part in only finitely many reproduction events in compact time intervals. In [13], coalescents with multiple collisions are described as metric measure spaces in the case without dust. In the case with dust, a negative result is proved in [13, Theorem 4] for the convergence of metric measure spaces associated with Λ-coalescents. To include the case with dust, we work with marked metric measure spaces. Marked metric measure spaces were introduced by Depperschmidt, Greven, and Pfaffelhuber [4] 2

3 and applied by the same authors to include types and mechanisms of mutation and selection into the tree-valued Fleming-Viot process [5]. In the present article, we consider only neutral and haploid reproduction and we do not work with types. We use the marks in a different way: We decompose the genealogical tree at each time such that we obtain a skeleton of internal nodes whose depths we encode by means of the marks. We define the tree-valued Ξ-Fleming-Viot process as a process with values in the space of equivalence classes of marked metric measure spaces. We obtain a process with càdlàg paths and jumps at the times of large reproduction events. We also study wether states in the tree-valued Ξ-Fleming-Viot process are purely atomic, contain atoms, or are non-atomic. We compare these properties at some times τ and τ to deduce the jump times of the process. In the setting with mutation and selection but without multiple reproduction events, Depperschmidt, Greven, and Pfaffelhuber [6] have shown by different methods that a. s., the states of the tree-valued Fleming-Viot process do not contain atoms. Moreover, in case that the Ξ-coalescent comes down from infinity, the construction from the Ξ-Lookdown model allows to read off a process which describes the genealogy at each time and whose state space is the space of measure-preserving isometry classes of compact metric measure spaces, endowed with the Gromov-Hausdorff-Prohorov metric. The induced topology is stronger than the Gromov-weak topology. We obtain a process with càdlàg paths where jumps occur at the times of large reproduction events and at the times when old families become extinct (Theorem 5). We call this process the evolving Ξ-coalescent. The matrix of the genealogical distances in the Lookdown model at a fixed time is a jointly exchangeable array. We point out the connection with the Aldous-Hoover- Kallenberg theory of exchangeable arrays, and with the Existence Theorem of Vershik [28]. We also use joint exchangeability to show in Theorem 3 and Theorem 4 that the tree-valued Ξ-Fleming-Viot process is the unique solution of a martingale problem which is a generalization of the martingale problem in [14]. We construct our family of probability measures on the Lookdown graph using partitions of N. In the case without dust, we use a flow of partitions which has been studied by Labbé [17] in the Lookdown model in the setting with multiple reproduction events. In the case with dust, we use a different family of partitions. We show uniform convergence in compact time intervals to the asymptotic frequencies of the blocks of the partitions in these families by adapting a technique from Donnelly and Kurtz [8]. Flows of partitions have also been studied by Foucart [12], and they are related to the flows of bridges of Bertoin and Le Gall [1]. The process we call the evolving Ξ-coalescent provides a connection to the tree length process of Pfaffelhuber, Wakolbinger, and Weisshaupt [22] as both processes contain jumps when old families become extinct a. s. In the context of measure-valued spatial Λ-Fleming-Viot processes with dust, a skeleton of internal nodes also appears in the work of Véber and Wakolbinger [27]. 1.1 Outline The further parts of this article are organized as follows: In section 2, we recall the theory of metric measure spaces and marked metric measure spaces. We discuss a connection be- 3

4 tween jointly exchangeable marked distance matrices and marked metric measure spaces in section 3. In section 4, a deterministic function which maps a point measure and an initial state to a process of distance matrices is considered, as well as integral equations for this process. For a Poisson random measure, this process is identified as a solution of a martingale problem in section 5. The tree-valued Ξ-Fleming-Viot process is studied in section 6. It is identified as the solution of a well-posed martingale problem in section 7. In section 8, we recall the Gromov-Hausdorff-Prohorov topology and introduce the process which describes the genealogies in this topology. The results from section 9 are applied in the proofs in section 6. 2 Marked metric measure spaces This section is about some parts of the theory of metric measure spaces and marked metric measure spaces from [4, 13]. A metric measure space (mm-space) is a triple (X,r,m) which consists of a complete and separable metric space (X,r) and a probability measure m on the Borel sigmaalgebra on (X,r). Two mm-spaces are defined to be equivalent if there exists a measurepreserving isometry between the supports of the two measures. A marked metric measure space (mmm-space) is a triple (X,r,µ) where (X,r) is a complete and separable metric space and µ is a probability measure on the Borel sigma-algebra on X R +. We endow X R + with the L -product metric d X R + of r and the Euclidean metric, i.e. d X R + ((x,u),(x,u )) = r(x,x ) u u for x,x X and u,u R +. In [4], a more general mark space than R + is considered. For sets Y and Z, we denote by π Y the projection from Y Z to Y. Two marked metric measure spaces (X,r,µ) and (X,r,µ ) are defined to be equivalent if there exists an isometry ϕ between the supports of the measures π X (µ) and π X (µ ) such that the isometry ϕ : supp π X (µ) R + supp π X (µ ) R +, (x, u) (ϕ(x), u) is measure-preserving. The Prohorov metric on the space of probability measures on a separable metric space (X,r) can be defined by d X P(m,m ) = inf ν inf{ε > 0 : ν{(x,y) X2 : r(x,y) > ε} < ε}, where the outer infimum is over all couplings ν of the probability measures m and m on X, cf. [9, Theorem 3.1.2]. We will often deduce from a coupling an upper bound for the Prohorov metric. The Gromov-Prohorov distance between two metric measure spaces (X,r,m) and (X,r,m ) is defined by d GP ((X,r,m),(X,r,m )) = inf{d Z P (ι(m),ι (m ))}. Here and in the next display, the infimum is over all complete and separable metric spaces (Z,d Z ) and isometric embeddings ι of (X,r) and ι of (X,r ) into (Z,d Z ). The marked Gromov-Prohorov distance between two marked metric measure spaces (X, r, µ) and (X,r,µ ) is defined by d MGP ((X,r,µ),(X,r,µ )) = inf{d Z R + P ( ι(µ), ι (µ ))}, 4

5 where we set ι = (ι,id), ι = (ι,id) with the identity map id : R + R +. Denote by D = {(r ij ) i,j N R N2 + : r ij = r ji,r ii = 0,r ij +r jk r ik for all i,j,k N} the cone of all semi-metrics on N which are written as matrices with index set N 2. We always write N = {1,2,...} and N 0 = {0,1,2,...}. For n N, we use the notation [n] = {1,...,n}. Define D = D R N +. Similarly as above, let and D n = {(r ij ) i,j [[n]] R n2 + : r ij = r ji,r ii = 0,r ij +r jk r ik for all i,j,k [n]} For n N, we define the restrictions D n = D n R n +. γ n : D k nd k D n, ρ (ρ i,j ) i,j [[n]] and γ n : D k nd k D n, (r,u) ((r i,j ) i,j [[n]],(u i ) i [[n]] ). In this article, we denote finite matrices (r ij ) i,j [[n]] or infinite matrices (r i,j ) i,j N by r, and vectors (u i ) i [[n]] or (u i ) i N by u. We will use marked distance matrices to decompose distances, which motivates the definition of the map ρ : D D, (r,u) ((u i +r ij +u j )1{i j}) i,j N. We endow D with the metric given by d(r,r ) = k=1 2 k γ k (r) γ k (r ) for r,r D. For a metric measure space (X,r,m), we set (X,r) : X N D, (x k ) k N (r(x i,x j )) i,j N. We define the distance matrix distribution of (X,r,m) on D by (X,r) (m N ), the pushforward measure under (X,r) of an infinite m-iid sequence in X. Similarly, for a marked metric measure space (X,r,µ), we set (X,r) : (X R + ) N D, (x k,u k ) k N ((r(x i,x j )) i,j N,(u i ) i N ) and we define the marked distance matrix distribution of (X,r,µ) on D by (X,r) (µ N ). Equivalent (marked) metric measure spaces have the same (marked) distance matrix distributions. Conversely, Gromov s reconstruction Theorem (see chapter 3 1 of [15], 2 and Theorem 1 of [4] for the setting of marked metric measure spaces) states that two (marked) metric measure spaces with the same (marked) distance matrix distribution are equivalent. For set-theoretical reasons, we define the set M = {(X,r,m) : (X,r,m) metric measure space with X R}. 5

6 For any metric measure space (X,r,m), we denote by X,r,m the intersection of M with the equivalence class of (X,r,m). As (X,r) is separable, X,r,m is nonempty. We define the set M = {X,r,m : (X,r,m) metric measure space}. The elements of M are characterized by their distance matrix distributions, these are well-defined throughrepresentants. Forχ M, we denote theassociated distance matrix distribution by ν χ. A sequence (χ n,n N) M converges to χ M in the Gromovweak topology if and only if the associated distance distributions (ν χn,n N) converge weakly to ν χ. It is shown in [13] that the Gromov-Prohorov distance d GP defines a complete and separable metric on M which metrizes the Gromov-weak topology on M. Similarly, we define the set M = {(X,r,µ) : (X,r,µ) marked metric measure space with X R}. For a marked metric measure space (X,r,µ), we denote by X,r,µ the intersection of M with the equivalence class of (X,r,µ) and we define the set M = {X,r,µ : (X,r,µ) marked metric measure space}. The elements of M are characterized by their marked distance matrix distributions. For χ M, we denote the associated marked distance matrix distribution by ν χ. A sequence (χ n,n N) M converges to χ M in the marked Gromov-weak topology if and only if the associated marked distance distributions (ν χn,n N) converge weakly to ν χ. It is shown in [4] that the marked Gromov-Prohorov distance d MGP defines a complete and separable metric on M which metrizes the marked Gromov-weak topology on M. We say χ M contains atoms or is purely atomic, respectively, if this holds for the probability measure µ in a representant (X,r,µ) of µ. We will also work with the space M 0 = {χ M : π R+ (µ) = δ 0 for all representants (X,r,µ) of χ} = {χ M : π R N + (ν χ ) = δ 0 } which is isometric to M. We recall the class of polynomials on (marked) metric measure spaces. For n N, let C n be the space of continuously differentiable functions φ : D n R with compact support. For φ C n, we denote also by φ the function φ γ n : D k n Dk R. For φ C n, we call the function Φ : M R, Φ(χ) = ν χ φ the polynomial associated with φ. The degree of Φ is the smallest possible k such that Φ = ν φ for some φ Ck. We set C = n N C n and denote by Π the space of all polynomials associated with some φ C. The class C n is convergence determining in D n and Π is convergence determining in M. Similarly, let C n be the space of continuously differentiable functions from Dn to R with compact support, and C = n N C n. For φ C n, we denote also by φ the function φ γ n : D k n Dk R. For φ C n, we call the function Φ : M R, Φ(χ) = ν χ φ 6

7 the polynomial associated with φ and we denote by Π the space of all polynomials associated with some φ C. The class C n is convergence determining in Dn and Π is convergence determining in M. We also need the following notation: For (r,u) D and a bijection p : N N, we set p(r,u) = ((r p(i),p(j) ) i,j N,(u p(i) ) i N ). For n N, (r,u) D n and a permutation p on [n], we set p(r,u) = ((r p(i),p(j) ) i,j [[n]],(u p(i) ) i [[n]] ). Remark 1 (Trees). A metric space (X,r) is 0-hyperbolic if it satisfies the four-point condition r(x,y)+r(z,t) max{r(x,z)+r(y,t),r(y,z)+r(x,t)} for all x,y,z,t X. Any 0-hyperbolic space is a subspace of a real tree. For details and alternative characterizations, cf. Evans [11] and the references therein. The closed subspace M t = {X,r,µ : (X,r,µ) is a 0-hyperbolic mmm-space} M is the space of equivalence classes of weighted real trees. The process introduced in section 6 may be called a tree-valued Fleming-Viot process if M is replaced by M t there. In particular, ultrametric spaces are 0-hyperbolic, hence they may be considered as trees, as mentioned in [14]. 3 Marked metric measure spaces from marked distance matrices We have seen how a marked distance matrix can be sampled from an element of M. In this section, we build a mmm-space from a suitable marked distance matrix. We show in Proposition 1 below that if a suitable random marked distance matrix is jointly exchangeable, it has the marked distance matrix distribution of the associated random element of M. A similar result is given in Lemma 8 of Vershik [28] for distance matrices. Foreach(r,u) D, thematrixrdefines asemi-metric r onn. Identifythepointswith r-distance zero to obtain a metric space and denote the completion of this metric space by (X,r). Assume that (r,u) is chosen such that the empirical distributions 1 n n i=1 δ (i,u i ) converge weakly to a probability measure µ on (X,r) R +. Denote by D the subspace of those (r,u) which satisfy this assumption. We define ψ : D M as the function which maps (r,u) to X,r,µ. For n N and (r,u) D n, the matrix r defines a semi-metric r on [n]. We identify the points with r-distance zero to obtain a metric space which we denote by (X,r). We define a probability measure µ = 1 n n i=1 δ (i,u i ) on X R +. We denote by ψ n : D n M the function which maps (r,u) to X,r,µ. Measurability of ψ follows as ψ n is continuous and as it holds lim n d MGP (ψ n (γ n (R)),ψ(R)) = 0 for R D by definition of D and d MGP. 7

8 A random variable R in D (in D) is called jointly exchangeable if for all bijections p : N N, the random variables p(r) and R are equal in distribution. For n N, the random variable R is said to be n-exchangeable if this holds for all p which satisfy p(i) = i for all i > n. We say a random variable R in D n is jointly exchangeable if p(r ) and R are equal in distribution for all permutations p on [n]. Proposition 1. Let (r,u) be a jointly exchangeable random variable with values in D. Let (r,u ) be a random variable with values in D and distribution ν ψ(r,u) conditionally given ψ(r,u). Then (r,u ) D a.s., and (r,u ) is distributed as (r,u). Proof. Let n N, φ C n, and let (X,r,µ) be a representant of ψ(r,u). We have E [ φ(r,u ) ] [ ] =E µ m (dx,du )φ((r(x i,x j )) i,j [[n]],(u i) i [[n]] ) 1 = lim m m m k 1 =1 =E [ φ(r,u) ]. 1 m m E [ φ((r ki,k j ) i,j [[n]],(u ki ) i [[n]] ) ] k n=1 The second equality follows from the definition of D, the Portmanteau Theorem, and dominated convergence. For the third equality, we use that summands where k 1,...,k n are not pairwise distinct vanish in the limit, and that for all other summands, the expectation in the second line equals by exchangeability the expectation in the third line. Similarly as above, let ρ D and let (X,r) be the completion of the metric space obtained from N, endowed with the semi-metric given by ρ. We define D as the space of those ρ D such that the empirical measures 1 n n i=1 δ i on (X,r) converge weakly to a probability measure m on (X,r). We define ψ 0 : D M as the measurable function which maps ρ to X,r,m. We remark that (r,u) D implies r D. 4 Deterministic Lookdown genealogies Schweinsberg [26] gives a Poisson construction of the coalescent with simultaneous multiple collisions. The Lookdown model is constructed from a Poisson random measure in the article of Birkner et al. [3] in the case with simultaneous multiple reproduction events. Labbé [17] reads off a flow of partitions from a deterministic point measure which describes a realization of the Lookdown model. In this section, we read off the Lookdown graph and a family of marked distance matrices from a deterministic point measure. From the family of distance matrices, we reconstruct the flow of partitions in section 6. The Lookdown graph was introduced by Pfaffelhuber and Wakolbinger [21] in the setting with binary reproduction events. Our definition of the Lookdown graph differs as we work with the metric completion, as we work in one-sided time, and as we include the ancestors of the individuals at time zero. We introduce in this section the decomposition of genealogical distances, essentially into the distances within a skeleton and lengths of external branches. We characterize the process of evolving (marked) distance matrices which describes the evolving genealogies by integral equations. Stochastic equations for the Lookdown model, in particular for the ancestral level and the type process, are also studied in the articles of Donnelly and Kurtz [8] and Birkner et al.[3]. 8

9 4.1 Genealogical distances and the Lookdown graph For n N, let P n be the space of partitions of [n]. Let P be the space of partitions of N and endow P with the metric d(π,π ) = 2 sup{k N:π [[k]]=π [[k]] }, π,π P, (1) where π [[k]] = {B [k] : B π}\{ } P k for k N. Furthermore, let P n be the space of those partitions in P such that the first n integers are not all in different blocks, that is P n = {π P : π [[n]] {{1},...,{n}}}. For π P, we will later also use the notations i π j : i and j are in the same block of π for i,j N, and π(i) = {B π : i B} for the block of π which contains i. Furthermore, we write π(0) = {i : {i} π} for the union of all singleton blocks of π. We denote the set of integers which are the minimal elements of any block of π by M π = {minb : B π}. Let ξ be a simple point measure on (0, ) P, that is, a purely atomic measure with all atoms of weight 1. Assume furthermore that ξ((0,t] P n ) < for all t (0, ) and n N. Denote the space of these measures ξ by N. A point measure ξ N encodes a realization of the Lookdown model as a population model. There are levels indexed by N, each level is occupied by one individual at each time t R +. Each point (t,π) of ξ N is interpreted as the following reproduction event at time t: Let i 1 < i 2 <... be the increasing enumeration of π(0) (N\M π ). For each j N, the individual on level i j moves to the j-th lowest level in π(0) if this level exists, else the individual dies. For each non-singleton block B π, the individual on level minb remains on its level and sets one offspring onto each level in B \minb. This is the reproduction mechanism studied by Birkner et al. [3]. We denote by (t,i) R + N the individual on level i at time t. We also call (t,i) an individual, taking the view that an individual lives for an infinitesimally short time in the time scale of the Lookdown model. For all 0 s t, each individual at time t has an ancestor at time s. We denote by A s (t,i) the level of the ancestor at time s of the individual on level i at time t such that s A s (t,i) is càdlàg. As in [3,8], this can also be defined in terms of integral equations. Let (t,i) R + N and let (M s,s [0,t]) be the càdlàg solution of the integral equation ( M s = i (M s minb)1{m s B} [t s,t] P B π +#((B \{minb}) [M s 1])1{{M s } π} ) ξ(d(t s) dπ), 9

10 with M 0 = i. By the assumption that ξ N, the solution must be piecewise constant, existence and uniqueness follow as the equation can be solved from one jump to the next. Then set A s (t,i) = M (t s) for s [0,t]. For purposes in further parts of this article, we define for (s,i) R + N and t s D t (s,i) = inf{j N : A s (t,j) = i}, with D t (s,i) = if i / A s (t,n). This is the lowest level occupied at time t by a descendant of the individual on level i at time s. This definition corresponds to the forward level process in [21]. There are two phenomena which can yield i / A s (t,n): Individuals can die at the time of a reproduction event encoded by π P with π(0) <, or it can hold lim t t D t (s,i) = for a time t (s,t] due to anaccumulation of jumps. We are interested in the genealogical distances between all individuals, also between individuals which live at different times. Let ρ = (ρ ij ) i,j N D. For i,j N, we interpret ρ ij as the genealogical distance between the individuals (0,i) and (0,j). We define the genealogical distance between the individual (s,i) R + N and the individual (t,j) R + N by { s+t 2sup{r s t : Ar (s,i) = A r (t,j)} if A 0 (s,i) = A 0 (t,j) ρ((s,i),(t,j)) = s+t+ρ A0 (s,i),a 0 (t,j) else. Then ρ is a semi-metric on R + N. We denote the matrix of genealogical distances of the individuals at each fixed time t R + by ρ(ξ,ρ,t) = (ρ((t,i),(t,j))) i,j N. We will refer to this construction through the map ρ : N D R + D, (ξ,ρ,t) ρ(ξ,ρ,t) and we write shortly ρ(t) = ρ(ξ,ρ,t) when there is no ambiguity. We will call the metric space obtained as the completion of the semi-metric space (R + N,ρ) the Lookdown graph associated with ξ and (ρ,0) D. In the following, we introduce the Lookdown graph associated with ξ and a general R D. This generalization will be used to study the tree-valued Fleming-Viot process in the setting with dust. To this aim, we decompose the genealogical distance between the individuals. In this article, we may restrict our attention to individuals which live at the same time. We assume that a decomposition of the genealogical distances ρ at time 0 is given by a marked distance matrix R = (r,u) D such that ρ = ρ(r) with ρ : D D defined as in section 2. At each time t R +, we define a marked distance matrix as follows: For i N, set R(ξ,R,t) = (r,u ) D u i = t sup{s (0,t] : ξ({s} [A s(t,i)]) > 0} 10

11 if for some s (0,t], it holds ξ({s} [A s (t,i)]) > 0, else set u i = t+u A 0 (t,i), with the notation [j] = {π P : {j} / π} for j N. Then define r ij = ((ρ(t)) ij u i u j)1{j i} for i,j N. The interpretation is that u i is the time back from the individual (t,i) until the ancestral lineage is involved in a reproduction event in which it belongs to a family of size larger than 1, if there is such an event, else u i is defined from the given marks at time 0. We will refer to this construction through the map and we write shortly R : N D R + D, (ξ,r,t) R(ξ,R,t) R(t) = (r(t),u(t)) = R(ξ,R,t) when there is no ambiguity. We will use later in particular that u i (t) t s implies A s (t,i) A s (t,j) for all i,j N and s,t R + with i j and s t. For all ξ N and ρ D, the path t ρ(ξ,ρ,t) is càdlàg as for all n N, the path t γ n (ρ(ξ,ρ,t)) has only finitely many jumps in compact time intervals by definition of N. Moreover, for all ξ N, R D, and i N, the path t u i (t) has only finitely many positive jumps in compact time intervals. It follows that t u(t) is càdlàg, and by definition of r(t) also t R(ξ,R,t). We denote the left limits at time t by ρ(ξ,ρ,t ) and R(ξ, R, t ), respectively. We want to embed the metric space obtained from (R + N,ρ) into a complete and separable metric space (Z,ρ) and assign to each individual (t,i) R + N an element z(t,i) Z such that ρ((t,i),z(t,i)) = u i (t) and ρ(z(t,i),z(t,j)) = r ij (t) for all j N. From the definition of u i (t), we can set if u i (t) < t. If u i (t) t, we set z(t,i) = (t u i (t),a t ui (t)(t,i)) R + N. z(t,i) = A 0 (t,i) N. Then we endow the space (R + N) N, that is, the disjoint union of R + N and N, with an extension of the semi-metric ρ on R + N which we define by for i,j N and ρ(i,j) = r ij ρ((t,i),j) = t+u A0 (t,i) +r A0 (t,i),j for (t,i) R + N and j N. We identify the elements with ρ-distance zero and take the completion, which we denote by (Z,ρ). We call (Z,ρ) the Lookdown graph associated with ξ N and R D. Note that for t (0, ) and i N, the limits z(t,i) := lim s t z(s,i) and (t,i) := lim s t (s,i) exist in Z. 11

12 Remark 2. If the marked distance matrix (r,u) D satisfies that the metric space obtained from N endowed with the semi-metric obtained from r is 0-hyperbolic, this property is also satisfied by the marked distance matrices R(ξ,(r,u),t) for all t R +. In this case, this allows to speak of tree-valued processes in sections 6 and Integral equations We give alternative characterizations of the family of distance matrices (ρ(ξ,ρ,t),t R + ) and the family of marked distance matrices (R(ξ,R,t),t R + ). We will analyze them for a Poisson random measure ξ in section 5 and apply them in section 7. Let n N. We denote by S n P n the set of systems of nonempty disjoint subsets of [n]. For σ S n and i [n], let min{j i : {j,i} B for some B σ} if i σ σ(i) = i B σ(#(b \{minb}) [i 1]) else Denote also by σ the maps σ : D n D n ρ (ρ σ(i),σ(j) ) i,j [[n]] and with and σ : D n D n (r,u) (r,u ) u i = u σ(i)1{i / σ} r ij = ( u σ(i) 1{i σ}+r σ(i),σ(j) +u σ(j) 1{j σ} ) 1{i j} for i,j [n]. The effect of a point (t,π) of ξ onthe individuals on the first n levels in the population model of subsection 4.1 is determined by the image of π under the map σ n : P k ns k S n σ {B [n] : B σ, B 2,B [n] }. Indeed, the genealogical distances between the individuals on the first n levels just before thispointaregivenbyγ n (ρ(t ))andbyγ n (ρ(t)) = σ n (π)(γ n (ρ(t )))attimet. Similarly, the marked genealogical distances are given by γ n (R(t )) at time t, and by γ n (R(t)) = σ n (π)(γ n (R(t ))) at time t. Remark 3. These jumps are not determined by π [[n]] : The level at time t to which an individual on level i [n] at time t is shifted can differ, depending on wether i is contained in σ n (π) or not. Moreover, i σ n (π) implies u i (t) = 0 which is not necessarily the case if i / σ n (π). We sometimes prefer to work with the element σ n (π) of the finite set S n rather than with π P. For σ S n, we write σ [[n]] = {B [n] : B σ} \{ }. Note that the consistency property σ k (π) [[n]] = σ n (π) for k n and π P l k Pl is satisfied. 12

13 Remark 4. Let n N. We say σ,σ S n are equivalent if {B σ : #B 2} = {B σ : #B 2}. We write σ σ in this case and denote by [n,σ] the equivalence class of σ. Between the jumps, the distances ((ρ(t)) ij,t R + ) for i,j [n] with i j grow linearly with slope 2. More formally, (ρ(ξ,ρ,t),t R + ) is the unique solution of the system of integral equations ( γ n (ρ(t)) = γ n (ρ)+ σn (π)(γ n (ρ(s ))) γ n (ρ(s )) ) ξ(ds dπ)+2 n t (2) (0,t] P for all n N, where 2 n = 2(1{i j}) i,j [[n]]. Due to the assumption that ξ((0,t] P n ) be finite for all t R + and n N, there are only finitely many times in bounded time intervals at which the the process (γ n (ρ(t)),t R + ) jumps. As the above integral equation can be solved from one jump to the next, it follows that (γ n (ρ(t)),t R + ) is the unique solution. For n N, let We define P (n) = {π P : {{1},...,{n}} π} = {π P : σ n (π) } P n. N dust = {ξ N : ξ((0,t] P (n) ) < for all t R +,n N}. Similarly as above, for ξ N dust, the family (R(ξ,R,t),t R + ) of marked distance matrices is the unique solution of the system of integral equations ( γ n (R(t)) = γ n (R)+ σn (π)(γ n (R(s ))) γ n (R(s )) ) ξ(ds dπ)+(0,1 n )t (3) (0,t] P for all n N, where 1 n denotes the vector in R n + whose entries are all 1. This follows as there are finitely many jumps on bounded time intervals and between the jumps, the matrix (r ij (t)) i,j [[n]] is constant in t and the entries of the vector (u i (t)) i [[n]] grow linearly with slope A model in two-sided time In the population model of subsection 4.1, we can use R as the time axis instead of R +. Let ξ be a simple point measure on R P which satisfies ξ([s,t] P n ) < for all s t and n N. Let again N be the set of levels. We interpret the points (t,π) of ξ in the same way as in subsection 4.1. There is no difference in the definition of the ancestral level A s (t,i), except that all s t are now allowed. For each t R, we define ˆR(ξ,t) = ((r ij) i,j N,(u i) i N ) with u i the depth of the first reproduction event, u i = t sup{s (,t] : ξ(s,[a s(t,i)]) > 0}, and the genealogical distance r ij of the individuals at these depths, r ij = ( 2t 2sup{s (,t] : A s (t,i) = A s (t,j)} u i u j) 1{i j} for i,j N. We impose the further assumption on ξ that these quantities be finite. Then (r,u ) D for all t R. We denote by ˆN the space of the point measures ξ which satisfy these assumptions. We refer to this construction through the map ˆR : ˆN R D, (ξ,t) (r,u ). 13

14 5 Evolution of genealogical distances in the Lookdown model Preservation of exchangeability for type configurations in the Lookdown model is shown in [8] and [3]. We discuss here preservation of exchangeability properties for genealogical marked distance matrices. We also consider invariance under permutations of only a subset of the indices. The argument in subsection 5.4 relies on the preservation of exchangeability properties in a single reproduction event, this is discussed in subsection 5.1 below. In subsection 5.2, a Poisson random measure ξ is defined which drives the integral equation from subsection 4.2. We state martingale problems for the processes ρ(t) = ρ(ξ,ρ,t) and R(t) = R(ξ,R,t) from subsections 4.1 and 4.2 in subsection Preservation of exchangeability properties in a single reproduction event For n N, a permutation p on [n], and a random element σ in S n, we define p(σ) = {p(b) : B σ} where p(b) = {p(i) : i B}. We say σ is exchangeable if σ and p(σ) have the same distribution for all permutations p on [n]. In particular, we say a random partition π in P n is exchangeable if p(π) is distributed as π for all permutations p on [n]. For a partition π P and a bijection p : N N, let p(π) = {p(b) : B π}. We say a random partition π in P is exchangeable if p(π) is distributed as π for all bijections p : N N. For example, if a random partition π in P or P k for k n is exchangeable, then σ n (π) is exchangeable. We denote the set of permutations on [n] by S n. For b,n N 0 with b + n 1, we write S b,n = {p S b+n : p(i) = i for all i [b]} with [0] =. We say a random variable R with values in D b+n is jointly (b,n)- exchangeable if the equality in distribution p(r) d = R holds for all p S b,n. Similarly, a random variable σ with values in S b+n is (b,n)-exchangeable if it holds p(σ) d = σ for all p S b,n. We say a random variable in D is jointly (b, )-exchangeable if γ b+n (R) is jointly (b, n)-exchangeable for all n N. Recall the equivalence relation from remark 4. Lemma 1. Let n N and b N 0. Let R be a jointly (b,n)-exchangeable random variable with values in D b+n and let σ be an independent (b,n)-exchangeable random variable with values in S b+n. Assume σ b (σ) a.s. if b > 0. Then σ(r) is jointly (b,n)-exchangeable. In the special case b = 0, Lemma 1 states that σ(r) is jointly exchangeable if R is jointly exchangeable and σ is exchangeable. 14

15 Proof. For all p S b+n, σ S b+n, and i,j [b+n], it holds Hence, there exists a map σ (p 1 (i)) = σ (p 1 (j)) p(σ )(i) = p(σ )(j). S b+n S b+n S b+n (σ,p) p σ,p which satisfies p(σ )(i) = p σ,p(σ (p 1 (i))) for all σ S b+n, p S b+n, and i [b + n]. This implies p 1 (σ (R )) = p(σ 1 ) ( p σ,p (R ) ) for all σ S b+n, p S b+n, and R D b+n. Let p S b,n, then it holds p(σ) = d σ. As σ b (σ) a.s., it holds p(σ)(i) = σ(p 1 (i)) = i for all i [b], hence p σ,p S b,n a.s. As σ and R are independent, it also holds p 1 σ,p(r) = d R. This implies that p 1 σ,p(r) and σ are independent. Altogether, we obtain σ(r) = d 1 p(σ) ( p σ,p(r) ), this implies the assertion. Lemma 2. Let n,b N with b 2. Let i,j [b] with i < j, and let σ = {{i,j}} S b+n. Let R be a jointly (b,n)-exchangeable random variable with values in D b+n. Then σ(r) is (b+1,n 1)-exchangeable. Proof. Let p S b+1,n 1. Then it holds for k [b+n] k for k < j σ(k) = i for k = j k 1 for k > j and σ(p 1 (k)) = { σ(k) for k b+1 p 1 (k) 1 for k > b+1. Let p S b,n such that p(p 1 (k) 1) = k 1 for all k [b + n] with k > b + 1. Then it holds σ(k) = p(σ(p 1 (k))) for all k [b + n], hence p 1 (σ(r)) = σ( p 1 (R)). The assertion follows as p(r) d = R. 5.2 The Ξ-Lookdown model In this subsection, we recall the Poisson random measure of Schweinsberg [26]: We define a Poisson random measure ξ on (0, ) P from a finite measure Ξ on the simplex = {x = (x 1,x 2,...) : x 1 x , x 1 1}, where x 1 = i N x i. We denote the space of finite measures on by M f ( ). The population model from section 4.1, driven by this Poisson random measure ξ, is the population model from [3]. Kingman s correspondence is a one-to-one correspondence between the exchangeable random partitions of N and the probability measures on : Every x can be interpreted as a partition of the unit interval into intervals of lengths x 1,x 2,..., and possibly another interval of length 1 x 1 which may be called the dust interval. Let U 1,U 2,... be iid uniform random variables on the unit interval. The paintbox construction of an 15

16 exchangeable random partition of N is as follows: Let i,j N be in the same block if U i and U j fall into the same interval of the partition of the unit interval, which is not the dust interval. Conversely, every exchangeable random partition can be represented in this way if x is a random point in the simplex. For Ξ M f ( ), we always decompose Ξ = Ξ 0 + aδ 0 with a = Ξ({0}). Kingman s correspondence defines a probability kernel κ(x,dπ) from to P. For i,j N with i j, we denote by π ij the partition in P which contains the block {i,j} and apart from that only singleton blocks. Let us denote by η Ξ the σ-finite measure on P defined by η Ξ (dπ) = κ(x,dπ) x 2 2 Ξ 0 (dx)+a 1 i<j δ πij (dπ), where x 2 2 = i x2 i. The Ξ-Lookdown model can be defined as the population model of section 4.1 for a Poisson random measure ξ on (0, ) P with intensity dt η Ξ (dπ). ForasetB N,wedenoteby B theasymptoticfrequency B = lim n 1 n #(B [n]) provided this limit exists. We denote the relative frequency of B among the first n integers by B n = 1 #(B [n]). n Remark 5. We may construct the Poisson random measure ξ in two steps. First, we define a Poisson random measure ζ on (0, ) with intensity dt x 2 2 Ξ 0 (dx). Then we define a random point measure ξ Ξ 0 on (0, ) P by ξ Ξ 0 (dt dπ) = ζ(dt dx)κ(x, dπ) and an independent Poisson random measure ξ K on (0, ) P with intensity dt a δ πij (dπ). 1 i<j By the properties of Poisson random measures, ξ Ξ 0 + ξ K is a Poisson random measure with intensity dt η Ξ (dπ). We define ξ = ξ Ξ 0 +ξ K, then it holds ζ = a.s., (t,π) (0, ) P: ξ{(t,π)}>0 δ (t, π ) where for π P, we denote by π the vector in of the asymptotic frequencies of the blocks of π if these exist. and For a partition π P, we write π 2 = π 1 = ( i M π π(i) i M π π(i) 2 ) 1/2, provided that the asymptotic frequencies of the blocks of π exist. The partition π is said to have proper frequencies if i M π π(i) = 1 and these asymptotic frequencies exist. 16

17 Remark 6. If Ξ is concentrated on {x : x 2 = 0} in the Ξ-Lookdown model, recall that the situation reduces to to the case without simultaneous reproduction events, the measureξisdetermined bythefinitemeasure Λ(dx 1 ) = Ξ(dx 1 )on[0,1]. Intheparticular case that Ξ = δ 0, only reproduction events which involve only two individuals occur. 5.3 Martingale problems The observations of this subsection will be applied in section 7. The martingale problems described here generalize the martingale problem in remark 2.20 of [14]. For a measurable space E, we denote by M b (E) the space of bounded measurable functions E R. Definition 1 (Martingale problem). Let E be a complete and separable metric space, D a subspace of the space of bounded continuous functions E R, and Ω : D M b (E) a linear operator. The law of a stochastic process (X(t),t R + ) with values in E and càdlàg paths is a solution of the martingale problem (Ω, D) if for all f D the process f(x(t)) t 0 Ωf(X(s))ds is a martingale with respect to the natural filtration of (X(t),t R + ). For martingale problems, we refer to the monograph of Ethier and Kurtz [9]. Let Ξ M f ( ). For n N, σ S n \{ }, r = #σ, and k 1,...,k r 1 the sizes of the subsets in σ in arbitrary order, we define the rate λ n,σ = i 1,...,i r pairwise distinct +a 1 i<j n Remark 7. Let n N. We introduce the notation λ [n,σ] = x k 1 i 1 x kr i r (1 x 1 ) n k 1... k r 1 Ξ x 2 0 (dx) 2 1{σ = {{i,j}}}+ 1{a > 0,r = 1,k 1 = 1}. σ S n :σ σ for σ S n, where the equivalence relation is defined in remark 4. For σ S n with {#B : B σ,#b 2} = {k 1,...,k r } for some k 1,...,k r 2 and s = n k 1... k r, it holds λ [n,σ] = λ n;k1,...,k r;s in the notation of Schweinsberg [26] on the right-hand side. Remark 8. In the case without simultaneous multiple reproduction events, the above rates simplify to λ n,σ = x k (1 x) n k 1 x 2Λ 0(dx)+a1{k = 2}+ 1{a > 0,k = 1} [0,1] for σ S n with σ = {B} and k = #B. The measure Λ is defined here as in remark 6 and we decompose Λ = Λ 0 +aδ 0 with a = Λ{0}. The rates λ n,σ for σ S n with #σ > 1 are equal to zero. For π P such that σ n (π) consists of one non-singleton block, it holds λ [n,σn(π)] = λ n,σn(π). 17 λ n,σ

18 The following rates are of particular importance, see also Schweinsberg [26] and the references therein. λ 2,{{1,2}} = x 2 Ξ 0 (dx) 2 +a = Ξ( ) < x 2 2 is the rate at which two fixed levels are in the same block in a reproduction event in the population model of subsections 4.1 and 4.2. Furthermore, for a = 0, Ξ 0 (dx) λ 1,{{1}} = x 1 + 1{a > 0} x 2 2 is the rate at which a fixed level is in a non-singleton block in a reproduction event in this population model. If λ 1,{{1}} is finite, we say the Ξ-Lookdown model contains dust, and we write Ξ M dust, else we say the Ξ-Lookdown model contains no dust, and we write Ξ M nd. Clearly, Ξ M dust implies Ξ{0} = 0. Let n N and S n 2 = {σ Sn : there exists B σ with #B 2}. As λ 2,{{1,2}} <, it also holds λ n,σ < for all σ S n 2. In case Ξ M dust, it follows from λ 1,{{1}} < that λ n,σ < for all σ S n \{ }. Thus, Ξ M dust also implies ξ N dust a.s. Conversely, in case Ξ M nd, it holds λ n,σ = for all σ S n with #B = 1 for all B σ. Thus Ξ M nd implies u(t) = 0 for all t (0, ) and ξ N \N dust a.s., where (r(t),u(t)) = R(ξ,R,t) for arbitrary R D. Wedenoteby1thevectorinR N whoseentriesareall1,andwewrite2 = 2(1{i j}) i,j N. In the following Propositions, we use for φ C and ρ D the notation φ,2 (ρ) = 2 and for φ C and (r,u) D the notations and i,j N i j r φ,2 (r,u) = 2 i,j N i j u φ,1 (r,u) = i N ρ ij φ(ρ), r ij φ(r,u) u i φ(r,u). Proposition 2. Let Ξ M f ( ) and let ξ be a Poisson random measure on (0, ) P with intensity dt η Ξ (dπ). Define an operator Ω 1 = Ω grow 1 +Ω coag 1 with domain C as follows: For n N, φ C n and ρ D, set Ω grow 1 φ(ρ) = φ,2 (ρ) and Ω coag 1 φ(ρ) = λ n,σ (φ(σ(ρ)) φ(ρ)). σ S2 n Then for each ρ D, the stochastic process (ρ(ξ,ρ,t),t R + ) solves the martingale problem (Ω 1,C ) with initial state ρ. 18

19 Proposition 3. Let Ξ M dust and let ξ be a Poisson random measure on (0, ) P with intensity dt η Ξ (dπ). Define an operator Ω 2 = Ω grow 2 +Ω coag 2 with domain C as follows: For n N, φ C n and (r,u) D, set Ω grow 2 φ(r,u) = u φ,1 (r,u) and Ω coag 2 φ(r,u) = σ S n \{ } λ n,σ (φ(σ(r,u)) φ(r,u)). Then for each R D, the stochastic process (R(ξ,R,t),t R + ) solves the martingale problem (Ω 2,C) with initial state R. Proof of Propositions 2 and 3. This follows from the integral representations (2) and (3), as for n N and σ S n \{ }, (Y σ (t),t R + ) = (ξ((0,t] (σ 1 n (σ) P)),t R + ) is a Poisson process with rate λ n,σ, and as Y σ1,...,y σk are independent for pairwise distinct σ 1,...,σ n S n \{ } by the coloring theorem for Poisson random measures. 5.4 Preservation of exchangeability in the Lookdown model In the following, Ξ is always a finite measure on, and ξ is a Poisson random measure on (0, ) P with intensity dtη Ξ (dπ). Let ζ be the Poisson random measure on (0, ) with intensity dt x 2 2 Ξ 0 (dx) coupled to ξ which was defined in remark 5. Furthermore, let R be a random variable with values in D, independent of ξ. Let ρ = ρ(r). We use the definitions from section 4. For n N and t R +, we define the maps λ t : D n D n, (r,u ) (r,u +1 n t) and λ t : Dn D n, ρ ρ +2 n t. The following Lemma builds on Lemma 1. We set P 0 =. Lemma 3. Let n N, b N 0, and t R +. Assume that γ b+n (R) is jointly (b,n)- exchangeable. Then, conditionally on the event {ξ((0,t] P b ) = 0} and given ζ, the marked distance matrix γ b+n (R(ξ,R,t)) is jointly (b,n)-exchangeable. Proof. First we consider the case Ξ M nd. Let (t 1,π 1 ),(t 2,π 2 ),... with 0 < t 1 < t 2 <... be the points of ξ in (0, ) P b+n a.s., and define a random variable L with values in N 0 such that t L t and t L+1 > t a.s., where we set t 0 = 0. Then L can be assumed to be measurable with respect to the sigma-algebra generated by ζ and the point measure ξ K ( P b+n ) on R +, with ξ K from remark 5. Conditionally on the event {ξ((0,t] P b ) = 0} and given (ζ,ξ K ( P b+n )), the sequence (π 1,π 2,...) consists of independent random partitions such that σ b+n (π i ) is (b,n)-exchangeable and σ b (π i ) for all i {1,...,L} a.s. This follows from the properties of Poisson random measures, 19

20 in particular as ξ( ((0, ) P b )) and ξ( ((0, ) (P b+n \ P b ))) are independent Poisson random measures. From the integral representation (2), we have γ b+n (ρ(ξ,ρ,t)) = λ t t L σ b+n (π L ) λ t L t L 1... σ b+n (π 1 ) λ t 1 (γ n (ρ)) a.s. on {L 1}, and γ b+n (ρ(ξ,ρ,t)) = λ t(γ b+n (ρ)) a.s. on {L = 0}. Lemma 1 implies that γ b+n (ρ(ξ,ρ,t))isjointly(b,n)-exchangeable conditionally ontheevent {ξ((0,t] P b ) = 0} and given ζ. This also holds for R(t), as u(t) = 0 a.s. A similar argument applies in case Ξ M dust. As ξ N dust a.s. in this case, we can define (t 1,π 1 ),(t 2,π 2 ),... with 0 < t 1 < t 2 <... to be the points of ξ in (0, ) P (b+n) a.s. We define a ζ-measurable random variable L in N 0 such that t L t and t L+1 > t, where t 0 = 0. Conditionally on the event {ξ((0,t] P b ) = 0} and given ζ, the random sequence (π 1,π 2,...) consists of independent random partitions such that σ b+n (π i ) is (b,n)-exchangeable and σ b (π i ) for all i {1,...,L} a.s. From the integral representation (3), we obtain γ b+n (R(ξ,R,t)) = λ t tl σ b+n (π L ) λ tl t L 1... σ b+n (π 1 ) λ t1 (γ n (R)) a.s. on {L 1}, and γ b+n (R(ξ,R,t)) = λ t (γ b+n (R)) a.s. on {L = 0}. Corollary 1. Assume R is jointly exchangeable. Let τ be a ζ-measurable and a.s. finite random time. Then R(τ) = R(ξ,R,τ) and R(τ ) = R(ξ,R,τ ) are jointly exchangeable. Proof. For k N, define by τ k = k τ/k a ζ-measurable random time which assumes countably many values. For n N, φ C n and any permutation p on [n], it holds by Lemma 3 with b = 0 E[φ(R(τ k ))] = j N 0 E[φ(R(j/k));τ k = j/k] = j N 0 E[φ(p(R(j/k)));τ k = j/k] = E[φ(p(R(τ k )))]. We let k tend to infinity, using that R(t) is càdlàg a.s, to obtain the assertion. To prove the assertion for R(τ ), we replace τ k with τ k = k τ/k. Lemma 4. Let n,b N with b 2. Assume Ξ{0} > 0 and that γ b+n (R) is jointly (b, n)-exchangeable. Let τ = inf{t > 0 : ξ((0,t] P b ) > 0}. Then, conditionallyonthe event{ζ({τ} ) = 0}, the markeddistancematrix γ b+n (R(τ)) is jointly (b+1,n 1)-exchangeable. Proof. We define a random variable L = ξ((0,τ) P b+n ). On the event {L 1}, let (t 1,π 1 ),...,(t L,π L ) with t 1 <... < t L be the points of ξ in (0,τ) P b+n a.s. Let π be the element of P such that ξ{(τ,π)} > 0 a.s. By the properties of Poisson random measures, the σ b+n (π 1 ),...,σ b+n (π L ) and σ b+n (π) are independent and (b,n)-exchangeable, and independent of (t 1,...,t L ), conditionally on the event {ζ({τ} ) = 0} and given 20

21 L. On the event {ζ({τ} ) = 0}, it holds σ b+n (π) = {{i,j}} for some i < j in [b] a.s. Furthermore, it holds σ b (π i ) for i {1,...,L} a.s. From the integral representation (2), we obtain γ b+n (ρ(τ)) = σ b+n (π) λ τ t L σ b+n (π L ) λ t L t L 1... σ b+n (π 1 ) λ t 1 (γ b+n (ρ)) a.s. on {L 1}, and γ b+n (ρ(τ)) = σ b+n (π) λ τ(γ b+n (ρ)) on {L = 0} a.s. The assertion follows from Lemmata 1 and 2, and as u(t) = 0 for all t R + a.s. We also consider exchangeability properties at stopping times with respect to two different filtrations. Let n N and b N 0. We define equivalence classes on D b+n by [R ] b,n = {p(r ) : p S b,n } for R D b+n. For t R +, we denote by H b,n t ([γ b+n (R(s))] b,n,s [0,t]). the sigma-algebra generated by A filtration is defined by H b,n = (H b,n t,t R + ). We set D b,n = {[R ] b,n : R D b+n }. Lemma 5. Let n N and b N 0. Assume that γ b+n (R) is jointly (b,n)-exchangeable. Let τ be a finite H b,n -stopping time. Then the marked distance matrix is jointly (b, n)-exchangeable. 1{ξ((0,τ] P b ) = 0}γ b+n (R(τ)) Proof. We show that for each t R +, the marked distance matrix 1{ξ((0,t] P b ) = 0}γ b+n (R(t)) is (b,n)-exchangeable conditionally given H b,n t. The assertion then follows for stopping times which assume countably many values, and by an approximation argument as in the proof of Corollary 1 for all finite stopping times. We enlarge the spaces D b+n and D b+n by a coffin state c. We define a probability kernel K from D b+n to D b+n such that it holds K(c,{c}) = 1, and for all y D b+n \{c}, the probability measure K(y, ) is the uniform distribution on the representants of y. Let τ = inf{t > 0 : ξ((0,t] P b ) > 0} and set R(t) = γ b+n (R(t)) for t < τ, and R(t) = c for t τ. By assumption, K is a regular conditional distribution of γ b+n (R) given [γ b+n (R)] b,n. For all t R +, Lemma 3 implies that K is a regular conditional distributionof R(t)given[ R(t)] b,n, whereweset[c] b,n = c. WeapplyTheorem2ofRogers and Pitman [24] to the probability kernel K and the Markov process ( R(t),t R + ) to obtain that for each t R +, the random variable R(t) has the same conditional distribution given H b,n t as given [ R(t)] b,n, this follows from equation (1) in [24]. This implies the assertion. Let n N. We define equivalence classes on D by [R ] n = {p(r ) : p is a bijection from N to N with p(i) = i for all i > n} for R D. For t R +, we denote by Jt n the sigma-algebra generated by ([R(s)] n,s [0,t]). A filtration is defined by J n = (Jt n,t R +). Let K be a probability kernel from {[R ] n : R D} to D such that for each R D, the probability measure K([R ] n, ) is the uniform distribution on the representants of [R ] n. 21