Existence of mark functions in marked metric measure spaces
EExistence of mark functions in marked metric measure spaces
Sandra Kliem ∗ Wolfgang L¨ohr ∗ October 22, 2018
Abstract
We give criteria on the existence of a so-called mark function in the context of marked metricmeasure spaces (mmm-spaces). If an mmm-space admits a mark function, we call it functionally-marked metric measure space (fmm-space). This is not a closed property in the usual marked Gromov-weak topology, and thus we put particular emphasis on the question under which conditions it carriesover to a limit. We obtain criteria for deterministic mmm-spaces as well as random mmm-spacesand mmm-space-valued processes. As an example, our criteria are applied to prove that the tree-valued Fleming-Viot dynamics with mutation and selection from [DGP12] admits a mark function atall times, almost surely. Thereby, we fill a gap in a former proof of this fact, which used a wrongcriterion.Furthermore, the subspace of fmm-spaces, which is dense and not closed, is investigated in detail.We show that there exists a metric that induces the marked Gromov-weak topology on this subspaceand is complete. Therefore, the space of fmm-spaces is a Polish space. We also construct a decom-position into closed sets which are related to the case of uniformly equicontinuous mark functions.
Key words: mark function; tree-valued Fleming-Viot process; mutation; marked metric measurespace; Gromov-weak topology; Prohorov metric; Lusin’s theorem.
MSC2000 subject classification.
Primary 60K35, Secondary 60J25, 60G17, 60G57.
Contents M fct I into closed sets and estimates on β . . . . . . . 9 Electronic Journal of Probability , , no. 73, pp. 1–24 ∗ Fakult¨at f¨ur Mathematik, Universit¨at Duisburg-Essen, Thea-Leymann-Str. 9, D-45127 Essen, Germany.E-mail: [email protected] , [email protected] a r X i v : . [ m a t h . P R ] J un Introduction
A metric (finite) measure spaces ( mm-space ) is a complete, separable metric space (
X, r ) together witha finite measure ν on it. Considering the space of (equivalence classes of) mm-spaces itself as a metricspace dates back to Gromov’s invention of the (cid:50) λ -metric in [Gro99, Chapter 3 ]. Motivated by Aldous’work on the Brownian continuum random tree ([Ald93]), it was realised in [GPW09] that the space ofmm-spaces is a useful state space for tree-valued stochastic processes, and Polish when equipped withthe Gromov-weak topology. That the Gromov-weak topology actually coincides with the one inducedby the (cid:50) λ -metric was shown in [L¨oh13].Important examples for the use of mm-spaces within probability theory are individual-based pop-ulations X with given mutual genealogical distances r between individuals. Here, r can for instancemeasure the time to the most recent common ancestor (MRCA) (cf. [DGP12, (2.7), Remark 3.3]), wherethe resulting metric space is ultrametric. Another possibility is the number of mutations back to theMRCA (cf. [KW15]), where the resulting space is not ultrametric. Finally, there is a sampling mea-sure ν on the space X which models population density. This means that the state of the process isan mm-space ( X, r, ν ). Such individual-based models are often formulated for infinite population size(with diffuse measures ν ) but obtained as the high-density limit of approximating models with finitepopulations (where ν is typically the uniform distribution on all individuals).For encoding more information about the individuals, such as an (allelic) type or location (which maychange over time), marked metric measure spaces (mmm-spaces) and the corresponding marked Gromov-weak topology (mGw-topology) have been introduced in [DGP11]. For a fixed complete, separable metricspace ( I, d ) of marks, the sampling measure ν is replaced by a measure µ on X × I , which modelspopulation density in combination with mark distribution.A natural question in this context is whether or not every point of the limiting population X hasa single mark almost surely, that is, does genetic distance zero imply the same type/location? Putdifferently, we ask ourselves if µ factorizes into a “population density” measure ν on X and a markfunction κ : X → I assigning each individual its mark. If this is the case, we call the mmm-spacefunctionally-marked (fmm-space). This property is often desirable, and one might want to consider thespace of fmm-spaces, rather than mmm-spaces, as the state space. Unfortunately, the subspace of fmm-spaces is not closed in the mGw-topology, which means that limits of finite-population models that areconstructed as fmm-spaces might not admit mark functions themselves. It is therefore of interest, if thespace of fmm-spaces with marked Gromov-weak topology is a Polish space (that is a “good” state space).Here, we show in Theorem 2.2 that this is indeed the case. We also produce criteria to enable one tocheck if an mmm-space admits a mark function. For limiting populations, they are given in terms of theapproximating mmm-spaces. We derive such criteria for deterministic spaces (Theorem 3.1), randomspaces (Theorem 3.7) and mmm-space-valued processes (Theorem 3.9 and Theorem 3.11).An important example of such a high-density limit of approximating models with finite populationsis the tree-valued Fleming-Viot dynamics. In the neutral case, it is constructed in [GPW13] using theformalism of mm-spaces. In [DGP12], (allelic) types – encoded as marks of mmm-spaces – are included,in order to model mutation and selection. For this process, the question of existence of a mark functionhas already been posed. In [DGP12, Remark 3.11] and [DGP13, Theorem 6] it is stated that thetree-valued Fleming-Viot process admits a mark function at all times, almost surely. The given proof,however, contains a gap, because it relies on the criterion claimed in [DGP13, Lemma 7.1], which iswrong in general, as we show in Example 4.1. We fill this gap by applying our criteria and showing inTheorem 4.3 that the claim is indeed true and the tree-valued Fleming-Viot process with mutation andselection (TFVMS) admits a mark function at all times, almost surely. We also show in Theorem 4.4 thatthe same arguments apply to the Λ-version of the TFVMS in the neutral case, that is where selection isnot present.Intuitively, the existence of a mark function in the case of the TFVMS holds because mutationsare large but rare in the approximating sequence of tree-valued Moran models. Hence, as genealogicaldistance becomes small, the probability that any mutation happened at all in the close past becomessmall as well (recall that distance equals time to the MRCA). In contrast, in [KW15], where evolving2hylogenies of trait-dependent branching with mutation and competition are under investigation, muta-tions happen at a high rate but are small which justifies the hope for the existence of a mark functionalso for the limiting model. Our criteria are also suited for this kind of situation. Outline.
The paper is organized as follows. In the subsections of the introduction we first introducenotations and basic results for the Prohorov metric for finite measures. Then, we give a short introductionto the space M I of marked metric measure spaces (mmm-spaces) with the marked Gromov-weak topology,as well as the marked Gromov-Prohorov metric d mGP on it. We continue with defining the so-calledfunctionally-marked metric measure spaces (fmm-spaces) M fct I ⊆ M I , and finally investigate the caseof equicontinuous mark functions as an illustrative example. We emphasize that the restriction of themarked Gromov-Prohorov metric d mGP to M fct I is not complete.In Section 2, we therefore show that there exists another metric on M fct I that induces the markedGromov-weak topology and is complete. As one sees in Subsection 1.4, the situation becomes easy ifwe restrict to a subspace of M I containing spaces with uniformly equicontinuous mark functions. Weintroduce in Subsection 2.2 several related subspaces capturing some aspect of equicontinuity, and obtaina decomposition of M fct I into closed sets. This decomposition is used to prove Polishness of M fct I , and inSection 3 to formulate criteria for the existence of mark functions.Section 3 gives criteria for the existence of mark functions. Based on the construction of the completemetric and the decomposition of M fct I , we derive in Subsection 3.1 criteria to check if an mmm-spaceadmits a mark function, especially in the case where it is given as a marked Gromov-weak limit. Wethen transfer the results in Subsection 3.2 to random mmm-spaces and in Subsection 3.3 to M I -valuedstochastic processes.To conclude, Section 4 gives examples. We first show that the criterion in [DGP13] is wrong ingeneral by means of counterexamples. Our criteria are then applied in Subsection 4.1 to prove theexistence of a mark function for the tree-valued Fleming-Viot dynamics with mutation and selection.To this goal, we verify the necessary assumptions for a sequence of approximating tree-valued Moranmodels. In Subsection 4.2 we show that a similar strategy applies if we replace the tree-valued Moranmodels by so-called tree-valued Λ-Cannings models. Finally, in Subsection 4.3, a future application toevolving phylogenies of trait-dependent branching with mutation and competition is indicated. In this paper, let all topological spaces be equipped with their Borel σ -algebras. We use the followingnotation throughout the article. Notation 1.1.
For a Polish space E , let M ( E ) respectively M f ( E ) denote the space of probabilityrespectively finite measures on the Borel σ -algebra B ( E ) on E . The space M f ( E ) is always equippedwith the topology of weak convergence, which is denoted by w −→ . We also use the distance in variationalnorm of µ, ν ∈ M f ( E ) , which is (cid:107) µ − ν (cid:107) := sup B ∈ B ( E ) (cid:12)(cid:12) µ ( B ) − ν ( B ) (cid:12)(cid:12) . (1.1) In particular, (cid:107) µ (cid:107) = µ ( E ) , and (cid:107) µ − ν (cid:107) = ν ( E ) − µ ( E ) if µ ≤ ν , that is µ ( A ) ≤ ν ( A ) for all A ∈ B ( E ) .For Y ∈ B ( E ) and µ ∈ M f ( E ) , denote by µ | Y ∈ M f ( E ) the restriction of µ to Y , that is µ | Y ( B ) := µ ( B ∩ Y ) for all B ∈ B ( E ) . Because µ | Y ≤ µ , we have (cid:107) µ | Y − µ (cid:107) = µ ( E \ Y ) .For ϕ : E → F measurable, with F some other Polish space, denote the image measure of µ under ϕ by ϕ ∗ µ := µ ◦ ϕ − . Finally, for the product space X := E × F , the canonical projection operators from X onto E and F are denoted by π E and π F , respectively. Definition 1.2 (Prohorov metric) . For finite measures µ , µ on a metric space ( E, r ) , the Prohorovmetric is defined as d Pr ( µ , µ ) := inf (cid:8) ε > µ i ( A ) ≤ µ − i ( A ε ) + ε ∀ A ∈ B ( E ) , i ∈ { , } (cid:9) , (1.2) where A ε := { x ∈ E : r ( A, x ) < ε } is the ε -neighbourhood of A .
3t is well-known that the Prohorov metric metrizes the weak convergence of measures if and only ifthe underlying metric space is separable. The following equivalent expression for the Prohorov metricturns out to be useful.
Remark 1.3 (coupling representation of the Prohorov metric) . Let (
E, r ) be a separable metric spaceand µ , µ ∈ M ( E ). For a finite measure ξ on E , we denote the marginals as ξ := ξ ( · × E ) and ξ := ξ ( E × · ). It is well-known (see, e.g., [EK05, Theorem III.1.2]) that d Pr ( µ , µ ) = inf (cid:8) ε > ∃ ξ ∈ M ( E ) with ξ ( N ε ) ≤ ε, ξ i = µ i , i = 1 , (cid:9) , (1.3)where N ε := { ( x, y ) ∈ E : r ( x, y ) ≥ ε } . We obtain from this equation d Pr ( µ , µ ) = inf (cid:8) ε > ∃ ξ (cid:48) ∈ M f ( E ) with ξ (cid:48) ( N ε ) = 0 , ξ (cid:48) i ≤ µ i , (cid:107) µ i − ξ (cid:48) i (cid:107) ≤ ε, i = 1 , (cid:9) . (1.4)Indeed, consider ξ (cid:48) := ξ | E \ N ε respectively ξ := ξ (cid:48) + (1 − (cid:107) ξ (cid:48) (cid:107) ) − (cid:0) ( µ − ξ (cid:48) ) ⊗ ( µ − ξ (cid:48) ) (cid:1) to obtain equalityin the above. Following the ideas of the proof of the representation (1.3) in [EK05], the representation(1.4) for the Prohorov metric d Pr ( µ , µ ) is easily seen to hold true for measures µ , µ ∈ M f ( E ) as well,which are not necessarily probability measures.From (1.4), we can easily deduce the following lemma, which we use below. Lemma 1.4 (rectangular lemma) . Let ( E, r ) be a separable, metric space, ε, δ > , and µ , µ ∈ M f ( E ) .Assume that d Pr ( µ , µ ) < δ and there is µ (cid:48) ≤ µ with (cid:107) µ − µ (cid:48) (cid:107) ≤ ε . Then ∃ µ (cid:48) ≤ µ : d Pr ( µ (cid:48) , µ (cid:48) ) < δ, (cid:107) µ − µ (cid:48) (cid:107) ≤ ε. (1.5) Proof.
According to (1.4), we find ξ ∈ M f ( E ) with marginals ξ i ≤ µ i , i = 1 , (cid:107) µ i − ξ i (cid:107) < δ , and ξ ( { r ≥ δ } ) = 0. Let L be a probability kernel from E to E (for existence see [Kle14, Theorems 8.36–8.38]) with ξ = µ ⊗ L and define ξ (cid:48) := ( µ (cid:48) ∧ ξ ) ⊗ L . Obviously, ξ (cid:48) ≤ µ (cid:48) and (cid:107) µ (cid:48) − ξ (cid:48) (cid:107) ≤ (cid:107) µ − ξ (cid:107) < δ .Now set µ (cid:48) := ξ (cid:48) + µ − ξ . (1.6)Then ξ (cid:48) ≤ µ (cid:48) , (cid:107) µ (cid:48) − ξ (cid:48) (cid:107) = (cid:107) µ − ξ (cid:107) < δ and thus d Pr ( µ , µ (cid:48) ) < δ by (1.4). Furthermore, µ (cid:48) ≤ µ and (cid:107) µ − µ (cid:48) (cid:107) = (cid:107) ξ − ξ (cid:48) (cid:107) ≤ (cid:107) µ − µ (cid:48) (cid:107) ≤ ε . In this subsection, we recall the space M I of marked metric measure spaces, and the marked Gromov-Prohorov metric d mGP , which induces the marked Gromov-weak topology on it. This space, ( M I , d mGP ),will be the basic space used in the rest of the paper. These concepts have been introduced in [DGP11],and are based on the corresponding non-marked versions introduced in [GPW09]. In contrast to [DGP11],we allow the measures of the marked metric measure spaces to be finite, that is do not restrict ourselvesto probability measures only. Because a sequence of finite measures converges weakly if and only iftheir total masses and the normalized measures converge, or the masses converge to zero, this straight-forward generalization requires only minor modifications (compare [LVW14, Section 2.1], where thisgeneralization is done for metric measure spaces without marks).In what follows, fix a complete, separable metric space ( I, d ), called the mark space . It is the samefor all marked metric measure spaces in M I . Definition 1.5 (mmm-spaces, M I ) . (i) An ( I -)marked metric measure space (mmm-space) is a triple ( X, r, µ ) such that ( X, r ) is a complete, separable metric space, and µ ∈ M f ( X × I ) , where X × I is equipped with the product topology. (ii) Let X i = ( X i , r i , µ i ) , i = 1 , , be two mmm-spaces, and ν i := µ i ( · × I ) the marginal of µ i on X i .For a map ϕ : X → X we use the notation ˜ ϕ : X × I → X × I, ( x, u ) (cid:55)→ ˜ ϕ ( x, u ) := ( ϕ ( x ) , u ) . (1.7)4 e call X and X equivalent if they are measure- and mark-preserving isometric, that is there isan isometry ϕ : supp( ν ) → supp( ν ) , such that ˜ ϕ ∗ µ = µ . (1.8)(iii) Finally, define M I := (cid:8) equivalence classes of mmm-spaces (cid:9) . (1.9) With a slight abuse of notation, we identify an mmm-space with its equivalence class and write X = ( X, r, µ ) ∈ M I for both mmm-spaces and equivalence classes thereof. Next, we recall the marked Gromov-weak topology from [DGP11, Section 2.2] that turns M I into aPolish space (cf. [DGP11, Theorem 2]). To this goal, we first recall Definition 1.6 (marked distance matrix distribution) . Let X := ( X, r, µ ) ∈ M I and R ( X,r ) := (cid:40) ( X × I ) N → R ( N ) + × I N , (cid:0) ( x k , u k ) k ≥ (cid:1) (cid:55)→ (cid:0)(cid:0) r ( x k , x l ) (cid:1) ≤ k Definition 1.8 (marked Gromov-Prohorov metric, d mGP ) . For X i = ( X i , r i , µ i ) ∈ M I , i = 1 , , set d mGP ( X , X ) := inf ( E,ϕ ,ϕ ) d Pr (cid:0) ( ˜ ϕ ) ∗ µ , ( ˜ ϕ ) ∗ µ (cid:1) , (1.13) where the infimum is taken over all complete, separable metric spaces ( E, r ) and isometric embeddings ϕ i : X i → E , and ˜ ϕ i is as in (1.7) , i = 1 , . The Prohorov metric d Pr is the one on M f ( E × I ) , basedon the metric ˜ r = r + d on E × I , metrizing the product topology. The metric d mGP is called the markedGromov-Prohorov metric . A direct consequence of the fact that d mGP induces the marked Gromov-weak topology is the followingcharacterization of marked Gromov-weak convergence obtained in [DGP11, Lemma 3.4]. Lemma 1.9 (embedding of marked Gromov-weakly converging sequences) . Let X n = ( X n , r n , µ n ) ∈ M I for n ∈ N ∪ {∞} . Then ( X n ) n ∈ N converges to X ∞ Gromov-weakly if and only if there is a complete,separable metric space ( E, r ) , and isometric embeddings ϕ n : X n → E , such that for ˜ ϕ n as in (1.7) , ( ˜ ϕ n ) ∗ µ n w −→ µ. (1.14)5 .3 Functionally-marked metric measure spaces (fmm-spaces) Consider an I -marked metric measure space X = ( X, r, µ ) ∈ M I . Since µ is a finite measure on thePolish space X × I , regular conditional measures exist (cf. [Kle14, Theorems 8.36–8.38]), and we write µ (d x, d u ) = ν (d x ) · K x (d u ) , (1.15)in short µ = ν ⊗ K , for the marginal ν := µ ( · × I ) ∈ M f ( X ), and a ( ν -a.s. unique) probability kernel K from X to I .In the present article we investigate criteria for the existence of a mark function for X , that is (cf.[DGP13, Section 3.3]) a measurable function κ : X → I such that µ (d x, d u ) = ν (d x ) · δ κ ( x ) (d u ) , (1.16)or equivalently, K x = δ κ ( x ) for ν -almost every x . Obviously, X admits a mark function if and only if K x is a Dirac measure for ν -almost every x . Recall that the complete, separable mark space ( I, d ) is fixedonce and for all. Definition 1.10 (fmm-spaces, M fct I ) . We call X = ( X, r, ν, κ ) an ( I -)functionally-marked metric mea-sure space ( fmm-space ) if ( X, r ) is a complete, separable metric space, ν ∈ M f ( X ) , and κ : X → I is measurable. We identify X with the marked metric measure space ( X, r, µ ) ∈ M I , where µ satisfies (1.16) . With a slight abuse of notation, we write ( X, r, ν, κ ) = ( X, r, µ ) if (1.16) is satisfied. Denote by M fct I ⊆ M I the space of (equivalence classes of ) fmm-spaces. A first, simple observation is that M fct I is a dense subspace of M I . Lemma 1.11. The subspace M fct I is dense in M I with marked Gromov-weak topology.Proof. For X = ( X, r, µ ) ∈ M I , define X n = ( X × I, r n , ν n , κ n ) ∈ M fct I with ν n = µ , κ n ( x, u ) = u , and r n (cid:0) ( x, u ) , ( y, v ) (cid:1) := r ( x, y ) + e − n ∧ d ( u, v ), for x, y ∈ X, u, v ∈ I . It is easy to see that X n → X in themarked Gromov-weak topology. It directly follows from Lemma 1.11 that the subspace M fct I is not closed in M I , meaning that if X n mGw −−−→ X is a marked Gromov-weakly converging sequence in M I , and all X n admit a mark function, this need notbe the case for X . In applications, however, the limit X is often not known explicitly, and it would beimportant to have (sufficient) criteria for the existence of a mark function in terms of the X n alone. Aneasy possibility is Lipschitz equicontinuity: if all X n admit a mark function that is Lipschitz continuouswith a common Lipschitz constant L > 0, the same is true for X (see [Pio11]). More generally, this holdsfor uniformly equicontinuous mark functions as introduced below. We briefly discuss the equicontinuouscase in this subsection, because it is straightforward and illustrates the main ideas.Recall that a modulus of continuity is a function h : R + → R + ∪ {∞} that is continuous in 0 andsatisfies h (0) = 0. A function f : X → I , where ( X, r ) is a metric space, is h -uniformly continuous if d (cid:0) f ( x ) , f ( y ) (cid:1) ≤ h (cid:0) r ( x, y ) (cid:1) for all x, y ∈ X . Note that for every modulus of continuity h , there existsanother modulus of continuity h (cid:48) ≥ h which is increasing and continuous with respect to the topologyof the one-point compactification of R + . Therefore, we can restrict ourselves without loss of generalityto moduli of continuity from H := (cid:8) h : R + → R + ∪ {∞} (cid:12)(cid:12) h (0) = 0 , h is continuous and increasing (cid:9) . (1.17)For h ∈ H and a metric space ( X, r ), we define A Xh := A ( X,r ) h := (cid:8) ( x i , u i ) i =1 , ∈ ( X × I ) : d ( u , u ) ≤ h ( r ( x , x )) (cid:9) ⊆ ( X × I ) . (1.18)Note that f : X → I is h -uniformly continuous if and only if (cid:0) ( x, f ( x )) , ( y, f ( y )) (cid:1) ∈ A Xh for all x, y ∈ X ,and that A Xh is a closed set in ( X × I ) with product topology.6 efinition 1.12 ( M hI ) . For h ∈ H , let M hI ⊆ M fct I be the space of marked metric measure spacesadmitting an h -uniformly continuous mark function. The next lemma states that a marked metric measure space ( X, r, µ ) admits an h -uniformly contin-uous mark function if and only if a pair of independent samples from µ is almost surely in A Xh . Further-more, if a sequence with h -uniformly continuous mark functions converges marked Gromov-weakly, thelimit space also admits an h -uniformly continuous mark function. Lemma 1.13 (uniform equicontinuity) . Fix a modulus of continuity h ∈ H . (i) M hI = (cid:8) ( X, r, µ ) ∈ M I : µ ⊗ ( A Xh ) = (cid:107) µ ⊗ (cid:107) (cid:9) . (ii) M hI is closed in the marked Gromov-weak topology.Proof. The mmm-space X = ( X, r, µ ) is in M hI if and only if supp( µ ) is the graph of an h -uniformlycontinuous function. This is clearly equivalent to µ ⊗ (cid:0) ( X × I ) \ A Xh (cid:1) = 0. Item (ii) is obvious from (i),because A Xh is a closed set.This preliminary result is quite restrictive because of the condition to have the same modulus ofcontinuity for all occurring spaces. In fact, the mark function of the tree-valued Fleming Viot dynamicconsidered in Subsection 4.1 is not even continuous.At the heart of the following generalisation to measurable mark functions lies the fact that measur-able functions are “almost continuous” by Lusin’s celebrated theorem (see for instance [Bog07, Theo-rem 7.1.13]). Here, we give a version tailored to our setup: Lusin’s theorem. Let X, Y be Polish spaces, µ a finite measure on X , and f : X → Y a measurablefunction. Then, for every ε > , there exists a compact set K ε ⊆ X such that µ ( X \ K ε ) < ε and f | K ε is continuous. The subspace M fct I is not closed in M I in the marked Gromov-weak topology, and hence the restrictionof the marked Gromov-Prohorov metric d mGP to M fct I is not complete. In this section, we show thatthere exists another metric on M fct I that induces the marked Gromov-weak topology and is complete.This shows that M fct I is a Polish space in its own right. For a measure ξ on I , we define β ξ := (cid:90) I (cid:90) I (1 ∧ d ( u, v )) ξ (d u ) ξ (d v ) . (2.1)Note that β ξ = 0 if and only if ξ is a Dirac measure. For X = ( X, r, µ ) ∈ M I , with µ = ν ⊗ K as in(1.15), we define β ( X ) := (cid:90) X β K x ν (d x ) = (cid:90) X × I (cid:90) I (1 ∧ d ( u, v )) K x (d v ) µ (cid:0) d( x, u ) (cid:1) . (2.2) Proposition 2.1 (characterization of M fct I as continuity points) . Let cont( β ) ⊆ M I be the set of conti-nuity points of β : M I → R + , where M I carries the marked Gromov-weak topology. Then cont( β ) = β − (0) = M fct I . (2.3)7 roof (first part). As seen before, X = ( X, r, ν ⊗ K ) ∈ M I admits a mark function if and only if K x is aDirac measure for ν -almost every x ∈ X , which is the case if and only if β ( X ) = 0. Hence β − (0) = M fct I .Because M fct I is dense in M I by Lemma 1.11, no X ∈ M I \ β − (0) can be a continuity point of β . Thuscont( β ) ⊆ β − (0).We defer the proof of the inclusion β − (0) ⊆ cont( β ) to Subsection 2.2, because it requires a technicalestimate on β derived in Proposition 2.7.In view of (2.3), we can use standard arguments to construct a complete metric on M fct I that metrizesmarked Gromov-weak topology. Namely consider the sets F m := β − (cid:0) [ m , ∞ ) (cid:1) ⊆ M I , m ∈ N , (2.4)where the closure is in the marked Gromov-weak topology. Then, due to Proposition 2.1, F m is disjointfrom M fct I , and M fct I = M I \ (cid:83) m ∈ N F m . Because F m is also closed by definition, we obtain M fct I = (cid:92) m ∈ N (cid:8) X ∈ M I : d mGP ( X , F m ) > (cid:9) . (2.5)We consider the metric d fGP on M fct I defined for X , Y ∈ M fct I by d fGP ( X , Y ) := d mGP ( X , Y ) + sup m ∈ N − m ∧ (cid:12)(cid:12)(cid:12) d mGP ( X , F m ) − d mGP ( Y , F m ) (cid:12)(cid:12)(cid:12) . (2.6) Theorem 2.2 ( M fct I is Polish) . The space M fct I of I -functionally-marked metric measure spaces withmarked Gromov-weak topology is a Polish space. Namely, d fGP is a complete metric on M fct I inducingthe marked Gromov-weak topology.Proof. First, we show that d fGP induces the marked Gromov-weak topology on M fct I . For m ∈ N , X ∈ M I ,define ρ m ( X ) := d mGP ( X , F m ) , (2.7)with F m defined in (2.4). Note that ρ m is a continuous function on M I . Let X n , X ∈ M fct I . Then ρ m ( X ) > m ∈ N because of (2.5). Therefore, by definition, d fGP ( X n , X ) −→ n →∞ d mGP ( X n , X ) −→ n →∞ ρ m ( X n ) −→ n →∞ ρ m ( X ) ∀ m ∈ N (2.8)hold. We have to show that the marked Gromov-weak convergence already implies (2.8). This, however,follows from the continuity of the ρ m .It remains to show that d fGP is a complete metric on M fct I . Consider a d fGP -Cauchy sequence ( X n ) n ∈ N in M fct I . By completeness of d mGP on M I , it converges marked Gromov-weakly to some X = ( X, r, µ ) ∈ M I . Furthermore, for every fixed m ∈ N , (2.6) implies that 1 /ρ m ( X n ) converges as n → ∞ , and hence d mGP ( X n , F m ) is bounded away from zero. Thus X (cid:54)∈ F m . Because M fct I = M I \ (cid:83) m ∈ N F m , this meansthat X ∈ M fct I , and by the first part of the proof d fGP ( X n , X ) −→ n →∞ B M I δ ( X ) := (cid:8) Y ∈ M I : d mGP ( X , Y ) < δ (cid:9) we denote the open δ -ball in M I with respect to d mGP .The following corollary gives formal criteria for a limiting space to admit a mark function, which areuseful only together with estimates on β . Corollary 2.3. Let ( X n ) n ∈ N be a sequence in M I which converges marked Gromov-weakly to X . Thenthe following four conditions are equivalent: (i) X ∈ M fct I . (ii) lim sup n →∞ ρ m ( X n ) > for all m ∈ N , with ρ m defined in (2.7) . For every δ > , lim sup n →∞ inf Y ∈ β − ([ δ, ∞ [) d mGP ( X n , Y ) > . (2.9)(iv) lim δ ↓ lim inf n →∞ sup Y ∈ B M Iδ ( X n ) β ( Y ) = 0 . (2.10) Proof. “ (i) ⇔ (ii) ”: We have ρ m ( X ) = lim n →∞ ρ m ( X n ), and ρ m ( X ) > m ∈ N if and only if X ∈ M fct I . “ (ii) ⇔ (iii) ”: follows directly from the definition of ρ m . “ (iii) ⇔ (iv) ”: Using monotonicity in δ we obtain(iii) ⇐⇒ ∀ δ > ∃ ε > ∀ ( Y n ) n ∈ N ⊆ M I with β ( Y n ) ≥ δ : lim sup n →∞ d mGP ( X n , Y n ) ≥ ε (2.11) ⇐⇒ ∀ δ > ∃ ε > ∀ ( Y n ) n ∈ N ⊆ M I : lim inf n →∞ β ( Y n ) < δ or lim sup n →∞ d mGP ( X n , Y n ) ≥ ε ⇐⇒ ∀ ε > ∃ δ > ∀ ( Y n ) n ∈ N ⊆ M I with Y n ∈ B M I δ ( X n ) : lim inf n →∞ β ( Y n ) < ε ⇐⇒ (iv) , where, in the third equivalence, we renamed δ to ε and ε to δ . M fct I into closed sets and estimates on β In this subsection, we derive some estimates on β and use them to complete the proof of Proposition 2.1.Furthermore, we construct a decomposition of M fct I into closed sets which are related to the sets M hI .As we have seen in Subsection 1.4, the situation becomes easy if we restrict to the uniformly equicon-tinuous case, that is to the subspace M hI for some h ∈ H as in Definition 1.12. We introduce in whatfollows several related subspaces capturing some aspect of equicontinuity. In analogy to the definitionof A Xh in (1.18), we use for a metric space ( X, r ), and δ, ε > 0, the notation A Xδ,ε := A ( X,r ) δ,ε := (cid:8) ( x i , u i ) i =1 , ∈ ( X × I ) : r ( x , x ) ≥ δ or d ( u , u ) ≤ ε (cid:9) ⊆ ( X × I ) . (2.12)Note that A Xδ,ε is a closed set. For every h ∈ H , using monotonicity and continuity of h , we observe that A Xh = (cid:92) δ> A Xδ,h ( δ ) . (2.13) Definition 2.4 ( M δ,εI , M δ,εI , M hI ) . Let δ, ε > and h ∈ H . We define M δ,εI := (cid:8) ( X, r, µ ) ∈ M I : µ ⊗ ( A Xδ,ε ) = (cid:107) µ ⊗ (cid:107) (cid:9) , (2.14) M δ,εI := (cid:8) ( X, r, µ ) ∈ M I : ∃ µ (cid:48) ∈ M f ( X × I ) : µ (cid:48) ≤ µ, (cid:107) µ − µ (cid:48) (cid:107) ≤ ε, ( X, r, µ (cid:48) ) ∈ M δ,εI (cid:9) , (2.15) and M hI := (cid:84) δ> M δ,h ( δ ) I . The intuition is that for spaces in M δ,h ( δ ) I , the measure behaves as if it admitted an h -uniformlycontinuous mark function when distances of order δ are observed. The same holds for the spaces in M δ,h ( δ ) I if we are additionally allowed to neglect a portion h ( δ ) of mass. Remark 2.5. (i) Clearly M hI ⊆ M hI . We will see in Lemma 2.8 that M hI ⊆ M fct I .(ii) The space M hI is much larger than M hI : while (cid:83) h ∈H M hI contains only mmm-spaces admitting auniformly continuous mark function, we will see in Lemma 2.8 that every element of M fct I is insome M hI .(iii) The spaces M δ,εI and M δ,εI are not contained in M fct I . For instance, consider I = R and X = (cid:0) { } , , δ (0 , + δ (0 ,ε ) ) ∈ M δ,εI ⊆ M δ,εI . 9e have the following stability of M δ,εI with respect to small perturbations in the marked Gromov-Prohorov metric. Lemma 2.6 (perturbation of M δ,εI ) . Let δ, ε > , X ∈ M δ,εI and ˆ X ∈ M I . Then δ (cid:48) := d mGP ( X , ˆ X ) < δ = ⇒ ˆ X ∈ M δ − δ (cid:48) , ε +2 δ (cid:48) I . (2.16) Proof. Let X = ( X, r, µ ), ˆ X = ( ˆ X, ˆ r, ˆ µ ). We may assume that X, ˆ X are subspaces of some separable,metric space ( E, r E ) such that d Pr ( µ, ˆ µ ) < δ (cid:48) . By definition of M δ,εI , there is µ (cid:48) ≤ µ with (cid:107) µ − µ (cid:48) (cid:107) ≤ ε and X (cid:48) := ( X, r, µ (cid:48) ) ∈ M δ,εI . Due to Lemma 1.4, we find ˆ µ (cid:48) ≤ ˆ µ with (cid:107) ˆ µ − ˆ µ (cid:48) (cid:107) ≤ ε and d Pr ( µ (cid:48) , ˆ µ (cid:48) ) < δ (cid:48) ,where ˆ X (cid:48) = ( ˆ X, ˆ r, ˆ µ (cid:48) ). By the coupling representation of the Prohorov metric, (1.4), we obtain a measure ξ on ( E × I ) with marginals ξ ≤ µ (cid:48) and ξ ≤ ˆ µ (cid:48) such that (cid:107) ˆ µ (cid:48) − ξ (cid:107) ≤ δ (cid:48) and ξ (cid:0)(cid:8)(cid:0) ( x, u ) , (ˆ x, ˆ u ) (cid:1) ∈ ( X × I ) × ( ˆ X × I ) : r E ( x, ˆ x ) + d ( u, ˆ u ) ≥ δ (cid:48) (cid:9)(cid:1) = 0 . (2.17)By definition, ( µ (cid:48) ) ⊗ is supported by A Xδ,ε . Therefore, the same is true for ξ ⊗ and we obtain (cid:107) ξ ⊗ (cid:107) = (cid:107) ξ ⊗ (cid:107) = ξ ⊗ (cid:0)(cid:8) ( x i , u i , ˆ x i , ˆ u i ) i =1 , ∈ (( X × I ) × ( ˆ X × I )) : ( x i , u i ) i =1 , ∈ A Xδ,ε (cid:9)(cid:1) (2.18) ≤ ξ ⊗ (cid:0)(cid:8) (ˆ x i , ˆ u i ) i =1 , ∈ ( ˆ X × I ) : r E (ˆ x , ˆ x ) ≥ δ − δ (cid:48) or d (ˆ u , ˆ u ) ≤ ε + 2 δ (cid:48) (cid:9)(cid:1) = ξ ⊗ ( A ˆ Xδ − δ (cid:48) ,ε +2 δ (cid:48) ) , where the inequality follows from (2.17) together with the triangle-inequality. Therefore, ( ˆ X, ˆ r, ξ ) ∈ M δ − δ (cid:48) , ε +2 δ (cid:48) I . Now the claim follows from (cid:107) ˆ µ − ξ (cid:107) ≤ (cid:107) ˆ µ − ˆ µ (cid:48) (cid:107) + (cid:107) ˆ µ (cid:48) − ξ (cid:107) ≤ ε + δ (cid:48) . Proposition 2.7 (estimates on β ) . Let δ, ε > and consider X = ( X, r, µ ) ∈ M I . Then the followinghold: (i) If µ (cid:48) ∈ M f ( X × I ) , then β ( X ) ≤ β (cid:0) ( X, r, µ (cid:48) ) (cid:1) + 2 (cid:107) µ − µ (cid:48) (cid:107) . (ii) If X ∈ M δ,εI , then β ( X ) ≤ ε (cid:107) µ (cid:107) . (iii) If X ∈ M δ,εI and ˆ X ∈ M I with d mGP ( X , ˆ X ) < δ , then β ( X ) ≤ ε (cid:0) (cid:107) µ (cid:107) + 2 (cid:1) and β ( ˆ X ) ≤ ( ε + 2 δ )(2 + (cid:107) µ (cid:107) + δ ) . (2.19) Proof. (i) follows directly from the definition.(ii) If x ∈ X and u, v ∈ I satisfy (cid:0) ( x, u ) , ( x, v ) (cid:1) ∈ A X δ,ε , then d ( u, v ) ≤ ε by definition of A X δ,ε . Thus β ( X ) = (cid:82) X × I (cid:82) I (1 ∧ d ( u, v )) K x (d v ) µ (cid:0) d( x, u ) (cid:1) ≤ ε (cid:107) µ (cid:107) .(iii) Combining (i) and (ii) yields β ( X ) ≤ ε + ε (cid:107) µ (cid:107) . Let δ (cid:48) = d mGP ( X , ˆ X ). By Lemma 2.6, we haveˆ X ∈ M δ − δ (cid:48) , ε +2 δ (cid:48) I and thus β ( ˆ X ) ≤ (2 + (cid:107) ˆ µ (cid:107) )( ε + 2 δ (cid:48) ) ≤ (2 + (cid:107) µ (cid:107) + δ )( ε + 2 δ ).In order to complete the proof of Proposition 2.1 with the help of Proposition 2.7, we first observethat, as a consequence of Lusin’s theorem, every functionally marked metric measure space is an elementof M hI for some h ∈ H . Together with Lemma 2.9 below, this means that we have a nice (thoughuncountable) decomposition of M fct I into closed sets. Lemma 2.8 (decomposition of M fct I ) . The following equality holds: M fct I = (cid:83) h ∈H M hI .Proof. We have M hI ⊆ β − (0) = M fct I for every h ∈ H . Indeed, the equality was shown in the first partof the proof of Proposition 2.1. To obtain the inclusion, that is β ( X ) = 0 for all X ∈ M hI , recall M hI fromDefinition 2.4 and choose ε = h (2 δ ) in Proposition 2.7(iii).10onversely, let X = ( X, r, ν, κ ) ∈ M fct I . According to Lusin’s theorem, we find for every ε > K ε ⊆ X , and a modulus of continuity h ε ∈ H , such that ν ( X \ K ε ) ≤ ε and κ | K ε is h ε -uniformly continuous. In particular, X ∈ M δ,h ε ( δ ) ∨ εI ∀ ε, δ > . (2.20)We may assume without loss of generality that ε (cid:55)→ h ε ( δ ) is decreasing and right-continuous for every δ > 0. We define h ( δ ) := inf (cid:8) ε > h ε ( δ ) < ε (cid:9) ∈ R + ∪ {∞} . (2.21)Clearly, h ( δ ) converges to 0 as δ ↓ h ε ∈ H . Furthermore, h h ( δ ) ( δ ) ≤ h ( δ ), and hence (2.20)with ε = h ( δ ) implies X ∈ M hI . Proof of Proposition (completion). We still have to show continuity of β in X ∈ β − (0). Due toLemma 2.8, there is h ∈ H with X ∈ M hI . Now Proposition 2.7 yields for δ > ˆ X ∈ B M Iδ ( X ) β ( ˆ X ) ≤ ( h (2 δ ) + 2 δ )(2 + (cid:107) µ (cid:107) + δ ), which converges to 0 as δ ↓ M hI is containedin M fct I . In fact, M hI is even Gromov-weakly closed, which will be used in the proof of Theorem 3.11below. Lemma 2.9 (closedness of M hI ) . For every δ, ε > , M δ,εI is marked Gromov-weakly closed in M I . Inparticular, M hI is closed for every h ∈ H .Proof. Fix ε, δ > X n ) n ∈ N be a sequence in M δ,εI converging marked Gromov-weakly to some X = ( X, r, µ ) ∈ M I . Using Lemma 1.9, we may assume that X n , n ∈ N , and X are subspaces of acommon separable, metric space ( E, r E ), such that µ n w −→ µ on E × I . By definition of M δ,εI , we find µ (cid:48) n ≤ µ n , (cid:107) µ (cid:48) n − µ n (cid:107) ≤ ε , such that ( µ (cid:48) n ) ⊗ is supported by A Eδ,ε for all n ∈ N . Since ( µ (cid:48) n ) n ∈ N is tight,we may assume, by passing to a subsequence, that µ (cid:48) n w −→ µ (cid:48) for some µ (cid:48) ∈ M f ( E ). Obviously, µ (cid:48) ≤ µ and (cid:107) µ − µ (cid:48) (cid:107) = lim n →∞ (cid:107) µ n (cid:107) − (cid:107) µ (cid:48) n (cid:107) ≤ ε . Because A Eδ,ε is closed, ( µ (cid:48) ) ⊗ is supported by A Eδ,ε and hence X ∈ M δ,εI . Based on the construction of the complete metric and the decomposition M fct I = (cid:83) h ∈H M hI into closedsets obtained in Section 2, we now derive criteria to check if a marked metric measure space admits amark function, especially in the case where it is given as a marked Gromov-weak limit. We then transferthe results to random mmm-spaces and M I -valued stochastic processes. Our main criterion for deterministic spaces is a direct consequence of the results in Section 2. Recallthat H is the set of moduli of continuity defined in (1.17). Theorem 3.1 (characterization of existence of a mark function in the limit) . Let ( X n ) n ∈ N be a sequencein M I with X n mGw −−−→ X ∈ M I . Then X ∈ M fct I if and only if there exists h ∈ H such that for every δ > X n ∈ M δ,h ( δ ) I for infinitely many n ∈ N . (3.1) In this case, X ∈ M hI . roof. First assume there is h ∈ H such that (3.1) is satisfied. Since M δ,h ( δ ) I is closed by Lemma 2.9,(3.1) implies that X ∈ M δ,h ( δ ) I for every δ , that is X ∈ M hI . By Lemma 2.8, M hI ⊆ M fct I .Conversely, assume X ∈ M fct I . Then, by Lemma 2.8, we find h ∈ H with X ∈ M hI . We claim that (3.1)holds with h replaced by ˆ h ( δ ) := h (3 δ ) + 2 δ . Indeed, fix δ > X ∈ M hI ⊆ M δ,h (3 δ ) I .Lemma 2.6 yields X n ∈ M δ, ˆ h ( δ ) I for all n with d mGP ( X , X n ) < δ .We will use Theorem 3.1 in the following form. Corollary 3.2. Let X n = ( X n , r n , ν n , κ n ) ∈ M fct I , X n mGw −−−→ X ∈ M I . Let Y n,δ ⊆ X n measurable for n ∈ N , δ > , and h ∈ H . Then X ∈ M fct I if the following two conditions hold for every δ > : lim inf n →∞ ν n ( X n \ Y n,δ ) ≤ h ( δ ) , (3.2) ∀ n ∈ N , x, y ∈ Y n,δ : r n ( x, y ) < δ = ⇒ d (cid:0) κ n ( x ) , κ n ( y ) (cid:1) ≤ h ( δ ) . (3.3) Proof. Let µ (cid:48) n := µ n | Y n,δ × I , where µ n = ν n ⊗ δ κ n . Then (3.3) implies ( X n , r n , µ (cid:48) n ) ∈ M δ,h ( δ ) I and (3.2)yields (cid:107) µ (cid:48) n − µ n (cid:107) ≤ h ( δ ) for infinitely many n . Hence we can apply Theorem 3.1. Remark 3.3. To obtain X ∈ M fct I , it is clearly enough to show in Theorem 3.1 and Corollary 3.2, (3.1)respectively (3.2) and (3.3) only for δ = δ m for a sequence ( δ m ) m ∈ N with δ m ↓ m → ∞ .We illustrate the rˆole of the exceptional set X n \ Y n,δ , and the importance of its dependence on δ ,with a simple example. Example 3.4. Consider X = [0 , 1] with Euclidean metric r , ν = λ + δ , where λ is Lebesgue-measure, and κ n ( x ) = ( nx ) ∧ 1. Obviously, X n = ( X, r, ν, κ n ) converges marked Gromov-weakly and the limit admitsthe mark function (0 , . To see this from Corollary 3.2, we choose h ( δ ) = δ and Y n,δ = { } ∪ [ δ ∨ n , Y n,δ independent of δ . Remark 3.5 (equicontinuous case) . If, in Corollary 3.2, Y n,δ = Y n does not depend on δ , then (3.3)means that κ n is h -uniformly continuous on Y n . Consequently, the mark function of X is in this case h -uniformly continuous. If we restrict to Y n = X n for all n , we recover part (ii) of Lemma 1.13. Corollary 3.6. Let X n = ( X n , r n , ν n , κ n ) ∈ M fct I and assume that X n converges to X = ( X, r, µ ) ∈ M I marked Gromov-weakly. Further assume that for n ∈ N , δ > , there are measurable sets Z n,δ ⊆ X n ,such that lim δ ↓ lim inf n →∞ (cid:18) ν n ( X n \ Z n,δ ) + (cid:90) Z n,δ (cid:16) ∧ diam (cid:0) κ n (cid:0) B X n δ ( x ) ∩ Z n,δ (cid:1)(cid:1)(cid:17) ν n (d x ) (cid:19) = 0 , (3.4) where diam is the diameter of a set. Then X admits a mark function, that is X ∈ M fct I .Proof. For δ > g ( δ ) := sup <δ (cid:48) ≤ δ lim inf n →∞ (cid:18) ν n ( X n \ Z n,δ (cid:48) ) + (cid:90) Z n,δ (cid:48) (cid:16) ∧ diam (cid:0) κ n (cid:0) B X n δ (cid:48) ( x ) ∩ Z n,δ (cid:48) (cid:1)(cid:1)(cid:17) ν n (d x ) (cid:19) . (3.5)By (3.4), lim δ ↓ g ( δ ) = 0 and g is increasing with (cid:107) g (cid:107) ∞ ≤ (cid:107) µ (cid:107) . Let h ∈ H be such that g ( δ ) ≤ h ( δ )2 (cid:0) ∧ h ( δ )) for all δ > 0. Then ν n (cid:0)(cid:8) x ∈ Z n,δ : diam (cid:0) κ n ( B X n δ ( x ) ∩ Z n,δ ) (cid:1) > h ( δ ) (cid:9)(cid:1) ≤ g ( δ )1 ∧ h ( δ ) ≤ h ( δ ) / . (3.6)Now apply Corollary 3.2 with Y n,δ := (cid:8) x ∈ Z n,δ : diam (cid:0) κ n (cid:0) B X n δ ( x ) ∩ Z n,δ (cid:1)(cid:1) ≤ h ( δ ) (cid:9) . (3.7)Then (3.3) follows from the definition of Y n,δ in (3.7), and ν n ( X n \ Y n,δ ) ≤ ν n ( X n \ Z n,δ ) + h ( δ ) / ≤ g ( δ ) + h ( δ ) / ≤ h ( δ ) holds by (3.6) and (3.5). 12 .2 Random fmm-spaces The following theorem is a randomized version of Theorem 3.1. It is our main criterion for M I -valuedrandom variables. Theorem 3.7 (random fmm-spaces as limits in distribution) . Let ( X n ) n ∈ N be a sequence of M I -valuedrandom variables which converges in distribution (w.r.t. marked Gromov-weak topology) to an M I -valuedrandom variable X . Further assume that for every ε > , there exists a modulus of continuity h ε ∈ H such that lim sup δ ↓ lim sup n →∞ P (cid:0)(cid:8) X n ∈ M δ,h ε ( δ ) I (cid:9)(cid:1) ≥ − ε. (3.8) Then X admits almost surely a mark function, that is X ∈ M fct I almost surely.If additionally X n = ( X n , r n , ν n , κ n ) ∈ M fct I almost surely for all n ∈ N , we can replace (3.8) byexistence of random measurable sets Y εn,δ ⊆ X n , n ∈ N , δ > , in addition to the h ε ∈ H , such that thefollowing two conditions hold for every ε > : lim sup δ ↓ lim sup n →∞ P (cid:0)(cid:8) ν n ( X n \ Y εn,δ ) ≤ h ε ( δ ) (cid:9)(cid:1) ≥ − ε. (3.9) ∀ n ∈ N , δ > , x, y ∈ Y εn,δ : r n ( x, y ) < δ = ⇒ d (cid:0) κ n ( x ) , κ n ( y ) (cid:1) ≤ h ε ( δ ) . (3.10) Remark 3.8. In (3.9), we need not worry about measurability of the “event” B n,δ := (cid:8) ν n ( X n \ Y εn,δ ) ≤ h ε ( δ ) (cid:9) due to the choice of Y εn,δ . The inequality (3.9) is to be understood in the sense of inner measure,that is we require that there are measurable sets C n,δ ⊆ B n,δ with lim sup δ ↓ lim sup n →∞ P ( C n,δ ) ≥ − ε . Proof. The second statement follows in the same way as Corollary 3.2. We divide the proof of the mainpart in two steps. First, we show X ∈ M fct I if, instead of (3.8), even P (cid:16) (cid:92) m ∈ N (cid:8) X n ∈ M δ m ,h ε ( δ m ) I for infinitely many n (cid:9)(cid:17) ≥ − ε (3.11)holds for a sequence δ m = δ m ( ε ) ↓ m → ∞ . In the second step, we show that, given (3.8), we canmodify h ε to ˆ h ε ∈ H such that (3.11) holds with h ε replaced by ˆ h ε . Step 1. By Skorohod’s representation theorem, we may assume that the X n are coupled such that theyconverge almost surely to X in the marked Gromov-weak topology. The inequality (3.11) implies thatwith probability at least 1 − ε , for all m ∈ N , X n ∈ M δ m ,h ε ( δ m ) I for infinitely many n . By Theorem 3.1 andRemark 3.3, this means that the probability that X admits a mark function is at least 1 − ε . Because ε is arbitrary, this implies X ∈ M fct I almost surely. Step 2. Let T ( ε, δ ) := lim sup n →∞ P (cid:0)(cid:8) X n ∈ M δ,h ε ( δ ) I (cid:9)(cid:1) in (3.8). Set δ := sup (cid:8) δ ∈ [0 , 1] : T ( ε/ , δ ) ≥ − ε/ h ε/ ( δ ) < (cid:9) . (3.12)By (3.8) and as h ε/ ∈ H , the set inside the supremum is non-empty. Next define recursively δ m := sup (cid:8) δ ∈ [0 , δ m − / 2] : T ( ε − ( m +1) , δ ) ≥ − ε − m and h ε − ( m +1) ( δ ) < /m (cid:9) (3.13)for m ∈ N , m ≥ 2. Again, the set inside the supremum is non-empty by (3.8) and as h ε − ( m +1) ∈ H .Moreover, δ m = δ m ( ε ) > δ m ↓ m → ∞ and h ε − ( m +1) ( δ m ) ≤ /m follows. We can therefore setˆ h ε ( δ m ) := h ε − ( m +1) ( δ m ) (3.14)13nd extend this to ˆ h ε ∈ H . Using Fatou’s lemma, we obtain P (cid:16) (cid:91) m ∈ N (cid:8) X n (cid:54)∈ M δ m , ˆ h ε ( δ m ) I eventually (cid:9)(cid:17) ≤ (cid:88) m ∈ N E (cid:0) lim inf n →∞ M I \ M δm, ˆ hε ( δm ) I ( X n ) (cid:1) (3.15) ≤ (cid:88) m ∈ N lim inf n →∞ P (cid:0)(cid:8) X n (cid:54)∈ M δ m , ˆ h ε ( δ m ) I (cid:9)(cid:1) = (cid:88) m ∈ N (cid:0) − T ( ε − ( m +1) , δ m ) (cid:1) ≤ (cid:88) m ∈ N ε − m = ε. Thus (3.11) holds with h ε replaced by ˆ h ε . Let J ⊆ R + be a (closed, open or half-open) interval and consider a stochastic process X = ( X t ) t ∈ J withvalues in M I and c`adl`ag paths, where M I is equipped with the marked Gromov-weak topology. We saythat X is an M fct I -valued c`adl`ag process if P (cid:0)(cid:8) X t , X t − ∈ M fct I for all t ∈ J (cid:9)(cid:1) = 1 , (3.16)where X t − is the left limit of X at t ( X (cid:96) − := X (cid:96) if (cid:96) is the left endpoint of J ). In the following, wegive sufficient criteria for X to be an M fct I -valued c`adl`ag process. We are particularly interested in thesituation where X is the limit of M fct I -valued processes X n .Unsurprisingly, if the set of P -measure smaller or equal to ε in Theorem 3.7 is independent of t , theresult is true for all t simultaneously, almost surely. The modulus of continuity may also depend on t ina continuous way; or be arbitrary if the limiting process has continuous paths: Theorem 3.9. Let J ⊆ R + be an interval, and X n = ( X nt ) t ∈ J , n ∈ N , a sequence of M I -valued c`adl`agprocesses converging in distribution to an M I -valued c`adl`ag process X = ( X t ) t ∈ J . Assume that for every t ∈ J , ε > , there exists h t,ε ∈ H such that lim sup δ ↓ lim sup n →∞ P (cid:0)(cid:8) X nt ∈ M δ,h t,ε ( δ ) I ∀ t ∈ J (cid:9)(cid:1) ≥ − ε. (3.17) Then X is an M fct I -valued c`adl`ag process, that is (3.16) is satisfied, if at least one of the following twoconditions holds: (i) X has continuous paths a.s. (ii) t (cid:55)→ h t,ε ( δ ) is continuous for every ε, δ > .If additionally X n is M fct I -valued almost surely for all n ∈ N , (3.17) can be replaced by existence ofrandom measurable sets Y nt,ε,δ ⊆ X nt , in addition to the h t,ε ∈ H , satisfying the following two conditionsfor every ε > : lim sup δ ↓ lim sup n →∞ P (cid:0)(cid:8) ν nt ( X nt \ Y nt,ε,δ ) ≤ h t,ε ( δ ) ∀ t ∈ J (cid:9)(cid:1) ≥ − ε, (3.18) ∀ n ∈ N , t ∈ J, δ > , x, y ∈ Y nt,ε,δ : r n ( x, y ) < δ = ⇒ d (cid:0) κ n ( x ) , κ n ( y ) (cid:1) ≤ h t,ε ( δ ) . (3.19) Proof. Due to the Skorohod representation theorem, we may assume that X n → X almost surely in theSkorohod topology. For condition (i) respectively (ii) we obtain(i) If X has continuous paths a.s., the convergence in Skorohod topology implies uniform convergenceof X nt ( ω ) on J a.s. with respect to d mGP . Hence we have X nt mGw −−−→ n →∞ X t for all t ∈ J , almost surely,and we can proceed as in the proof of Theorem 3.7.14ii) There are (random) continuous w n : J → J , converging to the identity uniformly on compacta,such that X nw n ( t ) → X t for all t ∈ J , almost surely. We can use the moduli of continuity ˆ h t,ε ( δ ) := h t,ε ( δ ) + δ and proceed as in the proof of Theorem 3.7. Note here that, due to continuity of h t,ε ( δ )in t , there is for every compact subinterval J of J an N J ,ε,δ ∈ N such that ˆ h t,ε ( δ ) ≥ h w n ( t ) ,ε ( δ ) forall n ≥ N J ,ε,δ and t ∈ J .The same arguments apply for left limits with w n − such that X nw n − ( t ) → X t − .To use Theorem 3.9, we have to check in (3.17) or (3.18) a condition for uncountably many t simulta-neously, which is often much more difficult than for every t individually. One situation, where it is easyto pass from individual t to all t simultaneously is the case where the moduli of continuity h t,ε actuallydo not depend on t and ε (see Corollary 3.13). The independence of ε , however, is a strong requirement.Therefore, we relax it to not blowing up too fast as ε ↓ 0, where the “too fast” is determined by thefollowing modulus of c`adl`agness of the limiting process. Definition 3.10 (modulus of c`adl`agness) . Let J be an interval, ( E, r ) a metric space, and e = ( e t ) t ∈ J ∈D E ( J ) a c`adl`ag path on J with values in E . Following [Bil68, (14.44)] , set w (cid:48)(cid:48) ( e, δ ) := sup t,t ,t ∈ J : t ≤ t ≤ t ,t − t ≤ δ min (cid:8) r ( e ( t ) , e ( t )) , r ( e ( t ) , e ( t )) (cid:9) . (3.20) We say that e admits w ∈ H as modulus of c`adl`agness if w (cid:48)(cid:48) ( e, δ ) ≤ w ( δ ) for all δ > . Theorem 3.11. Fix an interval J ⊆ R + . Let X = ( X t ) t ∈ J and X n = ( X nt ) t ∈ J , n ∈ N , be M I -valuedc`adl`ag processes such that X n converges in distribution to X . Furthermore, assume that there is a denseset Q ⊆ J and w ε , h ε ∈ H , such that for all ε > n →∞ P ( {X nt ∈ M δ,h ε ( δ ) I } ) ≥ − ε ∀ δ > , t ∈ Q, (3.21) P ( { t (cid:55)→ X t admits w ε as modulus of c`adl`agness w.r.t. d mGP } ) ≥ − ε, and (3.22)lim inf δ ↓ h ε · δ (cid:0) w ε ( δ ) (cid:1) = 0 . (3.23) Then X is an M fct I -valued c`adl`ag process, that is (3.16) holds. Recall the decomposition M I \ M fct I = (cid:83) m ∈ N F m with F m defined in (2.4). The basic idea of the proofis to use the following lemma about c`adl`ag paths to show that, almost surely, the path of X avoids F m .The assertion of the lemma follows easily using the triangle-inequality. Lemma 3.12. Let J be an interval, ( E, r ) a metric space, and e = ( e t ) t ∈ J ∈ D E ( J ) a c`adl`ag pathadmitting modulus of c`adl`agness w ∈ H . Let F ⊆ E be any set, δ > , and Q ⊆ J such that for all t ∈ J there is t , t ∈ Q with t ≤ t ≤ t ≤ t + δ . Then r ( e t , F ) > w ( δ ) ∀ t ∈ Q = ⇒ e t (cid:54)∈ F and e t − (cid:54)∈ F ∀ t ∈ J. (3.24) Proof of Theorem . Because M h ε I = (cid:84) δ> M δ,h ε ( δ ) I is closed by Lemma 2.9, the Portmanteau theoremand (3.21) imply P ( {X t (cid:54)∈ M h ε I } ) < ε ∀ t ∈ Q, ε > . (3.25)Due to the Skorohod representation theorem, we may assume that X n → X almost surely in Skorohodtopology. In order to simplify notation, we assume J = [0 , 1] and Q = (cid:83) k ∈ N Q k with Q k = { i − k : i =0 , . . . , k } . It is enough to show for every ε > , m ∈ N and F m as defined in (2.4) that p m := P (cid:0)(cid:8) ∃ t ∈ [0 , 1] : X t or X t − ∈ F m (cid:9)(cid:1) ≤ ε. (3.26)To show (3.26), fix ε > m ∈ N , and let X t = ( X t , r t , µ t ). Because X has c`adl`ag paths, we find K = K ( ε ) < ∞ such that P (cid:0)(cid:8) sup t ∈ [0 , (cid:107) µ t (cid:107) ≥ K − (cid:9)(cid:1) < ε. (3.27)15ccording to (3.23) and (3.25), we can choose k ∈ N big enough such that for h := h ε − k we have h (cid:0) w ε (2 − k ) (cid:1) < ( Km ) − − w ε (2 − k ) and P ( {X t (cid:54)∈ M hI } ) < ε − k . (3.28)Assume without loss of generality that w ε (2 − k ) ≤ 1. Now Proposition 2.7(iii) implies that, whenever X t ∈ M hI and (cid:107) µ t (cid:107) < K − 3, we have d mGP ( X t , F m ) > w ε (2 − k ) . (3.29)Combining (3.22) and Lemma 3.12, we obtain p m ≤ ε + P (cid:0)(cid:8) ∃ t ∈ Q k : d mGP ( X t , F m ) ≤ w ε (2 − k ) (cid:9)(cid:1) . (3.30)Using (3.27), (3.29), and (in the last step) (3.28), we conclude p m ≤ ε + 2 k sup t ∈ Q k P (cid:0)(cid:8) (cid:107) µ t (cid:107) < K − , X t (cid:54)∈ M hI (cid:9)(cid:1) ≤ ε. (3.31)Thus (3.26) holds for all ε > 0, and P ( {∃ t ∈ [0 , 1] : X t (cid:54)∈ M fct I } ) = sup m ∈ N p m = 0 follows.If, in Theorem 3.11, we can choose the modulus of continuity h ε = h ∈ H , independent of ε , suchthat (3.21) holds, we do not need to check (3.22) and (3.23). Corollary 3.13 ( ε -independent modulus of continuity) . Assume that X n = ( X nt ) t ∈ J converges in dis-tribution to an M I -valued c`adl`ag process X , and Q ⊆ J is dense. Then X is an M fct I -valued c`adl`agprocess if, for some h ∈ H , lim sup n →∞ P ( {X nt ∈ M hI } ) = 1 ∀ t ∈ Q. (3.32) Proof. Let h ∈ H be such that (3.32) is satisfied and set h ε := h . Then (3.23) is satisfied for every choiceof w ε ∈ H , ε > 0. For every c`adl`ag process, in particular for X , there exist moduli of c`adl`agness w ε suchthat (3.22) holds (cf. [Bil68, (14.6),(14.8) and (14.46)]). Thus, Theorem 3.11 yields the claim. The (neutral) tree-valued Fleming-Viot dynamics is constructed in [GPW13] using the formalism ofmetric measure spaces. In [DGP12], (allelic) types – encoded as marks of marked metric measure spaces– are included, in order to be able to model mutation and selection.In [DGP12, Remark 3.11] and [DGP13, Theorem 6] it is stated that the resulting tree-valued Fleming-Viot dynamics with mutation and selection (TFVMS) admits a mark function at all times, almostsurely. The given proof, however, contains a gap, because it relies on the criterion claimed in [DGP13,Lemma 7.1], which is wrong in general (see Example 4.1). The reason why the criterion may fail is alack of homogeneity of the measure ν , in the sense that there are parts with high and parts with lowmass density. Consequently, if we condition two samples to have distance less than ε , the probabilitythat they are from the high-density part tends to one as ε ↓ 0, and we do not “see” the low-densitypart. This phenomenon occurs if ν has an atom but is not purely atomic. We also give two non-atomicexamples, one a subset of Euclidean space, and the other one ultrametric. Example 4.1 (counterexamples) . In both examples, it is straight-forward to see that ( X, r, µ ), with µ = ν ⊗ K , satisfies the assumptions of [DGP13, Lemma 7.1], but does not admit a mark function. Themark space is I = { , } .(i) Let λ A be Lebesgue measure of appropriate dimension on a set A . Define X := [0 , ∪ [2 , , 3] is identified with [2 , × { } ⊆ R , ν := ( λ [0 , + λ [2 , ) and K x := (cid:40) ( δ + δ ) , x ∈ [0 , ,δ , x ∈ [2 , . (4.1)16ii) In this example think of a tree consisting of a left part with tertiary branching points and a rightpart with binary branching points. The leaves correspond to X := A ∪ B with A = { , , } N and B = { , } N , and we choose as a metric r (cid:0) ( x n ) n ∈ N , ( y n ) n ∈ N (cid:1) := max n ∈ N e − n · x n (cid:54) = y n . (4.2)Note that ( X, r ) is a compact, ultrametric space. The measure ν is constructed as follows: choosethe left respectively right part of the tree with probability each. Going deeper in the tree, at eachbranching point a branch is chosen uniformly. That is, let ν A and ν B be the Bernoulli measureson A and B with uniform marginals on { , , } and { , } , respectively. Define ν := ( ν A + ν B ) and K x := (cid:40) ( δ + δ ) , x ∈ A,δ , x ∈ B. (4.3) In the following, we prove the existence of a mark function for the TFVMS by verifying the assumptions ofTheorem 3.9 for a sequence of approximating tree-valued Moran models. Due to the Girsanov transformgiven in [DGP12, Theorem 2], it is enough to consider the neutral case, that is without selection.We briefly recall the construction of the tree-valued Moran model with mutation (TMMM) with finitepopulation U N = { , . . . , N } , N ∈ N , and types from the mark space I . For details and more formaldefinitions, see [DGP12, Subsections 2.1–2.3]. In the underlying Moran model with mutation (MMM),every pair of individuals “resamples” independently at rate γ > 0. Here, resampling means that one ofthe individuals (chosen uniformly at random among the two) is replaced by an offspring of the other one,and the offspring gets the same type as the parent. Furthermore, every individual mutates independentlyat rate ϑ ≥ 0, which means that it changes its type according to a fixed stochastic kernel β ( · , · ) on I .Denote the resulting type of individual x ∈ U N at time t ≥ κ Nt ( x ). To obtain the tree-valueddynamics, define the distance r Nt ( x, y ) between two individuals x, y ∈ U N at time t ≥ t + r N ( x, y ) otherwise. The TMMM is the resulting process X Nt = ( U N , r Nt , ν N , κ Nt ), withsampling measure ν N = N (cid:80) Nk =1 δ k . It is easy to check that, by definition, ( U N , r Nt ) is an ultrametricspace, provided that the initial metric space ( U N , r N ) is ultrametric. This explains the name tree-valued (cf. [DGP12, Remark 2.7]).Next recall the graphical construction of the MMM from [DGP12, Definition 2.2]. A resamplingevent is modeled by means of a family of independent Poisson point processes { η k,(cid:96) res : k, (cid:96) ∈ U N } on R + ,where each η k,(cid:96) res has rate γ/ 2. If t ∈ η k,(cid:96) res , draw an arrow from ( k, t ) to ( (cid:96), t ) to represent a resamplingevent at time t , where (cid:96) is an offspring of k . Similarly, model mutation times by a family of independentPoisson point processes { η k mut : k ∈ U N } , where each η k mut has rate ϑ . If t ∈ η k,(cid:96) res , draw a dot at ( k, t ) torepresent a mutation event changing the type of individual k (see Figure 1).Let ( M t ,Nt ) t ≥ t , M t ,Nt ⊆ U N with M t ,Nt = ∅ be the process that records the individuals of thepopulation at time t with an ancestor at a time t < s ≤ t involved in a mutation event. By a couplingargument, this process can be constructed by means of the Poisson point processes ( η k,(cid:96) res , η k mut , k, (cid:96) ∈ U N )as follows (compare Figures 1–2): M t ,Nt = M t ,Nt − ∪ { (cid:96) } if there is a resampling arrow from k ∈ M t ,Nt − to (cid:96) ∈ U N at time t,M t ,Nt − ∪ { k } if there is a mutation event at k ∈ U N at time t,M t ,Nt − \{ (cid:96) } if there is a resampling arrow from k / ∈ M t ,Nt − to (cid:96) ∈ U N at time t. (4.4)Let ξ Nt := N M t ,Nt + t be the proportion of individuals at time t + t, t ≥ t . 17 time axis1 2 3 4 5 6 7 8 9 10 = N b bb t 81 2 3 4 5 6 7 9 10 M t ,Nt ∅{ }{ }{ , }{ , }{ , , }{ , , , }{ , , , }{ , , , , }{ , , , , } b { , , }{ , } Figure 1: Graphical construction of the MMM for N = 10 for the time-period [ t , t ], and the resulting process( M t ,Ns ) s ∈ [ t ,t ] . Resampling arrows are drawn at points of η k,(cid:96) res , and mutation dots at points of η k mut . t N b bb t 81 2 3 4 5 6 7 9 10 b bb b Figure 2: Tracing the ancestor backwards in time in Figure 1: This dual construction is also known as thecoalescent backwards in time. Reverse the arrows to see for instance that 3 at time t is an ancestor of 8 at time t . The elements of M t ,Nt ⊆ U N are highlighted by boxes in the right part of the picture. Lemma 4.2. Let C := ϑ (2 ϑ + γ ) . Then for all a, δ > N →∞ P (cid:0) sup t ∈ [0 ,δ ] ξ Nt ≥ a (cid:1) ≤ Ca − δ . (4.5) Proof. By definition, (cid:0) ξ Nt (cid:1) t ≥ is a (continuous time) Markov jump process on [0 , 1] with ξ N = 0 andtransitions (cid:40) x (cid:55)→ x − /N at rate γ N x (1 − x ) ,x (cid:55)→ x + 1 /N at rate γ N x (1 − x ) + ϑN (1 − x ) . (4.6)This process converges weakly with respect to the Skorohod topology to the solution ( Z t ) t ≥ of thestochastic differential equation (SDE)d Z t = ϑ (1 − Z t )d t + (cid:112) γZ t (1 − Z t ) d B t , Z = 0 . (4.7)Indeed, to establish tightness use [EK05, Theorem III.9.4]. Note that, as [0 , 1] is compact, it suffices toshow the convergence of the generators applied to a set of appropriate test-functions. For existence anduniqueness of solutions to (4.7) reason as for the Bessel SDE in [RW00, (48.1) and below]. Moreover,18 t ∈ [0 , 1] is a bounded non-negative right-continuous submartingale. Hence, with Doob’s submartingaleinequality (see for instance [EK05, Proposition II.2.16(a)]), we obtain P (cid:0) sup t ∈ [0 ,δ ] Z t ≥ a (cid:1) = P (cid:0) sup t ∈ [0 ,δ ] Z t ≥ a (cid:1) ≤ a − E [ Z δ ] . (4.8)As Z t ∈ [0 , t ≥ E [ Z t ] ≤ ϑt and (4.9) E [ Z t ] = E (cid:2)(cid:90) t Z s ϑ (1 − Z s ) + γZ s (1 − Z s ) d s (cid:3) ≤ Ct . (4.10)Then lim sup N →∞ P (cid:0) sup t ∈ [0 ,δ ] ξ Nt ≥ a (cid:1) ≤ P (cid:0) sup t ∈ [0 ,δ ] Z t ≥ a (cid:1) ≤ Ca − δ (4.11)follows.As the construction of the TFVMS in [DGP12] is only given for a compact type-space I , we makethe same assumption. Note, however, that our proof itself does not use compactness and is thereforevalid for non-compact I , provided that the TFVMS is the limit of the corresponding Moran models, andthere exists a Girsanov transform allowing us to reduce to the neutral case. Theorem 4.3 (the TFVMS admits a mark-function) . Let I be compact and X = ( X t ) t ≥ be the tree-valued Fleming-Viot dynamics with mutation and selection as defined in [DGP12] . Then P ( X t ∈ M fct I for all t > 0) = 1 . (4.12) In particular, ( X t ) t> is an M fct I -valued c`adl`ag process.Proof. By [DGP12, Theorem 2], there exists a Girsanov transform that enables us to assume withoutloss of generality that selection is not present. In this case, according to [DGP12, Theorem 3], X is thelimit in distribution of TMMMs X N = ( X Nt ) t ≥ , as discussed above. Let X Nt = ( U N , r Nt , ν N , κ Nt ) with U N = { , . . . , N } and ν N the uniform distribution on U N . Let δ > r Nt ( x, y ) between two individuals x, y ∈ U N at time t ≥ δ/ r Nt ( x, y ) < δ , then x and y at time t have a common ancestor at time t − δ/ 2. Furtherrecall that ( M t ,Nt ) t ≥ t , with M t ,Nt ⊆ U N and M t ,Nt = ∅ , records the individuals of the population attime t with an ancestor at a time s ∈ ( t , t ] involved in a mutation event (cf. (4.4)).Fix an arbitrary time horizon T > i ∈ N , i ≤ T /δ . Using the notation of Theorem 3.9, for t ∈ [ iδ/ , ( i + 1) δ/ Y Nt,ε,δ := U N \ M ( i − δ/ ,Nt , independent of ε > 0. Set Y Nt,ε,δ := ∅ for t < δ/ h t,ε ∈ H . Indeed, if x, y ∈ Y Nt,ε,δ satisfy r Nt ( x, y ) < δ ,then they have a common ancestor at time t := ( i − δ/ ≤ t − δ/ 2, and after this point in timeno mutation occurred along their ancestral lineages. In particular, d ( κ Nt ( x ) , κ Nt ( y )) = 0, and (3.19)is obvious. Moreover, X N is M fct I -valued by construction, and X has continuous paths by [DGP12,Theorem 1]. According to Theorem 3.9, it is therefore enough to find moduli of continuity h t,ε ∈ H suchthat (3.18) holds for every ε > C > a > N →∞ P (cid:0) sup t ∈ [ iδ/ , ( i +1) δ/ ν N (cid:0) U N \ Y Nt,ε,δ (cid:1) ≥ a (cid:1) ≤ Ca − δ . (4.13)After summation over i ∈ { , . . . , (cid:98) T /δ (cid:99)} , we obtainlim sup N →∞ P (cid:0) sup t ∈ [ δ/ ,T ] ν N (cid:0) U N \ Y Nt,ε,δ (cid:1) ≥ a (cid:1) ≤ T Cδa − . (4.14)For ε > a := √ ε − T Cδ , together with (cid:107) ν N (cid:107) ≤ t < δ/ 2, tosee that (3.18) is satisfied for h t,ε ∈ H with h t,ε ( δ ) ≥ √ ε − T Cδ + [2 t, ∞ [ ( δ ) . (4.15)19 Figure 3: Tracing the ancestor backwards in time: The Λ-coalescent allows for one parent to have more than onechild. Λ -Fleming-Viot process Let Λ be a finite measure on [0 , k -tuple out of N blocks merges independently at rate λ N,k := (cid:90) y k − (1 − y ) N − k Λ(d y ) . (4.16)For fixed N , it is elementary to construct a finite, random (ultra-)metric measure space encoding therandom genealogy of the Λ-coalescent, where the distance is defined as the time to the MRCA (recallthe construction of Figures 1–2 and see Figure 3). In [GPW09, Theorem 4], existence and uniquenessof a Gromov-weak limit in distribution, as N → ∞ , is proven to be equivalent to the so-called “dust-free”-property, namely (cid:82) y − Λ(d y ) = ∞ . The resulting limit is called Λ-coalescent measure tree.Now, replace the tree-valued Moran models considered in Subsection 4.1 and [DGP12] by so-calledtree-valued Λ-Cannings models with Λ satisfying the dust-free-property. That is, leave the mutation- andselection-part as it is and change the resampling-part of the Moran models as follows: For k = 2 , . . . , N ,at rate (cid:0) Nk (cid:1) λ N,k a block of k individuals is chosen uniformly at random among the N individuals of thepopulation. Upon such a resampling event, all individuals in this block are replaced by an offspring ofa single individual which is chosen uniformly from this block. Note that the genealogy (disregardingtypes) of the resulting Λ-Cannings model with N individuals is dual to the Λ-coalescent starting with N blocks. We call any limit point (in path space) of the tree-valued Λ-Cannings processes, as N tendsto infinity and Λ is fixed, tree-valued Λ -Fleming-Viot process (TLFV). In the neutral case, existence anduniqueness of such a limit point follows as a special case of the forthcoming work [GKW15]. Here, weshow that, whenever limit points exist, all of them admit mark functions. Theorem 4.4 (the TLFV admits a mark-function) . Suppose there is no selection, that is α = 0 , and X = ( X t ) t ≥ is a tree-valued Λ -Fleming-Viot process with mutation. Then P ( X t ∈ M fct I for all t > 0) = 1 . (4.17) Proof. By passing to a subsequence if necessary, we may assume that the Λ-Cannings models convergein distribution to X . We proceed as in Subsection 4.1. Again, let ( M t ,Nt ) t ≥ t , M t ,Nt ⊆ U N with M t ,Nt = ∅ be the process that records the individuals of the population at time t with an ancestor at atime t < s ≤ t involved in a mutation event and ξ Nt := N M t ,Nt + t be the proportion of individuals attime t + t, t ≥ t . By definition,20 ξ Nt (cid:1) t ≥ is a (continuous time) Markov jump process on [0 , 1] with ξ N = 0 and generator (cid:0) Ω N f (cid:1) ( x ) = ϑN (1 − x ) (cid:0) f ( x + 1 /N ) − f ( x ) (cid:1) (4.18)+ N (cid:88) k =2 λ N,k ( Nx ) ∧ k (cid:88) m =0 (cid:18) N xm (cid:19)(cid:18) N (1 − x ) k − m (cid:19) × (cid:16) mk (cid:0) f ( x + ( k − m ) /N ) − f ( x ) (cid:1) + k − mk (cid:0) f ( x − m/N ) − f ( x ) (cid:1)(cid:17) , where x ∈ [0 , , N · x ∈ N ∪ { } , f ∈ C b ([0 , x + m,k,N ∈ [ x, x + ( k − m ) /N ], x − m,k,N ∈ [ x − m/N, x ] with (cid:0) Ω N f (cid:1) ( x ) = ϑN (1 − x ) (cid:0) f ( x + 1 /N ) − f ( x ) (cid:1) (4.19)+ N (cid:88) k =2 λ N,k ( Nx ) ∧ k (cid:88) m =0 (cid:18) N xm (cid:19)(cid:18) N (1 − x ) k − m (cid:19)(cid:16) f (cid:48)(cid:48) ( x + m,k,N )2 m ( k − m ) kN + f (cid:48)(cid:48) ( x − m,k,N )2 ( k − m ) m kN (cid:17) = ϑ (1 − x ) f (cid:48) ( x ) + O ( N − ) + x (1 − x ) N (cid:88) k =2 λ N,k ∆ N,k ( x ) , where, using (cid:0) ni (cid:1) = ni (cid:0) n − i − (cid:1) for i ≥ N,k ( x ) = ( Nx ) ∧ ( k − (cid:88) m =1 (cid:18) N x − m − (cid:19)(cid:18) N (1 − x ) − k − m − (cid:19)(cid:16) f (cid:48)(cid:48) ( x + m,k,N ) k − m k + f (cid:48)(cid:48) ( x − m,k,N ) m k (cid:17) . (4.20)Recall that (cid:80) km =0 (cid:0) (cid:96)m (cid:1)(cid:0) N − (cid:96)k − m (cid:1) = (cid:0) Nk (cid:1) and λ N,k = (cid:82) y k − (1 − y ) N − k Λ(d y ) with a finite measure Λ on [0 , N (cid:88) k =2 | ∆ N,k ( x ) | ≤ (cid:107) f (cid:48)(cid:48) (cid:107) ∞ N (cid:88) k =2 λ N,k k − (cid:88) m =0 (cid:18) N x − m (cid:19)(cid:18) N (1 − x ) − k − − m (cid:19) (4.21)= (cid:107) f (cid:48)(cid:48) (cid:107) ∞ (cid:90) N (cid:88) k =2 (cid:18) N − k − (cid:19) y k − (1 − y ) N − k Λ(d y )= (cid:107) f (cid:48)(cid:48) (cid:107) ∞ Λ([0 , . Therefore, (cid:0) Ω N f (cid:1) ( x ) = ϑ (1 − x ) f (cid:48) ( x ) + O ( N − ) + x (1 − x ) O (1) . (4.22)Use f ( x ) = x, x ∈ [0 , 1] in (4.19) to see that ( ξ Nt ) t ≥ is a non-negative right-continuous submartingalewith ξ N = 0 and E [ ξ Nt ] ≤ ϑt . Use f ( x ) = x to deduce from (4.22) that E (cid:2) ( ξ Nt ) (cid:3) ≤ Ct + O ( N − ) t. (4.23)Now reason as for the TFVMS in the proofs of Lemma 4.2 and Theorem 4.3 to complete the claim. In [KW15] the results of the present paper will be applied in a context of evolving genealogies to establishthe existence of a mark function with the help of Theorem 3.9. These genealogies are random markedmetric measure spaces, constructed as the limit of approximating particle systems. The individualbirth- respectively death-rates in the N th-approximating population depend on the present trait of theindividuals alive and are of order O ( N ). At each birth-event, mutation happens with a fixed probability.21ach individual is assigned mass 1 /N . The metric under consideration is genetic distance: in the N th-approximating population genetic distance is increased by 1 /N at each birth with mutation. Hence,genetic distance of two individuals is counted in terms of births with mutation backwards in time to theMRCA rather than in terms of the time to the MRCA.Because of the use of exponential times in the modeling of birth- and death-events in this thereforenon-ultrametric setup the analysis of the modulus of continuity of the trait-history of a particle incombination with the evolution of its genetic age plays a major role in establishing tightness of theapproximating systems and existence of a mark function. In [Kli14, Lemma 3.9], control on the modulusof continuity is obtained by transferring the model to the context of historical particle systems. In afirst step, time is related to genetic distance by means of the modulus of continuity. The extend of thechange of trait of an individual in a small amount of time (recall (3.9) and (3.3)) can then be controlledby means of the modulus of continuity of its trait-path in combination with a control on the height ofthe largest jump during this period of time. This can in turn be ensured by appropriate assumptions onthe mutation transition kernels of the approximating systems. Acknowledgements We are thankful to Anita Winter for discussions in the initial phase of the project, and to the referee forhelpful comments. The research of Sandra Kliem was supported by the DFG through the SPP PriorityProgramme 1590. References [Ald93] David Aldous. The continuum random tree III. Ann. Probab. , 21(1):248–289, 1993.[Bil68] Patrick Billingsley. Convergence of probability measures . John Wiley & Sons, Inc., New York-London-Sydney,1968.[Bog07] V. I. Bogachev. Measure Theory, Volume II . Springer, 2007.[DGP11] Andrej Depperschmidt, Andreas Greven, and Peter Pfaffelhuber. 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