The Brownian Web as a random $\mathbb R$-tree

Motivated by [G. Cannizzaro, M. Hairer, Comm. Pure Applied Math., '22], we provide a construction of the Brownian Web (see [T\'oth B., Werner W., Probab. Theory Related Fields, '98] and [L. R. G. Fontes, M. Isopi, C. M. Newman, and K. Ravishankar, Ann. Probab., '04]), i.e. a family of coalescing Brownian motions starting from every point in $\mathbb R^2$, as a random variable taking values in the space of (spatial) $\mathbb R$-trees. This gives a stronger topology than the classical one {(i.e.\ Hausdorff convergence on closed sets of paths)}, thus providing us with more continuous functions of the Brownian Web and ruling out a number of potential pathological behaviours. Along the way, we introduce a modification of the topology of spatial $\mathbb R$-trees in [T. Duquesne, J.-F. Le Gall, Probab. Theory Related Fields, '05] and [M. T. Barlow, D. A. Croydon, T. Kumagai, Ann. Probab. '17] which makes it a complete separable metric space and could be of independent interest. We determine some properties of the characterisation of the Brownian Web in this context (e.g.\ its box-counting dimension) and recover some which were determined in earlier works, such as duality, special points and convergence of the graphical representation of coalescing random walks.


Introduction
The Brownian Web is a random object that can be heuristically described as a collection of coalescing Brownian motions starting from every space-time point in R 2 , a typical realisation of which is displayed in Figure 1. Its study originated in the PhD thesis of Arratia [Arr79], who was interested in the Voter model [Lig05], its dual, given by a family of (backward) coalescing random walks, and their diffusive scaling limit. It was rediscovered by Tóth and Werner in [TW98], where they provided the first thorough construction, determined its main properties, and used it to introduce the so-called true self-repelling motion. A different characterisation was subsequently given in [FINR04] where, by means of a new topology, a sufficient condition for the convergence of families of coalescing random walks was derived. Later on, further generalisations via alternative approaches appeared, e.g. in [NT15] -motivated by the connection with Hastings-Levitov planar aggregation models -, in [BGS15] -where the optimal convergence condition was obtained and a family of coalescing Brownian motions on the Sierpinski gasket was built -, and in [GSW16] -where the Brownian Web was used to study the scaling limit of the genealogies of a population. For an account of further developments of the Brownian Web and the diverse contexts in which it emerged, we refer to the review paper [SSS17].
In most (but not all, see e.g. [GSW16]) of these works, the Brownian Web is viewed as a random (compact) collection of paths Win a suitable space of trajectories. Elements of W are pairs (t, π) with t ∈ R and π : R → R such that furthermore π(s) = π(t) for s ≥ t. In the present paper (similarly to [Ald93,GSW16]), we focus instead on another of its characterising features, namely its coalescence or tree structure, clearly apparent in Figure 1. The main motivation comes from the companion paper [CH23] in which such a structure is used to construct and study the Brownian Castle, a stochastic process whose value at a given point in R 2 equals that of a Brownian motion indexed by the Brownian Web. Since the characteristics of the Brownian Castle are given by backward (coalescing) Brownian motions, in what follows we will (mainly) consider the case in which paths in W run backward in time (the so-called backward Brownian Web [FINR04]).
To carry out this programme, we would like to view W as a metric space with metric given by the intrinsic distance, namely the distance between two points (t i , π i ) is given by t 1 + t 2 − 2τ , where τ is the largest time such that π 1 (s) = π 2 (s) for all s ≤ τ . More precisely, we view the Brownian Web as a (random) quadruplet ζ ↓ is a pointed locally compact R-tree, namely a connected locally compact metric space with no loops (see Definition 2.1) and such that M ↓ bw : T ↓ bw → R 2 is an embedding into R 2 . (In the above identification of T ↓ bw with a set of elements of the type (t, π), one would simply set M ↓ bw (t, π) = (t, π(t)).) The goals of the present article are: identify a "good" space in which the quadruplet ζ ↓ bw lives and in which we can uniquely characterise its law; determine a suitable topology under which such space is Polish and that allows for a manageable characterisation of its compact subsets; show that standard approximations to the Brownian Web converge in this (stronger) topology. Let us remark that the choice of the space and topology thereon is done in such a way that the Brownian Castle in [CH23] is continuous (in a suitable sense) as a map from such space to the set of càdlàg functions. First, we introduce the space T α sp , α ∈ (0, 1), whose elements are spatial R-trees, i.e. quadruplets of the form ζ = (T , * , d, M ) in which Introduction

Alternative topologies and relation to previous characterisations
Over the years, a variety of topologies on spaces of R-trees have been considered. The one outlined above is similar to those in [DLG05,BCK17], with the additional condition about the Hölder continuity and properness of the evaluation map. In [DGP11,KL15,ALW16,GSW16], the authors introduce so-called marked metric measure spaces, formed of triplets (T , d, µ) in which (T , d) is a metric space and µ is a locally finite measure on T × I, for I a complete and separable metric space. The measure µ should be thought of as equal to ν(dx) ⊗ δ κ(x) (du) for ν a locally finite measure on T and κ : T → I a "mark function". Upon taking I = R 2 , the mark function plays a similar role to the evaluation map M above. In our context a natural choice of measure ν would be the length measure, but this is only σ-finite and not locally finite as the above references require. In principle, we could artificially add a locally finite measure (necessarily with full support), but this would cause additional complications for no benefit in our setting.
With respect to the classical construction of the Brownian Web, in terms of a family of paths, notice that it is not always possible to view a family of paths as an R-tree (trivially, the paths might not be coalescing) and, conversely, there is no canonical way to associate a collection of paths to a generic spatial R-tree (think of the case in which segments in the tree backtrack so that they cannot be viewed as functions of one coordinate). In Definition 2.20 below, we define a subset D α sp ⊂ T α sp of "directed trees" for which the association is meaningful and prove that, as suggested by the heuristic description above, our topology is strictly finer than that in [FINR04] (see Proposition 2.25). While this ensures that many of the results obtained for the Brownian Web (existence of a dual, its properties, special points) can be translated to the present setting (see Section 3.3), convergence statements in T α sp do not follow from those previously established. This is remedied in Section 3.2, where a convergence criterion to ζ ↓ bw is derived. As shown in [CH23], there are two main advantages of the characterisation of the Brownian Web outlined above. First, it allows to preserve information on the intrinsic metric on the set of trajectories, which in turn is at the basis of the properties and the proof of the universality statement for the Brownian Castle in [CH23,Theorem 1.4]. Moreover, the R-tree structure (together with local compactness) automatically endows T ↓ bw with a σ-finite length measure (see [Eva08,Section 4.5.3]) that can be useful in many contexts and, for example, could provide a more direct construction of the marked Brownian Web of [FINR06]2. Moreover, this is the measure that gives the white noise arising in the construction of the Brownian Castle, which is also one reason why we do not attempt to distort the tree in a way that could potentially lead to better compactness properties.
At last, we mention that in [Ald93,Sec. 4.2], Aldous sketched the construction of a random R-tree corresponding to a mesh of the Brownian web as a limit of specific approximations with nice exchangeability properties. His work builds on a very general and rather "soft" construction, but provides relatively little information about the random tree obtained in this way. (His random trees are tree-like closed subsets of ℓ 1 , so for example even local compactness is not guaranteed.) On the other hand, the present paper provides a global construction of the Brownian Web (in space-time, as opposed to the one at fixed times of [GSW16]) as a random R-tree satisfying a number of useful properties.
Many fascinating random R-trees have been studied, such as Aldous's CRT in [Ald91a,Ald91b,Ald93], the Lévy and Stable trees of Le Gall and Duquesne and their connection to superprocesses [DLG05], and display interesting relations to important statistical mechanics models, e.g. the Brownian Map and random plane quadrangulations [LG13,Mie13], the scaling limit of the Uniform Spanning Tree and SLE [Sch00,BCK17], just to mention a few. As expected, the law of the Brownian Web as a random R-tree is different from those alluded to above (see Corollary 3.11 and Remark 3.13) but it would be interesting to explore further this new interpretation in light of the aforementioned works to see if extra properties of the Brownian Web itself or the Brownian Castle of [CH23] can be derived.

Outline of the paper
In Section 2, we collect all the preliminary results and constructions concerning R-trees which will be needed throughout the paper. After recalling their basic definitions and geometric features, we introduce, for α ∈ (0, 1), the spaces T α sp , of spatial R-trees, and their "directed" subset D α sp . We define a metric which makes them complete and separable, and identify a necessary and sufficient condition for a subset to be compact. In Section 2.4, we compare the metric above and that of [FINR04], and show that the former is stronger than the latter.
Section 3 is devoted to the Brownian Web and its periodic version [CMT19]. At first (Section 3.1), we provide another characterisation of its law on D α sp and determine some of its properties as an R-tree, such as box covering dimension and relation to [FINR04]. Then, we state and prove a convergence criterion (Section 3.2) and, in Section 3.3, we describe its dual and the so-called "special points".
At last, in Section 4 we first show how to make sense of the graphical construction of a system of coalescing backward random walks (and its dual) in the present context and conclude by deriving its scaling limit.
Let (T , d, * ) be a pointed metric space, i.e. (T , d) is as above and * ∈ T , and let M : T → R d be a map. For r > 0 and α ∈ (0, 1), we define the sup-norm and α-Hölder norm of M restricted to a ball of radius r as where B d ( * , r] ⊂ T is the closed ball of radius r centred at * , and, for δ > 0, the modulus of continuity as (1.1) In case T is compact, in all the quantities above, the suprema are taken over the whole space T and the dependence on r of the notation will be suppressed. Moreover, we say that a function M is (locally) little α-Hölder continuous if for all r > 0, lim δ→0 δ −α ω (r) (M, δ) = 0. Let I ⊆ R be a compact interval and D(I, R + ) be the space of càdàg functions on I with values in R + def = [0, ∞), endowed with the M1 topology that we now introduce. For f ∈ D(I, R + ), denote by Disc(f ) the set of discontinuities of f and by Γ f its completed graph, i.e. the graph of f to which all the vertical segments joining the points of discontinuity are added. Order Γ f by saying that (x 1 , t 1 ) ≤ (x 2 , t 2 ) if either t 1 < t 2 or t 1 = t 2 and |f (t − 1 ) − x 1 | ≤ |f (t − 1 ) − x 2 |. Let P f be the set of all parametric representations of Γ f , which is the set of all non-decreasing (with respect to the order on with ∥ · ∥ ∞ denoting the supremum norm) and d c M1 (f, g) to be a topologically equivalent metric with respect to which D(I, R + ) is complete (see [Whi02,Sec. 12.8] for an example of metric which makes the space complete).
which is well defined in view of Theorem 12.9.2 and eq. (9.1) in [Whi02]. At last, we will write a ≲ b if there exists a constant C > 0 such that a ≤ Cb and a ≈ b if a ≲ b and b ≲ a.
Given z ∈ T , the number of connected components of T \ {z} is the degree of z, deg(z) in short. A point of degree 1 is an endpoint, of degree 2, an edge point and if the degree is 3 or higher, a branch point. The following lemma is taken from [CMSP08, Lemma 3].
Lemma 2.4 Let (T , d) be an R-tree, z 0 ∈ T and let S be a dense subset of T . The following statements hold: Notice that the connected components of T \ {z} are themselves R-trees, i.e. subtrees of T , and they are called directions at z.
where I is an index set, the set of directions at z. For r > 0, we say that T i has length ≥ r if there exists w ∈ T i such that d(z, w) ≥ r. The R-tree T is r-locally finite at z if the set of all directions at z of length ≥ r is finite, and it is r-locally finite if it is r-locally finite at z for every z ∈ T .
An important notion for us in the context of R-trees, is that of end. To introduce it, we follow [Chi01, Chapter 2.3] (see also [Eva00, Section 2]). A subset L of an R-tree T is linear if it is isometric to an interval of R, which could be either bounded or unbounded. For z ∈ T , we write L z for an arbitrary linear subset of T having z as an endpoint and we say that L z is a T -ray from z if it is maximal for inclusion. We also say that rays L z and L z ′ are equivalent if there exists w ∈ T such that L z ∩ L z ′ is a ray from w. The equivalence classes of T -rays are the ends of T . Clearly, every endpoint determines an end for T and we will refer to them as closed ends, while the remaining ends will be called open. By [Chi01, Lemma 3.5], for every z ∈ T and every open end † of T , there exists a unique T -ray from z representing † which we will denote by z, †⟩. Moreover we say that † is an open end with (un-)bounded rays if for every z ∈ T , the mapῑ z : z, †⟩ → R + given bȳ We conclude this section by showing how the geometric structure of an R-tree is intertwined with its metric properties. The following statements summarise (or are easy consequences of) results in [Chi01,Theorem 4.14], [BBI01, Theorem 2.5.28] and [CMSP08, Theorem 2, Proposition 5].
Theorem 2.6 The completion of an R-tree is an R-tree and an R-tree is complete if and only if every open end has unbounded rays. Let (T , d) be a locally compact complete R-tree, then (a) T is proper, i.e. every closed bounded subset is compact, (b) T is r-locally finite and has r-finite branching for every r > 0, (c) T has countably many branch points and every point has at most countable degree.

Spatial R-trees
Now that we discussed geometric features of R-trees we are ready to study the metric properties of the space of R-trees. We will focus on the so-called α-spatial R-trees, which is a subset of the space of α-spatial metric spaces that we now introduce.
Definition 2.7 Let α ∈ (0, 1). The space of pointed α-spatial metric space M α sp is the set of equivalence classes3 of quadruplets ζ = (T , * , d, M ) where -(T , * , d) is a complete, separable pointed metric space such that every bounded closed subset is compact, -M , the evaluation map, is a locally little α-Hölder continuous proper4 map from We identify ζ and ζ ′ if there exists a bijective isometry φ : T → T ′ such that φ( * ) = * ′ and M ′ • φ ≡ M , in short (with a slight abuse of notation) φ(ζ) = ζ ′ . We denote by T α sp the subset of M α sp whose elements ζ are such that (T , * , d) is an R-tree5.

Remark 2.8
We will also consider situations in which the map M is R × T-valued, where T def = R/Z is the torus of size 1 endowed with the usual periodic metric d(x, y) = inf k∈Z |x − y + k|. We denote by M α sp,per the space of periodic pointed α-spatial metric spaces.
In what follows, we will denote subsets of M α sp and M α sp,per with the same notation except for the addition of a subscript "per", standing for periodic, in the latter case. It will always be immediate to see how the definitions, statements and proofs need to be adapted in order to hold not only for the space S we are considering but also for its periodic counterpart S per .
3The collection of all quadruplets as described here is not a set, but since every metric space T in which bounded closed subsets are compact has the cardinality of the continuum, one can see that the collection of equivalence classes is indeed set-sized.
5Note that by Theorem 2.6(a) in any complete locally compact R-tree, closed bounded subsets are compact.
Proof. The function b ζ is non-decreasing by construction, so that at every point r > 0 it admits left and right limits. To show it is càdlàg, it suffices to prove that lim s↓r b ζ (s) = b ζ (r).
Notice that, for every s > 0, since M is proper and Λ s is closed, there exists z s ∈ M −1 (Λ s ) such that b ζ (s) = d( * , z s ). Let s n be a sequence decreasing to r and, without loss of generality, assume M (z sn ) ∈ Λ sn \ Λ r . Using the properness of M once again, M −1 (Λ s 0 ) is compact so that {z sn } n ⊂ M −1 (Λ s 0 ) admits a converging subsequence. Letz be a limit point. By construction, d( * , z sn ) ≥ d( * , z r ) for all n, To turn M α sp into a Polish space, we proceed similarly to [BCK17], but we introduce two conditions taking into account the Hölder regularity and the properness of M respectively. Recall first that a correspondence C between two metric spaces (T , d), (T ′ , d ′ ) is a subset of T × T ′ such that for all z ∈ T there exists at least one z ′ ∈ T ′ for which (z, z ′ ) ∈ C and vice versa. The distortion of a correspondence C is given by and allows to give an alternative characterisation of the Gromov-Hausdorff metric. Let α ∈ (0, 1) and ζ = (T , * , d, M ), ζ ′ = (T ′ , * ′ , d ′ , M ′ ) ∈ M α sp . Let C be a correspondence between T and T ′ . We set where for every n, ψ n (x) def = ψ(2 n x) and ψ is a smooth function bounded above by 1, which is 1 on [1, 2] and 0 outside [2 −1 , 4] (so that in particular its first derivative ∂ψ n is uniformly bounded, modulo absolute constants, by 2 n ). We can now define where d M1 is the metric on the space of càdlàg functions given in (1.2) and In view of Lemma 2.9, the metric above is well-defined. Before proceeding let us comment on the summands at the right hand side of (2.4). As we pointed out above, once we take the infimum over all the correspondences, the first term gives us the Gromov-Hausdorff distance between (T , * , d) and (T ′ , * ′ , d ′ ) by [Eva08, Theorem 4.11], which is a natural way to compare different metric spaces. The second term just measures the sup-norm distance between the maps M and M ′ , while the third is a generalisation of the usual α-Hölder metric. To make a comparison, if T = T ′ , the latter can be easily seen to be equivalent to the more familiar sup z,w∈T d(z, w) −α ∥δ z,w M − δ z,w M ′ ∥. Now, in the present setting, we need to be able to measure the Hölder distance between functions which are not defined on the same space. The natural way to go beyond the same metric space case is to use the only way we have to "connect" T and T ′ , i.e. the correspondence C. Hence, we replace the supremum over z, w ∈ T with that over couples (z, z ′ ), (w, w ′ ) in C. On the other hand, we need to make sure we are comparing the increments of M and M ′ over points whose distance has the same order. The functions ψ m 's play exactly this role -they guarantee not only that this is the case but Proposition 2.10 For α ∈ (0, 1), let M α c (resp. T α c ) be the subset of M α sp (resp. T α sp ) consisting of compact metric spaces (resp. R-trees). Then, (M α c , ∆ c sp ) is a complete separable metric space and T α c is closed in M α c .
Proof. We begin by verifying that ∆ c sp is a metric. By definition, it is clearly non-negative and symmetric. Concerning the triangle inequality, it holds for the second summand in (2.5) by [Whi02, Theorem 12.3.1], while for the first we argue as follows. Let ζ 1 , ζ 2 , ζ 3 ∈ M α c , and C 1,2 , C 2,3 be two correspondences between T 1 , T 2 and T 2 , T 3 respectively. Then, upon choosing , which easily implies the triangle inequality for ∆ c sp . In order to show that the metric is positive definite, notice first that ∆ c sp (ζ, . Invoking [Whi02, Theorem 12.3.1] once more, we immediately have that b ζ ≡ b ζ ′ . Hence, we are left to show that there exists a bijective isometry such that φ(ζ) = ζ ′ , for which we argue similarly to [CHK12, Lemma 2.1]. Since ∆ c sp (ζ, ζ ′ ) = 0, for every ε > 0 there exists a correspondence C ε such that ∆ c, C ε sp (ζ, ζ ′ ) < ε. Let T be a countable dense subset of T and let φ ε : T → T ′ be such that (z, φ ε (z)) ∈ C ε . By construction, Since T ′ is compact, we can find a subsequence in ε such that for all z ∈ T , φ ε (z) converges to some element φ(z) ∈ T ′ . By (2.7), we immediately deduce that φ is a distance-preserving map on T such that, for all z ∈ T , M (z) = M ′ (φ(z)). Further, by reversing the roles of ζ and ζ ′ we can find a distance-preserving map ψ from T ′ to T . Since φ • ψ is an isometry from T ′ to itself and T ′ is compact, φ • ψ must be bijective (see [BBI01, Theorem 1.6.14]), which then implies that φ is itself bijective and satisfies φ(ζ) = ζ ′ . We now show completeness. Let {ζ n } n be a Cauchy sequence in (M α c , ∆ c sp ). As an immediate consequence, the sequence {(T n , * n , d n )} n is totally bounded in the space of (pointed) compact metric spaces. Moreover, it is not difficult to see that In order to construct the limit, we proceed as in the proof of [BBI01, Theorem 7.4.15]. Notice that, since the sequence {(T n , * n , d n )} n is totally bounded, by [BBI01,Theorem 7.4.15], for any ε > 0 there exists N (ε) such that, for all n, T n admits a finite ε-net of at most N (ε) elements. Now, we recursively set N 1 = N (1) and N k = N k−1 + N (1/k), and for each n we let S n = {z n i } i be a countable dense set of T n such that the first N k elements form a 1/k-net for T n and z n 0 def = * n . Then, passing at most to a subsequence, for every i, j, the limits lim n→∞ d n (z n i , z n j ), lim n→∞ M n (z n i ) can be shown to exist via a diagonal argument. LetT = {z i } i be an abstract countable set and define a semimetric d and a mapM on it, by imposing (2.9) We then set T to be the metric space obtained by taking the completion ofT ,T being the quotient space onT in which points at distance 0 are identified. T is a compact metric space and is the Gromov-Hausdorff limit of T n 's. It is also easy to see thatM is S ε is an ε-net for T . Define ζ ε n def = (S ε n , * n , d n , M n ) and ζ ε def = (S ε , * , d, M ). By the triangle inequality, we have (2.10) Thanks to Lemma 2.12 below, A 1 and A 3 are converging to 0 so that we only need to control A 2 . For the latter, let C n Then, (2.8) and (2.9) ensure that the assumptions of Lemma A.1 are satisfied, so that also A 2 converges to 0. At last, since ∆ c sp (ζ n , ζ) converges to 0, Lemma 2.15 immediately implies that b = b ζ . For separability, note that according to Lemmas 2.12 and 2.15 below, any element Hence a countable dense set of M α c can be obtained by considering the set of metric spaces with finitely many points whose respective distances are rationals, endowed with maps M which are Q 2 -valued.
To complete the proof of the statement, note that, as an immediate consequence of [Eva08, Lemma 4.22], T α c is a closed subset of M α c .

Remark 2.11
As pointed out in [BCK17, Remark 3.2], without the Hölder condition in the definition of ∆ c sp , the space of spatial metric spaces (R-trees in their setting) with the metric comprising only the first two summands at the right hand side of (2.4) is not complete while, if we did not assume the function M to be little Hölder continuous it would lack separability. Indeed, note that the space of α-Hölder continuous functions on, say, R d endowed with the usual Hölder metric is not separable. On the other hand, the subset of little α-Hölder continuous functions (corresponding to the assumption made in Definition 2.7) is the closure of smooth functions with respect to the usual Hölder metric and thus in particular it is separable.
Let δ > 0, T ⊂ T be such that * ∈ T and the Hausdorff distance between T and T is bounded above by δ ∈ (0, 1) and defineζ = (T, * , d, M ↾T ). Then Proof. The proof is provided in Appendix A.
Let us now turn to the non-compact case. As we will only ever work with R-trees, from now on, apart when explicitly stated, we will formulate and prove the results directly for T α sp . This in particular will allow us to use a number of statements from the literature which have been proved only for length spaces.
We begin by introducing a suitable metric on T α sp . For ζ = (T , * , d, M ) ∈ T α sp and any r > 0, let is the closed ball of radius r in T and M (r) is the restriction of M to T (r) . We define ∆ sp as the function on T α sp × T α sp given by (2.13) for all ζ, ζ ′ ∈ T α sp .

Remark 2.13
The presence of the term d M1 (b ζ , b ζ ′ ) in particular rules out the following type of example. Consider the R-tree given by one infinite branch e embedded into R 2 as [0, ∞) × {0}, as well as branches e n for n ≥ 1 that are embedded as [0, n] × {0} and merge with e at (n, 0). This tree lies in the completion of T α sp under ∆ sp , but does not lie in T α sp .
In general, properness guarantees that we cannot have a tree ζ = (T , * , d, M ) ∈ T α sp admitting a sequence of points z n ∈ T such that |M (z n )| ≤ 1 and d(z n , z m ) ≥ 1 for all n, m. Indeed, since bounded subsets of T are precompact (by the definition of local compactness), having a sequence such that d(z n , z m ) ≥ 1 for all (m, n) guarantees that lim n→∞ d(z n , * ) = +∞, so lim n→∞ |M (z n )| = ∞ by properness.
Theorem 2.14 For any α ∈ (0, 1), We will first show point (i) and separability, then state and prove two lemmas, one concerning the properness map while the other the relation between ∆ c sp and ∆ sp , and a characterisation of the compact subsets of T α sp . At last, we will see how to exploit them in order to show completeness.
Proof of Theorem 2.14(i). We begin by proving that ∆ sp is well-defined on T α sp × T α sp . Since T (r) and T ′ (r) are compact as both T and T ′ are locally compact by assumption, ζ (r) , ζ ′(r) ∈ T α c so that ∆ c sp (ζ (r) , ζ ′(r) ) makes sense. To see that the first summand in (2.13) is well-defined, we note that the map r → ∆ c sp (ζ (r) , ζ ′ (r) ) is càdlàg. Indeed, by the triangle inequality we have which, by switching the roles of r and r + ε, immediately implies that Since T (r) and T ′ (r) are closed by definition, it is not difficult to see that the Hausdorff distance between T (r+ε) and T (r) as well as that between T ′ (r) and T ′ (r+ε) is going to 0 as ε → 0. Hence, Lemma 2.12 implies that the right hand side is converging to 0. For the existence of left limits instead, letζ (r) def = (T (r) , * , d,M (r) ) be such that ) similarly. Then, replacing ζ (r) and ζ ′ (r) withζ (r) andζ ′ (r) in the argument for the right continuity, we can show that ∆ c sp (ζ (r−ε) , ζ ′ (r−ε) ) converges to ∆ c sp (ζ (r) ,ζ ′ (r) ).
We now prove that ∆ sp is indeed a metric. As in the proof of Proposition 2.10, we only need to focus on the first summand in (2.13) and show it satisfies the axioms of a metric. Positivity and symmetry clearly hold, while the triangle inequality follows by the fact that it holds for ∆ c sp . For positive definiteness, we argue as in [ADH13, , ζ ′ (r) ) = 0 for every r ≥ 0 and consequently there exists an isometry φ r from T (r) to T ′ (r) for which φ r (ζ) = ζ ′ . For every n, k positive integers, let {z n,k i } i≤N n,k be a 1/k-net of T (n) with z n,k 0 def = * and N n,k < ∞ be its cardinality. Notice that, since for every m ≥ n, φ m is distance preserving, the sequence {φ m (z n,k i )} m≥n is bounded for every n, k and i ≤ N n,k . Via a diagonal argument, it is then possible to find a subsequence such that φ(z n,k i ) def = lim m→∞ φ m (z n,k i ) exists for every n, k and i ≤ N n,k . Clearly, φ is distance preserving on {z n,k i } n,k,i and, for any n, k given To show separability, given ζ ∈ T α sp and r > 0, let R def = diam(M (T (r) )). Then, the definition of the metric implies ∆ sp (ζ, ζ (r) ) ≲ e −r ∨ e −R , so that any element of T α sp can be approximated arbitrarily well by elements in T α c . Since, in view of Proposition 2.10, the latter space is separable, and thanks to Lemma 2.16 convergence in ∆ c sp implies convergence in ∆ sp , separability on T α sp follows.

Lemma 2.15
Let α ∈ (0, 1), {ζ n = (T n , * n , d n , M n )} n∈N be in T α sp (resp. M α c ) and let ζ = (T , * , d, M ) be such that ∆ sp (ζ n , ζ) (resp. ∆ c (ζ n , ζ)), converge to 0 as n → ∞. Assume further that for every r > 0 there exists a finite constant Proof. We only show the statement for {ζ n = (T n , * n , d n , M n )} n∈N ⊂ T α sp , as the proof does not rely on the R-tree structure of the metric spaces and the case of M α c is the same but simpler.
We begin by proving that M is proper (which in the compact case is obvious). Let r > 0 be fixed and z ∈ T be such that M (z) ∈ Λ r . Then, by (2.14) and since ∆ sp (ζ n , ζ) → 0, there exists R > 0 such that z ∈ B d ( * , R] and for every n we have both that M −1 n (Λ r+1 ) ⊂ B dn ( * n , R] and that there exists a correspondence C R n between T (R) and T (R) n , ζ (R) ) → 0. Without loss of generality, we can take R > C ′ (r +1)+2, so that, in view of (2.14), for every n, which implies that M is proper. Hence, ζ ∈ T α sp and, by Lemma 2.9, b ζ is càdlàg. It remains to prove that b ζn converges to b ζ . [Whi02, Theorem 12.9.3 and Corollary 12.
n and ε n be as above. Notice that from which the conclusion follows.

Lemma 2.16
For any α ∈ (0, 1), the identity map from Let ε > 0 andm ∈ N be such that 2m α ω(M, 2 −m ) ≤ ε. Since ∆ c sp (ζ n , ζ) → 0, for n big enough, there exists a correspondence C n between T and T n such that Let r ≥ 0 be fixed. Our goal is to show that there exists a correspondence C r n such that the assumptions of Lemma A.1 are satisfied. Note that clearly, since ∆ c sp (ζ n , ζ) → 0 we have uniform bound on the modulus of continuity of M (r) n and M (r) so that (A.1) clearly holds. For (A.2), we proceed as in [BCK17,Proposition 3.4]. Namely, we define C r n as the correspondence which contains the pair (z, z n ) provided that either z ∈ T (r) , z n ∈ T (r) n and (z, z n ) ∈ C n , or z n ∈ T (r) n (resp. z ∈ T (r) ) and z (resp. z n ) is the closest point in To be more precise, since T and T n are R-trees and in particular length spaces, we will take z to be the point on the segment connecting * to z ′ such that d( * , z) = r and similarly for z n .
Assume (z, z n ) ∈ C r n \ C n is such that z n ∈ T (r) n and z ′ is as above. Then, by the first bound in (2.15) For (z, z n ), (w, w n ) ∈ C r n ∩ C n , thanks to (2.15) there is nothing to argue. If instead, say, (z, z n ) ∈ C r n \ C n is such that z n ∈ T (r) n and z ′ is as above, by (2.16), we obtain and we can argue similarly if also (w, w n ) ∈ C r n \ C n . The bound on the difference of the evaluation maps instead reads where we used the α-Hölder continuity of M . Collecting the previous bounds, we immediately see that (A.3) holds, so that by Lemma A.1 ∆ c sp (ζ (r) n , ζ (r) ) converges to 0 and the statement follows at once.
Proof. "⇐=" Let {ζ n = (T n , * n , d n , M n )} n ⊂ A be a sequence satisfying the three properties above.
We want to extract a converging subsequence for {ζ n } n and construct the corresponding limit point. The limit space is built as in [ADH13, Sections 5.2.1 and 5.2.2] so we briefly recall the construction and try to follow the notations therein as closely as possible. For r > 0, ℓ, k ∈ N and any n, Now, notice that, for every u, u ′ ∈ U , both the sequences {d n (z n u , z n u ′ )} n and {M n (z n u )} n are bounded -the second claim following by the first condition in (2.18). Hence, via a diagonal argument, upon passing to a subsequence, we can ensure that for every u, u ′ ∈ U they converge. LetT = {z u } u∈U be an abstract countable set and define a semimetric d and a mapM on it, by imposing We then set T to be the metric space obtained by taking the completion ofT ,T being the quotient space onT in which points at distance 0 are identified. [ADH13, Lemma 5.7] ensures that T is a length space, while we can see it is an R-tree as the four point condition (see [Eva08, Definition 3.9 and Theorem 3.40]) can be immediately shown to hold by (2.19) and the fact it holds for each of the T n 's. As in the first display in [ADH13, Section 5.2.2], we set U + ℓ k ,k to be the union of U j2 −k ,k for 0 ≤ j ≤ ℓ, and (2.20) [ADH13, Lemma 5.6] ensures that, for every ℓ, k, S + ℓ k ,k is a 2 −k -net for T (ℓ k ) , which in particular implies that T is locally compact. Moreover, by condition 2. and the second formula in (2.19), we have thatM is locally little α-Hölder continuous so that we can set M to be the unique locally little α-Hölder continuous extension ofM to T .
By condition 3. and Lemma 2.15, once we prove that ∆ sp (ζ n , ζ) converges to 0, To do so, let r > 0 and k ∈ N be fixed and define ℓ As we pointed out above S + ℓ,k and S n,+ ℓ,k are ε-nets for T (ℓ k ) and T (ℓ k ) n , respectively. Hence, by Lemma 2.12 and (2.18), all the A i 's, for i ̸ = 3, can be controlled by quantities which are vanishing as k → ∞, so that we only need to focus on A 3 . For this in turn, the second condition in (2.18) implies (A.1) while, upon choosing C r Hence, the assumptions of Lemma A.1 hold, so that also A 3 → 0.
"=⇒" Let A be relatively compact in T α sp . Then, property 1. holds by [BBI01,Theorem 7.4.15], while property 3. by [Whi02, Theorem 12.12.2] on the necessary condition for a set to be compact in the strong M 1 topology on the space of càdlàg functions. For the second property, notice that since A is totally bounded, for any ε > 0 and r > 0 there exist n ∈ N and {ζ k : k = 1, . . . n} such that A is contained in the union of the balls of radius e −r ε/4 centred at which implies that there exists a correspondence C between T (r) and T (r) and the first bound in (2.18) follows. For the others, let δ > 0 andn ∈ N the largest integer such that 2 −n ≤ δ. Then, (2.24) so that, once again, the second bound in (2.18) can be obtained by applying the triangle inequality and choosing the minimum δ for which sup k≤n δ −α ω (r) (M ζ k , δ) < ε/2 .
Proof of Theorem 2.14(ii). To prove completeness, it suffices to show that, if {ζ n } n is a Cauchy sequence in T α sp then the conditions of Proposition 2.17 are satisfied. Now, if {ζ n } n is Cauchy, then for every r > 0, {ζ (r) n } n is Cauchy with respect to ∆ c sp , which implies that the sequence converges so that 1. holds in view of [BBI01, Proposition 7.4.12], 2. can be seen to be satisfied by arguing as in (2.23) and (2.24), and 3. follows by the fact that D([−1, ∞), R + ) is complete with respect to d M1 .
We conclude this section with a lemma that will be useful in the construction and characterisation of the Brownian Web. It guarantees that, under certain conditions, we can build an α-spatial R-tree inductively, by "patching together" pieces of branches.

Lemma 2.18
Let α ∈ (0, 1) and ζ n = (T n , * n , d n , M n ) be a relatively compact sequence in T α sp . Assume that for every n < m ∈ N there exists an isometric embedding ι n,m of T n into T m such that ι n,m ( * n ) = * m , ι n,k = ι m,k • ι n,m for n < k < m and M m • ι n,m ≡ M n . Then, the sequence ζ n converges to ζ = (T , * , d, M ) and for every n ∈ N there exists an isometric embedding ι n of T n into T such that ι n ( * n ) = * , ι n = ι m • ι n,m for m > n and M • ι n ≡ M n . Moreover,T def = n ι n (T n ) is dense in T and M is the unique continuous extension ofM onT , the latter being defined by the relationM • ι n ≡ M n for all n.

Remark 2.19
A similar statement was given in [EPW06, Lemma 2.7]. The formulation is a bit different since we do not have a common ambient space and the trees we consider are spatial. One reason why we cannot directly reuse that result is that it is not clear a priori that relative compactness in T α sp implies relative compactness of the images in n T n /∼ with the natural equivalence relation induced by the consistency maps ι m,n (and part of our proof consists of showing that this is indeed the case). This is because the optimal correspondence between T n and T m may differ from the one given by ι m,n . Take for example the trees (T , * ) = ([0, 1], 1/3) and (T , * ) = ([0, 1/3], 1/3). Then, for the natural correspondence C suggested by our notation, one has dis C = 2/3, while the correspondenceC mapping x ∈T to 2/3 − x ∈ T is also an isometric embedding but has disC = 1/3. This shows that the condition in [EPW06, Lemma 2.7] assuming that the ζ n are Cauchy as subsets of a common space in the Hausdorff topology may a priori be stronger than the relative compactness assumed here. (A posteriori it is not, as demonstrated by the fact thatT is dense in T .) Proof. We will limit ourselves to the case of T n compact, the general case easily follows from the definition of the metric ∆ sp . We begin by constructing the limit space. LetT = ( n T n )/∼, where ∼ is the smallest equivalence relation such that z ∼ ι n,m (z) for every n ≤ m and every z ∈ T n , and let * ∈T be the equivalence class containing all the * n . We define d onT ×T by setting, for z ∈ T n , w ∈ T m with n ≤ m, d(z, w) def = d m (ι n,m (z), w), which is clearly a metric onT . For n ∈ N, let ι n : T n →T be the canonical embedding, which can be easily seen to be an isometry such ι n = ι m • ι n,m for all n < m, andT = n ι n (T n ). At last, letM :T → R 2 be the map defined asM (z) def = M n (ι −1 n (z)) for z ∈ ι n (T n ), so thatM • ι n = M n .
We will first show that (T , * , d) is totally bounded andM is little α-Hölder continuous. For the first, recall that, since the sequence ζ n is relatively compact in T α c , by point 1. of Proposition 2.17, for every ε > 0 there exists N (ε) ∈ N such that sup n N dn (T n , ε) ≤ N (ε) . (2.25) We now make the following claim.
(C) for every ε > 0 there exists n ε ∈ N such that for all n > n ε and z ∈ ι n (T n ) there is w ∈ ι nε (T nε ) for which d(z, w) < ε.
To prove (C), assume by contradiction that there existsε > 0 such that the claim fails, so that there exists a sequence z n with z n ∈ ι n (T n ) and such that d(z i , z j ) >ε for all i ̸ = j. Setting k = N (ε) + 1, this yields aε-separated set for T k whose cardinality is greater than N (ε), thus yielding a contradiction. We now show thatT is relatively compact, i.e. that for every ε > 0 it has a finite ε-net. Let ε > 0 be fixed, n ε be as in (C), and S ε be an ε-net for T nε , which can be chosen finite since T nε is compact. It then follows from (C) that ι nε (S ε ) is a 2ε-net forT .
To show the little Hölder continuity ofM , let z, w ∈T be such that d(z, w) ≤ δ. Then, there exist m, n, with n ≤ m, such that z ∈ ι n (T n ) ⊂ ι m (T m ) and w ∈ ι m (T m ), so that, in particular, and, since {ζ n } n is relatively compact by assumption, the right hand side converges to 0 as δ → 0 in view of point 2 of Proposition 2.17.
We are now ready to construct the limit point ζ. Let T be the completion ofT with respect to the metric d, M the unique little Hölder continuous extension ofM to T , and set ζ def = (T , * , d, M ), which, by the above discussion, belongs to T c sp . It remains to show that the sequence {ζ n } n converges to ζ. We apply Lemma A.1. Condition (A.1) is implied by (2.26), so that we only need to find a correspondence for which (A.2) holds.
For n > 0, we then set ε n = inf{ε > 0 : n ε < n}, with n ε as in (C), and we define C n = {(z, z ′ ) ∈ T n × T : d(ι n (z), z ′ ) ≤ ε n }. This is a correspondence by the definition of ε n and one has dis C n ≲ ε n . The second part of (A.2) follows from (2.26), so that the proof is complete.

Directed R-trees and the radial map
As mentioned in the introduction, we would like to view the backward Brownian Web as a flow. More specifically, at any time t and position x, we want to be able to follow a backward Brownian trajectory starting at x at time t. These trajectories will be encoded by the branches of our R-tree and should not be allowed to cross.
In the following definition we identify a subset of the space of α-spatial R-trees whose elements possess a notion of direction in time and satisfy a monotonicity assumption, both imposed at the level of the evaluation map M . Henceforth we use the following shorthand notation. Given an R-tree T , elements z 0 , z 1 ∈ T , and s ∈ [0, 1], we write z s for the unique element of z 0 , z 1 with d(z 0 , z s ) = s d(z 0 , z 1 ).

Remark 2.21
The first condition guarantees that geodesics are ∨-shaped and that the "time" coordinate moves at unit speed. Together with the first, the second condition enforces the statement that "characteristics cannot cross". They are still allowed (and forced, in our case) to coalesce but their spatial order must be preserved. The last requirement says that the tree is oriented and has a direction which corresponds to the direction of time, i.e. the characteristics move indeed backward in time.
First notice that it is not difficult to show that the properties in the previous definition are consistent with the equivalence relation in Definition 2.7, i.e. if there exists a bijective isometry φ such that φ • ζ = ζ ′ and ζ satisfies the conditions above then so does ζ ′ . In other words, the space D α sp is a well-defined subset of T α sp . In the next proposition, we study important properties of D α sp .

any ζ = (T , * , d, M ) ∈ D α sp is such that T has a unique open end † which satisfies
for all z ∈ T and w ∈ z, †⟩.
Proof. We first prove that D α sp is closed. Let {ζ n } n ⊂ D α sp be a sequence converging to ζ ∈ T α sp . Since ζ is directed if and only if, for every R > 0, ζ (R) is monotone and (3) holds for every s ∈ [0, R], and since ∆ c sp (ζ n,(R) , ζ (R) ) → 0 for every R > 0, it suffices to restrict to the compact case.
We start with monotonicity in time. Take z 0 , z 1 ∈ T , let C n be a sequence of correspondences such that lim n ∆ c, Cn sp (ζ n , ζ) → 0 and let z n i be such that (z n i , z i ) ∈ C n . For any s ∈ [0, 1], let z s ∈ T and z n s be defined as above. For every n, letz s (n) ∈ C n be a point for which (z n s , z s (n)) ∈ C n . Then, by the triangle inequality and the definition of correspondence, z s (n) belongs to a compact ball centred at, say, z 0 . Hence, the sequence {z s (n)} n is precompact. Now, for any limit pointz s ∈ T we have where f 0 (s) = s while f 1 (s) = 1 − s and in the last step we used the definition of z s and z n s . Now, all the terms at the right hand side are converging to 0, which implies thatz s must be such that d(z i ,z s ) = f i (s)d(z 0 , z 1 ). The pointz s is therefore unique and given by z s , so that the sequence {z s (n)} n converges to z s . It then follows that M t (z s ) = lim n→∞ M t (z s (n)) = lim n→∞ M n t (z n s ) .
Since furthermore lim n→∞ d n (z n 0 , z n 1 ) = d(z 0 , z 1 ) and lim n→∞ M n t (z n i ) = M t (z i ) by the definition of ∆ c, Cn sp , the claim follows. Regarding monotonicity in space, we perform the same construction, whence we get as required. For the last property, let s ∈ [0, R] and z n s ∈ T be such that (ι n * (s), z s (n)) ∈ C n . Then, arguing as above, there exists z s ∈ T such that d(z s (n), z s ) converges to 0 as n → ∞, which, as ( * n , * ) ∈ C n , satisfies d( * , z s ) = lim n→∞ d( * , z s (n)) = lim n→∞ d n ( * n , ι n * (s)) = s .
Further, by continuity of M , M t (z s (n)) and M t ( * n ) respectively converge to M t (z s ) and M t ( * ), we also have M t (z s ) = M t ( * ) − s .
Therefore, upon defining the map ι * (s) def = z s , we immediately have that ι * is an isometry and satisfies (2.29).
We now move to the second part of the statement. The third property in Definition 2.20 implies that any directed R-tree ζ = (T , * , d, M ) is unbounded, since M is continuous and T is complete. Therefore, T must have at least one unbounded open end. By property (3), the isometry ι * is such that L * def = ι * (R + ) is an unbounded ray in T . As ends are equivalence classes of rays, let † be the unique (necessarily) open end such that L * = * , †⟩. Now, (2.29) implies that (2.30) holds for z = * and any w ∈ * , †⟩. It then suffices to apply the monotonicity in time, i.e. property (1), to see that it must hold for any z ∈ T .
Thanks to the previous proposition, we can introduce, in the context of directed trees, the radial map. This is a map on the R-tree that allows to move along the rays.

Definition 2.24
Let α ∈ (0, 1], ζ = (T , * , d, M ) ∈ D α sp and † the open end with unbounded rays such that (2.30) holds. The radial map ϱ : T × R → T associated to ζ is uniquely defined by postulating that If instead ζ ∈D α sp , the radial mapρ is defined in the same way but with ∨ instead of ∧.
For the proof of the previous proposition we will need the following two lemmas. For the first, define and, for π ∈ Π, write π R ∈ Π for the stopped path such that

Lemma 2.27
Let K be a subset of Π and, for R > 0, let K R ⊂ Π be defined as If for all R > 0, the family of paths in K R is equicontinuous then K is relatively compact.
Proof. Our main ingredient then is the fact that, since |1 − tanh R| ≤ e −R , one has the bounds Writing π ± t for the path with σ π ± t = t and π ± t (s) = ±∞, it follows that for every π ∈ Π and every R ≥ 1 one has d Π (π, π R ) ≤ 2/R. If furthermore π ̸ ∈ Π R , then d Π (π, π + σπ ) ∧ d Π (π, π − σπ ) ≤ 2/R. It remains to note that, given ε > 0, we can cover K 4/ε with finitely many balls of radius ε/2 by Arzelà-Ascoli, so that K ∩ Π R is covered by the balls with same centres and radius ε. The complement of Π R on the other hand can be covered by finitely many balls of radius ε centred at elements of type π ± t for t ∈ εZ ∩ [−4ε −1 , 4ε −1 ].
The next lemma highlights the fact that if two directed trees are close then also the respective rays must be close in a suitable sense.
We are now ready for the proof of Proposition 2.25.
Let now {ζ n = (T n , * n , d n , M n )} n ⊂ D α sp be a sequence converging to ζ ∈ D α sp with respect to ∆ sp . In view of Proposition 2.17, the evaluation maps M n are uniformly proper and have uniformly bounded α-Hölder norm when restricted to balls of fixed size. Hence, arguing as above, we see that ∪ n K(ζ n ) is relatively compact in Π which, thanks to [SSS10,Lemma B.3], implies that the sequence {K(ζ n )} n is relatively compact in H with respect to the Hausdorff topology. It remains to show that K(ζ n ) converges to K(ζ) in H. By [SSS10, Lemma B.1], we need to prove that for every π z ∈ K(ζ) there exists a sequence π zn ∈ K(ζ n ) such that d Π (π z , π zn ) → 0, and that, if {π zn } n is a sequence such that π zn ∈ K(ζ n ), then any of its cluster points π belongs to K(ζ).
We begin with the first. Let z ∈ T and ε > 0. Pick C > 0 big enough so that z ∈ B d ( * , C] and sup n b ζ n (ε −1 ) ≤ C. Let n be sufficiently large so that there exists a correspondence C n between B d ( * , C] and B d n ( * n , C] with ∆ c, Cn sp (ζ (C) , ζ n, (C) ) < ε. Let z n ∈ B d n ( * n , C] with (z, z n ) ∈ C n and define π z and π zn as in (2.33). Since To estimate the distance between π z (s) and π zn (s) for s ≤ σ πz ∧ σ πz n , we first consider the case s ≥ −ε −1 . Since C is large enough so that ϱ n (z n , s) ∈ B d n ( * n , C], we can apply Lemma 2.28 and get |π z (s) − π zn (s)| = |M x (ϱ(z, s)) − M n x (ϱ n (z n , s))| ≲ ε + ∥M ∥ (C) α ε α .
(2.38) For s < −ε −1 we use again the last bound of (2.36). Combining these bounds, we obtain d Π (π z , π zn ) ≲ ε α . We now turn to the second. Let {π zn } n be a sequence such that π zn ∈ K(ζ n ), π be one of its cluster points. Passing at most to a subsequence (which we will still index by n) we can assume d Π (π zn , π) converges to 0. Our goal is to show that there exists z ∈ T such that π = π z .
LetR be big enough so that for all n, z n ∈ B dn ( * n ,R], which must exist since the evaluation maps M n are uniformly proper and π zn converges in d Π . Moreover, since T n is a directed R-tree, z n ∈ B dn ( * n ,R] implies that for every t ∈ [−R, M n t (z n )], ϱ n (z n , t) ∈ B dn ( * n ,R]. Let R >R and C n be a correspondence such that lim n ∆ c, Cn sp (ζ (R) , ζ n, (R) ) = 0. Since z n ∈ B dn ( * n , R] and T n is a directed R-tree, for every t ∈ [−R, M n t (z n )], ϱ n (z n , t) ∈ B dn ( * n , R]. Hence, for n ∈ N and t ≤ M n t (z n ) ∧ σ π , there is z t (n) ∈ T (R) such that (ϱ n (z n , t), z t (n)) ∈ C n . As ∆ c, Cn sp (ζ (R) , ζ n, (R) ) and d Π (π zz , π) converge to 0, for any s, t ≥ −R we have lim n |d(z t (n), z s (n)) − d n (ϱ n (z n , t), ϱ n (z n , s))| = 0 (2.39) For every t ∈ [−R, σ π ], the set {z t (n)} n ⊂ B d ( * , R] is bounded, and therefore relatively compact. Hence, via a diagonal argument, we can find a subsequence in n such that for every t in a countable dense subset D R π of [−R, σ π ] containing σ π , z t (n) converges to z t ∈ T (R) . Let z def = z σπ and note that by (2.40), for every t ∈ D R π M x (z t ) = π(t) and M t (z t ) = t, so that in particular M x (z) = π(0) and M t (z) = σ π . We now want to show that z t = ϱ(z t ), which in turn, since ζ is a directed R-tree and by the definition of the radial map ϱ, follows if we prove that d(z, z t ) = d(z, ϱ(z t )). To see this latter point, note that by (2.30) and (2.31), we have where the last step follows by (2.39). As a consequence, we have shown for every t ∈ D R π , π(t) = M x (z t ) = M x (ϱ(z, t)) = π z (t), so that the same equality holds for any R, as R was arbitrary, and t, by continuity of π and M . Therefore the proof is concluded.
In general, we cannot expect the map K to be injective. Indeed, there is no mechanism that a priori prevents different branches of the tree to be mapped via the evaluation map to the same path.
In the following definition, we introduce a (measurable) subset of D α sp whose elements satisfy a condition, the tree condition, which allows to distinguish two rays in the tree based on their images under the evaluation map.
We denote by D α sp (t), the subset of D α sp whose elements satisfy (t).
Condition (t), guarantees that different rays on the tree under study are mapped, via the evaluation map, to paths which cannot agree on any open interval up to the time they coalesce. Alternatively said, (t) is equivalent to requiring that M is segment-injective, i.e. that if two segments of positive length have the same image under M then they coincide. More precisely, the segment-injective property is It is not difficult to construct examples of directed trees for which (t) does not hold, while it clearly does if the evaluation map is injective. However, we cannot expect the evaluation map of the Brownian Web to be injective because of the presence of special points from which multiple trajectories depart (see Section 3.3). In the following lemma, the proof of which is immediate, we provide a less trivial example.

Lemma 2.30
Let α ∈ (0, 1) and ζ = (T , * , d, M ) ∈ D α sp . If there exists a dense subtree T of T such that (T, * , d, M ↾T ) satisfies (t) then so does ζ. Moreover, the subset of D α sp whose elements satisfy (t) is measurable with respect to the Borel σ-algebra generated by ∆ sp in (2.13).
Proof. The first part of the statement follows by Lemma 2.4 point 2. For the second, it suffices to observe that the set D α sp (t n −1 ) ⊂ T α sp , n ∈ N, whose elements are such that (t) holds for ε = n −1 , is closed and clearly D α We conclude this section by showing that on D α sp (t), K is indeed injective.

Remark 2.32 Even though the map K is injective on D α sp (t), it is not on any open subset of D α
sp and the set D α sp (t) is not closed. To see this, let ζ = (T , * , d, M ) ∈ D α sp (t). Now, add to T a branch of arbitrary finite length and impose that the image of the new branch via the evaluation map is contained in M (T ). We can clearly do so in such a way that the new spatial-tree ζ ′ is again directed. Now, ζ ′ does not satisfy condition (t), K(ζ) = K(ζ ′ ) and upon tuning the length of the extra branch, we can make it arbitrarily close to ζ.

Remark 2.33
In the periodic case, let Π per be the set of backward periodic paths endowed with the metric d per Π whose definition is the same as in (2.32) but in the second argument of the maximum the inner metric is replaced by the periodic one, i.e. for π 1 , π 2 ∈ Π per and t ≤ σ π 1 ∧ σ π 2 , we take inf k∈Z |π 1 (t) − π 2 (t) + k|. Let H per be the set of compact subsets of Π per with the Hausdorff metric. Then, Propositions 2.25 and 2.31 remain true, which means that the map K : D α sp,per → H per defined as in (2.34) is continuous and its restriction to D α sp,per (t) is injective.

The Brownian Web Tree and its dual
Here, we provide an alternative (and finer) characterisation of the Brownian Web so to be able to view it as a directed spatial R-tree.

An alternative characterisation of the Brownian Web
In this section, we will build both the standard (or planar) backward Brownian Web and its periodic (or cylindric) counterpart as given in [CMT19]. Since the two constructions are almost identical, we will mainly focus on the first and limit ourselves to indicate what needs to be modified in order to accommodate the second (see Remarks 3.1, 3.7, 3.9). Consider a standard probability space (Ω, A, P) supporting countably many independent standard Brownian motions {W ↓ k } k∈N starting at 0 and running backward in time, i.e. from 0 to −∞. Fix a countable dense set D def = {z k = (t k , x k ) : k ∈ N} of R 2 , with z 0 = (0, 0). Then, build inductively a family of coalescing backward Brownian motions {π ↓ z k } k∈N such that π ↓ z k starts at x k at time t k . As in [FINR04, Section 3], one way to do so is to set π ↓ z 0 (t) = W ↓ 0 (t) and then define π ↓ z k (t) = x k + W ↓ 0 (t − t k ) for all τ k ≤ t ≤ t k , where τ k is the largest value such that x k + W ↓ k (τ k − t k ) = π ↓ z ℓ (τ k ) for some ℓ < k, and for t ≤ τ k , π ↓ z k (t) = π ↓ z ℓ (t). The construction guarantees that even though ℓ may not be unique, the definition of π ↓ k is.
be the space defined as before but in which k is free to range over all of N. Now, for n ∈N def = N ∪ {∞}, consider the equivalence relation ∼ onT ↓ n (D), given by where i, j ≤ n and, in the definition of the ancestor metric d ↓ , Remark 3.1 The construction in the periodic setting is analogous. Indeed, it suffices to replace the family of backward Brownian motions {B ↓ k } k with a family of periodic ones defined via B ↓,per The construction above readily implies a number of properties each of the ζ ↓ n (D)'s enjoys. Indeed, for every n ∈ N finite, ζ ↓ n (D) is a spatial R-tree which is monotone in both space and time, and it further satisfies property (3) in Definition 2.20 as can be readily seen by setting ι * : . Moreover, as a consequence of the fact that Brownian trajectories are α ′ -Hölder continuous for any α ′ < 1 2 , M ↓, D n is little α-Hölder continuous for any α ∈ (α ′ , 1/2). In other words, for every n ∈ N, ζ ↓ n ( D) ∈ D α sp . In the next proposition, we will show that the sequence {ζ ↓ n (D)} n is not only tight in T α sp for any α < 1/2, but it actually converges to a unique limit in D α sp which can be explicitly characterised starting from ζ ↓ ∞ ( D).
Proposition 3.2 Let D be a countable dense of R 2 containing (0, 0) and, for n ∈N, Proof. We fix D once and for all for the duration of this proof and therefore suppress its dependence in the notation. By construction, the sequence {ζ ↓ n } n of α-spatial R-trees is such that for every n ∈ N, ζ ↓ n is embedded into ζ ↓ n+1 , and, as argued above ζ ↓ n ∈ D α sp . Hence, Lemma 2.18 and part 1. of Proposition 2.23 guarantee that, provided that the sequence is tight in T α sp , it converges to a unique ζ ↓ = (T ↓ , * ↓ , d ↓ , M ↓ ) ∈ D α sp which further satisfies (3.3), M ↓ is surjective and (t) holds.
Since every ζ ↓ n is canonically embedded in ζ ↓ ∞ = (T ↓ ∞ , * ↓ , d ↓ , M ↓ ∞ ), if we show that, almost surely, T ↓ ∞ (which is an R-tree and hence, by Point 2 in Theorem 2.6 so is its completion) is locally compact and M ↓ ∞ is proper and uniformly little α-Hölder continuous on bounded balls, then we have a bound uniform in n on both the size of the ε-nets of balls in T ↓ n and the local modulus of continuity of M ↓ n , so that tightness of the sequence follows readily from Proposition 2.17.
Let r ≥ 1. We start by introducing an event on which T ↓, (r) ∞ is enclosed between two paths. Let R > r, Q ± R be two squares of side 1 centred at (r + 1, ±(2R + 1)) and z ± = (t ± , x ± ) be two points in D ∩ Q ± R , respectively. By the non-crossing property of our coalescing paths, on the event Moreover, by the reflection principle, we have where E c R is the complement of E R in Ω, and C 1 is a positive constant independent of r and R. Now, in order to show that, almost surely, T ↓, (r) ∞ is relatively compact, note that we can brutally bound The following lemma implies (3.3) (and in fact that P(c > K) ≲ 1/ √ K) and consequently relative compactness. Lemma 3.3 There exists a constant C = C(r) > 0 such that uniformly over ε ∈ (0, 1] and K ≥ 1.
Proof. Let R > r and setR where ϱ is the radial map of T ↓ ∞ defined as in (2.31), and set η R (t 0 , t 1 ) to be the cardinality of Ξ R (t 0 , t 1 ). By the definition of T ↓ ∞ , η R (t 0 , t 1 ) has the same distribution as the quantitŷ η(t 0 , t 1 ; −R,R) of [FINR04, Definition 2.1], which is almost surely finite by [FINR04,Proposition 4.1].
We now focus on the Hölder continuity of the map M ↓ ∞ ↾T ↓, (r) ∞ . In this case, it suffices to show that (3.11) for any fixed α < 1/2 (then taking at most an even smaller α one deduces the little Hölder property). We claim that, on the event E R , M ↓ ∞ ↾T ↓, (r) ∞ is α-Hölder continuous provided that the paths π ↓ z , z ∈ D, restricted to the box Λ r,R def = [−r, r] × [−R,R] satisfy a suitable equi-Hölder continuity condition. The latter can be stated in terms of a modulus of continuity of the form (see also the proof of [SSS17, Theorem 6.2.3]) Rr ε 2α−1/2 e −ε 2α−1 and upon taking R = ε −1 , (3.12) follows.
We now want to show properness of M ↓ ∞ , which is a direct consequence of the following lemma.
Lemma 3.5 There exists a constant c > 0 independent of r such that for any K > 0 sufficiently large where b ζ ↓ ∞ is the properness map given in (2.2).
It remains to argue uniqueness and the properties of the limit. Uniqueness immediately follows since conditions 1-3 above imply conditions 1-4 in Theorem 1.1. On the other hand, we have just shown that ζ ↓ (D) satisfies 1-3 and, by Proposition 3.2 also the other claimed properties, so that the proof of the statement is concluded.
Remark 3.9 The theorem above remains true upon replacing conditions 1.-3. with 1 per ., 2 per . and 3 per ., obtained from the former by adding the word "periodic" before any instance of "Brownian motion", and taking the periodic version of all objects and spaces in the statement.
As a first property of the Brownian Web tree, which can be deduced by Theorem 3.8 and the results stated therein, we determine its Minkowski, also known as box-covering, dimension. Recall that the box-covering dimension of a (compact) metric space (T, d) is given by when this limit exists. (T ↓, (r) bw , ε) of the same order, for all r > 0. Now, the upper bound follows by the fact that, by Theorem 3.8, almost surely T ↓ bw satisfes (3.3) for all θ > 3/2. For the lower bound, we need to show that almost surely for all r > 0, κ > 0 there exists a random constant C = C(r, κ) > 0 such that N d ↓ bw (T ↓, (r) bw , ε) ≥ Cε κ− 3 2 . This in turn follows at once, provided we prove that for all δ > 0, r > 0 and κ > 0, there exists K > 0 such that We now fix δ, r, κ. To control the probability on the left-hand side of (3.18), by the reflection principle, we know that we can find t < 0 and x > 0 such that the event bw . Define L ε = ⌈t/ε⌉ + 1 and t ε k def = −kε, k = 0, . . . , L ε − 1. Therefore, arguing as in the proof of (3.9) we have where, for b < a, η x (a, b) is the cardinality of Ξ x (t 0 , t 1 ) defined as in (3.7), but with the interval [−R,R] replaced by [x/4, 3x/4]. Then, we have For the right-hand side, we notice that for anyK > 0 we have Now, the first summand is bounded above by where in the penultimate step we used that η x is negatively correlated [GSW16,Lemma C.4], so that [GSW16, Lemma C.5] implies that its variance is bounded above by twice its mean, and in the last step we exploited [SSS17, Proposition 6.2.7]. At this point, suitably choosingK = O(δ −1/2 ε −5/4 ) we see that the right-hand side of (3.20) is bounded above by δ/2 while the second summand in (3.19) vanishes.
Collecting the estimates obtained so far, (3.18), and consequently the lower bound, follow at once.

Remark 3.13
The previous corollary shows in particular that the law of the Brownian Web trees on the space of R-trees is, as expected, singular with respect to that of the scaling limit of the Uniform Spanning Tree in two dimensions. Indeed, the latter has Hausdorff dimension 5/8 [BCK17] and the Hausdorff dimension is always greater than or equal to the box-counting one (see e.g. [Edg98, Chapter 1]).
In the following Corollary, we establish the relation between the Brownian Web Tree of Definition 3.8 and the Brownian Web constructed in [FINR04], which is a simple consequence of Theorem 3.8 and the results in Section 2.4.
Corollary 3.14 Let ζ ↓ bw and ζ per,↓ bw be the backward and backward periodic Brownian Web trees of Theorem 3.8 and Remark 3.9, and K be the map defined in (2.34). Then, K(ζ ↓ bw ) is a backward Brownian Web according to [FINR04,  Proof. To prove the statement it suffices to verify that K(ζ ↓ bw ) and K(ζ ↓ bw,per ) satisfy

A convergence criterion to the Brownian Web tree
In this section, we want to derive a criterion that allows to conclude that the limit law for tight sequences of directed spatial R-trees is Θ ↓ bw .
Theorem 3.15 Let α ∈ (0, 1) and {ζ n } n be a tight sequence of random variables in D α sp with laws Θ n and assume that the following holds.
(I) For any k ∈ N and (deterministic) z 1 , . . . , z k ∈ R 2 there exist sequences z i n ∈ T n , i = 1, . . . , k such that lim n→∞ M n (z i n ) = z i almost surely and such that (M n (ϱ n (z i n , ·))) i converges in law to k coalescing backward Brownian motions.
We claim that ifT ⊊ T , then there exist a, b, t, h ∈ Q such that Indeed, let z ∈ T , z = (s, y) = M (z) and r > 0. Since ζ ∈ D α sp , M is locally α-Hölder continuous which implies that there exists C > 0 such that whereh is chosen in such a way that r ≥ Ch α . Let y + n , y − n , s n and h n be sequences in Q such that y ± n converges to y ± r, s n converges to s, h n converges to 0 and, for all n, y − n ≤ y − Ch α , y + n ≥ y + Ch α and s −h/2 ≤ s n − h n < s. Then, M (ϱ(z, s n )) ∈ {s n } × [y − n , y + n ]. Now, if for all a, b, t, h ∈ Q, (3.23) were an equality, then for all n, ϱ(z, s n − h n ) ∈ {ϱ(w, s n − h n ) : w ∈ (M ) −1 (I sn;y − n ,y + n )} ⊂T . But the sequence {ϱ(z, s n − h n )} n is Cauchy inT and since the latter is complete, ϱ(z, s) ∈T for every s sufficiently small, so that taking s to 0, we get that z ∈T .
The previous claim implies that the probability that T \T ̸ = ∅ is bounded above by the probability that there exist a, b, t, h ∈ Q such that (3.23) holds. Hence, if we show that for every a, b, t, h ∈ Q fixed, the probability of (3.23) is 0 we are done. Fix a, b, t, h ∈ Q, a < b. For N ∈ N, let and z N j = (t, x j ). By construction, there exist unique points z N j ∈T such that M (z N j ) = M (z N j ) = z N j for all j = 0, . . . , N . Hence, Moreover, since as soon as two rays in an R-tree touch, they coalesce (otherwise one could form a cycle), #{ϱ(w, t − h) : w ∈ M −1 (I t;a,b )} > #{ϱ(z N j , t − h) : j = 0, . . . , N } if and only if there exists i = 1, . . . , N such that #{ϱ(w, t − h) : w ∈ M −1 (I t,y N i ,ε )} ≥ 3, where, for i = 1, . . . , N , y N i denotes the mid-point of the interval (x N i−1 , x N i ). In other words, which converges to 0 as N → ∞ by (3.21), and the conclusion follows at once.
Remark 3.17 The first part of the proof above shows that, given that {ζ n } n is tight and satisfies conditions (I) and (II), the sequence {K(ζ n )} n converges to the Brownian Web. In light of Propositions 2.25 and 2.31, one might wonder whether tightness in D α sp of a sequence {ζ n } n ⊂ D α sp (t) together with convergence of {K(ζ n )} n in Π can directly imply convergence of {ζ n } n in D α sp . The answer is no as the following example shows. For all n odd, let ζ n be the directed tree given by one infinite branch e embedded into R 2 as {0} × (−∞, 0], while for n even let ζ n be the directed tree given by the same e together with a branch e n embedded as Clearly, the sequence {ζ n = (T n , * n , d n , M n )} n ⊂ D α sp (t) and is tight in D α sp but it does not converge -the odd subsequence is constant while the even one converges to the directed tree formed by two branches e ∪ e ∞ where e ∞ is embedded as {0} × [−1, 0] and the two branches meet at (0, −1). At the same time, {K(ζ n )} n converges in Π to the set {π (0,t) : t ≤ 0} for π (0,t) identically equal to 0 on (−∞, t].

The double Brownian Web tree and special points
A crucial aspect of the backward Brownian Web is that it comes naturally associated with a dual (see e.g. [TW98,FINR06]), which is given by a family of forward coalescing Brownian motions starting from every point in R 2 or R × T, in the periodic case. In the next theorem we will see how it is possible to devise such a duality in the present context and characterise the joint law of the Brownian Web Tree in Definition 3.10 and its dual. (ii) Almost surely, for any z ↓ ∈ T ↓ bw and z ↑ ∈ T ↑ bw , the paths M ↓ bw (ϱ ↓ (z ↓ , ·)) and

Remark 3.19
Here, given a random variable (X, Y ) on some product Polish space X × Y, we say that X is determined by Y if the conditional law of X given Y is almost surely given by a Dirac mass.
Proof. Throughout the proof, we will adopt the notation and conventions of Section 2.4.
Notice at first that, by Theorem 3.8, any D α sp ×D α sp -valued random variable for which (i) holds, almost surely belongs to D α sp (t) ×D α sp (t). Now, let (W ↓ , W ↑ ) be the H ×Ĥ-valued random variable constructed in [SSS17, Theorem 6.2.4] and K the map in (2.34). Since W ↓ is distributed as the backward Brownian Web, by Corollary 3.14, where the first equality is due to [SSS17, Theorem 6.2.4(a)] and the last is a consequence of Remark 2.26. Therefore, (W ↓ , W ↑ ) ∈ K(D α sp (t)) ×K(D α sp (t)) almost surely so that, by Proposition 2.25 and Remark 2.26, there exists a unique . By Proposition 3.2 and Theorem 3.8 we also have ζ ↓ bw ∈ D α sp (t) almost surely so that, since The definition of the map K in (2.33) and (2.34) combined with [SSS17, Theorem 6.2.4(b)] ensures that (ii) holds for (ζ W ↓ , ζ W ↑ ). The fact that ζ W ↑ is determined by ζ W ↓ is a direct consequence of the fact that this is known to be true for W ↓ and W ↑ and that K is invertible on D α (t).

Remark 3.20
In the periodic setting Theorem 3.18 remains true upon replacing all the objects and spaces appearing in the statement with their periodic counterparts. The proof follows the exact same lines but uses Remarks 3.9 and 2.33 instead of Theorem 3.8 and Proposition 2.25.
Definition 3.21 Let α < 1 2 . We define the double Brownian Web tree and double periodic Brownian Web tree as the D α sp ×D α sp and D α sp,per ×D α sp,per -valued random variables ζ ↓↑ bw def = (ζ ↓ bw , ζ ↑ bw ) and ζ per,↓↑ bw def = (ζ per,↓ bw , ζ per,↑ bw ) given by Theorem 3.18 and Remark 3.20. We will refer to ζ ↑ bw and ζ per,↑ bw as the forward (or dual) and forward periodic Brownian Web trees.

Remark 3.22
The proof of Theorem 3.18 heavily relies on the results of [FINR06] (summarised in [SSS17]). Clearly, it would have been possible to construct the double Brownian Web tree directly starting from a countable family of (independent) forward where {z ↓ i } i are the points in (M ↓ bw ) −1 (z) and |(M • bw ) −1 (z)| denotes the cardinality of (M • bw ) −1 (z). The relation (3.25) holds as well with the arrows ↑ , ↓ reversed and for their periodic counterpart.
Proof. As usual we will focus on the non-periodic case, the other being analogous.
We claim that for all z = (t, z) ∈ R 2 , |(M ↓ bw ) −1 (z)| = m b out (z) and the right-hand side of (3.25) coincides with m b in (z), where m b out (z) and m b in (z) are defined according to [FINR06,(3.11) and (3.10)] and respectively represent the number of distinct paths "leaving" and "entering" the point z for the backward Brownian Web (by removing the superscript b and reverting the arrows the same holds for the forward by duality).
Indeed, for every z ↓ ∈ (M ↓ bw ) −1 (z), denoting by ϱ ↓ the radial map associated to ζ ↓ bw , we have that (−∞, t] ∋ s → M ↓ bw,x (ϱ ↓ (z ↓ , s)) is a path from z. On the other hand, deg(z ↓ ) − 1 corresponds to the number of rays in the tree which coalesce at or reach z. Notice that, since almost surely ζ ↓ bw satisfies (t), the image of the rays coalescing or reaching z as well as that of the rays from points in (M ↓ bw ) −1 (z) are distinct so that the claim follows. Now, by Theorem 3.18 (K(ζ ↓ bw ),K(ζ ↑ bw )) is distributed as the double Brownian Web and almost surely ζ ↓↑ bw ∈ D α sp (t) ×D α sp (t). Since moreover the restriction of K to D α sp (t) is bijective on its image thanks to Proposition 2.25, (3.25) is a direct consequence of [FINR06, Proposition 3.10].
We are now ready to classify the different points in R 2 or in R × T based on the meaning they have for the (periodic) Brownian Web tree (and its dual) as we constructed it.
Thanks to the classification above, we can now prove one of the features that distinguishes the Brownian Web tree and its periodic version. In the next proposition, whose conclusion was first noted in [CMT19], we show that the periodic Brownian Web tree possesses a unique bi-infinite path connecting its two open ends with unbounded rays. Proof. Since T per,↓ bw and T per,↑ bw are periodic directed trees, we already know they have one open end with unbounded rays, and this is the one for which (2.27) holds (for the forward periodic Web see Remark 2.22). Denote them by † ↓ and † ↑ and let ϱ ↓ per and ϱ ↑ per be the radial maps introduced in Definition (2.24). Similarly to (3.7), for t 0 , t 1 ∈ R, t 0 < t 1 , we introduce and set η ↑ T (t 0 , t 1 ) and η ↓ T (t 1 , t 0 ) to be the cardinality of Ξ ↑ T (t 0 , t 1 ) and Ξ ↓ T (t 1 , t 0 ) respectively. We inductively define the sequence of stopping times These stopping times coincide (in distribution) with those in the proof of [CMT19, Theorem 3.1], where it is further showed that almost surely lim k→∞ τ k = +∞. Now, by definition, for every k ≥ 1, there must exist a point z k−1 ∈ T × {τ k−1 } such that |(M per,↑ bw ) −1 (z k−1 )| ≥ 2 and the distance of (at least) two elements in (M per,↑ bw ) −1 (z k−1 ) is 2(τ k − τ k−1 ). By (3.25) and Theorem 3.26, it follows that there exists exactly one point (M per,↓ bw ) −1 (z k−1 ) whose degree is greater or equal to 2. Denote it by z k . Then the map β ↓ : R → T per,↓ bw given by ϱ ↓ per (z k , s) , for s ∈ (τ k−1 , τ k ] ϱ ↓ per (z 0 , s) for s < 0.
is not only well-defined by Theorem 3.18(ii) but also uniquely defined since so is the choice of the point z k . The map β ↓ shows that there are exactly two open ends with unbounded rays, and β ↓ (R) is the unique linear subtree of T per,↓ bw satisfying the properties in [Chi01, Lemma 3.7(i)].

The Discrete Web Tree and convergence
In this section, we introduce the discrete web and its dual, and show that, as a couple, they converge to the Double Brownian Web Tree of Definition 3.21.

The Double Discrete Web Tree
We begin our analysis with the spatial tree representation of a family of coalescing backward random walks and its dual. The construction below will directly provide a coupling between forward and backwards paths under which one is determined by the other and the two satisfy the non-crossing property of Theorem 3.18(ii).
Let δ ∈ (0, 1] and (Ω, A, P δ ) be a standard probability space supporting four Poisson random measures, µ L γ , µ R γ ,μ L γ andμ R γ . The first two, µ L γ and µ R γ , live on S ↓ δ def = R × δZ, are independent and have both intensity γλ, where, for every k ∈ δZ, λ(dt, {k}) is a copy of the Lebesgue measure on R and throughout the section (4.1) The others live on S ↑ δ def = R × δ(Z + 1/2), and are obtained from the formers by setting, principle, the backward random walk π ↓,δ z δ defined above converges in law to a backward Brownian motion π ↓ z started at z. Let {z ± δ } δ ⊂ Q ± R ∩ (S δ ) be sequences converging to z ± . Denoting by E δ R the event E R in (3.4), but in which z ± is replaced by z ± δ , we see that the previous observation implies lim inf δ↓0 P δ (E δ R ) = P(E R ) (4.10) so that (3.5) holds. Moreover, the analog of [FINR04, Proposition 4.1] (see also [SSS17, pg 326]) for random walks ensures that for all R, r > 0 and a < b where η R (a, b) is the cardinality of Ξ R (a, b) given in (3.7) and E δ is the expectation with respect to P δ . Thanks to (4.10) and (4.11), we can argue as in Lemma 3.3 and obtain that there exists a constant C = C(r) > 0 independent of δ such that for all K > 0 lim sup so that by Borel-Cantelli (4.7) follows. As in Proposition 3.2, the uniform local Hölder continuity of the evaluation maps M ↓ so that the first two summands in (2.4) are controlled. For the other, let (z, z ′ ), (w, w ′ ) ∈ C δ andm ∈ N be the biggest integer for which 2 −m < √ δ. Then, for m >m + 2, we bound ψ m by 1, so that ∥ψ m (d(z, w))δ z,w M − ψ m (d(z ′ , w ′ ))δ z ′ ,w ′ M ∥ ≤ 2ω(M, 2 −m+2 ) ≲ 2 −mα (δ −α/2 ω(M, √ δ)) .