The Brownian Web As a Random $\Mathbb R $-Tree

$The Brownian Web As a Random $\Mathbb R $-Tree$

The Brownian Web as a random R-tree

February 9, 2021

G. Cannizzaro1,2 and M. Hairer1

1 Imperial College London, SW7 2AZ, UK 2 University of Warwick, CV4 7AL, UK Email: [email protected], [email protected]

Abstract Motivated by [CH21], we provide an alternative characterisation of the Brownian Web [TW98, FINR04], i.e. a family of coalescing Brownian motions starting from 2 every point in R simultaneously, and ﬁt it into the wider framework of random (spatial) R-trees. We determine some of its properties (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. Along the way, we introduce a modiﬁcation of the topology of spatial R-trees in [DLG05, BCK17] which makes it Polish and could be of independent interest.

Contents

1 Introduction 2 1.1 Outline of the paper ...... 5 2 Preliminaries 7 2.1 R-trees in a nutshell ...... 7 2.2 Spatial R-trees ...... 9 arXiv:2102.04068v1 [math.PR] 8 Feb 2021 2.3 Characteristic R-trees and the radial map ...... 18 2.4 Alternative topologies ...... 21 3 The Brownian Web Tree and its dual 26 3.1 An alternative characterisation of the Brownian Web ...... 26 3.2 A convergence criterion to the Brownian Web tree ...... 36 3.3 The double Brownian Web tree and special points ...... 38 Introduction 2

4 The Discrete Web Tree and convergence 43 4.1 The Double Discrete Web Tree ...... 43 4.2 Tightness and convergence ...... 46

1 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 R2, 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. Rediscovered by Tóth and Werner in [TW98], the authors 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, and in [BGS15], where the optimal convergence condition was obtained and a family of coalescing Brownian motions on the Sierpinski gasket were built. 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 (if not all) of these works, the Brownian Web is viewed as a random (compact) collection of paths Win a suitable space. The present paper aims at highlighting yet another of its characterising features, namely its coalescence or tree structure, clearly apparent in Figure 1. The main motivation comes from the companion paper [CH21] in which such a structure is used to construct and study the Brownian Castle, a stochastic process whose value at a given point equals that of a Brownian motion indexed by a 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 the set of points in the trajectories of paths in W as a metric space with metric given by the intrinsic distance, namely the ↓ distance between (ti, πi ) ∈ R × W, i = 1, 2, is the minimal time it takes to go from ↓ ↓ ↓ ↓ (t1, π1(t1)) to (t2, π2(t2)) moving along the trajectories of π1 and π2 at unit speed. Instead of working directly on R2, it turns out to be more convenient to first encode the points of the trajectories in an abstract space and then suitably embed the latter into R2. More precisely, ↓ def ↓ ↓ ↓ ↓ we will construct a (random) quadruplet ζbw = (Tbw, ∗bw, dbw,Mbw) whose elements we ↓ ↓ now describe. The first three form a pointed R-tree, which means that (Tbw, dbw) is a Introduction 3

Figure 1: A typical realisation of the Brownian web: coalescing Brownian trajectories emanate from every point of the plane simultaneously. Trajectories are coloured according to their creation time / age. Introduction 4

↓ ↓ connected metric space with no loops and ∗bw is an element of Tbw (see Deﬁnition 2.1). ↓ ↓ ↓ Morally, points in Tbw are of the form (t, π ) for π ∈ W and t ≤ σπ↓ , where σπ↓ is the ↓ ↓ starting time of π , and dbw is the ancestral distance deﬁned as

↓ ↓ ↓ def ↓ ↓ ↓ ↓ ↓ ↓ dbw((t, π ), (s, π˜ )) = (t + s) − 2τt,s(π , π˜ ) , for all (t, π ), (s, π˜ ) ∈ Tbw,

↓ ↓ ↓ ↓ ↓ ↓ def τt,s being the first time at which π and π˜ meet, i.e. τt,s(π , π˜ ) = sup{r < t ∧ s : ↓ ↓ ↓ ↓ 2 π (r) =π ˜ (r)}. Mbw is the evaluation map which embeds Tbw into R , and is given by ↓ ↓ ↓ 2 Tbw 3 (t, π ) 7→ (t, π (t)) ∈ R . The main task of the present paper is to identify a “good” space in which the quadruplet ↓ ζbw lives, which is Polish and allows for a manageable characterisation of its compact subsets. Elements of the form ζ = (T , ∗, d, M) are said to be spatial R-trees and have already been considered in the literature. Similar to [DLG05, BCK17], we endow the space of α spatial R-trees Tsp, α ∈ (0, 1), with a Gromov–Hausdorff-type topology (for an introduction in the case of general metric and length spaces we refer to the monograph [BBI01], and to [Eva08] for the specific case of R-trees) with an important caveat, namely our metric (see (2.10)) takes into account two extra conditions imposed at the level of the evaluation map M. More precisely, M is required to be both locally little α-Hölder continuous, i.e. 0 0 α limε→0 supz∈K supd(z,z0)≤ε kM(z) − M(z )k/d(z, z ) = 0 for every compact K, and proper, i.e. preimages of compacts are compact. As pointed out in [BCK17, Remark 3.2], without α the first property the space Tsp would not be Polish (the space Tsp of [DLG05, BCK17] lacks completeness). The second property prevents the existence of sequences of points that are order 1 distance apart in T but whose M-image is arbitrarily close in R2. In the weighted metric ∆sp in (2.10), necessary to consider the case of unbounded R-trees (as is ↓ Tbw) and which is in essence that of [BCK17], this is allowed. The topology introduced in Section 2.2 and briefly described above guarantees that a α sequence {ζn = (Tn, ∗n, dn,Mn)}n converges to ζ = (T , ∗, d, M) in Tsp provided that, morally, the metrics dn converge to d, which in the present context means that couples of distinct paths which are close also coalesce approximately at the same time, and the evaluation maps Mn converge to M in Hölder sense, which in turn ensures control over the sup-norm distance of paths and is somewhat similar in spirit to that of [FINR04]. Notice that it is not always possible to assign to a family of paths an R-tree (trivially, consider the case of paths which are not coalescing) and, conversely, there is no canonical way to associate a collection of paths to a generic spatial R-tree. However, we identify a subset α of Tsp for which this is the case 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), Introduction 5

α convergence statements in Tsp do not follow from those previously established. This is ↓ remedied in Section 3.2, where a convergence criterion to ζbw is derived. As shown in [CH21], 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 [CH21, Theorem 1.4]. Moreover, ↓ the R-tree structure automatically endows Tbw 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]. At last, the present paper fits the Brownian Web into the wider framework of random R-trees. Many fascinating objects belong to this realm and display interesting relations to important statistical mechanics models, 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], 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 3.12) 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 [CH21] can be derived.

1.1 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 α geometric features, we introduce, for α ∈ (0, 1), the spaces Tsp, of spatial R-trees, and its α “characteristic” subset Csp. We define a metric which makes them Polish 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 α (Section 3.1), we provide a characterisation of its law on Csp 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 introduce 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. Introduction 6

Notations d We will denote by | · |e the usual Euclidean norm on R , d ≥ 1, and adopt the short-hand def def 2 notations |x| = |x|e and kxk = |x|e for x ∈ R and R respectively. Let (T , d) be a metric space. We define the Hausdorff distance dH between two non-empty subsets A, B of T as def ε ε dH (A, B) = inf{ε: A ⊂ B and B ⊂ A} where Aε is the ε-fattening of A, i.e. Aε = {z ∈ T : ∃ w ∈ A s.t. d(z, w) < ε}. Let (T , d, ∗) be a pointed metric space, i.e. (T , d) is as above and ∗ ∈ T , and let M : T → Rd 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 |M(z) − M(w)| (r) def z (r) def e kMk∞ = sup |M( )|e , kMkα = sup α . z∈Bd(∗,r] z,w∈Bd(∗,r] d(z, w) d(z,w)≤1 where Bd(∗, r] ⊂ T is the closed ball of radius r centred at ∗, and, for δ > 0, the modulus of continuity as (r) def ω (M, δ) = sup |M(z) − M(w)|e . (1.1) z,w∈Bd(∗,r] d(z,w)≤δ 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 −α (r) function M is (locally) little α-Hölder continuous if for all r > 0, limδ→0 δ ω (M, δ) = 0. Let I ⊆ R be an interval and D(I, R+) be the space of càdàg functions on I with def values in R+ = [0, ∞), endowed with the M1 topology that we now introduce. For f ∈ D(I, X), denote by Disc(f) the set of discontinuities of f and by Γf its completed graph, i.e. the graph of f to which all the vertical segments joining the points of discontinuity are added. Order Γf by saying that (x1, t1) ≤ (x2, t2) if either t1 < t2 or t1 = t2 and − − |f(t1 ) − x1| ≤ |f(t1 ) − x2|. Let Pf be the set of all parametric representations of Γf , which is the set of all non-decreasing (with respect to the order on Γf ) functions σf : I → Γf . Then, if I is bounded, we set ˆc def dM1(f, g) = 1 ∨ inf kσf − σgk σf ,σg c and dM1(f, g) to be the topologically equivalent metric with respect to which D(I, R+) is complete (see [Whi02, Section 8] for more details). If instead I = [−1, ∞), we define Z ∞ def −t c (t) (t) dM1(f, g) = e (1 ∧ dM1(f , g )) dt (1.2) 0 where f (t) is the restriction of f to [−1, t]. 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. Preliminaries 7

Acknowledgements GC would like to thank the Hausdorff Institute in Bonn for the kind hospitality during the programme “Randomness, PDEs and Nonlinear Fluctuations”, where he carried out part of this work. GC gratefully acknowledges financial support via the EPSRC grant EP/S012524/1. MH gratefully acknowledges financial support from the Leverhulme trust via a Leadership Award, the ERC via the consolidator grant 615897:CRITICAL, and the Royal Society via a research professorship.

2 Preliminaries

In this section, we gather all the results on R-trees which will be necessary in the sequel. At ﬁrst, we summarise some of their geometric properties.

2.1 R-trees in a nutshell Let us begin by recalling the deﬁnition of R-tree given in [DLG05, Deﬁnition 2.1].

Deﬁnition 2.1 A metric space (T , d) is an R-tree if for every z1, z2 ∈ T

1. there is a unique isometric map fz1,z2 : [0, d(z1, z2)] → T such that fz1,z2 (0) = z1 and

fz1,z2 (d(z1, z2)) = z2, 2. for every continuous injective map q : [0, 1] → T such that q(0) = z1 and q(1) = z2, one has

q([0, 1]) = fz1,z2 ([0, d(z1, z2)]) .

A pointed R-tree is a triple (T , ∗, d) such that (T , d) is an R-tree and ∗ ∈ T .

Remark 2.2 We do not call such spaces rooted because, for the Brownian Web as we will construct it, the natural root should be thought of as a “point at inﬁnity” where all the paths starting from every point meet.

For an R-tree (T , d) and any two points z1, z2 ∈ T , we define the segment joining z1 and z2 as the range of the map fz1,z2 and denote it by z1, z2 . Notice that for every three points J K z1, z2, z3 ∈ T there exists a unique point w ∈ T such that z1, z3 ∩ z2, z3 = w, z3 . We call w, the projection of z2 onto z1, z3 , or equivalently theJ projectionK J of zK1 ontoJ z2,Kz3 . J K J K Definition 2.3 [CMSP08, Definition 2] Let (T , d) be an R-tree and r > 0. A segment z1, z2 ⊂ T has r-finite branching if the set of points w ∈ z1, z2 which are the projection J K J K of some point z ∈ T onto z1, z2 with d(z, w) ≥ r is finite. An R-tree T is said to have r-finite branching if every segmentJ K of T does. Preliminaries 8

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, z0 ∈ T and let S be a dense subset of T . The following statements hold:

1. If z ∈ T is not an endpoint for T , then there exists w ∈ S such that z ∈ z0, w . 2. If S is a subtree of T , then every point of T \ S is an endpoint for T . J K Notice that the connected components of T \{z} are themselves R-trees, i.e. subtrees of T , and they are called directions at z.

Definition 2.5 [CMSP08, Definition 1] Let (T , d) be an R-tree, z ∈ T and {Ti : i ∈ I}, where I is an index set, the set of directions at z. For r > 0, we say that Ti has length ≥ r if there exists w ∈ Ti 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]. 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 Lz for an arbitrary segment of T having z as an endpoint and we say that Lz is a T -ray from z if it is maximal for inclusion. We also say that rays Lz and Lz0 are equivalent if there exists w ∈ T such that Lz ∩ Lz0 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, †i. Moreover we say that † is an open end with (un-)bounded rays if J for every z ∈ T , the map ιz : z, †i → R+ given by J ιz(w) = d(z, w) , w ∈ z, †i (2.1) J is (un-)bounded. 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 Preliminaries 9

(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.

2.2 Spatial R-trees Now that we discussed geometric features of R-trees we are ready to study the metric properties of the space of all R-trees. We will focus on a speciﬁc subset of it, namely the space of α-spatial R-trees.

α Deﬁnition 2.7 Let α ∈ (0, 1). The space of pointed α-spatial R-trees Tsp is the set of equivalence classes of quadruplets ζ = (T , ∗, d, M) where

- (T , ∗, d) is a complete and locally compact pointed R-tree, - M, the evaluation map, is a locally little α-Hölder continuous proper1 map from T to R2, and we identify ζ and ζ0 if there exists a bĳective isometry ϕ : T → T 0 such that ϕ(∗) = ∗0 and M 0 ◦ ϕ ≡ M, in short (with a slight abuse of notation) ϕ(ζ) = ζ0.

Remark 2.8 We will also consider situations in which the map M is R × T-valued, def where T = R/Z is the torus of size 1 endowed with the usual periodic metric d(x, y) = infk∈Z |x − y + k|. Whenever this is the case, we will add an extra subscript “per”, standing for periodic, to the space under consideration, which will anyway always be a subset of α Tsp,per. In what follows, it is immediate to see how the deﬁnitions, statements and proofs need to be adapted in order to hold not only for the generic space Sbut also for its periodic counterpart Sper.

For any spatial tree ζ = (T , ∗, d, M), we introduce the properness map bζ : R → R+, that “quantiﬁes” the properness of M. For r < 0, bζ (r) = 0, while for r ≥ 0 we set

def bζ (r) = sup d(∗, z) , (2.2) z : M(z)∈Λr

def 2 2 per def where Λr = [−r, r] ⊂ R and in the periodic case Λr = Λr = [−r, r] × T.

α Lemma 2.9 Let α ∈ (0, 1). For all ζ = (T , ∗, d, M) ∈ Tsp the properness map is non-decreasing and càdlàg. 1 0 0 α Namely such that limε→0 supz∈K supd(z,z0)≤ε kM(z) − M(z )k/d(z, z ) = 0 for every compact K and the preimage of every compact set is compact. Preliminaries 10

Proof. The function bζ 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 lims↓r bζ (s) = bζ (r). Notice that, for every s > 0, since T is locally compact, M is continuous and Λs −1 is closed, there exists zs ∈ M (Λs) such that bζ (s) = d(∗, zs). Let sn be a sequence decreasing to r and, without loss of generality, assume M(zsn ) ∈ Λsn \Λr. Since M is proper, −1 −1 M (Λs0 ) is compact so that {zsn }n ⊂ M (Λs0 ) admits a converging subsequence. Let ¯z be a limit point. By construction, d(∗, zsn ) ≥ d(∗, zr) for all n, therefore d(∗,¯z) ≥ d(∗, zr). But M(¯z) ∈ Λr since M is continuous, so d(∗,¯z) ≤ d(∗, zr) as claimed. α To turn Tsp 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 0, d0) is a subset of T × T 0 such that for all z ∈ T there exists at least one z0 ∈ T 0 for which (z, z0) ∈ C and vice versa. The distortion of a correspondence C is given by dis C =def sup{|d(z, w) − d0(z0, w0)| : (z, z0), (w, w0) ∈ C} , and allows to give an alternative characterisation of the Gromov-Hausdorff metric (see [Eva08, Theorem 4.11], for the case of compact metric spaces). α α Now, let Tc be the subset of Tsp consisting of compact R-trees. Let ζ = (T , ∗, d, M) 0 0 0 0 0 α and ζ = (T , ∗ , d ,M ) be elements of Tc and C be a correspondence between T and T 0. We set

c, C 0 def 1 0 0 ∆sp (ζ, ζ ) = dis C+ sup kM(z) − M (z )k 2 (z,z0)∈ C nα 0 (2.3) + sup 2 sup kδz,wM − δz0,w0 M k n∈N (z,z0),(w,w0)∈ C 0 0 0 d(z,w),d (z ,w )∈An

def −n −(n−1) def where An = (2 , 2 ] for n ∈ N, and δz,wM = M(z) − M(w). In the above, we adopt the convention that if there exists no pair of couples (z, z0), (w, w0) ∈ C such that d(z, w) ∈ An, then the increment of M is removed and the supremum is taken among all 0 0 0 0 0 z , w such that d (z , w ) ∈ An and vice versa2. We can now deﬁne c 0 def c 0 ∆sp(ζ, ζ ) = ∆sp(ζ, ζ ) + dM1(bζ , bζ0 ) (2.4) where dM1 is the metric on the space of càdlàg functions given in (1.2) and

c 0 def c, C 0 ∆sp(ζ, ζ ) = inf ∆sp (ζ, ζ ) . (2.5) C: (∗,∗0)∈ C In view of Lemma 2.9, the metric above is well-deﬁned. 2If instead we adopted the more natural convention sup ∅ = 0, then the triangle inequality might fail, e.g. when comparing a generic spatial tree to the trivial tree made of only one point. Preliminaries 11

α c Proposition 2.10 For α ∈ (0, 1), (Tc , ∆sp) is a complete separable metric space.

c Proof. Notice that the definition of ∆sp in (2.4) comprises two independent summands. The c term dM1, which involves M, is a pseudometric by [Whi02, Theorem 12.3.1 and Sections 12.8 and 12.9], while the other term is shown to be a pseudometric by following the same steps as in [CHK12, Lemma 2.1] (see also [BCK17, Proposition 3.1] and [ADH13, Theorem 2.5(i)]). The proof of completeness is a simplified version of that of Theorem 2.13(ii) below, therefore we omit it and focus instead on separability. According to Lemmas 2.12 and 2.14 α (see also, for completeness, Lemma 2.15) below, any element ζ = (T , ∗, d, M) ∈ Tc can α ε ε ε be approximated in Tsp by ζ = (T , ∗, d, M), where T ⊂ T is a finite ε-net of T . We ε ˜ε def S ε can turn T into an R-tree by setting T = z,w∈T ε z, w , where for any z, w ∈ T the J K points in the edge z, w are those of T and set M ε to be the restriction of M to T˜ε. Then, c J K ε ˜ε ˜ε ˜ ε α clearly, the ∆sp-distance between ζ and ζ = (T , ∗, d, M ) ∈ Tc is going to 0 as ε goes to α 0. Therefore, a countable dense set in Tc can be obtained by considering the set of R-trees with finitely many endpoints and edge lengths, in which the distances between endpoints are rationals, endowed with maps M which are Q2-valued at the end- and branch points and linearly interpolated in between. Remark 2.11 As pointed out in [BCK17, Remark 3.2], without the Hölder condition in c the definition of ∆sp, the space of spatial pointed R-trees would not be complete while, if we did not assume the function M to be little Hölder continuous it would lack separability.

α Lemma 2.12 Let α ∈ (0, 1), ζ = (T , ∗, d, M) ∈ Tc . Let δ > 0, T ⊂ T be such that ∗ ∈ T and the Hausdorﬀ distance between T and T is bounded above by δ and deﬁne ¯ ζ = (T, ∗, d, MT ). Then c ¯ −α ∆sp(ζ, ζ) . (2δ) ω(M, 2δ) (2.6) Proof. Let Cδ be the correspondence given by {(z, z0) ∈ T × T : d(z, z0) ≤ δ}. Then, for every (z, z0), (w, w0) ∈ Cδ, we have

0 0 0 0 α α |d(z, w) − d(z , w )| ≤ 2δ , kM(z) − M(z )k ≤ kMkαd(z, z ) ≤ kMkαδ so that the ﬁrst two summands in (2.3) are controlled. For the other, let mT be the largest 0 0 0 0 integer for which there exist z , w ∈ T such that d(z , w ) ∈ AmT . By assumption, T is a δ-net for T and T is a length space, therefore the minimal distance between points in T −mT has to be less than 2δ, which implies that 2 ≤ 2δ. For m > mT and z, w ∈ T are such that d(z, w) ∈ Am, we have

−m −mα −α kδz,wMk ≤ ω(M, 2 ) ≤ 2 ((2δ) ω(M, 2δ)) . (2.7) Preliminaries 12

0 0 δ 0 0 If m ≤ mT , let (z, z ), (w, w ) ∈ C be such that d(z, w), d(z , w ) ∈ Am. Now, in case m −m satisﬁes 2 ≤ 2δ, then we apply the triangle inequality to the norm of δz,wM − δz0,w0 M −m and bound each of kδz,wMk and kδz0,w0 Mk as in (2.7). At last, in case 2 > 2δ we get

kδz,wM − δz0,w0 Mk ≤ kδz,z0 Mk + kδw,w0 Mk ≤ 2ω(M, δ) (2.8) −mα −α . 2 (δ ω(M, δ)) which implies the result.

α α We are now ready to introduce a metric on the whole of Tsp. For ζ = (T , ∗, d, M) ∈ Tsp and any r > 0, let ζ(r) =def (T (r), ∗, d, M (r)) (2.9) (r) def (r) where T = Bd(∗, r] is the closed ball of radius r in T and M is the restriction of M (r) α α to T . We deﬁne ∆sp as the function on Tsp × Tsp given by

Z +∞ 0 def −r h c (r) 0 (r) i ∆sp(ζ, ζ ) = e 1 ∧ ∆sp(ζ , ζ ) dr + dM1(bζ , bζ0 ) 0 (2.10) 0 =: ∆sp(ζ, ζ ) + dM1(bζ , bζ0 ).

0 α (r) 0 (r) for all ζ, ζ ∈ Tsp. Since T and T are R-trees and, in view of Theorem 2.6(a), (r) 0(r) α compact, ζ , ζ ∈ Tc so that the ﬁrst summand in (2.10) is well-deﬁned.

Theorem 2.13 For any α ∈ (0, 1),

α (i) ∆sp is a metric on Tsp α (ii) the space (Tsp, ∆sp) is Polish. We will ﬁrst show point (i) and separability, then state and prove two lemmas, one c concerning the properness map while the other the relation between ∆sp and ∆sp, and a α characterisation of the compact subsets of Tsp. At last, we will see how to exploit them in order to show completeness.

Proof of Theorem 2.13(i). As in the proof of Proposition 2.10, we only need to focus on the first summand in (2.10) and show it satisfies the axioms of a metric. Positivity and c symmetry clearly hold, while the triangle inequality follows by the fact that it holds for ∆sp. 0 α At last, positive definiteness can be shown by noticing that, for any ζ, ζ ∈ Tsp, the function c (r) 0 (r) r 7→ ∆sp(ζ , ζ ) is càdlàg (see [BCK17, Lemma 3.3]), and applying the same proof as in [BCK17, Proposition 3.4]. Preliminaries 13

α def (r) To show separability, given ζ ∈ Tsp and r > 0, let R = diam(M(T )). Then, the −r −R α deﬁnition of the metric implies ∆sp(ζ, ζr,R) . e ∨ e , so that any element of Tsp can be α approximated arbitrarily well by elements in Tc . Since, in view of Proposition 2.10, the c latter space is separable, and thanks to Lemma 2.15 convergence in ∆sp implies convergence in ∆sp, separability follows.

α Lemma 2.14 Let α ∈ (0, 1), {ζn = (Tn, ∗n, dn,Mn)}n∈N ⊂ Tsp and ζ = (T , ∗, d, M) be such that ∆sp(ζn, ζ) converges to 0 as n → ∞. Assume that for every r > 0 there exists a ﬁnite constant C0 = C0(r) > 0 such that

0 bζn (r) ≤ C , (2.11)

α uniformly over n ∈ N. Then, ζ ∈ Tsp and dM1(bζn , bζ ) converges to 0.

α Proof. In order to guarantee that ζ ∈ Tsp, we need to prove that M is proper. Let z ∈ T be such that M(z) ∈ Λr. Then, there exists R > 0 such that z ∈ Bd(∗,R]. Without loss of generality, we can take R > C0(r + 1) + 2, so that, in view of (2.11), for every −1 R n, all zn ∈ Mn (Λr+1) also belong to Bdn (∗n,R]. Now, let Cn be a correspondence def R (R) (R) c, Cn (R) (R) (R) between T and Tn such that εn = ∆sp (ζn , ζ ) → 0. Let zn ∈ Tn be such that (z, zn) ∈ CR. Then, |Mn(zn)| ≤ r + εn so that, thanks to (2.11),

0 d(z, ∗) ≤ bn(r + εn) + 2εn ≤ C (r + εn) + 2εn , which implies that M is proper.

It remains to prove that bζn converges to bζ .[Whi02, Theorem 12.9.3 and Corollary

12.5.1] ensure that it suﬃces to show that bζn (r) → bζ (r) for every r at which bζ is continuous. c 0 R Let r ∈ Disc(bζ ) , R > bζ (r) ∨ C (r) and Cn and εn be as above. Notice that

|b (r) − b (r)| = b (r) − sup d (∗ , z ) ≤ b (r) − sup d(∗, z) + ε . ζ ζn ζ n n n ζ n zn :Mn(zn)∈Λr zn :Mn(zn)∈Λr R R (z,zn)∈ Cn (z,zn)∈ Cn

Now, for (z, zn) ∈ Cn, if M(z) ∈ Λr−εn then Mn(zn) ∈ Λr, while if Mn(zn) ∈ Λr, then

M(z) ∈ Λr+εn which implies that

bζ (r − εn) − bζ (r) ≤ sup d(∗, z) − bζ (r) ≤ bζ (r + εn) − bζ (r) zn :Mn(zn)∈Λr (z,zn)∈ Cn from which the conclusion follows.

α c α Lemma 2.15 For any α ∈ (0, 1), the identity map from (Tc , ∆sp) to (Tsp, ∆sp) is continuous. Preliminaries 14

α c Proof. Let {ζn}n , ζ ⊂ Tc be such that ∆sp(ζn, ζ) converges to 0. In particular, dM1(bζn , bζ ) → 0 as n → ∞ so that we are left to show that ∆sp(ζn, ζ) converges to 0, which in turn can be proven by following the same strategy as in [BCK17, Proposition 3.4].

Proposition 2.16 Let α ∈ (0, 1) and A be an index set. A subset A = {ζa = (Ta, ∗a, da,Ma) : α a ∈ A} of Tsp is relatively compact if and only if for every r > 0 and ε > 0 there exist 1. a ﬁnite integer N(r; ε) such that uniformly over all a ∈ A,

(r) Nda (Ta , ε) ≤ N(r; ε) (2.12)

(r) (r) where Nda (Ta , ε) is the cardinality of the minimal ε-net in Ta with respect to the metric da, 2. a ﬁnite constant C = C(r) > 0 and δ = δ(r, ε) > 0 such that

(r) −α (r) sup kMak∞ ≤ C and sup δ ω (Ma, δ) < ε , (2.13) a∈A a∈A

3. a ﬁnite constant C0 = C0(r) > 0 such that (2.11) holds uniformly over a ∈ A.

Proof. “⇐=” Let {ζn = (Tn, ∗n, dn,Mn)}n ⊂ A be a sequence satisfying the three properties above. We want to extract a converging subsequence for {ζn}n and construct the corresponding −k limit point. For `, k ∈ N, let `k = `2 . In [ADH13, Section 5], the authors determine, for (`k) −k any n ∈ N, a subset of Tn which is a 2 -net for the latter and whose cardinality, thanks to condition 1., is ﬁnite and bounded above by some N`,k ∈ N uniformly over n ∈ N. Let n n n n S`,k = {zu : u ∈ U`,k} be such a net and S = {zu : u ∈ U}, where U`,k is the index set {u = (i, `, k) : i ≤ N`,k} and U the union of all U`,k. We also impose that for all `, k ∈ N, n n z(`,k,0) = ∗n. Notice that, by construction, S is a countable dense set of Tn for all n ∈ N. 0 n n In view of (2.13), passing at most to a subsequence, for every u, u ∈ U, limn→∞ dn(zu, zu0 ) n ˜ def and limn→∞ Mn(zu) exist. Let T = {zu : u ∈ U} be an abstract countable set and deﬁne a semimetric d and a map M˜ on it by imposing

def n n ˜ def n d(zu, zu0 ) = lim dn(zu, zu0 ) and M(zu) = lim Mn(zu) . (2.14) n→∞ n→∞

Identifying points at distance 0 in T˜ and taking the completion of the resulting space, we obtain T , which is a locally compact R-tree by [ADH13, Lemmas 5.6 and 5.7] and the proof of [CHK12, Lemma 3.5]. On the other hand, condition 2. and (2.14) guarantee that M˜ is locally little α-Hölder continuous so that we can set M to be the unique locally little α-Hölder continuous extension of M˜ to T . Preliminaries 15

In view of Lemma 2.14, it only remains to prove that ∆sp(ζn, ζ) converges to 0, where def def ζ = (T , ∗, d, M) and ∗ = z(0,k,`). def def Let r > 0 and k ∈ N be ﬁxed and set ` = d2kre and ε = 2−k. Take n big enough so that n n n −mα˜ sup |d(zu, zu0 ) − dn(zu, zu0 )| < ε , sup kM(zu) − Mn(zu)k < ε2 , (2.15) 0 u ,u ∈U`,k u∈U`,k

def where m˜ =m ¯ ∨ mn and m¯ ∈ N (resp. mn ∈ N) is the maximum integer for which 0 n n 0 there exist u , u ∈ U`,k such that d(zu, zu ) ∈ Am¯ (resp. dn(zu, zu0 ) ∈ Amn ). Set S`,k = (`k) {zu : u ∈ U`,k}, which, by [ADH13, Lemma 5.6], is a ε-net for T and deﬁne `,k def n `,k def ζn = (S`,k, ∗n, dn,Mn) and ζ = (S`,k, ∗, d, M). By the triangle inequality we have

c (r) (r) c (r) (`k) c (`k) `,k c `,k `,k ∆sp(ζ , ζn ) ≤∆sp(ζ , ζ ) + ∆sp(ζ , ζ ) + ∆sp(ζ , ζn ) 5 (2.16) c `,k (`k) c (`k) (r) X + ∆sp(ζn , ζn ) + ∆sp(ζn , ζn ) =: Ai . i=1

Thanks to Lemma 2.12 and (2.13), all the Ai’s, for i 6= 3, can be controlled in terms of quantities which are vanishing as k → ∞, so that we only need to focus on A3. Let def n Cn = {(zu, zu) : u ∈ U`,k} and, without loss of generality, assume m˜ =m ¯ . Then, for n n n n m ≤ mn, zu, zu0 , zu, zu0 such that d(zu, zu0 ), dn(zu, zu0 ) ∈ Am, the second bound in (2.15) implies −mα¯ −mα n kδz ,z 0 M − δzn,z Mnk ≤ ε2 ≤ ε2 u u u u0 while for m > mn we have

(`k) −m −mα mnα (r+1) −mn kδzu,zu0 Mk ≤ ω (M, 2 ) ≤ 2 (2 ω (M, 2 )) .

Since mn goes to infinity as n ↑ ∞, we have shown that, for any fixed r > 0, the term at the left hand side of (2.16) converges to 0, therefore also ∆sp(ζ, ζn) does. α “=⇒” Let Abe relatively compact in Tsp. Then, property 1. holds by [BBI01, Proposition 7.4.12], while property 3. holds by [Whi02, Theorems 12.9.3 and 12.12.2]. For the second property, notice that since Ais totally bounded, for any ε > 0 and r > 0 there exist n ∈ N −r and {ζk : k = 1, . . . n} such that Ais contained in the union of the balls of radius e ε/4 −r centred at ζk. Hence, if ζ ∈ B(ζk, e ε/4), then we have c (r) (r) ε ∆sp(ζ , ζk ) < 4 (2.17) (r) (r) which implies that there exists a correspondence C between T and Tk such that c, C (r) (r) (r) (r) ∆sp (ζ , ζk ) < ε/4. Since kMζ k∞ ≤ ε/2 + kMζk k∞ by the triangle inequality, (r) ε (r) sup kMζ k∞ ≤ 4 + max kMζk k∞ , (2.18) ζ∈A k=1,...,n Preliminaries 16 and the first bound in (2.13) follows. For the others, let δ > 0 and n¯ ∈ N the smallest integer such that 2−n¯ ≤ δ. Then,

nα sup 2 sup kδz,wMζ − δzk,wk Mζk k n>n¯ (z,zk),(w,wk)∈ C d(z,w),dk(zk,wk)∈An nα ε (2.19) ≤ sup 2 sup kδz,wMζ − δzk,wk Mζk k < 4 n∈N (z,zk),(w,wk)∈ C d(z,w),dk(zk,wk)∈An so that, once again, the second bound in (2.13) can be obtained by applying triangle −α (r) inequality and choosing the minimum δ for which supk≤n δ ω (Mζk , δ) < ε/2 .

Proof of Theorem 2.13(ii). For completeness, it suffices to show that, if {ζn}n is a Cauchy α sequence in Tsp then the conditions of Proposition 2.16 are satisfied. Now, if {ζn}n is (r) c Cauchy, then for every r > 0, {ζn }n is Cauchy with respect to ∆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 (2.18) and (2.19), and 3. follows by completeness of D([−1, ∞), R+) with respect to dM1.

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 2.17 Let α ∈ (0, 1) and ζn = (Tn, ∗n, dn,Mn) be a relatively compact sequence α in Tsp. Assume that for every n < m ∈ N there exists an isometric embedding ιn,m of Tn into Tm such that ιn,m(∗n) = ∗m, ιn,k = ιm,k ◦ ιn,m for n < k < m and Mm ◦ ιn,m ≡ Mn. Then, the sequence ζn converges to ζ = (T , ∗, d, M) and for every n ∈ N there exists an isometric embedding ιn of Tn into T such that ιn(∗n) = ∗, ιn = ιm ◦ ιn,m for m > n and ˜ def S M ◦ ιn ≡ Mn. Moreover, T = n ιn(Tn) is dense in T and M is the unique continuous ˜ ˜ ˜ extension of M on T , the latter being deﬁned by the relation M ◦ ιn ≡ Mn for all n.

Remark 2.18 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 α S that relative compactness in Tsp implies relative compactness of the images in n Tn/∼ with the natural equivalence relation induced by the consistency maps ιm,n. This is because the optimal correspondence between Tn and Tm may differ from the one given by ιm,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 notations, one has dis C = 2/3, while the correspondence C¯ mapping x ∈ T¯ to 2/3 − x ∈ T is also an isometric embedding but Preliminaries 17 has dis C¯ = 1/3. This shows that the condition in [EPW06, Lemma 2.7] assuming that the ζn 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 that T˜ is dense in T .)

Proof. We will limit ourselves to the case of Tn compact, the general case easily follows from the deﬁnition of the metric ∆sp. Let n < m < k and Cn,k be a correspondence between Tn and Tk. We can the obtain a correspondence Cn,m between Tn and Tm by setting

Cn,m = {(z,¯z) ∈ Tn × Tm : (z, ιm,k(¯z)) ∈ Cn,k} ∪ {(z, ιn,m(z)) : z ∈ Tn} , the second term being required to ensure Cn,m is indeed a correspondence. It is easy to c, C¯n,m c, Cn,k c c see that ∆sp (ζn, ζm) ≤ ∆sp (ζn, ζk), which then implies ∆sp(ζn, ζm) ≤ ∆sp(ζn, ζk). Since {ζn}n is relatively compact, it admits a Cauchy subsequence and in view of the last inequality the whole sequence is Cauchy. Hence, it converges to a unique ζ = (T , ∗, d, M) and there exists a sequence of correspondences Cm between T and Tm such c, Cm that ∆sp (ζm, ζ) → 0. In order to construct the isometries ιn and show they satisfy the properties stated, we ﬁrst ﬁx dense countable sets Dn ⊂ Tn with ∗n ∈ Dn and such that ιn,m Dn ⊂ Dm for every n ≤ m. We also write ιm,n for the inverse of ιn,m on its image in Dm. For n ≤ m, we then (m) choose a collection of maps ιn : Dn → T such that

(m) (m) (m) (ιn,m(z), ιn (z)) ∈ Cm ∀n ≤ m, z ∈ Dn , ιk = ιn ◦ ιk,n ∀k ≤ n ≤ m .

(m) (m) This is always possible: for every m, first fix ι1 , which determines the ιn on ι1,n(D1) for (m) all n ≤ m, then fix ι2 on D2 \ ι1,2(D1), etc. We now choose any enumeration {zk}k>0 S of D = n>0 and write nk ∈ N such that zk ∈ Dnk . This allows us to define maps ιn : Dn → T as follows. Let M1 ⊂ N be an infinite set such that the limit

def (m) ιn1 (z1) = lim ιn1 (z1) , m→∞ : m∈M1 exists. We then inductively define ιnk (zk) for every k ∈ N by the analogous formula, for some infinite set Mk ⊂ Mk−1. We claim that the maps ιn : Dn → T defined in this way are isometries satisfying the required consistency which is sufficient to complete the proof since Dn is dense in Tn.

Regarding consistency, if k ≤ ` is such that zk = ιn`,nk (z`), then

ι z = ι(m) z = ι(m) z = ι(m) z = ι z nk ( k) lim nk ( k) lim nk ( k) lim n` ( `) nm ( m) , m∈Mk m∈M` m∈M` Preliminaries 18

as required. To show that they are isometries, let k < ` be such that nk = n` = n. For every m ≥ n, we then have the bound (m) (m) |d(ιn(zk), ιn(z`)) − dn(zk, z`)| ≤ d(ιn(zk), ιn (zk)) + d(ιn(z`), ιn (z`)) (m) (m) + |d(ιn (zk), ιn (z`)) − dn(zk, z`)| .

Choosing m ∈ M`, we note that the first two terms converge to 0 as m → ∞ by the (m) definition of ιn. The last term on the other hand converges to 0 by the construction of ιn c, Cm combined with the fact that ∆sp (ζm, ζ) → 0. Similarly, one has |M(ι z ) − M z | ≤ |M(ι z ) − M(ι(m) z )| + |M(ι(m) z ) − M z | nk ( k) nk ( k) nk ( k) nk ( k) nk ( k) nk ( k) , and both terms vanish in the limit as m ∈ Mk converges to ∞. ˜ ¯ Finally, let T be given by the union of all ιn(Tn) and denote by T its closure in T . ¯ By the very definition of Gromov–Hausdorff distance, it is clear that Tn converges to T , which then implies the last part of the statement.

2.3 Characteristic 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 z0, z1 ∈ T , and s ∈ [0, 1], we write zs for the unique element of z0, z1 with d(z0, zs) = s d(z0, z1). J K Definition 2.19 For α ∈ (0, 1), we define the space of characteristic α-spatial R-trees, α α Csp ⊂ Tsp consisting of those elements ζ = (T , ∗, d, M), whose evaluation map M satisfies the following additional conditions.

(1) Monotonicity in time, i.e. for every z0, z1 ∈ T and s ∈ [0, 1] one has

Mt(zs) = (Mt(z0) − s d(z0, z1)) ∨ (Mt(z1) − (1 − s) d(z0, z1)) . (2.20) (2) Monotonicity in space, i.e. for every s < t, interval I = (a, b) and any four elements z0,¯z0, z1,¯z1 such that Mt(z0) = Mt(¯z0) = t, Mt(z1) = Mt(¯z1) = s, Mx(z0) < Mx(¯z0), and M( z0, z1 ),M( ¯z0,¯z1 ) ⊂ [s, t] × (a, b) , we have J K J K Mx(zs) ≤ Mx(¯zs) (2.21) for every s ∈ [0, 1]. Preliminaries 19

(3) For all z = (t, x) ∈ R2, M −1({t} × [x − 1, x + 1]) 6= ∅. Note that (2) also makes sense in the periodic case if we restrict to intervals (a, b) that do not wrap around the whole torus.

Remark 2.20 The ﬁrst condition guarantees that geodesics are ∨-shaped and that the “time” coordinate moves at unit speed. 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 map M is suﬃciently spread so that the vicinity of any point contains a backward characteristic one can follow. We do not impose the map M to be surjective since this is not true for the type of discrete approximation we want to consider. Clearly, the choice of 1 is completely arbitrary.

ˆ α α Remark 2.21 We denote by Csp the subspace of Tsp defined in exactly the same way but def ˆ α with ∨ replaced by ∧ in (1). Note that ζ = (T , ∗, d, M) 7→ −ζ = (T , ∗, d, −M) ∈ Csp is an isometric involution. 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 = ζ0 and ζ satisfies the conditions above then so does ζ0. In other α α words, the space Csp is a well-defined subset of Tsp. Before studying further properties of α α characteristic R-trees, we note that Csp is closed in Tsp.

α α n α Lemma 2.22 For every α ∈ (0, 1), Csp is a closed subset of Tsp. Moreover, let {ζ }n ⊂ Tsp be a sequence whose elements are monotone in both space and time. Assume that the α 2 sequence converges to ζ ∈ Tsp and that for every z = (t, x) ∈ R there exists nz ∈ N such n −1 α that for all n ≥ nz, (M ) ({t} × [x − 1, x + 1]) 6= ∅. Then ζ ∈ Csp.

n α Proof. Let {ζ }n ∈ Tsp be a sequence whose elements are monotone in both space and α (R) time and let ζ ∈ Tsp be its limit. Since ζ is monotone if and only if ζ is monotone for c n,(R) (R) every R and since ∆sp(ζ , ζ ) → 0 for every R > 0, we ﬁrst restrict ourselves to the compact case and show monotonicity of the limit. We start with monotonicity in time. Take z0, z1 ∈ T , let Cn be a sequence of c, Cn n n n correspondences such that limn ∆sp (ζ , ζ) → 0 and let zi be such that (zi , zi) ∈ Cn. For n n n any s ∈ [0, 1], we choose ¯zs ∈ T such that (zs ,¯zs ) ∈ Cn. It then follows from the tree n property and the deﬁnition of distortion that d(¯zs , zs) ≤ 2 dis Cn, so that in particular

n n n Mt(zs) = lim Mt(¯zs ) = lim Mt (zs ) . n→∞ n→∞ Preliminaries 20

n n n n Since furthermore limn→∞ d(z0 , z1 ) = d(z0, z1) and limn→∞ Mt (zi ) = Mt(zi) by the c, Cn deﬁnition of ∆sp , the claim follows. Regarding monotonicity in space, we perform the same construction, whence we get

n n n n Mx(zs) = lim Mt (zs ) ≤ lim Mt (zs0 ) = Mx(zs0 ) , n→∞ n→∞ as required. 2 For the last property, let z = (t, x) ∈ R . For any n ≥ nz, by assumption, there exists zn ∈ (M n)−1({t} × [x − 1, x + 1]), and, by (2.11), there exists R > 0 such that n n n n d (∗ , z ) ≤ R uniformly in n. Now, ∆sp(ζ , ζ) → 0, hence, for any n ≥ nz there exists R R (R) n,(R) c, Cn n,(R) (R) a correspondence Cn between T and T for which ∆sp (ζ , ζ ) → 0. Let (R) n R (R) zn ∈ T be such that (zn, z ) ∈ Cn . Notice that, the sequence {zn}n ⊂ T converges along subsequences so we can pick z ∈ T (R) to be a limit point. Then

The third property in Deﬁnition 2.19 implies that any characteristic 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. One of these open ends will play for us a distinguished role.

α Proposition 2.23 Let α ∈ (0, 1) and ζ = (T , ∗, d, M) ∈ Csp. Then, T has a unique open end † such that for every z ∈ T and every w ∈ z, †i, one has J Mt(w) = Mt(z) − d(z, w) . (2.22)

α Proof. Let ζ = (T , ∗, d, M) ∈ Csp and ﬁx z ∈ T . We want to construct an unbounded def T -ray from z such that (2.22) holds. Set z0 = z and (t0, x0) = M(z). Assume that we are given elements {zj}j≤n ⊂ T which are collinear (i.e. zj ∈ z, zn for 1 ≤ j ≤ n), such def J K that, setting (tj, xj) = M(zj), we have ti+1 − ti > 1, and such that (2.22) holds for every w ∈ z, zn . J K Preliminaries 21

As an easy consequence of (3) in Deﬁnition 2.19, there exists wn+1 ∈ T such that M(wn+1) ∈ B((tn − 2, xn), 1] and zn+1 ∈ zn, wn+1 for which necessarily tn+1 ≥ tn + 1 and such that Mt(w) = Mt(zn) − d(zn, w) forJ every wK ∈ zn, zn+1 . Then, we have J K Mt(w) = Mt(zn) − d(zn, w) = Mt(z) − d(z, zn) − d(zn, w) = Mt(z) − d(z, w) , where the last step follows from the fact that z, zn ∩ zn, zn+1 = {zn} by the induction hypothesis and property 1). This yields a (necessarilyJ K unbounded)J K T -ray from z and we set S † to be the open end it represents, i.e. z, †i = n≥0 z, zn . The uniqueness of † follows immediately from property 1. (The timeJ coordinate JMt mustK converge to −∞ along any unbounded ray which forces any two to coalesce at some point by considering any geodesic linking them.)

Thanks to the previous proposition, we can introduce, in the context of characteristic trees, the radial map. This is a map on the R-tree that allows to move along the rays. Let α ζ = (T , ∗, d, M) ∈ Csp and † the open end for which (2.22) holds. For z ∈ T we deﬁne %(z, ·) : (−∞,Mt(z)] → T as

def −1 %(z, s) = ιz (Mt(z) − s) , for s ∈ (−∞,Mt(z)] (2.23) where ιz was given in (2.1).

ˆ α Remark 2.24 If ζ ∈ Csp (see Remark 2.21), then, for z ∈ T the radial map %(z, ·) is def −1 deﬁned on [Mt(z), +∞) as %(z, s) = ιz (s − Mt(z)). 2.4 Alternative topologies Before detailing our alternative construction of the Brownian Web, we show how the 2 topology introduced above relates to that of [FINR04]. To describe the latter, let ﬁrst Rc be the completion of R2 with respect to the metric

def tanh(x1) tanh(x2) %((t1, x1), (t2, x2)) = | tanh(t1) − tanh(t2)| ∨ − 1 + |t1| 1 + |t2|

2 for all (t1, x1), (t2, x2) ∈ R . (See [NRS15, Fig. 3] for a cartoon illustrating the geometry 2 2 of the resulting compactiﬁcation of R .) A backward path π in Rc with starting time 2 σπ ∈ [−∞, ∞] is a continuous map R 3 t 7→ (t, π(t)) ∈ Rc with π(t) = π(σπ) for all t ≥ σπ. We deﬁne a metric d on the space Π of such paths by

def tanh(π1(t)) tanh(π2(t)) d(π1, π2) = | tanh(σπ ) − tanh(σπ )| ∨ sup − (2.24) 1 2 1 + |t| 1 + |t| t≤σπ1 ∧σπ2 Preliminaries 22

for all π1, π2 ∈ Π. Since (Π, d) is a Polish space, so is the space Hof compact subsets of Π endowed with the Hausdorff metric. α Let α ∈ (0, 1), ζ = (T , ∗, d, M) ∈ Csp, and %, ζ’s radial map defined according to (2.23). For z ∈ T , define

def πz(t) = Mx(%(z, t)) , for all t ≤ Mt(z). (2.25)

Since πz ∈ Π by continuity of M, we have a map

α def Csp 3 ζ 7→ K(ζ) = {πz : z ∈ T } ⊂ Π . (2.26)

α Proposition 2.25 Let α ∈ (0, 1). For every ζ ∈ Csp, K(ζ) is compact and the map α ζ 7→ K(ζ) is continuous from Csp to H. ˆ ˆ Remark 2.26 Defining Π and H in the same way, except that now π(t) = π(σπ) for all t ≤ σπ and ≤ is replaced by ≥ in the right-hand side of (2.24), we also have a map ˆ ˆ α ˆ ˆ K : Csp → H given by K(ζ) = −K(−ζ). For the proof of the previous proposition we will need the following two lemmas. For the first, define

R def 2 Π = {π ∈ Π: ∃ t ≤ σπ s.t. (t, π(t)) ∈ [−R,R] } , and, for π ∈ Π, write πR ∈ Π for the stopped path such that  π(R) if t ≥ R, R  σπR = σπ , π (t) = π(−R) if t ≤ −R,  π(t) otherwise.

Lemma 2.27 Let K be a subset of Π and, for R > 0, let KR ⊂ Π be deﬁned as

def KR = {πR : π ∈ K ∩ ΠR} . (2.27)

If for all R > 0, the family of paths in KR 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

x ≥ R ⇒ %((t, x), (t, ∞)) ≤ e−R ∀t , x ≤ −R ⇒ %((t, x), (t, −∞)) ≤ e−R ∀t , (2.28) Preliminaries 23

2 |t| ≥ R ⇒ %((t, x), (t, y)) ≤ ∀x, y . R

± ± Writing π for the path with σ ± = t and π (s) = ±∞, it follows that for every t πt t π ∈ Π and every R ≥ 1 one has d(π, πR) ≤ 2/R. If furthermore π 6∈ ΠR, then + − d(π, πσπ ) ∧ d(π, πσπ ) ≤ 2/R. It remains to note that, given ε > 0, we can cover K4/ε with ﬁnitely many balls of radius ε/2 by Arzelà–Ascoli, so that K ∩ ΠR is covered by the balls with same centres and radius ε. The complement of ΠR on the other hand can be covered by ﬁnitely many balls of radius ± −1 −1 ε centred at elements of type πt for t ∈ εZ ∩ [−4ε , 4ε ].

The next lemma highlights the fact that if two characteristic trees are close then also the respective rays must be close in a suitable sense which will be made explicit in the statement below.

α Lemma 2.28 Let α ∈ (0, 1) and ζ1, ζ2 ∈ Csp. Let r > 0 and assume there exists a (r) (r) c, C (r) (r) correspondence C between T1 and T2 such that ∆sp (ζ1 , ζ2 ) < ε for some ε > 0. Let (z1, z2) ∈ C and deﬁne a new correspondence C C as

def C C = C∪ {(%1(z1, s),%2(z2, s): − r ≤ s ≤ M1,t(z1) ∧ M2,t(z2)} (2.29)

Then, 1 (r) α dis C C + sup kM1(z) − M2(¯z)k . ε + kM1kα ε 2 (z,¯z)∈C C

Proof. Let (z1, z2) ∈ C be as in the statement and −r ≤ s ≤ M1,t(z1) ∧ M2,t(z2). Let zs ∈ T1 be such that (zs,%2(z2, s)) ∈ C. Notice that for any (w1, w2) ∈ C, by the triangle c, C (r) (r) inequality and the assumption ∆sp (ζ1 , ζ2 ) < ε, we have

|d1(%1(z1, s), w1) − d2(%2(z2, s), w2)| ≤ d1(zs,%1(z1, s)) + dis C ≤ d1(zs,%1(z1, s)) + 2ε which means that we only need to focus on d1(zs,%1(z1, s)). Now, if %(z1, s) belongs to the ray starting at zs, by (2.22), we have d1(zs,%1(z1, s)) = M1,t(zs) − M1,t(%1(z1, s)) = M1,t(zs) − s ≤ M2,t(%2(z2, s)) + ε − s ≤ ε .

Otherwise,

d1(zs,%1(z1, s)) = d1(zs, z1) − d1(z1,%1(z1, s)) ≤ d2(%2(z2, s), z2) + ε − d1(z1,%1(z1, s))

= M2,t(z2) − s + ε − M1,t(z1) + s ≤ 2ε . Preliminaries 24

Therefore, we immediately conclude that dis C C < 4ε. Concerning the bound on the evaluation maps, we have

kM1(%1(z1, s)) − M2(%2(z2, s))k ≤kM1(%1(z1, s)) − M1(zs)k (r) α + kM1(zs) − M2(%2(z2, s))k . kM1kα ε + ε where we exploited the Hölder continuity of M1, the bound on d1(zs,%1(z1, s)) and the fact that (zs,%2(z2, s)) ∈ C. The conclusion follows at once.

We are now ready for the proof of Proposition 2.25.

α Proof of Proposition 2.25. Let ζ = (T , ∗, d, M) ∈ Csp and K(ζ) be as in (2.26). By −1 definition, M (ΛR) ⊂ Bd(∗, bζ (R)] and, since T is a tree, if z ∈ Bd(∗, bζ (R)] then %(z, s) ∈ Bd(∗, bζ (R)] for all s ∈ [−R,Mt(z)]. Moreover, M is α-Hölder continuous on R Bd(∗, bζ (R)], therefore K(ζ) as defined in (2.27) consists of equicontinuous paths and Lemma 2.27 implies that K(ζ) ∈ H. n n n n n α α Let now {ζ = (T , ∗ , d ,M )}n ⊂ Csp be a sequence converging to ζ ∈ Csp with n respect to ∆sp. In view of Proposition 2.16, the evaluation maps M are uniformly proper and have uniformly bounded α-Hölder norm when restricted to balls of fixed size. Hence, n arguing as above, we see that ∪nK(ζ ) is relatively compact in Π which, thanks to [SSS10, n Lemma B.3], implies that the sequence {K(ζ )}n is relatively compact in H with respect to the Hausdorff topology. It remains to show that K(ζn) converges to K(ζ) in H. By [SSS10, Lemma B.1], it n suffices to prove that for every πz ∈ K(ζ) there exists a sequence πzn ∈ K(ζ ) such that d(πz, πzn ) → 0. Let z ∈ T and ε > 0. Pick C > 0 big enough so that z ∈ Bd(∗,C] and −1 supn bζn (ε ) ≤ C. Let n be sufficiently large so that there exists a correspondence Cn n c, Cn (C) n, (C) n between Bd(∗,C] and Bdn (∗ ,C] with ∆sp (ζ , ζ ) < ε. Let zn ∈ Bdn (∗ ,C] with n (z, zn) ∈ Cn and define πz and πzn as in (2.25). Since |Mt(z) − Mt (zn)| < ε, it follows that

| tanh(σπz ) − tanh(σπzn )| < ε.

To estimate the distance between πz(s) and πzn (s) for s ≤ σπz ∧ σπzn , we ﬁrst consider −1 n n the case s ≥ −ε . Since C is large enough so that % (zn, s) ∈ Bdn (∗ ,C], we can apply Lemma 2.28 and get

n (C) α |πz(s) − πzn (s)| = |Mx(%(z, s)) − Mx (%n(zn, s))| . ε + kMkα ε . (2.30) For s < −ε−1 we use again the last bound of (2.28). Combining these bounds, we obtain α d(πz, πzn ) . ε and 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 Preliminaries 25 to the same path and, as we will see below, we cannot expect the evaluation map of the Brownian Web to be injective because of presence special points from which multiple trajectories depart (see Section 3.3). α In the following deﬁnition, we introduce a (measurable) subset of Csp whose elements satisfy a condition, the tree condition, which allows to set two rays in the tree apart based on their image via the evaluation map.

α Deﬁnition 2.29 Let α ∈ (0, 1). We say that ζ = (T , ∗, d, M) ∈ Csp satisﬁes the tree condition if

(t) for all z1, z2 ∈ T , if M(z1) = M(z2) = (t, x) and there exists ε > 0 such that M(%(z1, s)) = M(%(z2, s)) for all s ∈ [t − ε, t], then z1 = z2.

α α We denote by Csp(t), the subset of Csp whose elements satisfy (t). Condition (t), guarantees that diﬀerent rays on the tree under study are mapped, via the evaluation map, to paths which are almost everywhere distinct. It is not diﬃcult to construct examples of characteristic trees for which (t) does not hold, while it clearly does if the evaluation map is injective. In the following Lemma, whose proof is immediate, we provide a less trivial example.

α Lemma 2.30 Let α ∈ (0, 1) and ζ = (T , ∗, d, M) ∈ Csp. If there exists a dense subtree α T of T such that (T, ∗, d, MT ) satisﬁes (t) then so does ζ. Moreover, the subset of Csp whose elements satisfy (t) is measurable with respect to the Borel σ-algebra generated by ∆sp in (2.10).

Proof. The second part of the statement is immediate while the ﬁrst follows by Lemma 2.4 point 2.

α We conclude this section by showing that on Csp(t), K is indeed injective.

α Proposition 2.31 Let α ∈ (0, 1) and Csp(t) be given as in Deﬁnition 2.29. Then, the map α K in (2.26) is injective on Csp(t).

0 α Proof. For the last part of the statement, let ζ, ζ ∈ Csp be such that (t) holds and K(ζ) ≡ K(ζ0). Then, for all z ∈ T there exists a unique element ϕ(z) ∈ T 0 such that 0 0 0 0 πz ≡ πϕ(z) and therefore not only M(z) = M (ϕ(z)), but M(%(z, s)) = M (% (z , s)) for all s. To show that ϕ is the required isomorphism, assume by contradiction that there exist 0 0 z1, z2 ∈ T such that d(z1, z2) 6= d (ϕ(z1), ϕ(z2)) and let s,¯ s¯ ≤ Mt(z1) ∧ Mt(z2) be the 0 0 0 0 ﬁrst times at which %(z1, s¯) = %(z2, s¯) and % (ϕ(z1), s¯ ) = % (ϕ(z2), s¯ ) respectively. Since The Brownian Web Tree and its dual 26

0 0 d(z1, z2) 6= d (ϕ(z1), ϕ(z2)) we have s¯ 6=s ¯ so that, without loss of generality, we can assume s¯ > s¯0. Since T is a tree, we must have

0 0 0 0 0 M (% (ϕ(z1), s))) = M(%(z1, s)) = M(%(z2, s)) = M (% (ϕ(z2), s))) ∀s ∈ [s¯ , s¯] ,

0 0 0 which, by (t), implies that % (ϕ(z1), s¯) = % (ϕ(z2), s¯). Hence, d(z1, z2) = d (ϕ(z1), ϕ(z2)) and we reach the required contradiction.

Remark 2.32 In the periodic case, let Πper be the set of backward periodic paths endowed with the metric dper whose definition is the same as in (2.24) but in the second argument of the maximum the inner metric is replaced by the periodic one, i.e. for π1, π2 ∈ Πper and t ≤ σπ1 ∧ σπ2 , we take infk∈Z |π1(t) − π2(t) + k|. Let Hper be the set of compact subsets of Πper with the Hausdorff metric. Then, Propositions 2.25 and 2.31 remain true, which α means that the map K : Csp,per → Hper defined as in (2.26) is continuous and its restriction α to Csp,per(t) is injective.

3 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 a characteristic spatial R-tree. 3.1 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 {Wk }k∈N starting at 0 and running backward in time, i.e. from def 2 0 to −∞. Fix a countable dense set D = {zk = (tk, xk) : k ∈ N} of R , with z0 = (0, 0). {π↓ } Then, build inductively a family of coalescing backward Brownian motions zk k∈N such π↓ x t that zk starts at k at time k. As in [FINR04, Section 3], one way to do so is to set π↓ t = W ↓ t π↓ t = x + W ↓ t − t τ ≤ t ≤ t τ z0 ( ) 0 ( ) and then define zk ( ) k 0 ( k) for all k k, where k x + W ↓ τ − t = π↓ τ ` < k t ≤ τ is the largest value such that k k ( k k) z` ( k) for some , and for k, π↓ t = π↓ t ` zk ( ) z` ( ). The construction guarantees that even though may not be unique, the ↓ definition of πk is. n ∈ ˜↓ =def { t, π↓ : t ≤ t , k ≤ n} ˜↓ For every N, let Tn (D) ( zk ) k and T∞(D) be the space def defined as before but in which k is free to range over all of N. Now, for n ∈ N¯ = N ∪ {∞}, The Brownian Web Tree and its dual 27

˜↓ consider the equivalence relation ∼ on Tn (D), given by

t, π↓ ∼ t, π↓ π↓ s = π↓ s ∀ s ≤ t ( zi ) ( zj ) if and only if zi ( ) zj ( ) (3.1)

↓ def ↓ ↓ ↓ ↓,D for t ≤ ti ∧ tj and i, j ≤ n. We now introduce ζn(D) = (Tn (D), ∗ , d ,Mn ), as

↓ def ˜ Tn (D) = Tn(D)/ ∼ , ↓ def ↓ ∗ = (0, π0) , (3.2) d↓ t, π↓ , s, π↓ =def t + s − 2τ ↓ π↓ , π↓ , (( zi ) ( zj )) ( ) t,s( zi zj ) M ↓,D s, π↓ = M ↓,D s, π↓ ,M ↓,D s, π↓ =def s, π↓ s , n (( zi )) ( n,t (( zi )) n,x (( zi ))) ( zi ( ))

i, j ≤ n d↓ τ ↓ π↓ , π↓ =def {r < where and, in the deﬁnition of the ancestor metric , t,s( zi zj ) sup t ∧ s : π↓ r = π↓ r } zi ( ) zj ( ) .

Remark 3.1 The construction in the periodic setting is analogous. Indeed, it suffices to ↓ replace the family of backward Brownian motions {Bk}k with a family of periodic ones de- ↓,per def ↓ per def fined via Bk = Bk mod 1, take a countable dense set D = {wk = (sk, yk) : k ∈ N} × {πper,↓} ζper,↓ per = per,↓ , ∗↓, d↓,M per,Dper,↓ of R T, build wk k∈N as before and define n (D ) (Tn (D) n ) as in (3.2).

↓ The construction above readily implies a number of properties each of the ζn(D)’s ¯ ↓ enjoys. Indeed, for every n ∈ N, ζn(D) is a spatial R-tree which is monotone in both space and time according to Deﬁnition 2.19 and, as a consequence of the fact that Brownian 1 trajectories are α-Hölder continuous for any α < 2 , it is immediate to see that, at least for n ↓ α ↓ ﬁnite, ζn(D) ∈ Tsp. In the next proposition, we will show that the sequence {ζn(D)}n is not α α only tight in Tsp for any α < 1/2, but it actually converges to a unique limit in Csp which is ↓ a characteristic spatial R-tree and can be explicitly characterised starting from ζ∞(D).

Proposition 3.2 Let D be a countable dense of R2 containing (0, 0) and, for n ∈ N¯, let ↓ def ↓ ↓ ↓ ↓,D 1 ζn(D) = (Tn (D), ∗ , d ,Mn ) be deﬁned according to (3.2). Then, for every α < 2 the ↓ α ↓ def ↓ ↓ ↓ ↓,D sequence {ζn(D}n∈N converges in Tsp to a unique limit ζ (D) = (T (D), ∗ , d ,M ), ↓ ↓ ↓,D where T (D) is the completion of T∞(D) and M is the unique continuous extension of ↓,D ↓ M∞ to all of T (D). 3 Moreover, almost surely, for any ﬁxed θ > 2 and all r > 0 there exists a constant c = c(r) > 0 depending only on r such that for all ε > 0

↓, (r) −θ Nd↓ (T (D), ε) ≤ cε (3.3) The Brownian Web Tree and its dual 28

↓, (r) where Nd↓ (T (D), ε) is deﬁned as in (2.12), i.e. it is the cardinality of the minimal ε-net in T ↓, (r)(D) with respect to d↓. At last, almost surely M ↓,D is surjective and (t) holds.

Proof. We ﬁx D once and for all for the duration of this proof and therefore suppress its ↓ dependence in the notations. By construction, the sequence {ζn}n of α-spatial R-trees is such ↓ ↓ ↓ that for every n ∈ N, ζn is embedded into ζn+1, ζn is monotone in both space and time and for 2 ↓ −1 every z = (t, x) ∈ R there exists nz such that for all n ≥ nz, (Mn) ({t} × [x − 1, x + 1]) is not empty by the density of D. Lemmas 2.17 and 2.22 guarantee that, provided α that the sequence is tight in Tsp, it converges to a unique characteristic α-spatial R-tree ζ↓ = (T ↓, ∗↓, d↓,M ↓) which satisﬁes the properties in the statement. ↓ ↓ ↓ ↓ ↓ ↓ Since every ζn 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 Tn and the local modulus of continuity of Mn, so that tightness of the sequence follows readily from Proposition 2.16. ↓, (r) Let r ≥ 1. We start by introducing an event on which T∞ is enclosed between ± two paths. Let R > r, QR be two squares of side 1 centred at (r + 1, ±(2R + 1)) and ± ± ± ± z = (t , x ) be two points in D ∩ QR, respectively. By the non-crossing property of our coalescing paths, on the event

def ↓ ↓ ± ER = { sup |π0(s)| ≤ R, sup |πz± (s) − x | ≤ R} (3.4) 0≥s≥−r t±≥s≥−r

↓ ↓, (r) any element (s, πz ) ∈ T∞ with z = (t, x) ∈ D is necessarily such that s ∈ [−r, t ∧ r] ↓ ↓ ↓ and πz− (s) < πz (s) < πz+ (s). Moreover, by the reﬂection principle, we have √ 2 c r − R (E ) ≤ C1 e 2r (3.5) P R R

c where ER is the complement of ER in Ω, and C1 is a positive constant independent of r and R. ↓, (r) Now, in order to show that, almost surely, T∞ is relatively compact, note that

↓, (r) −θ X ↓, (r) −n θ(n−1) P(Nd(T∞ , ε) ≥ Kε ∀ε ∈ (0, 1]) ≤ P(Nd(T∞ , 2 ) ≥ K2 ) . n≥1

Hence, the following lemma together with Borel–Cantelli imply (3.3) (and consequently relative compactness). The Brownian Web Tree and its dual 29

Lemma 3.3 There exists a constant C = C(r) > 0 such that

↓, (r) −3/2 C P(Nd(T∞ , ε) > Kε ) ≤ √ (3.6) K uniformly over ε ∈ (0, 1] and K ≥ 1.

˜ def Proof. Let R > r and set R = 3R + 1. For t0, t1 ∈ R, t0 > t1, we deﬁne n o def ↓ ↓ ˜ ˜ ΞR(t0, t1) = %(z, t1) : M∞,t(z) > t0 and M∞,x(%(z, t1)) ∈ [−R, R] (3.7)

↓ where % is the radial map of T∞ defined as in (2.23), and set ηR(t0, t1) to be the cardinality ↓ R of ΞR(t0, t1). By the definition of T∞, η (t0, t1) has the same distribution as the quantity ˜ ˜ ηˆ(t0, t1; −R, R) of [FINR04, Definition 2.1] (see in particular the comment below), which is almost surely finite by [FINR04, Proposition 4.1]. ε Consider the numbers Lε and the times tk given by

def l8rm ε def ε Lε = + 1, t = r − k , k = 0,...,Lε − 1, (3.8) ε k 4 where, for x ∈ R, dxe is the least integer greater than x. We now claim that, on the event ER, Lε ↓, (r) X R ε ε Nd(T∞ , ε) ≤ η (tk, tk+1) . (3.9) k=1 ↓ ↓, (r) Indeed, if (t, πz ) ∈ T∞ for some t ∈ R and z ∈ D, then by deﬁnition of the metric ↓ ↓ ↓ t ∈ [−r, r] and πz− (t) < πz (t) < πz+ (t), since we are on ER. Then, there exists ε ε k ∈ {0,...,Lε − 1} such that t ∈ [tk+1, tk] and, consequently, a unique element z ∈ ↓, (r) ΞR(tk+1, tk+2), necessarily belonging to T∞ , such that, by the coalescing property, ↓ ε ↓ ↓ %((t, πz ), tk+2) = z. Since d ((t, πz ), z) ≤ ε/2 < ε, (3.9) follows. Therefore, we obtain

Lε ↓, (r) c n X R ε ε o P(Nd(T∞ , ε) ≥ N) ≤ P ER ∪ η (tk, tk+1) > N (3.10) k=1

Lε √ 2 c −1 X R ε ε r − R LεR ≤ (E ) + N [η (t , t )] ≤ C1 e 2r + C2 √ , P R E k k+1 R εN k=1 for some constant C2 > 0, where the last inequality√ follows from [SSS17, Proposition 2.7]. Setting N = Kε−3/2, it suﬃces to choose R = K to obtain (3.6). The Brownian Web Tree and its dual 30

↓ ↓, (r) We now focus on the Hölder continuity of the map M∞T∞ . In this case, it suﬃces to show that

↓ ↓ 0 0 ↓, (r) ↓ 0 α limsup P( sup{kM∞(z) − M∞(z )k : z, z ∈ T∞ s.t. d (z, z ) ≤ ε} ≤ ε ) = 1 , (3.11) ε→0 for some α < 1/2 (then taking at most an even smaller α one deduces the little Hölder ↓ ↓, (r) property). We claim that, on the event ER, M∞T∞ is α-Hölder continuous provided ↓ def ˜ ˜ that the paths πz , z ∈ D, restricted to the box Λr,R = [−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 2.3])

def ↓ ↓ ↓ ↓ Ψ ↓ (ε) = sup{|π (s) − π (t)| : z ∈ D,M (s, π ) ∈ Λr,R , t ∈ [s, s + ε]} T∞,R,r z z ∞ z

α ↓ ↓ ↓, (r) for ε ∈ (0, 1). Indeed, on ER, assume Ψ ↓ (ε) ≤ ε /2 and let (s, π ), (t, π 0 ) ∈ T∞,R,r z z T∞ ↓ ↓ ↓ ↓ ↓ ↓ ↓ be such that d ((s, πz ), (t, πz0 )) ≤ ε. Then, necessarily, M∞(s, πz ), M∞(s, πz0 ) ∈ Λr,R and ↓ ↓ ↓ ↓ ↓ ↓ both s − τs,t(πz , πz0 ) and t − τs,t(πz , πz0 ) ≤ ε. Therefore, by the coalescing property,

↓ ↓ ↓ ↓ ↓ ↓ kM∞(s, πz ) − M∞(s, πz0 )k = |πz (s) − πz0 (t)| ∨ |t − s| ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ α ≤ |πz (τs,t(πz , πz0 )) − πz (s)| + |πz0 (τs,t(πz , πz0 )) − πz0 (t)| ∨ |t − s| ≤ ε .

The following lemma concludes the proof of (3.11).

Lemma 3.4 There exists a constant C = C(r) > 0 such that

α ε C −ε2α−1 Ψ ↓ (ε) > ≤ e (3.12) P T∞,R,r 2α+ 1 2 ε 2 uniformly over ε ∈ (0, 1].

Proof. We proceed similarly to what done in the proofs of [FINR04, Proposition B.1 and ε def α B.3] and in [SSS17, Theorem 2.3]. We introduce the grid Gr,R = {(n ε, m ε /4) : m, n ∈ ε − Z}∩Λr,R. For any z0 = (t0, x0) ∈ Gr,R, we deﬁne the rectangles Rz0 = [t0+ε/4, t0+ε/2]× α α + α α [x0 − 7ε /32, x0 − 5ε /32] and Rz0 = [t0 + ε/4, t0 − ε/2] × [x0 + 5ε /32, x0 + 7ε /32] ± ± ↓ and consider two points z0 ∈ D∩ Rz0 . Let π ± be the backward Brownian motions starting z0 ± from z0 respectively. α ↓ Assume now that Ψ ↓ (ε) > ε /2, then there exists a path π , z ∈ D such that T∞,R,r z ↓ ↓ α ↓ |πz (s) − πz (t)| > ε /2, for some s for which (πz (s), s) ∈ Λr,R and t ∈ [s − ε, s]. Then, ε ↓ α pick the closest point z0 = (t0, x0) ∈ GR,r, for which |πz (s) − x0| ≤ ε /8 and |s − t0| ≤ ε. The Brownian Web Tree and its dual 31

↓ By the coalescing property of our paths, it follows that necessarily one between π ± must z0 be such that sup |π↓ (t − h) − x | ≥ εα/32 . z± 0 0 h∈[0,2ε] 0

ε def ↓ α Let ER,r(z0) = {suph∈[0,2ε] |π ± (t0 − h) − x0| ≤ ε /32}, then, again by the reﬂection z0 principle we have

c ε c c X ε c P(ER ∪ (ER,r) ) ≤ P(ER) + P((ER,r(z0)) ) ε z0∈G √ R,r 2 2r + 1 − R Rr −ε2α−1 ≤ C e 2r+1 + C e 1 R 3 ε2α−1/2 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 r P b ↓ (r) ≥ K ≤ c√ (3.13) ζ∞ 4 K where b ↓ is the properness map given in (2.2). ζ∞

˜± Proof. Let R > 1 and consider two squares QR of side 1 centred at (r + 1, ±(r + R + 1)). ± ˜± ± ˜± Let z˜ = (t , x˜ ) be two points respectively belonging to QR ∩ D, and without loss of ˜+ ˜− ˜ ↓ ± generality, assume t = t = t. Let πz˜± be the two paths starting from z˜ . For K > 4r, we introduce the event n o ˜K def ↓ ↓ ↓ ↓ ↓ K ER,r = sup π − (s) < −r , inf π + (s) > r , τ (π + , π − ) > r − , (3.14) z˜ ˜ z˜ z˜ z˜ −r≤s≤t˜ −r≤s≤t 2

↓ ˜+ ˜− ˜K where in τ we omitted the subscript since we imposed t = t . Notice that on ER,r, for any ↓ ↓ ↓ ↓ point (s, πz ) ∈ T∞ such that M∞(s, πz ) ∈ Λr, by the coalescing property, the trajectories ↓ ↓ ↓ ↓ of both π0 and πz , after time s must be conﬁned between those of πz˜+ and πz˜− . Therefore, ↓ ↓ ↓ ↓ ↓ ↓ d ((πz , s), (π0, 0)) ≤ 2r − 2τ (πz˜+ , πz˜− ) < K, so that one has

K c b ↓ (r) ≥ K ≤ ((E˜ ) ) , P ζ∞ P R,r The Brownian Web Tree and its dual 32 independently of the choice of R. The reflection principle, combined with standard tail estimates on the first time a Brownian motion hits a specified level, yield a bound of the type √ r R2 R + r + 1 ˜K c − r P((ER,r) ) ≤ C e + C √ , (3.15) R K √ for some constant C > 0, and (3.13) follows at once, upon choosing R =def 4 K.

Since M ↓ is proper, D, which is dense in R2, is contained in M ↓,D(T ↓), surjectivity ↓ follows. At last, (t) is a direct consequence of the fact that, almost surely, it holds for ζ∞(D) by construction and Lemma 2.30.

↓,D ↓ Remark 3.6 Almost surely, the map M is continuous and proper on T∞(D). Moreover, it is bĳective on its image (endowed with the usual Euclidean topology) by construction, ↓,D ↓ ↓,D ↓ hence M : T∞(D) → M (T∞(D)) is a homeomorphism.

↓ Remark 3.7 The previous proposition remains true if instead of the sequence ζn(D) we take per,↓ per per ζn (D ), D being a countable dense set of R × T. The proof is actually simpler since it is not necessary to introduce the event in (3.4). In the periodic setting, the convergence α per,↓ per per,↓ ↓ ↓ per,↓,Dper α happens in Tsp,per, the limit ζ (D ) = (Tn (D), ∗ , d ,M ) belongs to Csp,per per,↓,Dper per,↓ and M (Tn (D)) = R × T. The next theorem introduces and uniquely characterises the law on the space of characteristic trees of the random variable which in the sequel we will refer to as the Brownian Web tree.

1 α ↓ ↓ ↓ ↓ ↓ Theorem 3.8 Let α < 2 . There exists a Csp-valued random variable ζbw = (Tbw, ∗bw, dbw,Mbw) with radial map %↓, whose law is uniquely characterised by the following properties

1. for any deterministic point w = (s, y) ∈ R2 there exists almost surely a unique point ↓ ↓ w ∈ Tbw such that Mbw(w) = w, 2 2. for any deterministic n ∈ N and w1 = (s1, y1), . . . , wn = (sn, yn) ∈ R , the joint ↓ ↓ distribution of (Mbw,x(% (wi, ·)))i=1,...,n, where w1,..., wn are the points determined in 1., is that of n coalescing backward Brownian motions starting at w1, . . . , wn, 3. for any deterministic countable dense set D such that 0 ∈ D, let w be the point determined in 1. associated to w ∈ D and ∗˜↓ that associated to 0. Deﬁne ˜↓ ˜↓ ↓ ↓ ˜ ↓,D ζ∞(D) = (T∞(D), ∗˜ , d , M∞ ) as ˜↓ def ↓ 0 T∞(D) = {% (w, t) : w = (s, y) ∈ D , t ≤ s} (3.16) ˜ ↓,D ↓ def ↓ M∞ (% (w, t)) = Mbw(% (w, t)) The Brownian Web Tree and its dual 33

↓ ˜↓ ˜↓ and d to be the ancestral metric in (3.2). Let T (D) be the completion of T∞(D) ↓ ˜ ↓,D ˜ ↓,D under d , M be the unique little α-Hölder continuous extension of M∞ and ˜↓ def ˜↓ ↓ ↓ ˜ ↓,D ˜↓ law ↓ ζ (D) = (T (D), ∗ , d , M ). Then, ζ (D) = ζbw.

↓ ↓ Moreover, almost surely, ζbw satisﬁes (3.3) for any ﬁxed θ > 3/2, Mbw is surjective and (t) holds.

Proof. Let D be a countable dense set of R2 containing 0. Thanks to Proposition 3.2, for 1 ↓ α α < 2 , ζ (D) almost surely belongs to Csp so, if we show that it satisﬁes properties 1.-3. above, then the existence part of the statement follows. In order to prove 1., let w = (s, y) ∈/ D, and consider two sequences of points ± ± ± ± zn = (tn , xn ) ∈ D for which there exist two constants c > 0 such that c c y − 1 < x− < y < x+ < y − 2 n2 n n n2 − − 3 + + 3 s < tn < s + |xn | and s < tn < s + |xn | .

↓ ± For every n ∈ , let π ± be the two backward Brownian motions starting at zn respectively. N zn ↓ ↓ ↓ ↓ ↓ Denote by τn = τ − + (π − , π + ) and Xn = π − (τn) = π + (τn) the time and spatial point tn ,tn zn zn zn zn 2 ± at which they coalesce. Deﬁne ∆n as the triangular region in R with vertices zn and − + (τn,Xn), the base being given by the segment joining zn and zn , while the sides by the ↓ ↓ paths (r, π − (r)) − , (r, π + (r)) + . zn tn ≥r≥τn zn tn ≥r≥τn In the proof of [FINR03, Proposition 3.1] the authors show that the event

def n ↓ ↓ −1/4o En = π − (s) < y < π + (s) , τn ≥ s − 1/n , |Xn − y| < n zn zn occurs inﬁnitely often. Hence, for any sequence zm = (tm, xm) ∈ D converging to w, for all n ∈ N large enough there exists mn ∈ N such that, for all m ≥ mn, zm ∈ ∆n. The coalescing property then implies that for every m1, m2 ≥ mn,

↓ ↓ ↓ d ((tm , π ), (tm , π )) ≤ (tm + tm ) − 2τn ≤ (tm − s) + (tm − s) + 2/n . 1 zm1 2 zm2 1 2 1 2

↓ In other words, for any zm = (tm, xm) ∈ D converging to w, (tm, πzm )m∈N is Cauchy in ↓ ↓ ↓,D T∞(D) therefore it converges in T (D) to a unique point w which, by continuity of M , is necessarily such that M D(w) = w. ↓ ↓ ↓ Moreover, by construction we know that % ((tm, πzm ), t) = (πzm , t) for all t ≤ tm and, ↓ since at τn the ray starting at w must have coalesced with that starting at (πzn , tn), we must ↓ ↓ ↓ have % (w, t) = % ((πzn , tn), t) for any t ≤ τn. Hence, the sequence of paths (−∞, tm] 3 ↓,D ↓ ↓ ↓,D ↓ ↓,D ↓ t 7→ Mx (% ((tm, πzm ), t)) = Mx (t, πzm ) converges to (−∞, s] 3 t 7→ Mx (% (w, t)) The Brownian Web Tree and its dual 34

↓,D ↓ ↓ in Π, where Π is given as in Appendix ??. Since (Mx (% ((tm, πzm ), t)))t≤tm is distributed ↓,D ↓ according to a backward Brownian motion starting at zm, (Mx (% (w, t))t≤s is itself distributed according to a backward Brownian motion, but starting at w. 2 For 2., let w1, . . . , wn be n deterministic points in R and w1,..., wn be the points ↓ in T (D) determined by applying 1. Thanks to the last part of the proof of 1., if zmi = ↓,D ↓ (tm , xm ) is a sequence in D converging to wi, i ∈ [n], then the paths (M (·, π ))i∈[n] i i x zmi ↓,D ↓ n converge to (Mx (% (w1, ·))i∈[n] in Π . Since the first are distributed as coalescing backward Brownian motions starting from (zm1 , . . . , zmn ), it is easy to see that the limit will be also distributed according to coalescing Brownian motions starting from (w1, . . . , wn). We now prove 3., for which we proceed as follows. Let D0 be another countable dense set in R2 containing (0, 0). We want to determine a suitable coupling of ζ↓(D) and ζ˜↓(D0) under which they are almost surely equal. We first construct ζ↓(D0) as in (3.2) and ˜↓ 0 ˜↓ 0 ↓ ↓ ˜ ↓,D0 ↓ Proposition 3.2. Then, we build ζ∞(D ) = (T∞(D ), ∗˜ , d , M∞ ) inside ζ (D) according to (3.16). Obviously ζ↓(D0) and ζ˜↓(D0) are equal in distribution, and the latter is such ˜↓ 0 ↓ ↓ ↓ ↓,D ˜↓ 0 ˜ ↓,D0 that T (D ) ⊆ T (D), ∗˜ = ∗ and M T (D ) = M . Therefore, if we are able to show that T˜↓(D0) coincides with T ↓(D), we are done. We claim that if z ∈ D and z ∈ T ↓(D) is the unique point such that M ↓,D(z) = z (which holds by 1.) then z also belongs to T˜↓(D0). Notice that if this is the case then for all z ∈ D, if z ∈ T ↓(D) is the unique point such that M ↓,D(z) = z then z ∈ T˜↓(D0). It follows that all the rays starting from these z’s are contained in T˜↓(D0) and hence also the closure of their union, which by construction is T ↓(D). We turn to the proof of the claim. ↓ ↓,D Let z ∈ Dand z ∈ T (D) be the unique point such that M (z) = z. Let wn = (sn, yn) be a sequence in D0 converging to z in R2. By 1., we know that for all n there exists a ↓ ↓,D ˜↓ 0 ↓ unique point wn in T (D) such that M (wn) = wn and since T∞(D ) ⊆ T (D) and, ˜↓ 0 by construction, there is a unique point in T∞(D ) whose image is wn, it follows that ˜↓ 0 ↓,D wn ∈ T∞(D ). Now, the map M is proper and the sequence {wn}n is bounded, therefore the sequence {wn}n is also bounded and it converges along subsequences. Fix one of these subsequences (that, with a slight abuse of notation, will still be indexed by n) and notice ↓,D that by continuity of M and uniqueness of z, we necessarily have that (wn)n converges to ↓ ˜↓ 0 z in T (D). Now, since {wn}n converges, it is Cauchy and since it is contained in T∞(D ), the limit must belong to T˜↓(D0). ↓ It remains to argue uniqueness and the properties of the limit. Let ζbw be as in the statement and, for a given countable dense set D = {zn = (tn, xn) : n ∈ N} with z0 = 0, ↓ let ζbw(D) be constructed as in (3.2) and Proposition 3.2. Notice that, thanks to the proof of 3. above, the distribution of ζ↓(D) is independent of the choice of D. Now, by 1. and ↓ ↓ def ↓ ↓ 2. define ζn = (Tn, ∗, dbw,Mbw) as follows, Tn = {% (zn, t) : Mbw(zn) = zn and t ≤ tn} ↓ ↓ so that Tn ⊂ Tbw. By construction, ζn and ζn(D) (in (3.2)) are equal in law for every n. ↓ Moreover, 3. and Lemma 2.17 guarantee that the sequence {ζn}n converges to ζbw, while The Brownian Web Tree and its dual 35

↓ ↓ ↓ the convergence of {ζn(D)}n to ζ (D) is implied by Proposition 3.2. Therefore, ζbw and ζ↓(D) are equal in law. In particular, by Proposition 3.2 also the other claimed properties ↓ of ζbw hold and the proof is concluded.

Remark 3.9 The theorem above remains true upon replacing conditions 1.-3. with 1per., 2per. and 3per., 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.

1 Definition 3.10 Let α < 2 . We define backward Brownian Web Tree and periodic backward α α ↓ ↓ ↓ ↓ ↓ Brownian Web tree, the Csp and Csp,per random variables ζbw = (Tbw, ∗bw, dbw,Mbw) per,↓ per,↓ per,↓ per,↓ per,↓ and ζbw = (Tbw , ∗bw , dbw ,Mbw ) whose distributions is uniquely characterised by properties 1.-3. in Theorem 3.8 and 1per., 2per. and 3per. in Remark 3.9. We will denote their ↓ per,↓ respective laws by Θbw(dζ) and Θbw (dζ). 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 def log Nd(T, ε) dimbox(T ) = lim (3.17) ε→0 log ε−1 when this limit exists.

↓ ↓ ↓ ↓ ↓ Corollary 3.11 Let ζbw = (Tbw, ∗bw, dbw,Mbw) be the Brownian Web tree of Deﬁnition 3.10. ↓ 3 Then, almost surely Tbw has box-covering dimension 2 .

Proof. According to (3.17), it suﬃces to determine almost sure upper and lower bounds ↓, (r) for N↓ (Tbw , ε) of the same order, for all r > 0. Now, the upper bound follows by the dbw ↓ fact that, by Theorem 3.8, almost surely Tbw satisfes (3.3). For the lower bound, it suﬃces −3/2 ↓, (r) to show that there are at least O(ε ) points in Tbw at distance ε. This is turn can be deduced using ideas similar to those in the proof of Proposition 3.2 and Lemma 3.3.

Remark 3.12 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 those of the Brownian Tree of Aldous and the scaling limit of the Uniform Spanning Tree in two dimensions. Indeed, the first has Hausdorff dimension 2 [Ald91a], while the other 5/8 [BCK17] and the Hausdorff dimension is always greater than or equal to the box-counting one (see e.g. [Edg98, Chapter 1]). The Brownian Web Tree and its dual 36

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.

↓ per,↓ Corollary 3.13 Let ζbw and ζbw be the backward and backward periodic Brownian Web ↓ trees of Theorem 3.8 and Remark 3.9, and K be the map deﬁned in (2.26). Then, K(ζbw) ↓ is a backward Brownian Web according to [FINR04, Theorem 2.1] and K(ζbw,per) is a backward cylindric Brownian Web according to [CMT19, Theorem 2.3].

↓ ↓ Proof. To prove the statement it suﬃces to verify that K(ζbw) and K(ζbw,per) satisfy (o), (i) and (ii) in [FINR04, Theorem 2.1] and [CMT19, Theorem 2.3], respectively. This is in turn an immediate consequence of the deﬁnition of K and properties 1.-3. in Theorem 3.8 and 1per.-3per. in Remark 3.9. 3.2 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 characteristic spatial R-trees is Θbw.

α Theorem 3.14 Let α ∈ (0, 1) and {ζn}n be a tight sequence of random variables in Csp with laws Θn and assume that the following holds. 1 k 2 i (I) For any k ∈ N and (deterministic) z , . . . , z ∈ R there exist sequences zn ∈ Tn, i = i i i 1, . . . , k such that limn→∞ Mn(zn) = zn almost surely and such that (Mn(%n(zn, ·)))i converges in law to k coalescing backward Brownian motions. (II) For every h > 0

1 −1 ε→0 limsup sup Θn #{%(w, t − h) : w ∈ M (It,x,ε)} ≥ 3 −→ 0 (3.18) ε n→∞ (t,x)∈R2

def where It,x,ε = {t} × (x − ε, x + ε). ↓ Then Θn converges weakly to Θbw. Remark 3.15 In view of Corollary 3.13, the Brownian Web tree and the Brownian Web are strictly connected so that it is not surprising that the convergence criterion stated above is extremely similar to [FINR04, Theorem 2.2]. As a matter of fact, requiring the sequence ζn to be made of characteristic trees, allows us to talk about paths, while the fact that we are dealing with trees enforces the non-crossing condition. That said, even though Proposition 2.25 guarantees continuity of the map assigning to any characteristic tree a α compact subset of Π, the inverse map is not continuous, even when restricted to Csp(t), so α that we cannot infer convergence in Csp from [FINR04]. The Brownian Web Tree and its dual 37

Proof. Let K be the map defined in (2.26). Notice at first that (I) implies that {K(ζn)}n satisfies [FINR04, Theorem 2.2(I1)] and that, since

−1 −1 #{%n(w, t−h) : w ∈ Mn (It,x,ε)} ≥ #{Mn(%n(w, t−h)) : w ∈ Mn (It,x,ε)} , (3.19)

(II) implies [FINR04, Theorem 2.2(B2)], so that {K(ζn)}n converges in law to the backward Brownian Web. Since the sequence ζn is tight by assumption, it converges along some α subsequence. Let ζ be a limit point, % its radial map and denote by Θ its law on Csp. Since K is continuous by Proposition 2.25, K(ζn) converges to K(ζ), which by the above is a α backward Brownian Web. Since K is injective on Csp(t) by Proposition 2.25, it remains to show that ζ satisfies (t). If (t) fails then, with positive probability, one can find t, x, ε, h ∈ Q with h > 0 such that the inequality in (3.19) is strict, so the proof is complete if we show that for any fixed values this happens with probability 0. Fix t, x, ε, h ∈ Q with h > 0 and, for N ∈ N, let

N def jε N x = x + , z = (t, xj) , for j = −N,...,N. j N j

N N N For i = 1 − N,...,N, let yi denote the mid-point of of the interval (xi−1, xi ). By our N assumptions we know that almost surely, there exist unique points zj ∈ T such that N N N N M(zj ) = zj for all j, so that, in particular #{%(zj , t − h) : |j| ≤ N} = #{M(%(zj , t − h)) : |j| ≤ N}. Hence,

−1 −1 Θ #{%(w, t − h) : w ∈ M (It,x,ε)} > #{M(%(w, t − h)) : w ∈ M (It,x,ε)} −1 N ≤ lim Θ #{%(w, t − h) : w ∈ M (It,x,ε)} > #{%(zj , t − h) : |j| ≤ N} . N→∞

−1 N Moreover, by space monotonicity, #{%(w, t − h) : w ∈ M (It,x,ε)} > #{%(zj , t − h) : −1 |j| ≤ N} can only happen if there exists i such that #{%(w, t−h) : w ∈ M (I N ε )} ≥ t,yi , N 3. In other words,

−1 N Θ(#{%(w, t − h) : w ∈ M (It,x,ε)} > #{%(zj , t − h) : |j| ≤ N}) N X −1 ≤ Θ #{%(w, t − h) : w ∈ M (I N ε )} ≥ 3 t,yi , N i=1−N −1 N sup Θ #{%(w, t − h) : w ∈ M (I ε )} ≥ 3 . t,y, N (t,y)∈R2 −1 N limsup sup Θ #{%(w, t − h) : w ∈ M (I ε )} ≥ 3 . n n t,y, N , n→∞ (t,y)∈R2 which converges to 0 as N → ∞ by (3.18), and the claim follows. The Brownian Web Tree and its dual 38

3.3 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 R2 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.

α ˆ α ↓↑ def Theorem 3.16 Let α < 1/2. There exists a Csp × Csp-valued random variable ζbw = ↓ ↑ • • • • • • (ζbw, ζbw), ζbw = (Tbw, ∗bw, dbw,Mbw), ∈ {↓, ↑}, whose law is uniquely characterised by the following properties

↑ def ↑ ↑ ↑ ↑ ↓ (i) Both −ζbw = (Tbw, ∗bw, dbw, −Mbw) and ζbw are distributed as the backward Brownian Web tree in Deﬁnition 3.10.

↓ ↓ ↑ ↑ ↓ ↓ ↓ (ii) Almost surely, for any z ∈ Tbw and z ∈ Tbw, the paths Mbw(% (z , ·)) and ↑ ↑ ↑ ↓ ↑ ↓ Mbw(% (z , ·)) do not cross, where % (resp. % ) is the radial map of ζbw (resp. ↑ ζbw). ↓↑ α ˆ α ↑ ↓ Moreover, almost surely ζbw ∈ Csp(t) × Csp(t) and ζbw is determined by ζbw and vice-versa. ↓ ˆ ↑ Finally, (K(ζbw), K(ζbw)) is distributed according to the double Brownian Web of [SSS17, Theorem 2.1].

Remark 3.17 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 notations and conventions of Section 2.4. α ˆ α Notice at first that, by Theorem 3.8, any Csp × Csp-valued random variable for which (i) α ˆ α holds, almost surely belongs to Csp(t) × Csp(t). Now, let (W ↓,W ↑) be the H × Hˆ-valued random variable constructed in [SSS17, Theorem 2.1] and K the map in (2.26). Since W ↓ is distributed as the backward Brownian ↓ law ↓ ↑ law ↓ law ↓ ˆ ↓ Web, by Corollary 3.13, W = K(ζbw) and W = −W = −K(ζbw) = K(−ζbw), where the first equality is due to [SSS17, Theorem 2.1(a)] and the last is a consequence of ↓ ↑ α ˆ ˆ α Remark 2.26. Therefore, (W ,W ) ∈ K(Csp(t)) × K(Csp(t)) almost surely so that, by α ˆ α Proposition 2.25 and Remark 2.26, there exists a unique couple (ζW ↓ , ζW ↑ ) ∈ Csp(t) × Csp(t) ˆ ↓ ↑ such that (K(ζW ↓ ), K(ζW ↑ )) = (W ,W ). By Proposition 3.2 and Theorem 3.8 we also ↓ α law ↓ law ↓ have ζbw ∈ Csp(t) almost surely so that, since K(ζW ↓ ) = K(ζbw) and K(−ζW ↑ ) = K(ζbw), (ζW ↓ , ζW ↑ ) satisfies (i). The definition of the map K in (2.25) and (2.26) combined The Brownian Web Tree and its dual 39

with [SSS17, Theorem 2.1(b)] ensures that (ii) holds for (ζW ↓ , ζW ↑ ). The fact that ζW ↑ is ↓ determined by ζW ↓ is a direct consequence of the fact that this is known to be true for W and W ↑ and that K is invertible on Cα(t). 0 α ˆ α We argue uniqueness. Let (ζ, ζ ) be another random variable in Csp × Csp which satisﬁes (i) and (ii). Now, (t) holds for both ζ and ζ0, while (i), (ii) and (2.26) ensure that (K(ζ), Kˆ (ζ0)) satisﬁes [SSS17, Theorem 2.1 (a)-(b)]. Hence, the conclusion follows by the uniqueness part of [SSS17, Theorem 2.1] and Proposition 2.25.

Remark 3.18 In the periodic setting Theorem 3.16 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 2.32 instead of Theorem 3.8 and Proposition 2.25.

1 Deﬁnition 3.19 Let α < 2 . We deﬁne the double Brownian Web tree and double periodic α ˆ α α ˆ α ↓↑ def Brownian Web tree as the Csp × Csp and Csp,per × Csp,per-valued random variables ζbw = ↓ ↑ per,↓↑ def per,↓ per,↑ (ζbw, ζbw) and ζbw = (ζbw , ζbw ) given by Theorem 3.16 and Remark 3.18. We will refer ↑ per,↑ to ζbw and ζbw as the forward (or dual) and forward periodic Brownian Web trees. ↓↑ ↓ ↑ per,↓↑ ↓ ↑ ↓ We denote their laws by Θbw(d(ζ ×ζ )) and Θbw (d(ζ ×ζ )), with marginals Θbw(dζ), ↑ per,↓ per,↑ Θbw(dζ) and Θbw (dζ), Θbw (dζ) respectively.

Remark 3.20 The proof of Theorem 3.16 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 and backward standard Brownian motion, turning it into a perfectly coalescing / reflecting system (see [STW00, Section 3.1.1]) and follow the same procedure as in (3.2), Proposition 3.2 and Theorem 3.8. As a first consequence of the duality the Brownian Web tree enjoys we show that each ↑ ↓ of the R-trees Tbw and Tbw has a unique open end with unbounded rays. This end should be thought of as the point at (±)∞ where all the Brownian motions coalesce. As we will see in Proposition 3.25, the periodic Brownian Web tree, instead, has (exactly) two open ends with unbounded rays which are connected by a unique bi-infinite edge.

↑ ↓ Proposition 3.21 Let ζbw and ζbw be respectively the forward and backward Brownian Web ↑ ↓ trees. Then, almost surely, the R-trees Tbw and Tbw have precisely one open end with unbounded rays, which we denote by †↑ and †↓ respectively.

↑ Proof. We prove the result for Tbw, the other being analogous by duality. Notice that the statement follows if we show that for every r > 0 almost surely there exists a compact The Brownian Web Tree and its dual 40

↑ ↑, (r) 0 c 0 K ⊂ Tbw with Tbw ⊂ K, such that for all z, z ∈ K the path connecting z and z does not ↑, (r) intersect Tbw . Thanks to the double Brownian Web tree we are able to exhibit an explicit compact set for which the latter claim holds. Let r > 0 be fixed, D be a countable dense set 2 ↓ ↓ in R containing 0 and recall that, with probability one, ζbw = ζ (D). Ñ Using the same notations and conventions as in the proof of Proposition 3.2, let ER,r def ↓ ↓ ↓ def ↓ ↓ be defined according to (3.14). Set τ = τ (πz˜+ , πz˜− ), X = πz˜+ (τn) = πz˜− (τn) and let 2 ± ∆N be the triangular region of R with vertices z˜ and (τ, X), base given by the segment + − ↓ ↓ joining z˜ and z˜ , and sides formed by the paths (s, π − (s))t˜−≥s≥τ , (s, π + (s)) + . On z˜ zn tn ≥s≥τ Ñ ↑ def ↑ −1 ER,r, ∆N is compact and the properness of Mbw guarantees that so is KN = (Mbw) (∆N ). By point (ii) in Theorem 3.16 paths in the forward and backward Web trees do not cross, ↑, (r) c therefore Tbw ⊂ KN and the path connecting any two points in KN cannot intersect ↑, (r) Tbw . Hence, it remains to argue that there is an almost surely finite N for which the ↓ Ñ realisation of ζbw belongs to ER,r. This in turn is a direct consequence of (3.15) and a standard application of Borel–Cantelli.

• −1 per,• −1 We are now interested in deriving properties of the inverse maps (Mbw) and (Mbw ) , • • for ∈ {↑, ↓}, and how these are related to the degrees of points in the R-trees Tbw and per,• Tbw . We begin with the following proposition, which is a translation in the language of the present paper of [FINR06, Proposition 3.10].

↓↑ ↑ ↓ ↓↑,per ↑,per ↓,per Proposition 3.22 Let ζbw = (ζbw, ζbw) and ζbw = (ζbw , ζbw ) be the double and double periodic Brownian Web trees. Then, almost surely for every point z = (t, x) ∈ R2

↓ −1 |(Mbw) (z)| ↑ −1 X ↓ |(Mbw) (z)| − 1 = (deg(zi ) − 1) (3.20) i=1

↓ ↓ −1 • −1 where {zi }i are the points in (Mbw) (z) and |(Mbw) (z)| denotes the cardinality of • −1 (Mbw) (z). The relation (3.20) 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. 2 ↓ −1 b We claim that for all z = (t, z) ∈ R , |(Mbw) (z)| = mout(z) and the right-hand side b b b of (3.20) coincides with min(z), where mout(z) and min(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). ↓ ↓ −1 ↓ ↓ Indeed, for every z ∈ (Mbw) (z), denoting by % the radial map associated to ζbw, we ↓ ↓ ↓ ↓ have that (−∞, t] 3 s 7→ Mbw,x(% (z , s)) is a path from z. On the other hand, deg(z ) − 1 The Brownian Web Tree and its dual 41 corresponds to the number of rays in the tree which coalesce at or reach z. Notice that, since ↓ almost surely ζbw satisﬁes (t), the image of the rays coalescing or reaching z as well as that ↓ −1 of the rays from points in (Mbw) (z) are distinct so that the claim follows. ↓ ˆ ↑ Now, by Theorem 3.16 (K(ζbw), K(ζbw)) is distributed as the double Brownian Web ↓↑ α ˆ α α and almost surely ζbw ∈ Csp(t) × Csp(t). Since moreover the restriction of K to Csp(t) is bĳective on its image thanks to Proposition 2.25, (3.20) is a direct consequence of [FINR06, Proposition 3.10].

We are now ready to classify the diﬀerent points in R2 or in R × T based on the meaning they have for the (periodic) Brownian Web tree (and its dual) as we constructed it.

↓↑ ↑ ↓ • Deﬁnition 3.23 Let ζbw = (ζbw, ζbw) be the double Brownian Web tree. For ∈ {↑, ↓}, the 2 • 2 type of a point z ∈ R for ζbw is (i, j) ∈ N , where

• −1 |(Mbw) (z)| X • • −1 i = (deg(zi) − 1) and j = |(Mbw) (z)| . i=1

• • −1 • −1 ↑ ↓ Above, {zi : i ∈ {1,..., |(Mbw) (z)|}} = (Mbw) (z). We deﬁne Si,j (resp. Si,j) as the subset of R2 containing all points of type (i, j) for the forward (resp. backward) Brownian per,↓↑ per,↑ per,↓ Web tree. For the periodic Brownian Web ζbw = (ζbw , ζbw ), the deﬁnition is the same as above and the set of all of points in R × T of type (i, j) for the backward (resp. forward) per,↓ per,↑ periodic Brownian Web tree, will be denoted by Si,j (resp. Si,j ).

↓ ↓,per Theorem 3.24 For the backward and backward periodic Brownian Web trees ζbw and ζbw , almost surely, every z ∈ R2 (resp. R × T) is of one of the following types, all of which occur: (0, 1), (1, 1), (2, 1), (0, 2), (1, 2) and (0, 3). Moreover, almost surely, for every t ∈ R ↓ 2 ↓ - S0,1 has full Lebesgue measure on R and S0,1 ∩ {t} × R has full Lebesgue measure in {t} × R, ↓ ↓ ↓ ↓ - S1,1 and S0,2 have Hausdorﬀ dimension 3/2 while S1,1 ∩ {t} × R and S0,2 ∩ {t} × R are both countable and dense in {t} × R, ↓ ↑ ↑ - S1,2 has Hausdorﬀ dimension 1, S2,1 and S0,3 are countable and dense while ↓ ↓ ↓ S2,1 ∩ {t} × R, S1,2 ∩ {t} × R and S0,3 ∩ {t} × R have each cardinality at most 1. ↓ ↓ ↓ For deterministic times t, S2,1 ∩ {t} × R, S1,2 ∩ {t} × R and S0,3 ∩ {t} × R are almost surely empty. Upon reversing all arrows, the properties above hold for the forward and forward periodic Brownian Web trees.

Proof. Arguing as in the proof of Proposition 3.22, the statement follows immediately by [FINR06, Theorems 3.11, 3.13 and 3.14]. The Brownian Web Tree and its dual 42

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 we show that the periodic Brownian Web tree possesses a unique bi-inﬁnite path connecting its two open ends with unbounded rays.

per,• per,• per,• per,• per,• • Proposition 3.25 For ∈ {↓, ↑}, let ζbw = (Tbw , ∗bw , dbw ,Mbw ) be the periodic backward and forward Brownian Web trees of Deﬁnition 3.19. Then, almost surely, each per,↓ per,↑ Tbw and Tbw has exactly two open ends with unbounded rays and a unique bi-inﬁnite edge connecting them.

per,↓ per,↑ Proof. Since Tbw and Tbw are periodic characteristic trees, we already know they have one open end with unbounded rays, and this is the one for which (2.20) holds (for the ↓ ↑ ↓ ↑ forward periodic Web see Remark 2.21). Denote them by † and † and let %per and %per be the radial maps introduced in (2.23) and Remark 2.24, respectively. Similarly to (3.7), for t0, t1 ∈ R, t0 < t1, we introduce Ξ↑ (t , t ) =def {%↑ (z, t ) : z ∈ per,↑ M per,↑(z) ≤ t } T 0 1 per 1 Tbw and t,bw 0 Ξ↓ (t , t ) =def {%↓ (z, t ) : z ∈ per,↓ M per,↓(z) ≥ t } T 1 0 per 0 Tbw and t,bw 1 η↑ (t , t ) η↓ (t , t ) Ξ↑ (t , t ) Ξ↓ (t , t ) and set T 0 1 and T 1 0 to be the cardinality of T 0 1 and T 1 0 respectively. We inductively deﬁne the sequence of stopping times τ =def inf{t > 0 : η↑ (0, t) = 1} 1 T τ =def inf{t > τ : η↑ (τ , t) = 1} . k k−1 T k−1 These stopping times coincide (in distribution) with those in the proof of [CMT19, Theorem 3.1], where it is further showed that almost surely limk→∞ τk = +∞. Now, by deﬁnition, for every k ≥ 1, there must exist a point zk−1 ∈ T × {τk−1} such per,↑ −1 per,↑ −1 that |(Mbw ) (zk−1)| ≥ 2 and the distance of (at least) two elements in (Mbw ) (zk−1) is 2(τk − τk−1). By (3.20) and Theorem 3.24, it follows that there exists exactly one point per,↓ −1 (Mbw ) (zk−1) whose degree is greater or equal to 2. Denote it by zk. Then the map ↓ per,↓ β : R → Tbw given by

( ↓ def % (zk, s) , for s ∈ (τk−1, τk] β↓ s = per ( ) ↓ %per(z0, s) for s < 0. is not only well-deﬁned by Theorem 3.16(ii) but also uniquely deﬁned since so is the choice ↓ of the point zk. The map β shows that there are exactly two open ends with unbounded ↓ per,↓ rays, and β (R) is the unique linear subtree of Tbw satisfying the properties in [Chi01, Lemma 3.7(i)]. The Discrete Web Tree and convergence 43

4 The Discrete Web Tree and convergence

In this section, we introduce the discrete web and its dual and show that the couple converges to the Double Brownian Web Tree of Deﬁnition 3.19.

4.1 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 3.16(ii). Let δ ∈ (0, 1] and (Ω, A, Pδ) be a standard probability space supporting four Poisson L R L R L R ↓ def random measures, µγ , µγ , µˆγ and µˆγ . The ﬁrst two, µγ and µγ , live on Dδ = 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 1 γ = γ(δ) =def . (4.1) 2δ2

↑ def The others live on Dδ = R × δ(Z + 1/2), and are obtained from the formers by setting, for ↑ every measurable A ⊂ Dδ

L def R R def L µˆγ (A) = µγ (A − δ/2) and µˆγ (A) = µγ (A + δ/2) . (4.2)

Here, A ± δ/2 is the translate of A in the spatial direction, i.e. A ± δ/2 =def {z ± (0, δ/2) : z ∈ A}. • From now on, we will adopt the convention of writing z ∈ µγ, • ∈ {R,L}, if z is an • L R L R event of the given realisation of µγ. We represent the Poisson points of µγ , µγ , µˆγ and µˆγ L R with arrows as follows. If z ∈ µγ (resp. µγ ) then we draw an arrow from z to z − δ (resp. L R z + δ), and similarly for µˆγ and µˆγ , as shown in Figure 2. We also deﬁne

T L R µγ = {z − δ : z ∈ µγ } ∪ {z + δ : z ∈ µγ } , (4.3)

T T and similarly for µˆγ . (Here, T stands for ‘tip’ since µγ denotes the collection of all tips of arrows.) ↓,δ Let us now introduce two families of random walks. We deﬁne {πz (s)}s≤t, for ↓ z = (t, y) ∈ Dδ, as the random walk going backwards in time, “following the arrows” L R ↑ ↑,δ determined by µγ and µγ , and, for z = (t, y) ∈ Dδ, {πz (s)}s≥t as the forward random walk which follows those of µˆL and µˆR, as shown in Figure 2. (By convention, if z is the ↓,δ ↑,δ start of an arrow, then πz and πz start by going downwards / upwards.) These are almost The Discrete Web Tree and convergence 44

y t

0 yˆ

Figure 2: Graphical representation of the realisation of the Poisson processes µL and µR, L R ↓ ↑ and their dual µˆ and µˆ which respectively live on Dδ and Dδ. The red and blue lines ↓,δ ↑,δ illustrate the restrictions of the backward and forward paths π(t,y) and π(0,yˆ) to the interval [0, t].

L R ↓ surely well-deﬁned µγ and µγ are disjoint with probability one and, for all z ∈ Dδ and ↑ ↓,δ ↑,δ zˆ ∈ Dδ, πz is càglàd (or càdlàg if we run time backwards from +∞ to −∞), while πzˆ is ↓,δ ↑,δ càdlàg. Moreover, {πz }z and {πzˆ }zˆ are coalescing families of paths starting from every ↓ ↑ point in Dδ and Dδ respectively, which do not cross.

L R Deﬁnition 4.1 Let δ ∈ (0, 1], γ as in (4.1), µγ and µγ be two independent Poisson random ↓ L R ↓,δ measures on δ of intensity γλ, µˆ and µˆ be given as in (4.2) and {πz } ↓ and D z∈Dδ ↑,δ {πzˆ } ↑ be the families of coalescing random walks introduced above. We deﬁne the zˆ∈Dδ ↓↑ def ↓ ↑ Double Discrete Web Tree as the couple ζδ = (ζδ , ζδ ), in which

↓ def ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ - ζδ = (Tδ , ∗δ, dδ,Mδ ) is given by setting Tδ = Dδ, ∗δ = (0, 0), Mδ the canonical inclusion, and

↓ 0 0 ↓,δ ↓,δ dδ(z, z¯) = t + t − 2 sup{s ≤ t ∧ t : πz (s) = πz¯ (s)} . (4.4)

↑ def ↑ ↑ ↑ ↑ ↑ - ζδ = (Tδ , ∗δ, dδ,Mδ ) is built similarly, but with ∗δ = (0, δ/2) and the supremum in 0 ↑,δ ↑,δ (4.4) replaced by inf{s ≥ t ∨ t : πz (s) = πz¯ (s)}. ↓ Notice that neither the Discrete Web Tree ζδ nor its dual are characteristic spatial R-trees. Indeed, even though they satisfy the conditions of Deﬁnition 2.19 and Remark 2.21 the evaluation maps are discontinuous. 2 ↓ To circumvent this technical issue, we introduce two connected subsets of R , Sδ and ↑ • • • Sδ , obtained by interpolating the Poisson points of µγ and µˆγ, ∈ {L, R}, and which will • represent the image of modiﬁed evaluation maps. Fix a realisation of µγ, • ∈ {L, R}, The Discrete Web Tree and convergence 45

T ↓ ↓ and consider µγ as in (4.3). Given z = (t, x) ∈ Dδ, we then define z as follows. Let ↓ R L T ↓ ↓ t = sup{s < t : (s, x) ∈ µγ ∪ µγ ∪ µγ } and set z = (t , x + cδ), where c = 1 if ↓ R ↓ L ↓ (t , x) ∈ µ , c = −1 if (t , x) ∈ µ , and c = 0 otherwise. We then define Sδ as the union of ↓ R L T ↓ all closed line segments joining z to z with z ∈ µγ ∪ µγ ∪ µγ . Given z = (t, x) ∈ Dδ and ↑ ↑ ↑ R L T ˜ ↓ ↓ setting z = (t , x) with t = inf{s ≥ t : (s, x) ∈ µγ ∪ µγ ∪ µγ }, we write Mδ (z) ∈ Sδ for the unique element on the line segment joining z↑ to z↓ with the same time coordinate as ↑ z. The set Sδ is defined similarly, but with time reversed. It is immediate to see that, almost ↓ ↑ surely, the sets Sδ and Sδ are well-defined and connected. With the previous construction at hand we are ready for the following definition.

Definition 4.2 In the same setting as Definition 4.1, we define the Interpolated Double ↓↑ def ↓ ↑ • def • • • • ˜ ˜ ˜ ˜ ˜ • Discrete Web Tree as the couple ζδ = (ζδ , ζδ ) in which ζδ = (Tδ , ∗δ, dδ, Mδ), ∈ {↑, ↓}, • • • • ˜ • and (Tδ , ∗δ, dδ) coincides with that of ζδ, while the evaluation map Mδ is defined as just described.

Proposition 4.3 For any δ ∈ (0, 1] and α ∈ (0, 1), almost surely the interpolated double ˜↓↑ α ˆ α ˜ • Discrete Web tree ζδ in Deﬁnition 4.2 belongs to Csp × Csp and the evaluation maps Mδ, • • ∈ {↑, ↓} are bĳective on Sδ. Moreover, it satisﬁes the following two properties

˜↑ law ˜↓ ˜↑ def ↑ ↑ ↑ ˜ ↑ (iδ) −ζδ + δ/2 = ζδ where −ζδ + δ/2 = (Tδ , ∗δ, dδ, −Mδ + δ/2)

↓ ↓ ↑ ↑ (iiδ) almost surely, for every z ∈ Tδ and z ∈ Tδ there exists c ∈ {+1, −1} such that ˜ ↑ ↑ ˜ ↓ ↓ for all Mδ,t(z ) ≤ s1 < s2 ≤ Mδ,t(z )

Y ˜ ↑ ↑ ↑ ˜ ↓ ↓ ↓ (Mδ,x(% (z , si)) − Mδ,x(% (z , si)) + cδ) ≥ 0 i=1,2

At last, almost surely, for • ∈ {↑, ↓}

˜ • • sup kMδ(z) − Mδ(z)k ≤ δ (4.5) • z∈Tδ

• where Mδ are the evaluation maps of the double Discrete Web Tree in Deﬁnition 4.1.

Proof. The proof of the statement is an immediate consequence of basic properties of ↓ ↑ Poisson random measures and the deﬁnition of the sets Sδ and Sδ . The Discrete Web Tree and convergence 46

4.2 Tightness and convergence ˜↓↑ We are now ready to show that the family {ζδ }δ is tight.

↓↑ α ˆ α Proposition 4.4 Let α ∈ (0, 1) and, for δ ∈ (0, 1], let Θδ be the law on Csp × Csp of the ˜↓↑ ˜↓ ˜↑ • Interpolated Double Discrete Web Tree ζδ = (ζδ , ζδ ) of Deﬁnition 4.2 and denote by Θδ • ↓↑ • ˜ 1 α ˆ α with ∈ {↑, ↓} the law of ζδ. Then, for any α ∈ (0, 2 ) the family Θδ is tight in Csp × Csp. 3 Furthermore, for any θ > 2 and r > 0, the following holds

↓ (r) −θ lim liminf Θδ ∀ ε ∈ (0, 1] , Nd(T , ε) ≤ Kε = 1 . (4.6) K↑∞ δ↓0

˜↑ law ˜↓ Proof. Let us point out that since by Proposition 4.3(iδ), −ζδ + δ/2 = ζδ , it suﬃces to ↓ α show that the family {Θδ}δ is tight in Csp. In view of Proposition 2.16 and Lemma 2.22 we need to prove that for every r > 0, the limits (4.6), and

↓ (r) α lim liminf Θδ sup{kM(z) − M(w)k : z, w ∈ T , d(z, w) ≤ ε} ≤ ε = 1 , (4.7) ε↓0 δ↓0 ↓ lim liminf Θδ(bζ (r) ≤ K) = 1 , (4.8) K↑∞ δ↓0 hold. These can be shown by following the same strategy and estimates as in the proof of Proposition 3.2, so that below we will adopt the notations and conventions therein. 2 Notice at ﬁrst that, for any z = (t, x) in a countable dense set D of R , if {zδ}δ is such ↓ that for all δ ∈ (0, 1], zδ ∈ Dδ and {zδ}δ converges to z, then, by Donsker’s invariance π↓,δ principle, the backward random walk zδ deﬁned above converges in law to a backward ↓ Brownian motion πz started at z. ± ± ± δ Let {zδ }δ ⊂ QR ∩ (Dδ) be sequences converging to z . Denoting by ER, the event ER ± ± in (3.4), but in which z is replaced by zδ , we see that the previous observation implies

δ liminf Pδ(ER) = P(ER) (4.9) δ↓0 so that (3.5) holds. Moreover, the analog of [FINR04, Proposition 4.1] (see also [SSS17, pg 46]) for random walks ensures that for all R, r > 0 and a < b

R R limsup Eδ[η (a, b)] ≤ E[η (a, b)] (4.10) δ↓0

R where η (a, b) is the cardinality of ΞR(a, b) given in (3.7) and Eδ is the expectation with respect to Pδ. Thanks to (4.9) and (4.10), we can argue as in Lemma 3.3 and obtain that The Discrete Web Tree and convergence 47 there exists a constant C = C(r) > 0 independent of δ such that for all K > 0

(r) −θ C limsup Pδ(Nd(T , ε) > Kε ) ≤ √ δ↓0 K so that by Borel-Cantelli (4.6) follows. ↓ As in Proposition 3.2, the uniform local Hölder continuity of the evaluation maps Mδ can be reduced to properties of the paths π↓,δ. For ﬁxed R and r, let

δ def ↓,δ ↓,δ ↓ ↓,δ Ψ (ε) = sup{|πz (s) − πz (t)| : z ∈ Dδ,Mδ (s, πz ) ∈ Λr,R, t ∈ [s − ε, s]}

δ α ↓,δ ↓,δ ↓, (r) If Ψ (ε) ≤ ε /4 for every ε ≥ 4δ, then for every (s, πz ), (t, πz0 ) ∈ Tδ such that ↓ ↓,δ ↓,δ dδ((s, πz ), (t, πz0 )) ≤ ε, we have α ↓ ↓,δ ↓ ↓,δ ε α |M (s, π ) − M (t, π 0 )| ≤ 2δ + ≤ ε . δ,x z δ,x z 2 where we exploited the triangle inequality and (4.5). Therefore, (4.7) follows at once if

↓ δ α limsup liminf Θδ Ψ (ε) ≤ ε /4 = 1 . (4.11) ε→0 δ→0 This in turn follows from the same arguments as in the proof of Lemma 3.4, together with +,δ −,δ ± ↓ + the fact that if {z0 }δ and {z0 }δ are sequences of points in Rz0 ∩ (Dδ) converging to z0 − and z0 ∈ D respectively, then

↓,δ α ε liminf δ sup |π ±,δ (t0 − h) − x0| ≤ ε /32 = (E (z0)) . P z P R,r δ↓0 h∈[0,2ε] 0

Finally, (4.8) can be proved by proceeding as in Lemma 3.5 and adapting the deﬁnition ˜K of the event ER,r in (3.14) as done for ER above.

In the following theorem we show that the Interpolated Double Discrete Web tree converges in law to the Double Brownian Web Tree.

↓↑ α ˆ α Theorem 4.5 Let α ∈ (0, 1/2) and, for δ ∈ (0, 1], Θδ be the law on Csp × Csp of the ˜↓↑ ↓↑ Interpolated Double Discrete Web Tree ζδ in Deﬁnition 4.2. Then, as δ ↓ 0, Θδ converges ↓↑ α ˆ α to Θbw weakly on Csp × Csp.

↓↑ ↓ ↑ α ˆ α Proof. Thanks to Proposition 4.4, the sequence {ζδ = (ζδ , ζδ )}δ is tight in Csp × Csp. ↓↑ ↓ ↓ Moreover, Proposition 4.3 (iδ) and (iiδ) imply that any limit point ζ = (ζ , ζ ) must be law such that −ζ↑ = ζ↓ and the non-crossing property holds. In view of Theorem 3.16, the The Discrete Web Tree and convergence 48

↓ ↓ statement then follows once we show that ζδ → ζbw in law as δ → 0. To do so, we will apply Theorem 3.14, for which we need to verify the validity of (I) and (II). 1 k 2 i i i i Clearly, for any z , . . . , z ∈ R , if {zδ}δ is such that zδ ∈ Dδ and zδ → z as δ → 0, ↓,δ then (π i (·))i converges in law to a family of coalescing Brownian motions starting at zδ z1, . . . , zk. Since furthermore (4.5) holds, (I) follows. ↓ For (II), our construction implies that, for any t, x ∈ R, h, ε > 0, #{%δ(w, t − h) w ∈ ˜ ↓ −1 law ↓ (Mδ ) (It,x,ε)} =η ˆδ(t, t + h; x − ε, x + ε), where %δ is the radial map of ζδ and ηˆδ was deﬁned in [FINR04, Deﬁnition 2.1]3. For the latter, the statement was shown in the proof of [FINR04, Theorem 6.1].

References

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