The Brownian Web as a random \mathbb R-tree
TThe Brownian Web as a random R -tree February 9, 2021
G. Cannizzaro , and M. Hairer Imperial College London, SW7 2AZ, UK University of Warwick, CV4 7AL, UKEmail: [email protected], [email protected]
Abstract
Motivated by [CH21], we provide an alternative characterisation of the BrownianWeb [TW98, FINR04], i.e. a family of coalescing Brownian motions starting fromevery 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) andrecover some which were determined in earlier works, such as duality, special points andconvergence 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] whichmakes it Polish and could be of independent interest. Contents R -trees in a nutshell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Spatial R -trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Characteristic R -trees and the radial map . . . . . . . . . . . . . . . . . . . . . . . 182.4 Alternative topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 a r X i v : . [ m a t h . P R ] F e b ntroduction The Brownian Web is a random object that can be heuristically described as a collectionof coalescing Brownian motions starting from every space-time point in R , a typicalrealisation of which is displayed in Figure 1. Its study originated in the PhD thesis ofArratia [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 Tóthand Werner in [TW98], the authors provided the first thorough construction, determinedits main properties and used it to introduce the so-called true self-repelling motion. Adifferent characterisation was subsequently given in [FINR04] where, by means of a newtopology, a sufficient condition for the convergence of families of coalescing random walkswas 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 ofcoalescing Brownian motions on the Sierpinski gasket were built. For an account of furtherdevelopments of the Brownian Web and the diverse contexts in which it emerged, we referto the review paper [SSS17].In most (if not all) of these works, the Brownian Web is viewed as a random (compact)collection of paths (cid:87) in a suitable space. The present paper aims at highlighting yet anotherof its characterising features, namely its coalescence or tree structure , clearly apparent inFigure 1. The main motivation comes from the companion paper [CH21] in which such astructure is used to construct and study the Brownian Castle, a stochastic process whosevalue at a given point equals that of a Brownian motion indexed by a Brownian web. Sincethe characteristics of the Brownian Castle are given by backward (coalescing) Brownianmotions, in what follows we will (mainly) consider the case in which paths in (cid:87) runbackward 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 trajectoriesof paths in (cid:87) as a metric space with metric given by the intrinsic distance , namely thedistance between ( t i , π ↓ i ) ∈ R × (cid:87) , i = 1 , , is the minimal time it takes to go from ( t , π ↓ ( t )) to ( t , π ↓ ( t )) moving along the trajectories of π ↓ and π ↓ at unit speed. Insteadof working directly on R , it turns out to be more convenient to first encode the points of thetrajectories in an abstract space and then suitably embed the latter into R . More precisely,we will construct a (random) quadruplet ζ ↓ bw def = ( T ↓ bw , ∗ ↓ bw , d ↓ bw , M ↓ bw ) whose elements wenow describe. The first three form a pointed R -tree, which means that ( T ↓ bw , d ↓ bw ) is antroduction Figure 1: A typical realisation of the Brownian web: coalescing Brownian trajectoriesemanate from every point of the plane simultaneously. Trajectories are coloured accordingto their creation time / age.ntroduction connected metric space with no loops and ∗ ↓ bw is an element of T ↓ bw (see Definition 2.1).Morally, points in T ↓ bw are of the form ( t, π ↓ ) for π ↓ ∈ (cid:87) and t ≤ σ π ↓ , where σ π ↓ is thestarting time of π ↓ , and d ↓ bw is the ancestral distance defined as d ↓ bw (( t, π ↓ ) , ( s, ˜ π ↓ )) def = ( t + s ) − τ ↓ t,s ( π ↓ , ˜ π ↓ ) , for all ( t, π ↓ ) , ( s, ˜ π ↓ ) ∈ T ↓ bw , τ ↓ t,s being the first time at which π ↓ and ˜ π ↓ meet, i.e. τ ↓ t,s ( π ↓ , ˜ π ↓ ) def = sup { r < t ∧ s : π ↓ ( r ) = ˜ π ↓ ( r ) } . M ↓ bw is the evaluation map which embeds T ↓ bw into R , and is given by T ↓ bw (cid:51) ( t, π ↓ ) (cid:55)→ ( t, π ↓ ( t )) ∈ R .The main task of the present paper is to identify a “good” space in which the quadruplet ζ ↓ bw 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 alreadybeen considered in the literature. Similar to [DLG05, BCK17], we endow the space ofspatial R -trees T α sp , α ∈ ( , ) , with a Gromov–Hausdorff-type topology (for an introductionin the case of general metric and length spaces we refer to the monograph [BBI01], andto [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 evaluationmap M . More precisely, M is required to be both locally little α -Hölder continuous, i.e. lim ε → sup z ∈ K sup d ( z , z (cid:48) ) ≤ ε (cid:107) M ( z ) − M ( z (cid:48) ) (cid:107) /d ( z , z (cid:48) ) α = 0 for every compact K , and proper,i.e. preimages of compacts are compact. As pointed out in [BCK17, Remark 3.2], withoutthe first property the space T α sp would not be Polish (the space T sp of [DLG05, BCK17]lacks completeness). The second property prevents the existence of sequences of pointsthat are order distance apart in T but whose M -image is arbitrarily close in R . In theweighted metric ∆ sp in (2.10), necessary to consider the case of unbounded R -trees (as is T ↓ bw ) and which is in essence that of [BCK17], this is allowed.The topology introduced in Section 2.2 and briefly described above guarantees that asequence { ζ n = ( T n , ∗ n , d n , M n ) } n converges to ζ = ( T , ∗ , d, M ) in T α sp provided that,morally, the metrics d n converge to d , which in the present context means that couplesof distinct paths which are close also coalesce approximately at the same time, and theevaluation maps M n converge to M in Hölder sense, which in turn ensures control over thesup-norm distance of paths and is somewhat similar in spirit to that of [FINR04]. Noticethat it is not always possible to assign to a family of paths an R -tree (trivially, considerthe case of paths which are not coalescing) and, conversely, there is no canonical way toassociate a collection of paths to a generic spatial R -tree. However, we identify a subsetof T α sp for which this is the case and prove that, as suggested by the heuristic descriptionabove, our topology is strictly finer than that in [FINR04] (see Proposition 2.25). Whilethis 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),ntroduction convergence statements in T α sp do not follow from those previously established. This isremedied in Section 3.2, where a convergence criterion to ζ ↓ bw is derived.As shown in [CH21], there are two main advantages of the characterisation of theBrownian Web outlined above. First, it allows to preserve information on the intrinsicmetric on the set of trajectories, which in turn is at the basis of the properties and the proofof the universality statement for the Brownian Castle in [CH21, Theorem 1.4]. Moreover,the R -tree structure automatically endows T ↓ bw with a σ -finite length measure (see [Eva08,Section 4.5.3]) that can be useful in many contexts and, for example, could provide a moredirect 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 relationsto 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 tosuperprocesses [DLG05], the Brownian Map and random plane quadrangulations [LG13,Mie13], the scaling limit of the Uniform Spanning Tree and SLE [Sch00, BCK17], just tomention a few. As expected, the law of the Brownian Web as a random R -tree is differentfrom those alluded to above (see Corollary 3.11 and Remark 3.12) but it would be interestingto explore further this new interpretation in light of the aforementioned works to see if extraproperties of the Brownian Web itself or the Brownian Castle of [CH21] can be derived. In Section 2, we collect all the preliminary results and constructions concerning R -treeswhich will be needed throughout the paper. After recalling their basic definitions andgeometric features, we introduce, for α ∈ ( , ) , the spaces T α sp , of spatial R -trees, and its“characteristic” subset C α sp . We define a metric which makes them Polish and identify anecessary and sufficient condition for a subset to be compact. In Section 2.4, we comparethe metric above and that of [FINR04], and show that the former is stronger than the latter.Section 3 is devoted to the Brownian Web and its periodic version [CMT19]. At first(Section 3.1), we provide a characterisation of its law on C α sp and determine some of itsproperties 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 introduceits dual and the so-called “special points”.At last, in Section 4 we first show how to make sense of the graphical construction ofa system of coalescing backward random walks (and its dual) in the present context andconclude by deriving its scaling limit.ntroduction We will denote by | · | e the usual Euclidean norm on R d , d ≥ , and adopt the short-handnotations | x | def = | x | e and (cid:107) x (cid:107) def = | x | e for x ∈ R and R respectively. Let ( T , d ) be a metricspace. We define the Hausdorff distance d H between two non-empty subsets A, B of T as d H ( A, B ) def = inf { ε : A ε ⊂ B and B ε ⊂ A } where A ε is the ε -fattening of A , i.e. A ε = { 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 → R d be a map. For r > and α ∈ ( , ) , we define the sup -norm and α -Höldernorm of M restricted to a ball of radius r as (cid:107) M (cid:107) ( r ) ∞ def = sup z ∈ B d ( ∗ ,r ] | M ( z ) | e , (cid:107) M (cid:107) ( r ) α def = sup z , w ∈ B d ( ∗ ,r ] d ( z , w ) ≤ | M ( z ) − M ( w ) | e d ( z , w ) α . where B d ( ∗ , r ] ⊂ T is the closed ball of radius r centred at ∗ , and, for δ > , the modulusof continuity as ω ( r ) ( M, δ ) def = sup z , w ∈ B d ( ∗ ,r ] d ( z , w ) ≤ δ | M ( z ) − M ( w ) | e . (1.1)In case T is compact, in all the quantities above, the suprema are taken over the whole space T and the dependence on r of the notation will be suppressed. Moreover, we say that afunction M is (locally) little α -Hölder continuous if for all r > , lim δ → δ − α ω ( r ) ( M, δ ) = 0 .Let I ⊆ R be an interval and D ( I, R + ) be the space of càdàg functions on I withvalues in R + def = [ , ∞ ) , endowed with the M1 topology that we now introduce. For f ∈ D ( I, (cid:88) ) , denote by Disc( f ) the set of discontinuities of f and by Γ f its completedgraph, i.e. the graph of f to which all the vertical segments joining the points of discontinuityare added. Order Γ f by saying that ( x , t ) ≤ ( x , t ) if either t < t or t = t and | f ( t − ) − x | ≤ | f ( t − ) − x | . Let P f be the set of all parametric representations of Γ f , whichis the set of all non-decreasing (with respect to the order on Γ f ) functions σ f : I → Γ f .Then, if I is bounded, we set ˆ d cM1 ( f, g ) def = 1 ∨ inf σ f ,σ g (cid:107) σ f − σ g (cid:107) and d cM1 ( f, g ) to be the topologically equivalent metric with respect to which D ( I, R + ) iscomplete (see [Whi02, Section 8] for more details). If instead I = [ − , ∞ ) , we define d M1 ( f, g ) def = (cid:90) ∞ e − t (1 ∧ d cM1 ( f ( t ) , g ( t ) ) ) d t (1.2)where f ( t ) is the restriction of f to [ − , t ] .At last, we will write a (cid:46) b if there exists a constant C > such that a ≤ Cb and a ≈ b if a (cid:46) b and b (cid:46) a .reliminaries 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. GCgratefully acknowledges financial support via the EPSRC grant EP/S012524/1. MH gratefullyacknowledges financial support from the Leverhulme trust via a Leadership Award, the ERC via theconsolidator grant 615897:CRITICAL, and the Royal Society via a research professorship.
In this section, we gather all the results on R -trees which will be necessary in the sequel. Atfirst, we summarise some of their geometric properties. 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 z , z ∈ T
1. there is a unique isometric map f z , z : [ , d ( z , z )] → T such that f z , z ( ) = z and f z , z ( d ( z , z )) = z ,2. for every continuous injective map q : [ , ] → T such that q ( ) = z and q ( ) = z ,one has q ([ , ]) = f z , z ([ , d ( z , z )]) . 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 willconstruct it, the natural root should be thought of as a “point at infinity” where all the pathsstarting from every point meet.For an R -tree ( T , d ) and any two points z , z ∈ T , we define the segment joining z and z as the range of the map f z , z and denote it by (cid:74) z , z (cid:75) . Notice that for every three points z , z , z ∈ T there exists a unique point w ∈ T such that (cid:74) z , z (cid:75) ∩ (cid:74) z , z (cid:75) = (cid:74) w , z (cid:75) . Wecall w , the projection of z onto (cid:74) z , z (cid:75) , or equivalently the projection of z onto (cid:74) z , z (cid:75) . Definition 2.3 [CMSP08, Definition 2] Let ( T , d ) be an R -tree and r > . A segment (cid:74) z , z (cid:75) ⊂ T has r -finite branching if the set of points w ∈ (cid:74) z , z (cid:75) which are the projectionof some point z ∈ T onto (cid:74) z , z (cid:75) with d ( z , w ) ≥ r is finite. An R -tree T is said to have r -finite branching if every segment of T does.reliminaries Given z ∈ T , the number of connected components of T \ { z } is the degree of z , deg( z ) in short. A point of degree is an endpoint , of degree , an edge point and if the degree is or higher, a branch point . The following lemma is taken from [CMSP08, Lemma 3]. Lemma 2.4
Let ( T , d ) be an R -tree, z ∈ T and let S be a dense subset of T . Thefollowing statements hold:1. If z ∈ T is not an endpoint for T , then there exists w ∈ S such that z ∈ (cid:74) z , w (cid:75) .2. If S is a subtree of T , then every point of T \ S is an endpoint for T . Notice that the connected components of T \ { z } are themselves R -trees, i.e. subtreesof T , and they are called directions at z . Definition 2.5 [CMSP08, Definition 1] Let ( T , d ) be an R -tree, z ∈ T and { T i : i ∈ I } ,where I is an index set, the set of directions at z . For r > , we say that T i has length ≥ r if there exists w ∈ T i such that d ( z , w ) ≥ r . The R -tree T is r -locally finite at z if the setof 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, wefollow [Chi01, Chapter 2.3]. A subset L of an R -tree T is linear if it is isometric to aninterval of R , which could be either bounded or unbounded. For z ∈ T , we write L z for anarbitrary segment of T having z as an endpoint and we say that L z is a T -ray from z ifit is maximal for inclusion. We also say that rays L z and L z (cid:48) are equivalent if there exists w ∈ T such that L z ∩ L z (cid:48) 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 closedends , 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 † whichwe will denote by (cid:74) z , †(cid:105) . Moreover we say that † is an open end with (un-)bounded rays iffor every z ∈ T , the map ι z : (cid:74) z , †(cid:105) → R + given by ι z ( w ) = d ( z , w ) , w ∈ (cid:74) z , †(cid:105) (2.1)is (un-)bounded.We conclude this section by showing how the geometric structure of an R -tree isintertwined with its metric properties. The following statements summarise (or are easyconsequences 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 andonly if every open end has unbounded rays. Let ( T , d ) be a locally compact complete R -tree, then reliminaries (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 > ,(c) T has countably many branch points and every point has at most countable degree. R -trees Now that we discussed geometric features of R -trees we are ready to study the metricproperties of the space of all R -trees. We will focus on a specific subset of it, namely thespace of α -spatial R -trees. Definition 2.7
Let α ∈ ( , ) . The space of pointed α -spatial R -trees T α sp is the set ofequivalence 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 proper map from T to R ,and we identify ζ and ζ (cid:48) if there exists a bijective isometry ϕ : T → T (cid:48) such that ϕ ( ∗ ) = ∗ (cid:48) and M (cid:48) ◦ ϕ ≡ M , in short (with a slight abuse of notation) ϕ ( ζ ) = ζ (cid:48) . Remark 2.8
We will also consider situations in which the map M is R × T -valued,where T def = R / Z is the torus of size endowed with the usual periodic metric d ( x, y ) = inf k ∈ Z | x − y + k | . Whenever this is the case, we will add an extra subscript “ per ”, standingfor periodic , to the space under consideration, which will anyway always be a subset of T α sp , per . In what follows, it is immediate to see how the definitions, statements and proofsneed to be adapted in order to hold not only for the generic space (cid:83) but also for its periodiccounterpart (cid:83) per .For any spatial tree ζ = ( T , ∗ , d, M ) , we introduce the properness map b ζ : R → R + ,that “quantifies” the properness of M . For r < , b ζ ( r ) = 0 , while for r ≥ we set b ζ ( r ) def = sup z : M ( z ) ∈ Λ r d ( ∗ , z ) , (2.2)where Λ r def = [ − r, r ] ⊂ R and in the periodic case Λ r = Λ per r def = [ − r, r ] × T . Lemma 2.9
Let α ∈ ( , ) . For all ζ = ( T , ∗ , d, M ) ∈ T α sp the properness map isnon-decreasing and càdlàg. Namely such that lim ε → sup z ∈ K sup d ( z , z (cid:48) ) ≤ ε (cid:107) M ( z ) − M ( z (cid:48) ) (cid:107) /d ( z , z (cid:48) ) α = 0 for every compact K andthe preimage of every compact set is compact. reliminaries Proof.
The function b ζ is non-decreasing by construction, so that at every point r > itadmits left and right limits. To show it is càdlàg, it suffices to prove that lim s ↓ r b ζ ( s ) = b ζ ( r ) .Notice that, for every s > , since T is locally compact, M is continuous and Λ s is closed, there exists z s ∈ M − ( Λ s ) such that b ζ ( s ) = d ( ∗ , z s ) . Let s n be a sequencedecreasing to r and, without loss of generality, assume M ( z s n ) ∈ Λ s n \ Λ r . Since M is proper, M − ( Λ s ) is compact so that { z s n } n ⊂ M − ( Λ s ) admits a converging subsequence. Let ¯ z be a limit point. By construction, d ( ∗ , z s n ) ≥ d ( ∗ , z r ) for all n , therefore d ( ∗ , ¯ z ) ≥ d ( ∗ , z r ) .But M ( ¯ z ) ∈ Λ r since M is continuous, so d ( ∗ , ¯ z ) ≤ d ( ∗ , z r ) as claimed.To turn T α sp into a Polish space, we proceed similarly to [BCK17], but we introduce twoconditions taking into account the Hölder regularity and the properness of M respectively.Recall first that a correspondence (cid:67) between two metric spaces ( T , d ) , ( T (cid:48) , d (cid:48) ) is a subsetof T × T (cid:48) such that for all z ∈ T there exists at least one z (cid:48) ∈ T (cid:48) for which ( z , z (cid:48) ) ∈ (cid:67) and vice versa. The distortion of a correspondence (cid:67) is given by dis (cid:67) def = sup {| d ( z , w ) − d (cid:48) ( z (cid:48) , w (cid:48) ) | : ( z , z (cid:48) ) , ( w , w (cid:48) ) ∈ (cid:67) } ,and allows to give an alternative characterisation of the Gromov-Hausdorff metric (see [Eva08,Theorem 4.11], for the case of compact metric spaces).Now, let T α c be the subset of T α sp consisting of compact R -trees. Let ζ = ( T , ∗ , d, M ) and ζ (cid:48) = ( T (cid:48) , ∗ (cid:48) , d (cid:48) , M (cid:48) ) be elements of T α c and (cid:67) be a correspondence between T and T (cid:48) . We set ∆ c , (cid:67) sp ( ζ, ζ (cid:48) ) def = 12 dis (cid:67) + sup ( z , z (cid:48) ) ∈ (cid:67) (cid:107) M ( z ) − M (cid:48) ( z (cid:48) ) (cid:107) + sup n ∈ N nα sup ( z , z (cid:48) ) , ( w , w (cid:48) ) ∈ (cid:67) d ( z , w ) ,d (cid:48) ( z (cid:48) , w (cid:48) ) ∈ (cid:65) n (cid:107) δ z , w M − δ z (cid:48) , w (cid:48) M (cid:48) (cid:107) (2.3)where (cid:65) n def = ( − n , − ( n − ) ] for n ∈ N , and δ z , w M def = M ( z ) − M ( w ) . In the above, weadopt the convention that if there exists no pair of couples ( z , z (cid:48) ) , ( w , w (cid:48) ) ∈ (cid:67) such that d ( z , w ) ∈ (cid:65) n , then the increment of M is removed and the supremum is taken among all z (cid:48) , w (cid:48) such that d (cid:48) ( z (cid:48) , w (cid:48) ) ∈ (cid:65) n and vice versa . We can now define ∆ csp ( ζ, ζ (cid:48) ) def = ∆ csp ( ζ, ζ (cid:48) ) + d M1 ( b ζ , b ζ (cid:48) ) (2.4)where d M1 is the metric on the space of càdlàg functions given in (1.2) and ∆ csp ( ζ, ζ (cid:48) ) def = inf (cid:67) : ( ∗ , ∗ (cid:48) ) ∈ (cid:67) ∆ c , (cid:67) sp ( ζ, ζ (cid:48) ) . (2.5)In view of Lemma 2.9, the metric above is well-defined. If 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. reliminaries For α ∈ ( , ) , ( T α c , ∆ csp ) is a complete separable metric space.Proof. Notice that the definition of ∆ csp in (2.4) comprises two independent summands. Theterm d cM1 , which involves M , is a pseudometric by [Whi02, Theorem 12.3.1 and Sections12.8 and 12.9], while the other term is shown to be a pseudometric by following the samesteps as in [CHK12, Lemma 2.1] (see also [BCK17, Proposition 3.1] and [ADH13, Theorem2.5(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(see also, for completeness, Lemma 2.15) below, any element ζ = ( T , ∗ , d, M ) ∈ T α c canbe approximated in T α sp by ζ ε = ( T ε , ∗ , d, M ) , where T ε ⊂ T is a finite ε -net of T . Wecan turn T ε into an R -tree by setting ˜ T ε def = (cid:83) z , w ∈ T ε (cid:74) z , w (cid:75) , where for any z , w ∈ T ε thepoints in the edge (cid:74) z , w (cid:75) are those of T and set M ε to be the restriction of M to ˜ T ε . Then,clearly, the ∆ csp -distance between ζ ε and ˜ ζ ε = ( ˜ T ε , ∗ , d, ˜ M ε ) ∈ T α c is going to as ε goes to . Therefore, a countable dense set in T α c can be obtained by considering the set of R -treeswith finitely many endpoints and edge lengths, in which the distances between endpointsare rationals, endowed with maps M which are Q -valued at the end- and branch pointsand linearly interpolated in between. Remark 2.11
As pointed out in [BCK17, Remark 3.2], without the Hölder condition inthe definition of ∆ csp , the space of spatial pointed R -trees would not be complete while, ifwe did not assume the function M to be little Hölder continuous it would lack separability.
Lemma 2.12
Let α ∈ ( , ) , ζ = ( T , ∗ , d, M ) ∈ T α c . Let δ > , T ⊂ T be such that ∗ ∈ T and the Hausdorff distance between T and T is bounded above by δ and define ¯ ζ = ( T, ∗ , d, M (cid:22) T ) . Then ∆ csp ( ζ, ¯ ζ ) (cid:46) ( δ ) − α ω ( M, δ ) (2.6) Proof.
Let (cid:67) δ be the correspondence given by { ( z , z (cid:48) ) ∈ T × T : d ( z , z (cid:48) ) ≤ δ } . Then, forevery ( z , z (cid:48) ) , ( w , w (cid:48) ) ∈ (cid:67) δ , we have | d ( z , w ) − d ( z (cid:48) , w (cid:48) ) | ≤ δ , (cid:107) M ( z ) − M ( z (cid:48) ) (cid:107) ≤ (cid:107) M (cid:107) α d ( z , z (cid:48) ) α ≤ (cid:107) M (cid:107) α δ α so that the first two summands in (2.3) are controlled. For the other, let m T be the largestinteger for which there exist z (cid:48) , w (cid:48) ∈ T such that d ( z (cid:48) , w (cid:48) ) ∈ (cid:65) m T . By assumption, T is a δ -net for T and T is a length space, therefore the minimal distance between points in T has to be less than δ , which implies that − m T ≤ δ . For m > m T and z , w ∈ T are suchthat d ( z , w ) ∈ (cid:65) m , we have (cid:107) δ z , w M (cid:107) ≤ ω ( M, − m ) ≤ − mα ( ( δ ) − α ω ( M, δ ) ) . (2.7)reliminaries If m ≤ m T , let ( z , z (cid:48) ) , ( w , w (cid:48) ) ∈ (cid:67) δ be such that d ( z , w ) , d ( z (cid:48) , w (cid:48) ) ∈ (cid:65) m . Now, in case m satisfies − m ≤ δ , then we apply the triangle inequality to the norm of δ z , w M − δ z (cid:48) , w (cid:48) M and bound each of (cid:107) δ z , w M (cid:107) and (cid:107) δ z (cid:48) , w (cid:48) M (cid:107) as in (2.7). At last, in case − m > δ we get (cid:107) δ z , w M − δ z (cid:48) , w (cid:48) M (cid:107) ≤ (cid:107) δ z , z (cid:48) M (cid:107) + (cid:107) δ w , w (cid:48) M (cid:107) ≤ ω ( M, δ ) (2.8) (cid:46) − mα ( δ − α ω ( M, δ )) which implies the result.We are now ready to introduce a metric on the whole of T α sp . For ζ = ( T , ∗ , d, M ) ∈ T α sp and any r > , let ζ ( r ) def = ( T ( r ) , ∗ , d, M ( r ) ) (2.9)where T ( r ) def = B d ( ∗ , r ] is the closed ball of radius r in T and M ( r ) is the restriction of M to T ( r ) . We define ∆ sp as the function on T α sp × T α sp given by ∆ sp ( ζ, ζ (cid:48) ) def = (cid:90) + ∞ e − r (cid:104) ∧ ∆ csp ( ζ ( r ) , ζ (cid:48) ( r ) ) (cid:105) d r + d M1 ( b ζ , b ζ (cid:48) ) =: ∆ sp ( ζ, ζ (cid:48) ) + d M1 ( b ζ , b ζ (cid:48) ) . (2.10)for all ζ, ζ (cid:48) ∈ T α sp . Since T ( r ) and T (cid:48) ( r ) are R -trees and, in view of Theorem 2.6(a),compact, ζ ( r ) , ζ (cid:48) ( r ) ∈ T α c so that the first summand in (2.10) is well-defined. Theorem 2.13
For any α ∈ ( , ) ,(i) ∆ sp is a metric on T α sp (ii) the space ( T α sp , ∆ sp ) is Polish. We will first show point (i) and separability, then state and prove two lemmas, oneconcerning the properness map while the other the relation between ∆ csp and ∆ sp , and acharacterisation of the compact subsets of T α sp . At last, we will see how to exploit them inorder to show completeness. Proof of Theorem 2.13(i).
As in the proof of Proposition 2.10, we only need to focus onthe first summand in (2.10) and show it satisfies the axioms of a metric. Positivity andsymmetry clearly hold, while the triangle inequality follows by the fact that it holds for ∆ csp .At last, positive definiteness can be shown by noticing that, for any ζ, ζ (cid:48) ∈ T α sp , the function r (cid:55)→ ∆ csp ( ζ ( r ) , ζ (cid:48) ( r ) ) is càdlàg (see [BCK17, Lemma 3.3]), and applying the same proof asin [BCK17, Proposition 3.4].reliminaries To show separability, given ζ ∈ T α sp and r > , let R def = diam( M ( T ( r ) )) . Then, thedefinition of the metric implies ∆ sp ( ζ, ζ r,R ) (cid:46) e − r ∨ e − R , so that any element of T α sp can beapproximated arbitrarily well by elements in T α c . Since, in view of Proposition 2.10, thelatter space is separable, and thanks to Lemma 2.15 convergence in ∆ csp implies convergencein ∆ sp , separability follows. Lemma 2.14
Let α ∈ ( , ) , { ζ n = ( T n , ∗ n , d n , M n ) } n ∈ N ⊂ T α sp and ζ = ( T , ∗ , d, M ) besuch that ∆ sp ( ζ n , ζ ) converges to as n → ∞ . Assume that for every r > there exists afinite constant C (cid:48) = C (cid:48) ( r ) > such that b ζ n ( r ) ≤ C (cid:48) , (2.11) uniformly over n ∈ N . Then, ζ ∈ T α sp and d M1 ( b ζ n , b ζ ) converges to .Proof. In order to guarantee that ζ ∈ T α sp , we need to prove that M is proper. Let z ∈ T be such that M ( z ) ∈ Λ r . Then, there exists R > such that z ∈ B d ( ∗ , R ] . Withoutloss of generality, we can take R > C (cid:48) ( r + 1 ) + 2 , so that, in view of (2.11), for every n , all z n ∈ M − n ( Λ r +1 ) also belong to B d n ( ∗ n , R ] . Now, let (cid:67) Rn be a correspondencebetween T ( R ) and T ( R ) n such that ε n def = ∆ c , (cid:67) Rn sp ( ζ ( R ) n , ζ ( R ) ) → . Let z n ∈ T ( R ) n be such that ( z , z n ) ∈ (cid:67) R . Then, | M n ( z n ) | ≤ r + ε n so that, thanks to (2.11), d ( z , ∗ ) ≤ b n ( r + ε n ) + 2 ε n ≤ C (cid:48) ( r + ε n ) + 2 ε n , which implies that M is proper.It remains to prove that b ζ n converges to b ζ . [Whi02, Theorem 12.9.3 and Corollary12.5.1] ensure that it suffices to show that b ζ n ( r ) → b ζ ( r ) for every r at which b ζ is continuous.Let r ∈ Disc( b ζ ) c , R > b ζ ( r ) ∨ C (cid:48) ( r ) and (cid:67) Rn and ε n be as above. Notice that | b ζ ( r ) − b ζ n ( r ) | = (cid:12)(cid:12)(cid:12) b ζ ( r ) − sup z n : M n ( z n ) ∈ Λ r ( z , z n ) ∈ (cid:67) Rn d n ( ∗ n , z n ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) b ζ ( r ) − sup z n : M n ( z n ) ∈ Λ r ( z , z n ) ∈ (cid:67) Rn d ( ∗ , z ) (cid:12)(cid:12)(cid:12) + ε n . Now, for ( z , z n ) ∈ (cid:67) n , if M ( z ) ∈ Λ r − ε n then M n ( z n ) ∈ Λ r , while if M n ( z n ) ∈ Λ r , then M ( z ) ∈ Λ r + ε n which implies that b ζ ( r − ε n ) − b ζ ( r ) ≤ sup z n : M n ( z n ) ∈ Λ r ( z , z n ) ∈ (cid:67) n d ( ∗ , z ) − b ζ ( r ) ≤ b ζ ( r + ε n ) − b ζ ( r ) from which the conclusion follows. Lemma 2.15
For any α ∈ ( , ) , the identity map from ( T α c , ∆ csp ) to ( T α sp , ∆ sp ) is continuous. reliminaries Proof.
Let { ζ n } n , ζ ⊂ T α c be such that ∆ csp ( ζ n , ζ ) converges to . In particular, d M1 ( b ζ n , b ζ ) → as n → ∞ so that we are left to show that ∆ sp ( ζ n , ζ ) converges to , which in turn can beproven by following the same strategy as in [BCK17, Proposition 3.4]. Proposition 2.16
Let α ∈ ( , ) and A be an index set. A subset (cid:65) = { ζ a = ( T a , ∗ a , d a , M a ) : a ∈ A } of T α sp is relatively compact if and only if for every r > and ε > there exist1. a finite integer N ( r ; ε ) such that uniformly over all a ∈ A , (cid:78) d a ( T ( r ) a , ε ) ≤ N ( r ; ε ) (2.12) where (cid:78) d a ( T ( r ) a , ε ) is the cardinality of the minimal ε -net in T ( r ) a with respect to themetric d a ,2. a finite constant C = C ( r ) > and δ = δ ( r, ε ) > such that sup a ∈ A (cid:107) M a (cid:107) ( r ) ∞ ≤ C and sup a ∈ A δ − α ω ( r ) ( M a , δ ) < ε , (2.13)
3. a finite constant C (cid:48) = C (cid:48) ( r ) > such that (2.11) holds uniformly over a ∈ A .Proof. “ ⇐ = ” Let { ζ n = ( T n , ∗ n , d n , M n ) } n ⊂ (cid:65) be a sequence satisfying the threeproperties above.We want to extract a converging subsequence for { ζ n } n and construct the correspondinglimit point. For (cid:96), k ∈ N , let (cid:96) k = (cid:96) − k . In [ADH13, Section 5], the authors determine, forany n ∈ N , a subset of T ( (cid:96) k ) n which is a − k -net for the latter and whose cardinality, thanksto condition 1., is finite and bounded above by some N (cid:96),k ∈ N uniformly over n ∈ N . Let S n(cid:96),k = { z nu : u ∈ U (cid:96),k } be such a net and S n = { z nu : u ∈ U } , where U (cid:96),k is the index set { u = ( i, (cid:96), k ) : i ≤ N (cid:96),k } and U the union of all U (cid:96),k . We also impose that for all (cid:96), k ∈ N , z n ( (cid:96),k, ) = ∗ n . Notice that, by construction, S n is a countable dense set of T n for all n ∈ N .In view of (2.13), passing at most to a subsequence, for every u, u (cid:48) ∈ U , lim n →∞ d n ( z nu , z nu (cid:48) ) and lim n →∞ M n ( z nu ) exist. Let ˜ T def = { z u : u ∈ U } be an abstract countable set and definea semimetric d and a map ˜ M on it by imposing d ( z u , z u (cid:48) ) def = lim n →∞ d n ( z nu , z nu (cid:48) ) and ˜ M ( z u ) def = lim n →∞ M n ( z nu ) . (2.14)Identifying points at distance in ˜ T and taking the completion of the resulting space, weobtain T , which is a locally compact R -tree by [ADH13, Lemmas 5.6 and 5.7] and theproof 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 .reliminaries In view of Lemma 2.14, it only remains to prove that ∆ sp ( ζ n , ζ ) converges to , where ζ def = ( T , ∗ , d, M ) and ∗ def = z ( ,k,(cid:96) ) .Let r > and k ∈ N be fixed and set (cid:96) def = (cid:100) k r (cid:101) and ε def = 2 − k . Take n big enough sothat sup u ,u (cid:48) ∈ U (cid:96),k | d ( z u , z u (cid:48) ) − d n ( z nu , z nu (cid:48) ) | < ε , sup u ∈ U (cid:96),k (cid:107) M ( z u ) − M n ( z nu ) (cid:107) < ε − ˜ mα , (2.15)where ˜ m def = ¯ m ∨ m n and ¯ m ∈ N (resp. m n ∈ N ) is the maximum integer for whichthere exist u , u (cid:48) ∈ U (cid:96),k such that d ( z u , z u (cid:48) ) ∈ (cid:65) ¯ m (resp. d n ( z nu , z nu (cid:48) ) ∈ (cid:65) m n ). Set S (cid:96),k = { z u : u ∈ U (cid:96),k } , which, by [ADH13, Lemma 5.6], is a ε -net for T ( (cid:96) k ) and define ζ (cid:96),kn def = ( S n(cid:96),k , ∗ n , d n , M n ) and ζ (cid:96),k def = ( S (cid:96),k , ∗ , d, M ) . By the triangle inequality we have ∆ csp ( ζ ( r ) , ζ ( r ) n ) ≤ ∆ csp ( ζ ( r ) , ζ ( (cid:96) k ) ) + ∆ csp ( ζ ( (cid:96) k ) , ζ (cid:96),k ) + ∆ csp ( ζ (cid:96),k , ζ (cid:96),kn ) + ∆ csp ( ζ (cid:96),kn , ζ ( (cid:96) k ) n ) + ∆ csp ( ζ ( (cid:96) k ) n , ζ ( r ) n ) =: (cid:88) i =1 A i . (2.16)Thanks to Lemma 2.12 and (2.13), all the A i ’s, for i (cid:54) = 3 , can be controlled in termsof quantities which are vanishing as k → ∞ , so that we only need to focus on A . Let (cid:67) n def = { ( z u , z nu ) : u ∈ U (cid:96),k } and, without loss of generality, assume ˜ m = ¯ m . Then, for m ≤ m n , z u , z u (cid:48) , z nu , z nu (cid:48) such that d ( z u , z u (cid:48) ) , d n ( z nu , z nu (cid:48) ) ∈ (cid:65) m , the second bound in (2.15)implies (cid:107) δ z u , z u (cid:48) M − δ z nu , z nu (cid:48) M n (cid:107) ≤ ε − ¯ mα ≤ ε − mα while for m > m n we have (cid:107) δ z u , z u (cid:48) M (cid:107) ≤ ω ( (cid:96) k ) ( M, − m ) ≤ − mα (2 m n α ω ( r +1 ) ( M, − m n ) ) . Since m n goes to infinity as n ↑ ∞ , we have shown that, for any fixed r > , the term at theleft hand side of (2.16) converges to , therefore also ∆ sp ( ζ, ζ n ) does.“ = ⇒ ” Let (cid:65) be relatively compact in T α sp . Then, property 1. holds by [BBI01, Proposition7.4.12], while property 3. holds by [Whi02, Theorems 12.9.3 and 12.12.2]. For the secondproperty, notice that since (cid:65) is totally bounded, for any ε > and r > there exist n ∈ N and { ζ k : k = 1 , . . . n } such that (cid:65) is contained in the union of the balls of radius e − r ε/ centred at ζ k . Hence, if ζ ∈ B ( ζ k , e − r ε/ ) , then we have ∆ csp ( ζ ( r ) , ζ ( r ) k ) < ε (2.17)which implies that there exists a correspondence (cid:67) between T ( r ) and T ( r ) k such that ∆ c , (cid:67) sp ( ζ ( r ) , ζ ( r ) k ) < ε/ . Since (cid:107) M ζ (cid:107) ( r ) ∞ ≤ ε/ (cid:107) M ζ k (cid:107) ( r ) ∞ by the triangle inequality, sup ζ ∈ (cid:65) (cid:107) M ζ (cid:107) ( r ) ∞ ≤ ε + max k =1 ,...,n (cid:107) M ζ k (cid:107) ( r ) ∞ , (2.18)reliminaries and the first bound in (2.13) follows. For the others, let δ > and ¯ n ∈ N the smallestinteger such that − ¯ n ≤ δ . Then, sup n> ¯ n nα sup ( z , z k ) , ( w , w k ) ∈ (cid:67) d ( z , w ) ,d k ( z k , w k ) ∈ (cid:65) n (cid:107) δ z , w M ζ − δ z k , w k M ζ k (cid:107)≤ sup n ∈ N nα sup ( z , z k ) , ( w , w k ) ∈ (cid:67) d ( z , w ) ,d k ( z k , w k ) ∈ (cid:65) n (cid:107) δ z , w M ζ − δ z k , w k M ζ k (cid:107) < ε (2.19)so that, once again, the second bound in (2.13) can be obtained by applying triangleinequality and choosing the minimum δ for which sup k ≤ n δ − α ω ( r ) ( M ζ k , δ ) < ε/ . Proof of Theorem 2.13(ii).
For completeness, it suffices to show that, if { ζ n } n is a Cauchysequence in T α sp then the conditions of Proposition 2.16 are satisfied. Now, if { ζ n } n isCauchy, then for every r > , { ζ ( r ) n } n is Cauchy with respect to ∆ csp , which implies thatthe sequence converges so that 1. holds in view of [BBI01, Proposition 7.4.12], 2. can beseen to be satisfied by arguing as in (2.18) and (2.19), and 3. follows by completeness of D ([ − , ∞ ) , R + ) with respect to d M1 .We conclude this section with a lemma that will be useful in the construction andcharacterisation of the Brownian Web. It guarantees that, under certain conditions, we canbuild an α -spatial R -tree inductively, by “patching together” pieces of branches. Lemma 2.17
Let α ∈ ( , ) and ζ n = ( T n , ∗ n , d n , M n ) be a relatively compact sequencein T α sp . Assume that for every n < m ∈ N there exists an isometric embedding ι n,m of T n into T m such that ι n,m ( ∗ n ) = ∗ m , ι n,k = ι m,k ◦ ι n,m for n < k < m and M m ◦ ι n,m ≡ M n .Then, the sequence ζ n converges to ζ = ( T , ∗ , d, M ) and for every n ∈ N there exists anisometric embedding ι n of T n into T such that ι n ( ∗ n ) = ∗ , ι n = ι m ◦ ι n,m for m > n and M ◦ ι n ≡ M n . Moreover, ˜ T def = (cid:83) n ι n ( T n ) is dense in T and M is the unique continuousextension of ˜ M on ˜ T , the latter being defined by the relation ˜ M ◦ ι n ≡ M n for all n . Remark 2.18
A similar statement was given in [EPW06, Lemma 2.7]. The formulation isa bit different since we do not have a common ambient space and the trees we consider arespatial. One reason why we cannot directly reuse that result is that it is not clear a priorithat relative compactness in T α sp implies relative compactness of the images in (cid:83) n T n / ∼ with the natural equivalence relation induced by the consistency maps ι m,n . This is becausethe optimal correspondence between T n and T m may differ from the one given by ι m,n .Take for example the trees ( T , ∗ ) = ([ , ] , / ) and ( ¯ T , ¯ ∗ ) = ([ , / ] , / ) . Then, forthe natural correspondence (cid:67) suggested by our notations, one has dis (cid:67) = 2 / , while thecorrespondence ¯ (cid:67) mapping x ∈ ¯ T to / − x ∈ T is also an isometric embedding butreliminaries has dis ¯ (cid:67) = 1 / . This shows that the condition in [EPW06, Lemma 2.7] assuming that the ζ n are Cauchy as subsets of a common space in the Hausdorff topology may a priori bestronger than the relative compactness assumed here. (A posteriori it is not, as demonstratedby the fact that ˜ T is dense in T .) Proof.
We will limit ourselves to the case of T n compact, the general case easily followsfrom the definition of the metric ∆ sp .Let n < m < k and (cid:67) n,k be a correspondence between T n and T k . We can the obtain acorrespondence (cid:67) n,m between T n and T m by setting (cid:67) n,m = { ( z , ¯ z ) ∈ T n × T m : ( z , ι m,k ( ¯ z )) ∈ (cid:67) n,k } ∪ { ( z , ι n,m ( z )) : z ∈ T n } , the second term being required to ensure (cid:67) n,m is indeed a correspondence. It is easy tosee that ∆ c , ¯ (cid:67) n,m sp ( ζ n , ζ m ) ≤ ∆ c , (cid:67) n,k sp ( ζ n , ζ k ) , which then implies ∆ csp ( ζ n , ζ m ) ≤ ∆ csp ( ζ n , ζ k ) .Since { ζ n } n is relatively compact, it admits a Cauchy subsequence and in view of thelast inequality the whole sequence is Cauchy. Hence, it converges to a unique ζ = ( T , ∗ , d, M ) and there exists a sequence of correspondences (cid:67) m between T and T m suchthat ∆ c , (cid:67) m sp ( ζ m , ζ ) → .In order to construct the isometries ι n and show they satisfy the properties stated, wefirst fix dense countable sets (cid:68) n ⊂ T n with ∗ n ∈ (cid:68) n and such that ι n,m (cid:68) n ⊂ (cid:68) m for every n ≤ m . We also write ι m,n for the inverse of ι n,m on its image in (cid:68) m . For n ≤ m , we thenchoose a collection of maps ι ( m ) n : (cid:68) n → T such that ( ι n,m ( z ) , ι ( m ) n ( z ) ) ∈ (cid:67) m ∀ n ≤ m, z ∈ (cid:68) n , ι ( m ) k = ι ( m ) n ◦ ι k,n ∀ k ≤ n ≤ m . This is always possible: for every m , first fix ι ( m ) , which determines the ι ( m ) n on ι ,n ( (cid:68) ) forall n ≤ m , then fix ι ( m ) on (cid:68) \ ι , ( (cid:68) ) , etc. We now choose any enumeration { z k } k> of (cid:68) = (cid:83) n> and write n k ∈ N such that z k ∈ (cid:68) n k . This allows us to define maps ι n : (cid:68) n → T as follows. Let M ⊂ N be an infinite set such that the limit ι n ( z ) def = lim m →∞ : m ∈ M ι ( m ) n ( z ) ,exists. We then inductively define ι n k ( z k ) for every k ∈ N by the analogous formula, forsome infinite set M k ⊂ M k − .We claim that the maps ι n : (cid:68) n → T defined in this way are isometries satisfying therequired consistency which is sufficient to complete the proof since (cid:68) n is dense in T n .Regarding consistency, if k ≤ (cid:96) is such that z k = ι n (cid:96) ,n k ( z (cid:96) ) , then ι n k ( z k ) = lim m ∈ M k ι ( m ) n k ( z k ) = lim m ∈ M (cid:96) ι ( m ) n k ( z k ) = lim m ∈ M (cid:96) ι ( m ) n (cid:96) ( z (cid:96) ) = ι n m ( z m ) ,reliminaries as required. To show that they are isometries, let k < (cid:96) be such that n k = n (cid:96) = n . Forevery m ≥ n , we then have the bound | d ( ι n ( z k ) , ι n ( z (cid:96) ) ) − d n ( z k , z (cid:96) ) | ≤ d ( ι n ( z k ) , ι ( m ) n ( z k ) ) + d ( ι n ( z (cid:96) ) , ι ( m ) n ( z (cid:96) ) )+ | d ( ι ( m ) n ( z k ) , ι ( m ) n ( z (cid:96) ) ) − d n ( z k , z (cid:96) ) | . Choosing m ∈ M (cid:96) , we note that the first two terms converge to as m → ∞ by thedefinition of ι n . The last term on the other hand converges to by the construction of ι ( m ) n combined with the fact that ∆ c , (cid:67) m sp ( ζ m , ζ ) → . Similarly, one has | M ( ι n k ( z k ) ) − M n k ( z k ) | ≤ | M ( ι n k ( z k ) ) − M ( ι ( m ) n k ( z k ) ) | + | M ( ι ( m ) n k ( z k ) ) − M n k ( z k ) | ,and both terms vanish in the limit as m ∈ M k converges to ∞ .Finally, let ˜ T be given by the union of all ι n ( T n ) and denote by ¯ T its closure in T .By the very definition of Gromov–Hausdorff distance, it is clear that T n converges to ¯ T ,which then implies the last part of the statement. R -trees and the radial map As mentioned in the introduction, we would like to view the backward Brownian Web asa flow. More specifically, at any time t and position x , we want to be able to follow abackward Brownian trajectory starting at x at time t . These trajectories will be encoded bythe 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 whoseelements possess a notion of direction in time and satisfy a monotonicity assumption, bothimposed at the level of the evaluation map M . Henceforth we use the following shorthandnotation. Given an R -tree T , elements z , z ∈ T , and s ∈ [ , ] , we write z s for theunique element of (cid:74) z , z (cid:75) with d ( z , z s ) = s d ( z , z ) . Definition 2.19
For α ∈ ( , ) , we define the space of characteristic α -spatial R -trees , C α sp ⊂ T α sp consisting of those elements ζ = ( T , ∗ , d, M ) , whose evaluation map M satisfiesthe following additional conditions.(1) Monotonicity in time , i.e. for every z , z ∈ T and s ∈ [ , ] one has M t ( z s ) = ( M t ( z ) − s d ( z , z ) ) ∨ ( M t ( z ) − ( − s ) d ( z , z ) ) . (2.20)(2) Monotonicity in space , i.e. for every s < t , interval I = ( a, b ) and any four elements z , ¯ z , z , ¯ z such that M t ( z ) = M t ( ¯ z ) = t , M t ( z ) = M t ( ¯ z ) = s , M x ( z ) < M x ( ¯ z ) ,and M ( (cid:74) z , z (cid:75) ) , M ( (cid:74) ¯ z , ¯ z (cid:75) ) ⊂ [ s, t ] × ( a, b ) , we have M x ( z s ) ≤ M x ( ¯ z s ) (2.21)for every s ∈ [ , ] .reliminaries (3) For all z = ( t, x ) ∈ R , M − ( { t } × [ x − , x + 1 ]) (cid:54) = ∅ .Note that (2) also makes sense in the periodic case if we restrict to intervals ( a, b ) that donot 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 coalescebut their spatial order must be preserved. The last requirement says that the map M issufficiently spread so that the vicinity of any point contains a backward characteristic onecan follow. We do not impose the map M to be surjective since this is not true for thetype of discrete approximation we want to consider. Clearly, the choice of is completelyarbitrary. Remark 2.21
We denote by ˆ C α sp the subspace of T α sp defined in exactly the same way butwith ∨ replaced by ∧ in (1). Note that ζ = ( T , ∗ , d, M ) (cid:55)→ − ζ def = ( T , ∗ , d, − M ) ∈ ˆ C α sp isan isometric involution.First notice that it is not difficult to show that the properties in the previous definitionare consistent with the equivalence relation in Definition 2.7, i.e. if there exists a bijectiveisometry ϕ such that ϕ ◦ ζ = ζ (cid:48) and ζ satisfies the conditions above then so does ζ (cid:48) . In otherwords, the space C α sp is a well-defined subset of T α sp . Before studying further properties ofcharacteristic R -trees, we note that C α sp is closed in T α sp . Lemma 2.22
For every α ∈ ( , ) , C α sp is a closed subset of T α sp . Moreover, let { ζ n } n ⊂ T α sp be a sequence whose elements are monotone in both space and time. Assume that thesequence converges to ζ ∈ T α sp and that for every z = ( t, x ) ∈ R there exists n z ∈ N suchthat for all n ≥ n z , ( M n ) − ( { t } × [ x − , x + 1 ]) (cid:54) = ∅ . Then ζ ∈ C α sp .Proof. Let { ζ n } n ∈ T α sp be a sequence whose elements are monotone in both space andtime and let ζ ∈ T α sp be its limit. Since ζ is monotone if and only if ζ ( R ) is monotone forevery R and since ∆ csp ( ζ n, ( R ) , ζ ( R ) ) → for every R > , we first restrict ourselves to thecompact case and show monotonicity of the limit.We start with monotonicity in time. Take z , z ∈ T , let (cid:67) n be a sequence ofcorrespondences such that lim n ∆ c , (cid:67) n sp ( ζ n , ζ ) → and let z ni be such that ( z ni , z i ) ∈ (cid:67) n . Forany s ∈ [ , ] , we choose ¯ z ns ∈ T such that ( z ns , ¯ z ns ) ∈ (cid:67) n . It then follows from the treeproperty and the definition of distortion that d ( ¯ z ns , z s ) ≤ dis (cid:67) n , so that in particular M t ( z s ) = lim n →∞ M t ( ¯ z ns ) = lim n →∞ M nt ( z ns ) . reliminaries Since furthermore lim n →∞ d ( z n , z n ) = d ( z , z ) and lim n →∞ M nt ( z ni ) = M t ( z i ) by thedefinition of ∆ c , (cid:67) n sp , the claim follows.Regarding monotonicity in space, we perform the same construction, whence we get M x ( z s ) = lim n →∞ M nt ( z ns ) ≤ lim n →∞ M nt ( z ns (cid:48) ) = M x ( z s (cid:48) ) ,as required.For the last property, let z = ( t, x ) ∈ R . For any n ≥ n z , by assumption, thereexists z n ∈ ( M n ) − ( { t } × [ x − , x + 1 ]) , and, by (2.11), there exists R > such that d n ( ∗ n , z n ) ≤ R uniformly in n . Now, ∆ sp ( ζ n , ζ ) → , hence, for any n ≥ n z there existsa correspondence (cid:67) Rn between T ( R ) and T n, ( R ) for which ∆ c , (cid:67) Rn sp ( ζ n, ( R ) , ζ ( R ) ) → . Let z n ∈ T ( R ) be such that ( z n , z n ) ∈ (cid:67) Rn . Notice that, the sequence { z n } n ⊂ T ( R ) convergesalong subsequences so we can pick z ∈ T ( R ) to be a limit point. Then | M t ( z ) − t | ≤ | δ z , z n M t | + | M t ( z n ) − t | ≤ (cid:107) δ z , z n M (cid:107) + ∆ c , (cid:67) Rn sp ( ζ n, ( R ) , ζ ( R ) ) which implies that M t ( z ) = t and | M x ( z ) − x | ≤ | δ z , z n M x | + | M x ( z n ) − M nx ( z n ) | + | M nx ( z n ) − x |≤ (cid:107) δ z , z n M (cid:107) + ∆ c , (cid:67) Rn sp ( ζ n, ( R ) , ζ ( R ) ) + 1 from which the conclusion follows.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 adistinguished role. Proposition 2.23
Let α ∈ ( , ) and ζ = ( T , ∗ , d, M ) ∈ C α sp . Then, T has a unique openend † such that for every z ∈ T and every w ∈ (cid:74) z , †(cid:105) , one has M t ( w ) = M t ( z ) − d ( z , w ) . (2.22) Proof.
Let ζ = ( T , ∗ , d, M ) ∈ C α sp and fix z ∈ T . We want to construct an unbounded T -ray from z such that (2.22) holds. Set z = z and ( t , x ) def = M ( z ) . Assume that we aregiven elements { z j } j ≤ n ⊂ T which are collinear (i.e. z j ∈ (cid:74) z , z n (cid:75) for ≤ j ≤ n ), suchthat, setting ( t j , x j ) def = M ( z j ) , we have t i +1 − t i > , and such that (2.22) holds for every w ∈ (cid:74) z , z n (cid:75) .reliminaries As an easy consequence of (3) in Definition 2.19, there exists w n +1 ∈ T such that M ( w n +1 ) ∈ B (( t n − , x n ) , ] and z n +1 ∈ (cid:74) z n , w n +1 (cid:75) for which necessarily t n +1 ≥ t n + 1 and such that M t ( w ) = M t ( z n ) − d ( z n , w ) for every w ∈ (cid:74) z n , z n +1 (cid:75) . Then, we have M t ( w ) = M t ( z n ) − d ( z n , w ) = M t ( z ) − d ( z , z n ) − d ( z n , w ) = M t ( z ) − d ( z , w ) ,where the last step follows from the fact that (cid:74) z , z n (cid:75) ∩ (cid:74) z n , z n +1 (cid:75) = { z n } by the inductionhypothesis and property ) . This yields a (necessarily unbounded) T -ray from z and we set † to be the open end it represents, i.e. (cid:74) z , †(cid:105) = (cid:83) n ≥ (cid:74) z , z n (cid:75) . The uniqueness of † followsimmediately from property . (The time coordinate M t must converge to −∞ along anyunbounded ray which forces any two to coalesce at some point by considering any geodesiclinking them.)Thanks to the previous proposition, we can introduce, in the context of characteristictrees, the radial map . This is a map on the R -tree that allows to move along the rays. Let ζ = ( T , ∗ , d, M ) ∈ C α sp and † the open end for which (2.22) holds. For z ∈ T we define (cid:37) ( z , · ) : ( −∞ , M t ( z )] → T as (cid:37) ( z , s ) def = ι − z ( M t ( z ) − s ) , for s ∈ ( −∞ , M t ( z )] (2.23)where ι z was given in (2.1). Remark 2.24 If ζ ∈ ˆ C α sp (see Remark 2.21), then, for z ∈ T the radial map (cid:37) ( z , · ) isdefined on [ M t ( z ) , + ∞ ) as (cid:37) ( z , s ) def = ι − z ( s − M t ( z )) . Before detailing our alternative construction of the Brownian Web, we show how thetopology introduced above relates to that of [FINR04]. To describe the latter, let first R c bethe completion of R with respect to the metric (cid:37) (( t , x ) , ( t , x )) def = | tanh( t ) − tanh( t ) | ∨ (cid:12)(cid:12)(cid:12) tanh( x ) | t | − tanh( x ) | t | (cid:12)(cid:12)(cid:12) for all ( t , x ) , ( t , x ) ∈ R . (See [NRS15, Fig. 3] for a cartoon illustrating the geometryof the resulting compactification of R .) A backward path π in R c with starting time σ π ∈ [ −∞ , ∞ ] is a continuous map R (cid:51) t (cid:55)→ ( t, π ( t )) ∈ R c with π ( t ) = π ( σ π ) for all t ≥ σ π . We define a metric d on the space Π of such paths by d ( π , π ) def = | tanh( σ π ) − tanh( σ π ) | ∨ sup t ≤ σ π ∧ σ π (cid:12)(cid:12)(cid:12) tanh( π ( t )) | t | − tanh( π ( t )) | t | (cid:12)(cid:12)(cid:12) (2.24)reliminaries for all π , π ∈ Π . Since ( Π , d ) is a Polish space, so is the space (cid:72) of compact subsets of Π endowed with the Hausdorff metric.Let α ∈ ( , ) , ζ = ( T , ∗ , d, M ) ∈ C α sp , and (cid:37) , ζ ’s radial map defined accordingto (2.23). For z ∈ T , define π z ( t ) def = M x ( (cid:37) ( z , t )) , for all t ≤ M t ( z ) . (2.25)Since π z ∈ Π by continuity of M , we have a map C α sp (cid:51) ζ (cid:55)→ K ( ζ ) def = { π z : z ∈ T } ⊂ Π . (2.26) Proposition 2.25
Let α ∈ ( , ) . For every ζ ∈ C α sp , K ( ζ ) is compact and the map ζ (cid:55)→ K ( ζ ) is continuous from C α sp to (cid:72) . Remark 2.26
Defining ˆΠ and ˆ (cid:72) in the same way, except that now π ( t ) = π ( σ π ) for all t ≤ σ π and ≤ is replaced by ≥ in the right-hand side of (2.24), we also have a map ˆ K : ˆ C α sp → ˆ (cid:72) given by ˆ K ( ζ ) = − K ( − ζ ) .For the proof of the previous proposition we will need the following two lemmas. Forthe first, define Π R def = { π ∈ Π : ∃ t ≤ σ π s.t. ( t, π ( t )) ∈ [ − R, R ] } ,and, for π ∈ Π , write π R ∈ Π for the stopped path such that σ π R = σ π , π R ( t ) = π ( R ) if t ≥ R , π ( − R ) if t ≤ − R , π ( t ) otherwise. Lemma 2.27
Let K be a subset of Π and, for R > , let K R ⊂ Π be defined as K R def = { π R : π ∈ K ∩ Π R } . (2.27) If for all
R > , the family of paths in K R is equicontinuous then K is relatively compact.Proof. Our main ingredient then is the fact that, since | − tanh R | ≤ e − R , one has thebounds x ≥ R ⇒ (cid:37) (( t, x ) , ( t, ∞ )) ≤ e − R ∀ t , x ≤ − R ⇒ (cid:37) (( t, x ) , ( t, −∞ )) ≤ e − R ∀ t , (2.28)reliminaries | t | ≥ R ⇒ (cid:37) (( t, x ) , ( t, y )) ≤ R ∀ x, y . Writing π ± t for the path with σ π ± t = t and π ± t ( s ) = ±∞ , it follows that for every π ∈ Π and every R ≥ one has d ( π, π R ) ≤ /R . If furthermore π (cid:54)∈ Π R , then d ( π, π + σ π ) ∧ d ( π, π − σ π ) ≤ /R .It remains to note that, given ε > , we can cover K /ε with finitely many balls of radius ε/ by Arzelà–Ascoli, so that K ∩ Π R is covered by the balls with same centres and radius ε . The complement of Π R on the other hand can be covered by finitely many balls of radius ε centred at elements of type π ± t for t ∈ ε Z ∩ [ − ε − , ε − ] .The next lemma highlights the fact that if two characteristic trees are close then also therespective rays must be close in a suitable sense which will be made explicit in the statementbelow. Lemma 2.28
Let α ∈ ( , ) and ζ , ζ ∈ C α sp . Let r > and assume there exists acorrespondence (cid:67) between T ( r ) and T ( r ) such that ∆ c , (cid:67) sp ( ζ ( r ) , ζ ( r ) ) < ε for some ε > .Let ( z , z ) ∈ (cid:67) and define a new correspondence C (cid:67) as C (cid:67) def = (cid:67) ∪ { ( (cid:37) ( z , s ) , (cid:37) ( z , s ) : − r ≤ s ≤ M ,t ( z ) ∧ M ,t ( z ) } (2.29) Then, dis C (cid:67) + sup ( z , ¯ z ) ∈ C (cid:67) (cid:107) M ( z ) − M ( ¯ z ) (cid:107) (cid:46) ε + (cid:107) M (cid:107) ( r ) α ε α Proof.
Let ( z , z ) ∈ (cid:67) be as in the statement and − r ≤ s ≤ M ,t ( z ) ∧ M ,t ( z ) . Let z s ∈ T be such that ( z s , (cid:37) ( z , s )) ∈ (cid:67) . Notice that for any ( w , w ) ∈ (cid:67) , by the triangleinequality and the assumption ∆ c , (cid:67) sp ( ζ ( r ) , ζ ( r ) ) < ε , we have | d ( (cid:37) ( z , s ) , w ) − d ( (cid:37) ( z , s ) , w ) | ≤ d ( z s , (cid:37) ( z , s )) + dis (cid:67) ≤ d ( z s , (cid:37) ( z , s )) + 2 ε which means that we only need to focus on d ( z s , (cid:37) ( z , s )) . Now, if (cid:37) ( z , s ) belongs to theray starting at z s , by (2.22), we have d ( z s , (cid:37) ( z , s )) = M ,t ( z s ) − M ,t ( (cid:37) ( z , s )) = M ,t ( z s ) − s ≤ M ,t ( (cid:37) ( z , s )) + ε − s ≤ ε . Otherwise, d ( z s , (cid:37) ( z , s )) = d ( z s , z ) − d ( z , (cid:37) ( z , s )) ≤ d ( (cid:37) ( z , s ) , z ) + ε − d ( z , (cid:37) ( z , s )) = M ,t ( z ) − s + ε − M ,t ( z ) + s ≤ ε . reliminaries Therefore, we immediately conclude that dis C (cid:67) < ε . Concerning the bound on theevaluation maps, we have (cid:107) M ( (cid:37) ( z , s )) − M ( (cid:37) ( z , s )) (cid:107) ≤(cid:107) M ( (cid:37) ( z , s )) − M ( z s ) (cid:107) + (cid:107) M ( z s ) − M ( (cid:37) ( z , s )) (cid:107) (cid:46) (cid:107) M (cid:107) ( r ) α ε α + ε where we exploited the Hölder continuity of M , the bound on d ( z s , (cid:37) ( z , s )) and the factthat ( z s , (cid:37) ( z , s )) ∈ (cid:67) . The conclusion follows at once.We are now ready for the proof of Proposition 2.25. Proof of Proposition 2.25.
Let ζ = ( T , ∗ , d, M ) ∈ C α sp and K ( ζ ) be as in (2.26). Bydefinition, M − ( Λ R ) ⊂ B d ( ∗ , b ζ ( R )] and, since T is a tree, if z ∈ B d ( ∗ , b ζ ( R )] then (cid:37) ( z , s ) ∈ B d ( ∗ , b ζ ( R )] for all s ∈ [ − R, M t ( z )] . Moreover, M is α -Hölder continuous on B d ( ∗ , b ζ ( R )] , therefore K ( ζ ) R as defined in (2.27) consists of equicontinuous paths andLemma 2.27 implies that K ( ζ ) ∈ (cid:72) .Let now { ζ n = ( T n , ∗ n , d n , M n ) } n ⊂ C α sp be a sequence converging to ζ ∈ C α sp withrespect to ∆ sp . In view of Proposition 2.16, the evaluation maps M n are uniformly properand have uniformly bounded α -Hölder norm when restricted to balls of fixed size. Hence,arguing as above, we see that ∪ n K ( ζ n ) is relatively compact in Π which, thanks to [SSS10,Lemma B.3], implies that the sequence { K ( ζ n ) } n is relatively compact in (cid:72) with respectto the Hausdorff topology.It remains to show that K ( ζ n ) converges to K ( ζ ) in (cid:72) . By [SSS10, Lemma B.1], itsuffices to prove that for every π z ∈ K ( ζ ) there exists a sequence π z n ∈ K ( ζ n ) such that d ( π z , π z n ) → . Let z ∈ T and ε > . Pick C > big enough so that z ∈ B d ( ∗ , C ] and sup n b ζ n ( ε − ) ≤ C . Let n be sufficiently large so that there exists a correspondence (cid:67) n between B d ( ∗ , C ] and B d n ( ∗ n , C ] with ∆ c , (cid:67) n sp ( ζ ( C ) , ζ n, ( C ) ) < ε . Let z n ∈ B d n ( ∗ n , C ] with ( z , z n ) ∈ (cid:67) n and define π z and π z n as in (2.25). Since | M t ( z ) − M nt ( z n ) | < ε , it follows that | tanh( σ π z ) − tanh( σ π zn ) | < ε .To estimate the distance between π z ( s ) and π z n ( s ) for s ≤ σ π z ∧ σ π zn , we first considerthe case s ≥ − ε − . Since C is large enough so that (cid:37) n ( z n , s ) ∈ B d n ( ∗ n , C ] , we can applyLemma 2.28 and get | π z ( s ) − π z n ( s ) | = | M x ( (cid:37) ( z , s )) − M nx ( (cid:37) n ( z n , s )) | (cid:46) ε + (cid:107) M (cid:107) ( C ) α ε α . (2.30)For s < − ε − we use again the last bound of (2.28). Combining these bounds, we obtain d ( π z , π z n ) (cid:46) ε α and the proof is concluded.In general, we cannot expect the map K to be injective. Indeed, there is no mechanismthat a priori prevents different branches of the tree to be mapped via the evaluation mapreliminaries to the same path and, as we will see below, we cannot expect the evaluation map of theBrownian Web to be injective because of presence special points from which multipletrajectories depart (see Section 3.3).In the following definition, we introduce a (measurable) subset of C α sp whose elementssatisfy a condition, the tree condition , which allows to set two rays in the tree apart basedon their image via the evaluation map. Definition 2.29
Let α ∈ ( , ) . We say that ζ = ( T , ∗ , d, M ) ∈ C α sp satisfies the treecondition if( t ) for all z , z ∈ T , if M ( z ) = M ( z ) = ( t, x ) and there exists ε > such that M ( (cid:37) ( z , s )) = M ( (cid:37) ( z , s )) for all s ∈ [ t − ε, t ] , then z = z .We denote by C α sp ( t ) , the subset of C α sp whose elements satisfy ( t ).Condition ( t ), guarantees that different rays on the tree under study are mapped, viathe evaluation map, to paths which are almost everywhere distinct. It is not difficult toconstruct examples of characteristic trees for which ( t ) does not hold, while it clearly doesif the evaluation map is injective. In the following Lemma, whose proof is immediate, weprovide a less trivial example. Lemma 2.30
Let α ∈ ( , ) and ζ = ( T , ∗ , d, M ) ∈ C α sp . If there exists a dense subtree T of T such that ( T, ∗ , d, M (cid:22) T ) satisfies ( t ) then so does ζ . Moreover, the subset of C α sp 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 2.4point 2.We conclude this section by showing that on C α sp ( t ) , K is indeed injective. Proposition 2.31
Let α ∈ ( , ) and C α sp ( t ) be given as in Definition 2.29. Then, the map K in (2.26) is injective on C α sp ( t ) .Proof. For the last part of the statement, let ζ, ζ (cid:48) ∈ C α sp be such that ( t ) holds and K ( ζ ) ≡ K ( ζ (cid:48) ) . Then, for all z ∈ T there exists a unique element ϕ ( z ) ∈ T (cid:48) such that π z ≡ π ϕ ( z ) and therefore not only M ( z ) = M (cid:48) ( ϕ ( z )) , but M ( (cid:37) ( z , s )) = M (cid:48) ( (cid:37) (cid:48) ( z (cid:48) , s )) for all s . To show that ϕ is the required isomorphism, assume by contradiction that there exist z , z ∈ T such that d ( z , z ) (cid:54) = d (cid:48) ( ϕ ( z ) , ϕ ( z )) and let ¯ s, ¯ s (cid:48) ≤ M t ( z ) ∧ M t ( z ) be thefirst times at which (cid:37) ( z , ¯ s ) = (cid:37) ( z , ¯ s ) and (cid:37) (cid:48) ( ϕ ( z ) , ¯ s (cid:48) ) = (cid:37) (cid:48) ( ϕ ( z ) , ¯ s (cid:48) ) respectively. Sincehe Brownian Web Tree and its dual d ( z , z ) (cid:54) = d (cid:48) ( ϕ ( z ) , ϕ ( z )) we have ¯ s (cid:54) = ¯ s (cid:48) so that, without loss of generality, we can assume ¯ s > ¯ s (cid:48) . Since T is a tree, we must have M (cid:48) ( (cid:37) (cid:48) ( ϕ ( z ) , s ))) = M ( (cid:37) ( z , s )) = M ( (cid:37) ( z , s )) = M (cid:48) ( (cid:37) (cid:48) ( ϕ ( z ) , s ))) ∀ s ∈ [ ¯ s (cid:48) , ¯ s ] ,which, by ( t ), implies that (cid:37) (cid:48) ( ϕ ( z ) , ¯ s ) = (cid:37) (cid:48) ( ϕ ( z ) , ¯ s ) . Hence, d ( z , z ) = d (cid:48) ( ϕ ( z ) , ϕ ( z )) and we reach the required contradiction. Remark 2.32
In the periodic case, let Π per be the set of backward periodic paths endowedwith the metric d per whose definition is the same as in (2.24) but in the second argument ofthe maximum the inner metric is replaced by the periodic one, i.e. for π , π ∈ Π per and t ≤ σ π ∧ σ π , we take inf k ∈ Z | π ( t ) − π ( t ) + k | . Let (cid:72) per be the set of compact subsetsof Π per with the Hausdorff metric. Then, Propositions 2.25 and 2.31 remain true, whichmeans that the map K : C α sp , per → (cid:72) per defined as in (2.26) is continuous and its restrictionto C α sp , per ( t ) is injective. Here, we provide an alternative (and finer) characterisation of the Brownian Web so to beable to view it a characteristic spatial R -tree. 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 arealmost identical, we will mainly focus on the first and limit ourselves to indicate what needsto be modified in order to accommodate the second (see Remarks 3.1, 3.7, 3.9).Consider a standard probability space ( Ω , (cid:65) , P ) supporting countably many independentstandard Brownian motions { W ↓ k } k ∈ N starting at and running backward in time, i.e. from to −∞ . Fix a countable dense set (cid:68) def = { z k = ( t k , x k ) : k ∈ N } of R , with z = ( , ) .Then, build inductively a family of coalescing backward Brownian motions { π ↓ z k } k ∈ N suchthat π ↓ z k starts at x k at time t k . As in [FINR04, Section 3], one way to do so is to set π ↓ z ( t ) = W ↓ ( t ) and then define π ↓ z k ( t ) = x k + W ↓ ( t − t k ) for all τ k ≤ t ≤ t k , where τ k is the largest value such that x k + W ↓ k ( τ k − t k ) = π ↓ z (cid:96) ( τ k ) for some (cid:96) < k , and for t ≤ τ k , π ↓ z k ( t ) = π ↓ z (cid:96) ( t ) . The construction guarantees that even though (cid:96) may not be unique, thedefinition of π ↓ k is.For every n ∈ N , let ˜ T ↓ n ( (cid:68) ) def = { ( t, π ↓ z k ) : t ≤ t k , k ≤ n } and ˜ T ↓∞ ( (cid:68) ) be the spacedefined as before but in which k is free to range over all of N . Now, for n ∈ ¯ N def = N ∪ {∞} ,he Brownian Web Tree and its dual consider the equivalence relation ∼ on ˜ T ↓ n ( (cid:68) ) , given by ( t, π ↓ z i ) ∼ ( t, π ↓ z j ) if and only if π ↓ z i ( s ) = π ↓ z j ( s ) ∀ s ≤ t (3.1)for t ≤ t i ∧ t j and i, j ≤ n . We now introduce ζ ↓ n ( (cid:68) ) def = ( T ↓ n ( (cid:68) ) , ∗ ↓ , d ↓ , M ↓ , (cid:68) n ) , as T ↓ n ( (cid:68) ) def = ˜ T n ( (cid:68) ) / ∼ , ∗ ↓ def = ( , π ↓ ) ,d ↓ (( t, π ↓ z i ) , ( s, π ↓ z j )) def = ( t + s ) − τ ↓ t,s ( π ↓ z i , π ↓ z j ) ,M ↓ , (cid:68) n (( s, π ↓ z i )) = ( M ↓ , (cid:68) n,t (( s, π ↓ z i )) , M ↓ , (cid:68) n,x (( s, π ↓ z i ))) def = ( s, π ↓ z i ( s )) , (3.2)where i, j ≤ n and, in the definition of the ancestor metric d ↓ , τ ↓ t,s ( π ↓ z i , π ↓ z j ) def = sup { r The construction in the periodic setting is analogous. Indeed, it suffices toreplace the family of backward Brownian motions { B ↓ k } k with a family of periodic ones de-fined via B ↓ , per k def = B ↓ k mod 1 , take a countable dense set (cid:68) per def = { w k = ( s k , y k ) : k ∈ N } of R × T , build { π per , ↓ w k } k ∈ N as before and define ζ per , ↓ n ( (cid:68) per ) = ( T per , ↓ n ( (cid:68) ) , ∗ ↓ , d ↓ , M per , (cid:68) per , ↓ n ) as in (3.2).The construction above readily implies a number of properties each of the ζ ↓ n ( (cid:68) ) ’senjoys. Indeed, for every n ∈ ¯ N , ζ ↓ n ( (cid:68) ) is a spatial R -tree which is monotone in both spaceand time according to Definition 2.19 and, as a consequence of the fact that Browniantrajectories are α -Hölder continuous for any α < , it is immediate to see that, at least for n finite, ζ ↓ n ( (cid:68) ) ∈ T α sp . In the next proposition, we will show that the sequence { ζ ↓ n ( (cid:68) ) } n is notonly tight in T α sp for any α < / , but it actually converges to a unique limit in C α sp which isa characteristic spatial R -tree and can be explicitly characterised starting from ζ ↓∞ ( (cid:68) ) . Proposition 3.2 Let (cid:68) be a countable dense of R containing ( , ) and, for n ∈ ¯ N , let ζ ↓ n ( (cid:68) ) def = ( T ↓ n ( (cid:68) ) , ∗ ↓ , d ↓ , M ↓ , (cid:68) n ) be defined according to (3.2) . Then, for every α < thesequence { ζ ↓ n ( (cid:68) } n ∈ N converges in T α sp to a unique limit ζ ↓ ( (cid:68) ) def = ( T ↓ ( (cid:68) ) , ∗ ↓ , d ↓ , M ↓ , (cid:68) ) ,where T ↓ ( (cid:68) ) is the completion of T ↓∞ ( (cid:68) ) and M ↓ , (cid:68) is the unique continuous extension of M ↓ , (cid:68) ∞ to all of T ↓ ( (cid:68) ) .Moreover, almost surely, for any fixed θ > and all r > there exists a constant c = c ( r ) > depending only on r such that for all ε > (cid:78) d ↓ ( T ↓ , ( r ) ( (cid:68) ) , ε ) ≤ cε − θ (3.3)he Brownian Web Tree and its dual where (cid:78) d ↓ ( T ↓ , ( r ) ( (cid:68) ) , ε ) is defined as in (2.12) , i.e. it is the cardinality of the minimal ε -netin T ↓ , ( r ) ( (cid:68) ) with respect to d ↓ .At last, almost surely M ↓ , (cid:68) is surjective and ( t ) holds.Proof. We fix (cid:68) once and for all for the duration of this proof and therefore suppress itsdependence in the notations. By construction, the sequence { ζ ↓ n } n of α -spatial R -trees is suchthat for every n ∈ N , ζ ↓ n is embedded into ζ ↓ n +1 , ζ ↓ n is monotone in both space and time and forevery z = ( t, x ) ∈ R there exists n z such that for all n ≥ n z , ( M ↓ n ) − ( { t } × [ x − , x + 1 ]) is not empty by the density of (cid:68) . Lemmas 2.17 and 2.22 guarantee that, providedthat the sequence is tight in T α sp , it converges to a unique characteristic α -spatial R -tree ζ ↓ = ( T ↓ , ∗ ↓ , d ↓ , M ↓ ) which satisfies the properties in the statement.Since every ζ ↓ n is canonically embedded in ζ ↓∞ = ( T ↓∞ , ∗ ↓ , d ↓ , M ↓∞ ) , if we show that,almost surely, T ↓∞ (which is an R -tree and hence, by Point 2 in Theorem 2.6 so is itscompletion) is locally compact and M ↓∞ is proper and uniformly little α -Hölder continuouson bounded balls, then we have a bound uniform in n on both the size of the ε -nets of ballsin T ↓ n and the local modulus of continuity of M ↓ n , so that tightness of the sequence followsreadily from Proposition 2.16.Let r ≥ . We start by introducing an event on which T ↓ , ( r ) ∞ is enclosed betweentwo paths. Let R > r , Q ± R be two squares of side centred at ( r + 1 , ± ( R + 1 )) and z ± = ( t ± , x ± ) be two points in (cid:68) ∩ Q ± R , respectively. By the non-crossing property of ourcoalescing paths, on the event E R def = { sup ≥ s ≥− r | π ↓ ( s ) | ≤ R , sup t ± ≥ s ≥− r | π ↓ z ± ( s ) − x ± | ≤ R } (3.4)any element ( s, π ↓ z ) ∈ T ↓ , ( r ) ∞ with z = ( t, x ) ∈ (cid:68) is necessarily such that s ∈ [ − r, t ∧ r ] and π ↓ z − ( s ) < π ↓ z ( s ) < π ↓ z + ( s ) . Moreover, by the reflection principle, we have P ( E cR ) ≤ C √ rR e − R r (3.5)where E cR is the complement of E R in Ω , and C is a positive constant independent of r and R . Now, in order to show that, almost surely, T ↓ , ( r ) ∞ is relatively compact, note that P ( (cid:78) d ( T ↓ , ( r ) ∞ , ε ) ≥ Kε − θ ∀ ε ∈ ( , ] ) ≤ (cid:88) n ≥ P ( (cid:78) d ( T ↓ , ( r ) ∞ , − n ) ≥ K θ ( n − ) ) . Hence, the following lemma together with Borel–Cantelli imply (3.3) (and consequentlyrelative compactness).he Brownian Web Tree and its dual There exists a constant C = C ( r ) > such that P ( (cid:78) d ( T ↓ , ( r ) ∞ , ε ) > Kε − / ) ≤ C √ K (3.6) uniformly over ε ∈ ( , ] and K ≥ .Proof. Let R > r and set ˜ R def = 3 R + 1 . For t , t ∈ R , t > t , we define Ξ R ( t , t ) def = (cid:110) (cid:37) ( z , t ) : M ↓∞ ,t ( z ) > t and M ↓∞ ,x ( (cid:37) ( z , t )) ∈ [ − ˜ R, ˜ R ] (cid:111) (3.7)where (cid:37) is the radial map of T ↓∞ defined as in (2.23), and set η R ( t , t ) to be the cardinalityof Ξ R ( t , t ) . By the definition of T ↓∞ , η R ( t , t ) has the same distribution as the quantity ˆ η ( t , t ; − ˜ R, ˜ R ) of [FINR04, Definition 2.1] (see in particular the comment below), whichis almost surely finite by [FINR04, Proposition 4.1].Consider the numbers L ε and the times t εk given by L ε def = (cid:108) rε (cid:109) + 1 , t εk def = r − k ε , k = 0 , . . . , L ε − , (3.8)where, for x ∈ R , (cid:100) x (cid:101) is the least integer greater than x . We now claim that, on the event E R , (cid:78) d ( T ↓ , ( r ) ∞ , ε ) ≤ L ε (cid:88) k =1 η R ( t εk , t εk +1 ) . (3.9)Indeed, if ( t, π ↓ z ) ∈ T ↓ , ( r ) ∞ for some t ∈ R and z ∈ (cid:68) , then by definition of the metric t ∈ [ − r, r ] and π ↓ z − ( t ) < π ↓ z ( t ) < π ↓ z + ( t ) , since we are on E R . Then, there exists k ∈ { , . . . , L ε − } such that t ∈ [ t εk +1 , t εk ] and, consequently, a unique element z ∈ Ξ R ( t k +1 , t k +2 ) , necessarily belonging to T ↓ , ( r ) ∞ , such that, by the coalescing property, (cid:37) (( t, π ↓ z ) , t εk +2 ) = z . Since d ↓ (( t, π ↓ z ) , z ) ≤ ε/ < ε , (3.9) follows. Therefore, we obtain P ( (cid:78) d ( T ↓ , ( r ) ∞ , ε ) ≥ N ) ≤ P (cid:16) E cR ∪ (cid:110) L ε (cid:88) k =1 η R ( t εk , t εk +1 ) > N (cid:111)(cid:17) (3.10) ≤ P ( E cR ) + N − L ε (cid:88) k =1 E [ η R ( t εk , t εk +1 )] ≤ C √ rR e − R r + C L ε R √ εN ,for some constant C > , where the last inequality follows from [SSS17, Proposition 2.7].Setting N = Kε − / , it suffices to choose R = √ K to obtain (3.6).he Brownian Web Tree and its dual We now focus on the Hölder continuity of the map M ↓∞ (cid:22) T ↓ , ( r ) ∞ . In this case, it sufficesto show that limsup ε → P ( sup {(cid:107) M ↓∞ ( z ) − M ↓∞ ( z (cid:48) ) (cid:107) : z , z (cid:48) ∈ T ↓ , ( r ) ∞ s.t. d ↓ ( z , z (cid:48) ) ≤ ε } ≤ ε α ) = 1 , (3.11)for some α < / (then taking at most an even smaller α one deduces the little Hölderproperty). We claim that, on the event E R , M ↓∞ (cid:22) T ↓ , ( r ) ∞ is α -Hölder continuous providedthat the paths π ↓ z , z ∈ (cid:68) , restricted to the box Λ r,R def = [ − r, r ] × [ − ˜ R, ˜ R ] satisfy a suitableequi-Hölder continuity condition. The latter can be stated in terms of a modulus of continuityof the form (see also the proof of [SSS17, Theorem 2.3]) Ψ T ↓∞ ,R,r ( ε ) def = sup {| π ↓ z ( s ) − π ↓ z ( t ) | : z ∈ (cid:68) , M ↓∞ ( s, π ↓ z ) ∈ Λ r,R , t ∈ [ s, s + ε ] } for ε ∈ ( , ) . Indeed, on E R , assume Ψ T ↓∞ ,R,r ( ε ) ≤ ε α / and let ( s, π ↓ z ) , ( t, π ↓ z (cid:48) ) ∈ T ↓ , ( r ) ∞ be such that d ↓ (( s, π ↓ z ) , ( t, π ↓ z (cid:48) )) ≤ ε . Then, necessarily, M ↓∞ ( s, π ↓ z ) , M ↓∞ ( s, π ↓ z (cid:48) ) ∈ Λ r,R andboth s − τ ↓ s,t ( π ↓ z , π ↓ z (cid:48) ) and t − τ ↓ s,t ( π ↓ z , π ↓ z (cid:48) ) ≤ ε . Therefore, by the coalescing property, (cid:107) M ↓∞ ( s, π ↓ z ) − M ↓∞ ( s, π ↓ z (cid:48) ) (cid:107) = | π ↓ z ( s ) − π ↓ z (cid:48) ( t ) | ∨ | t − s |≤ (cid:16) | π ↓ z ( τ ↓ s,t ( π ↓ z , π ↓ z (cid:48) )) − π ↓ z ( s ) | + | π ↓ z (cid:48) ( τ ↓ s,t ( π ↓ z , π ↓ z (cid:48) )) − π ↓ z (cid:48) ( t ) | (cid:17) ∨ | t − s | ≤ ε α . The following lemma concludes the proof of (3.11). Lemma 3.4 There exists a constant C = C ( r ) > such that P (cid:16) Ψ T ↓∞ ,R,r ( ε ) > ε α (cid:17) ≤ Cε α + e − ε α − (3.12) uniformly over ε ∈ ( , ] .Proof. We proceed similarly to what done in the proofs of [FINR04, Proposition B.1 andB.3] and in [SSS17, Theorem 2.3]. We introduce the grid G εr,R def = { ( n ε, m ε α / ) : m, n ∈ Z }∩ Λ r,R . For any z = ( t , x ) ∈ G εr,R , we define the rectangles R − z = [ t + ε/ , t + ε/ ] × [ x − ε α / , x − ε α / ] and R + z = [ t + ε/ , t − ε/ ] × [ x + 5 ε α / , x + 7 ε α / ] and consider two points z ± ∈ (cid:68) ∩ R ± z . Let π ↓ z ± be the backward Brownian motions startingfrom z ± respectively.Assume now that Ψ T ↓∞ ,R,r ( ε ) > ε α / , then there exists a path π ↓ z , z ∈ (cid:68) such that | π ↓ z ( s ) − π ↓ z ( t ) | > ε α / , for some s for which ( π ↓ z ( s ) , s ) ∈ Λ r,R and t ∈ [ s − ε, s ] . Then,pick the closest point z = ( t , x ) ∈ G εR,r , for which | π ↓ z ( s ) − x | ≤ ε α / and | s − t | ≤ ε .he Brownian Web Tree and its dual By the coalescing property of our paths, it follows that necessarily one between π ↓ z ± mustbe such that sup h ∈ [ , ε ] | π ↓ z ± ( t − h ) − x | ≥ ε α / . Let E εR,r ( z ) def = { sup h ∈ [ , ε ] | π ↓ z ± ( t − h ) − x | ≤ ε α / } , then, again by the reflectionprinciple we have P ( E cR ∪ ( E εR,r ) c ) ≤ P ( E cR ) + (cid:88) z ∈ G εR,r P (( E εR,r ( z )) c ) ≤ C √ r + 1 R e − R r +1 + C Rrε α − / e − ε α − and upon taking R = ε − , (3.12) follows.We now want to show properness of M ↓∞ , which is a direct consequence of the followinglemma. Lemma 3.5 There exists a constant c > independent of r such that for any K > sufficiently large P (cid:16) b ζ ↓∞ ( r ) ≥ K (cid:17) ≤ c r √ K (3.13) where b ζ ↓∞ is the properness map given in (2.2) .Proof. Let R > and consider two squares ˜ Q ± R of side centred at ( r + 1 , ± ( r + R + 1 )) .Let ˜ z ± = ( ˜ t ± , ˜ x ± ) be two points respectively belonging to ˜ Q ± R ∩ (cid:68) , and without loss ofgenerality, assume ˜ t + = ˜ t − = ˜ t . Let π ↓ ˜ z ± be the two paths starting from ˜ z ± . For K > r ,we introduce the event ˜ E KR,r def = (cid:110) sup − r ≤ s ≤ ˜ t π ↓ ˜ z − ( s ) < − r , inf − r ≤ s ≤ ˜ t π ↓ ˜ z + ( s ) > r , τ ↓ ( π ↓ ˜ z + , π ↓ ˜ z − ) > r − K (cid:111) , (3.14)where in τ ↓ we omitted the subscript since we imposed ˜ t + = ˜ t − . Notice that on ˜ E KR,r , for anypoint ( s, π ↓ z ) ∈ T ↓∞ such that M ↓∞ ( s, π ↓ z ) ∈ Λ r , by the coalescing property, the trajectoriesof both π ↓ and π ↓ z , after time s must be confined between those of π ↓ ˜ z + and π ↓ ˜ z − . Therefore, d ↓ (( π ↓ z , s ) , ( π ↓ , )) ≤ r − τ ↓ ( π ↓ ˜ z + , π ↓ ˜ z − ) < K , so that one has P (cid:16) b ζ ↓∞ ( r ) ≥ K (cid:17) ≤ P (( ˜ E KR,r ) c ) ,he Brownian Web Tree and its dual independently of the choice of R . The reflection principle, combined with standard tailestimates on the first time a Brownian motion hits a specified level, yield a bound of the type P (( ˜ E KR,r ) c ) ≤ C √ rR e − R r + C R + r + 1 √ K , (3.15)for some constant C > , and (3.13) follows at once, upon choosing R def = √ K .Since M ↓ is proper, (cid:68) , which is dense in R , is contained in M ↓ , (cid:68) ( T ↓ ) , surjectivityfollows. At last, ( t ) is a direct consequence of the fact that, almost surely, it holds for ζ ↓∞ ( (cid:68) ) by construction and Lemma 2.30. Remark 3.6 Almost surely, the map M ↓ , (cid:68) is continuous and proper on T ↓∞ ( (cid:68) ) . Moreover,it is bijective on its image (endowed with the usual Euclidean topology) by construction,hence M ↓ , (cid:68) : T ↓∞ ( (cid:68) ) → M ↓ , (cid:68) ( T ↓∞ ( (cid:68) )) is a homeomorphism. Remark 3.7 The previous proposition remains true if instead of the sequence ζ ↓ n ( (cid:68) ) we take ζ per , ↓ n ( (cid:68) per ) , (cid:68) per being a countable dense set of R × T . The proof is actually simpler sinceit is not necessary to introduce the event in (3.4). In the periodic setting, the convergencehappens in T α sp , per , the limit ζ per , ↓ ( (cid:68) per ) = ( T per , ↓ n ( (cid:68) ) , ∗ ↓ , d ↓ , M per , ↓ , (cid:68) per ) belongs to C α sp , per and M per , ↓ , (cid:68) per ( T per , ↓ n ( (cid:68) )) = R × T .The next theorem introduces and uniquely characterises the law on the space ofcharacteristic trees of the random variable which in the sequel we will refer to as the Brownian Web tree . Theorem 3.8 Let α < . There exists a C α sp -valued random variable ζ ↓ bw = ( T ↓ bw , ∗ ↓ bw , d ↓ bw , M ↓ bw ) with radial map (cid:37) ↓ , whose law is uniquely characterised by the following properties1. for any deterministic point w = ( s, y ) ∈ R there exists almost surely a unique point w ∈ T ↓ bw such that M ↓ bw ( w ) = w ,2. for any deterministic n ∈ N and w = ( s , y ) , . . . , w n = ( s n , y n ) ∈ R , the jointdistribution of ( M ↓ bw ,x ( (cid:37) ↓ ( w i , · ))) i =1 ,...,n , where w , . . . , w n are the points determinedin 1., is that of n coalescing backward Brownian motions starting at w , . . . , w n ,3. for any deterministic countable dense set (cid:68) such that ∈ (cid:68) , let w be the pointdetermined in 1. associated to w ∈ (cid:68) and ˜ ∗ ↓ that associated to . Define ˜ ζ ↓∞ ( (cid:68) ) = ( ˜ T ↓∞ ( (cid:68) ) , ˜ ∗ ↓ , d ↓ , ˜ M ↓ , (cid:68) ∞ ) as ˜ T ↓∞ ( (cid:68) ) def = { (cid:37) ↓ ( w , t ) : w = ( s, y ) ∈ (cid:68) (cid:48) , t ≤ s } ˜ M ↓ , (cid:68) ∞ ( (cid:37) ↓ ( w , t )) def = M bw ( (cid:37) ↓ ( w , t )) (3.16)he Brownian Web Tree and its dual and d ↓ to be the ancestral metric in (3.2) . Let ˜ T ↓ ( (cid:68) ) be the completion of ˜ T ↓∞ ( (cid:68) ) under d ↓ , ˜ M ↓ , (cid:68) be the unique little α -Hölder continuous extension of ˜ M ↓ , (cid:68) ∞ and ˜ ζ ↓ ( (cid:68) ) def = ( ˜ T ↓ ( (cid:68) ) , ∗ ↓ , d ↓ , ˜ M ↓ , (cid:68) ) . Then, ˜ ζ ↓ ( (cid:68) ) law = ζ ↓ bw .Moreover, almost surely, ζ ↓ bw satisfies (3.3) for any fixed θ > / , M ↓ bw is surjective and ( t )holds.Proof. Let (cid:68) be a countable dense set of R containing . Thanks to Proposition 3.2, for α < , ζ ↓ ( (cid:68) ) almost surely belongs to C α sp 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 ) / ∈ (cid:68) , and consider two sequences of points z ± n = ( t ± n , x ± n ) ∈ (cid:68) for which there exist two constants c ± > such that y − c n < x − n < y < x + n < y − c n s < t − n < s + | x − n | and s < t + n < s + | x + n | . For every n ∈ N , let π ↓ z ± n be the two backward Brownian motions starting at z ± n respectively.Denote by τ n = τ ↓ t − n ,t + n ( π ↓ z − n , π ↓ z + n ) and X n = π ↓ z − n ( τ n ) = π ↓ z + n ( τ n ) the time and spatial pointat which they coalesce. Define ∆ n as the triangular region in R with vertices z ± n and ( τ n , X n ) , the base being given by the segment joining z − n and z + n , while the sides by thepaths ( r, π ↓ z − n ( r )) t − n ≥ r ≥ τ n , ( r, π ↓ z + n ( r )) t + n ≥ r ≥ τ n .In the proof of [FINR03, Proposition 3.1] the authors show that the event E n def = (cid:110) π ↓ z − n ( s ) < y < π ↓ z + n ( s ) , τ n ≥ s − /n , | X n − y | < n − / (cid:111) occurs infinitely often. Hence, for any sequence z m = ( t m , x m ) ∈ (cid:68) converging to w , forall n ∈ N large enough there exists m n ∈ N such that, for all m ≥ m n , z m ∈ ∆ n . Thecoalescing property then implies that for every m , m ≥ m n , d ↓ (( t m , π ↓ z m ) , ( t m , π ↓ z m )) ≤ ( t m + t m ) − τ n ≤ ( t m − s ) + ( t m − s ) + 2 /n . In other words, for any z m = ( t m , x m ) ∈ (cid:68) converging to w , ( t m , π ↓ z m ) m ∈ N is Cauchy in T ↓∞ ( (cid:68) ) therefore it converges in T ↓ ( (cid:68) ) to a unique point w which, by continuity of M ↓ , (cid:68) ,is necessarily such that M (cid:68) ( w ) = w .Moreover, by construction we know that (cid:37) ↓ (( t m , π ↓ z m ) , t ) = ( π ↓ z m , t ) for all t ≤ t m and,since at τ n the ray starting at w must have coalesced with that starting at ( π ↓ z n , t n ) , we musthave (cid:37) ↓ ( w , t ) = (cid:37) ↓ (( π ↓ z n , t n ) , t ) for any t ≤ τ n . Hence, the sequence of paths ( −∞ , t m ] (cid:51) t (cid:55)→ M ↓ , (cid:68) x ( (cid:37) ↓ (( t m , π ↓ z m ) , t )) = M ↓ , (cid:68) x ( t, π ↓ z m ) converges to ( −∞ , s ] (cid:51) t (cid:55)→ M ↓ , (cid:68) x ( (cid:37) ↓ ( w , t )) he Brownian Web Tree and its dual in Π , where Π is given as in Appendix ?? . Since ( M ↓ , (cid:68) x ( (cid:37) ↓ (( t m , π ↓ z m ) , t ))) t ≤ t m is distributedaccording to a backward Brownian motion starting at z m , ( M ↓ , (cid:68) x ( (cid:37) ↓ ( w , t )) t ≤ s is itselfdistributed according to a backward Brownian motion, but starting at w .For 2., let w , . . . , w n be n deterministic points in R and w , . . . , w n be the pointsin T ↓ ( (cid:68) ) determined by applying 1. Thanks to the last part of the proof of 1., if z m i = ( t m i , x m i ) is a sequence in (cid:68) converging to w i , i ∈ [ n ] , then the paths ( M ↓ , (cid:68) x ( · , π ↓ z mi )) i ∈ [ n ] converge to ( M ↓ , (cid:68) x ( (cid:37) ↓ ( w , · )) i ∈ [ n ] in Π n . Since the first are distributed as coalescingbackward Brownian motions starting from ( z m , . . . , z m n ) , it is easy to see that the limit willbe also distributed according to coalescing Brownian motions starting from ( w , . . . , w n ) .We now prove 3., for which we proceed as follows. Let (cid:68) (cid:48) be another countabledense set in R containing ( , ) . We want to determine a suitable coupling of ζ ↓ ( (cid:68) ) and ˜ ζ ↓ ( (cid:68) (cid:48) ) under which they are almost surely equal. We first construct ζ ↓ ( (cid:68) (cid:48) ) as in (3.2) andProposition 3.2. Then, we build ˜ ζ ↓∞ ( (cid:68) (cid:48) ) = ( ˜ T ↓∞ ( (cid:68) (cid:48) ) , ˜ ∗ ↓ , d ↓ , ˜ M ↓ , (cid:68) (cid:48) ∞ ) inside ζ ↓ ( (cid:68) ) accordingto (3.16). Obviously ζ ↓ ( (cid:68) (cid:48) ) and ˜ ζ ↓ ( (cid:68) (cid:48) ) are equal in distribution, and the latter is suchthat ˜ T ↓ ( (cid:68) (cid:48) ) ⊆ T ↓ ( (cid:68) ) , ˜ ∗ ↓ = ∗ ↓ and M ↓ , (cid:68) (cid:22) ˜ T ↓ ( (cid:68) (cid:48) ) = ˜ M ↓ , (cid:68) (cid:48) . Therefore, if we are ableto show that ˜ T ↓ ( (cid:68) (cid:48) ) coincides with T ↓ ( (cid:68) ) , we are done. We claim that if z ∈ (cid:68) and z ∈ T ↓ ( (cid:68) ) is the unique point such that M ↓ , (cid:68) ( z ) = z (which holds by 1.) then z alsobelongs to ˜ T ↓ ( (cid:68) (cid:48) ) . Notice that if this is the case then for all z ∈ (cid:68) , if z ∈ T ↓ ( (cid:68) ) is theunique point such that M ↓ , (cid:68) ( z ) = z then z ∈ ˜ T ↓ ( (cid:68) (cid:48) ) . It follows that all the rays startingfrom these z ’s are contained in ˜ T ↓ ( (cid:68) (cid:48) ) and hence also the closure of their union, which byconstruction is T ↓ ( (cid:68) ) . We turn to the proof of the claim.Let z ∈ (cid:68) and z ∈ T ↓ ( (cid:68) ) be the unique point such that M ↓ , (cid:68) ( z ) = z . Let w n = ( s n , y n ) be a sequence in (cid:68) (cid:48) converging to z in R . By 1., we know that for all n there exists aunique point w n in T ↓ ( (cid:68) ) such that M ↓ , (cid:68) ( w n ) = w n and since ˜ T ↓∞ ( (cid:68) (cid:48) ) ⊆ T ↓ ( (cid:68) ) and,by construction, there is a unique point in ˜ T ↓∞ ( (cid:68) (cid:48) ) whose image is w n , it follows that w n ∈ ˜ T ↓∞ ( (cid:68) (cid:48) ) . Now, the map M ↓ , (cid:68) is proper and the sequence { w n } n is bounded, thereforethe sequence { w n } n is also bounded and it converges along subsequences. Fix one of thesesubsequences (that, with a slight abuse of notation, will still be indexed by n ) and noticethat by continuity of M ↓ , (cid:68) and uniqueness of z , we necessarily have that ( w n ) n converges to z in T ↓ ( (cid:68) ) . Now, since { w n } n converges, it is Cauchy and since it is contained in ˜ T ↓∞ ( (cid:68) (cid:48) ) ,the limit must belong to ˜ T ↓ ( (cid:68) (cid:48) ) .It remains to argue uniqueness and the properties of the limit. Let ζ ↓ bw be as in thestatement and, for a given countable dense set (cid:68) = { z n = ( t n , x n ) : n ∈ N } with z = 0 ,let ζ ↓ bw ( (cid:68) ) be constructed as in (3.2) and Proposition 3.2. Notice that, thanks to the proofof 3. above, the distribution of ζ ↓ ( (cid:68) ) is independent of the choice of (cid:68) . Now, by 1. and2. define ζ n = ( T n , ∗ , d ↓ bw , M ↓ bw ) as follows, T n def = { (cid:37) ↓ ( z n , t ) : M ↓ bw ( z n ) = z n and t ≤ t n } so that T n ⊂ T ↓ bw . By construction, ζ n and ζ ↓ n ( (cid:68) ) (in (3.2)) are equal in law for every n .Moreover, 3. and Lemma 2.17 guarantee that the sequence { ζ n } n converges to ζ ↓ bw , whilehe Brownian Web Tree and its dual the convergence of { ζ ↓ n ( (cid:68) ) } n to ζ ↓ ( (cid:68) ) is implied by Proposition 3.2. Therefore, ζ ↓ bw and ζ ↓ ( (cid:68) ) are equal in law. In particular, by Proposition 3.2 also the other claimed propertiesof ζ ↓ bw hold and the proof is concluded. Remark 3.9 The theorem above remains true upon replacing conditions . - . with per . , per . and per . , obtained from the former by adding the word “periodic” before any instanceof “Brownian motion”, and taking the periodic version of all objects and spaces in thestatement. Definition 3.10 Let α < . We define backward Brownian Web Tree and periodic backwardBrownian Web tree , the C α sp and C α sp , per random variables ζ ↓ bw = ( T ↓ bw , ∗ ↓ bw , d ↓ bw , M ↓ bw ) and ζ per , ↓ bw = ( T per , ↓ bw , ∗ per , ↓ bw , d per , ↓ bw , M per , ↓ bw ) whose distributions is uniquely characterised byproperties . - . in Theorem 3.8 and per . , per . and per . in Remark 3.9. We will denote theirrespective laws by Θ ↓ bw (d ζ ) and Θ per , ↓ bw (d ζ ) .As a first property of the Brownian Web tree, which can be deduced by Theorem 3.8and 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 ) isgiven by dim box ( T ) def = lim ε → log (cid:78) d ( T, ε )log ε − (3.17)when this limit exists. Corollary 3.11 Let ζ ↓ bw = ( T ↓ bw , ∗ ↓ bw , d ↓ bw , M ↓ bw ) be the Brownian Web tree of Definition 3.10.Then, almost surely T ↓ bw has box-covering dimension .Proof. According to (3.17), it suffices to determine almost sure upper and lower boundsfor (cid:78) d ↓ bw ( T ↓ , ( r )bw , ε ) of the same order, for all r > . Now, the upper bound follows by thefact that, by Theorem 3.8, almost surely T ↓ bw satisfes (3.3). For the lower bound, it sufficesto show that there are at least O ( ε − / ) points in T ↓ , ( r )bw at distance ε . This is turn can bededuced 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 Webtrees on the space of R -trees is, as expected, singular with respect to those of the BrownianTree of Aldous and the scaling limit of the Uniform Spanning Tree in two dimensions.Indeed, the first has Hausdorff dimension [Ald91a], while the other / [BCK17] andthe Hausdorff dimension is always greater than or equal to the box-counting one (seee.g. [Edg98, Chapter 1]).he Brownian Web Tree and its dual In the following Corollary, we establish the relation between the Brownian Web Treeof Definition 3.8 and the Brownian Web constructed in [FINR04], which is a simpleconsequence of Theorem 3.8 and the results in Section 2.4. Corollary 3.13 Let ζ ↓ bw and ζ per , ↓ bw be the backward and backward periodic Brownian Webtrees of Theorem 3.8 and Remark 3.9, and K be the map defined in (2.26) . Then, K ( ζ ↓ bw ) is a backward Brownian Web according to [FINR04, Theorem 2.1] and K ( ζ ↓ bw , per ) is abackward cylindric Brownian Web according to [CMT19, Theorem 2.3].Proof. To prove the statement it suffices to verify that K ( ζ ↓ bw ) and K ( ζ ↓ bw , per ) satisfy ( o ) , ( i ) and ( ii ) in [FINR04, Theorem 2.1] and [CMT19, Theorem 2.3], respectively. This is inturn an immediate consequence of the definition of K and properties 1.-3. in Theorem 3.8and per . - per . in Remark 3.9. In this section, we want to derive a criterion that allows to conclude that the limit law fortight sequences of characteristic spatial R -trees is Θ ↓ bw . Theorem 3.14 Let α ∈ ( , ) and { ζ n } n be a tight sequence of random variables in C α sp with laws Θ n and assume that the following holds.(I) For any k ∈ N and (deterministic) z , . . . , z k ∈ R there exist sequences z in ∈ T n , i =1 , . . . , k such that lim n →∞ M n ( z in ) = z in almost surely and such that ( M n ( (cid:37) n ( z in , · ))) i converges in law to k coalescing backward Brownian motions.(II) For every h > ε limsup n →∞ sup ( t,x ) ∈ R Θ n (cid:0) { (cid:37) ( w , t − h ) : w ∈ M − ( I t,x,ε ) } ≥ (cid:1) ε → −→ (3.18) where I t,x,ε def = { t } × ( x − ε, x + ε ) .Then Θ n converges weakly to Θ ↓ bw . Remark 3.15 In view of Corollary 3.13, the Brownian Web tree and the Brownian Webare strictly connected so that it is not surprising that the convergence criterion stated aboveis extremely similar to [FINR04, Theorem 2.2]. As a matter of fact, requiring the sequence ζ n to be made of characteristic trees, allows us to talk about paths, while the fact thatwe are dealing with trees enforces the non-crossing condition. That said, even thoughProposition 2.25 guarantees continuity of the map assigning to any characteristic tree acompact subset of Π , the inverse map is not continuous, even when restricted to C α sp ( t ) , sothat we cannot infer convergence in C α sp from [FINR04].he Brownian Web Tree and its dual Proof. Let K be the map defined in (2.26). Notice at first that (I) implies that { K ( ζ n ) } n satisfies [FINR04, Theorem 2.2(I1)] and that, since { (cid:37) n ( w , t − h ) : w ∈ M − n ( I t,x,ε ) } ≥ { M n ( (cid:37) n ( w , t − h )) : w ∈ M − n ( I t,x,ε ) } , (3.19)(II) implies [FINR04, Theorem 2.2(B2)], so that { K ( ζ n ) } n converges in law to the backwardBrownian Web. Since the sequence ζ n is tight by assumption, it converges along somesubsequence. Let ζ be a limit point, (cid:37) its radial map and denote by Θ its law on C α sp . Since K is continuous by Proposition 2.25, K ( ζ n ) converges to K ( ζ ) , which by the above is abackward Brownian Web. Since K is injective on C α sp ( t ) by Proposition 2.25, it remains toshow that ζ satisfies ( t ).If ( t ) fails then, with positive probability, one can find t, x, ε, h ∈ Q with h > suchthat the inequality in (3.19) is strict, so the proof is complete if we show that for any fixedvalues this happens with probability . Fix t, x, ε, h ∈ Q with h > and, for N ∈ N , let x Nj def = x + jεN , z Nj = ( t, x j ) , for j = − N, . . . , N .For i = 1 − N, . . . , N , let y Ni denote the mid-point of of the interval ( x Ni − , x Ni ) . By ourassumptions we know that almost surely, there exist unique points z Nj ∈ T such that M ( z Nj ) = z Nj for all j , so that, in particular { (cid:37) ( z Nj , t − h ) : | j | ≤ N } = { M ( (cid:37) ( z Nj , t − h )) : | j | ≤ N } . Hence, Θ (cid:0) { (cid:37) ( w , t − h ) : w ∈ M − ( I t,x,ε ) } > { M ( (cid:37) ( w , t − h )) : w ∈ M − ( I t,x,ε ) } (cid:1) ≤ lim N →∞ Θ (cid:0) { (cid:37) ( w , t − h ) : w ∈ M − ( I t,x,ε ) } > { (cid:37) ( z Nj , t − h ) : | j | ≤ N } (cid:1) . Moreover, by space monotonicity, { (cid:37) ( w , t − h ) : w ∈ M − ( I t,x,ε ) } > { (cid:37) ( z Nj , t − h ) : | j | ≤ N } can only happen if there exists i such that { (cid:37) ( w , t − h ) : w ∈ M − ( I t,y Ni , εN ) } ≥ . In other words, Θ( { (cid:37) ( w , t − h ) : w ∈ M − ( I t,x,ε ) } > { (cid:37) ( z Nj , t − h ) : | j | ≤ N } ) ≤ N (cid:88) i =1 − N Θ (cid:16) { (cid:37) ( w , t − h ) : w ∈ M − ( I t,y Ni , εN ) } ≥ (cid:17) (cid:46) N sup ( t,y ) ∈ R Θ (cid:0) { (cid:37) ( w , t − h ) : w ∈ M − ( I t,y, εN ) } ≥ (cid:1) (cid:46) N limsup n →∞ sup ( t,y ) ∈ R Θ n (cid:0) { (cid:37) ( w , t − h ) : w ∈ M − n ( I t,y, εN ) } ≥ (cid:1) ,which converges to as N → ∞ by (3.18), and the claim follows.he Brownian Web Tree and its dual A crucial aspect of the backward Brownian Web is that it comes naturally associated with adual (see e.g. [TW98, FINR06]), which is given by a family of forward coalescing Brownianmotions starting from every point in R or R × T , in the periodic case. In the next theoremwe will see how it is possible to devise such a duality in the present context and characterisethe joint law of the Brownian Web tree in Definition 3.10 and its dual. Theorem 3.16 Let α < / . There exists a C α sp × ˆ C α sp -valued random variable ζ ↓↑ bw def = ( ζ ↓ bw , ζ ↑ bw ) , ζ • bw = ( T • bw , ∗ • bw , d • bw , M • bw ) , • ∈ {↓ , ↑} , whose law is uniquely characterised bythe following properties(i) Both − ζ ↑ bw def = ( T ↑ bw , ∗ ↑ bw , d ↑ bw , − M ↑ bw ) and ζ ↓ bw are distributed as the backward BrownianWeb tree in Definition 3.10.(ii) Almost surely, for any z ↓ ∈ T ↓ bw and z ↑ ∈ T ↑ bw , the paths M ↓ bw ( (cid:37) ↓ ( z ↓ , · )) and M ↑ bw ( (cid:37) ↑ ( z ↑ , · )) do not cross, where (cid:37) ↓ (resp. (cid:37) ↑ ) is the radial map of ζ ↓ bw (resp. ζ ↑ bw ).Moreover, almost surely ζ ↓↑ bw ∈ C α sp ( t ) × ˆ C α sp ( t ) and ζ ↑ bw is determined by ζ ↓ bw and vice-versa.Finally, ( K ( ζ ↓ bw ) , ˆ K ( ζ ↑ bw )) 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 (cid:88) × (cid:89) ,we say that X is determined by Y if the conditional law of X given Y is almost surely givenby a Dirac mass. Proof. Throughout the proof, we will adopt the notations and conventions of Section 2.4.Notice at first that, by Theorem 3.8, any C α sp × ˆ C α sp -valued random variable for which (i)holds, almost surely belongs to C α sp ( t ) × ˆ C α sp ( t ) .Now, let ( W ↓ , W ↑ ) be the (cid:72) × ˆ (cid:72) -valued random variable constructed in [SSS17,Theorem 2.1] and K the map in (2.26). Since W ↓ is distributed as the backward BrownianWeb, by Corollary 3.13, W ↓ law = K ( ζ ↓ bw ) and W ↑ law = − W ↓ law = − K ( ζ ↓ bw ) = ˆ K ( − ζ ↓ bw ) , wherethe first equality is due to [SSS17, Theorem 2.1(a)] and the last is a consequence ofRemark 2.26. Therefore, ( W ↓ , W ↑ ) ∈ K ( C α sp ( t )) × ˆ K ( ˆ C α sp ( t )) almost surely so that, byProposition 2.25 and Remark 2.26, there exists a unique couple ( ζ W ↓ , ζ W ↑ ) ∈ C α sp ( t ) × ˆ C α sp ( t ) such that ( K ( ζ W ↓ ) , ˆ K ( ζ W ↑ )) = ( W ↓ , W ↑ ) . By Proposition 3.2 and Theorem 3.8 we alsohave ζ ↓ bw ∈ C α sp ( t ) almost surely so that, since K ( ζ W ↓ ) law = K ( ζ ↓ bw ) and K ( − ζ W ↑ ) law = K ( ζ ↓ bw ) , ( ζ W ↓ , ζ W ↑ ) satisfies (i). The definition of the map K in (2.25) and (2.26) combinedhe Brownian Web Tree and its dual with [SSS17, Theorem 2.1(b)] ensures that (ii) holds for ( ζ W ↓ , ζ W ↑ ) . The fact that ζ W ↑ isdetermined by ζ W ↓ is a direct consequence of the fact that this is known to be true for W ↓ and W ↑ and that K is invertible on C α ( t ) .We argue uniqueness. Let ( ζ, ζ (cid:48) ) be another random variable in C α sp × ˆ C α sp whichsatisfies (i) and (ii). Now, ( t ) holds for both ζ and ζ (cid:48) , while (i), (ii) and (2.26) ensure that ( K ( ζ ) , ˆ K ( ζ (cid:48) )) satisfies [SSS17, Theorem 2.1 (a)-(b)]. Hence, the conclusion follows by theuniqueness 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 theobjects and spaces appearing in the statement with their periodic counterparts. The prooffollows the exact same lines but uses Remarks 3.9 and 2.32 instead of Theorem 3.8 andProposition 2.25. Definition 3.19 Let α < . We define the double Brownian Web tree and double periodicBrownian Web tree as the C α sp × ˆ C α sp and C α sp , per × ˆ C α sp , per -valued random variables ζ ↓↑ bw def = ( ζ ↓ bw , ζ ↑ bw ) and ζ per , ↓↑ bw def = ( ζ per , ↓ bw , ζ per , ↑ bw ) given by Theorem 3.16 and Remark 3.18. We will referto ζ ↑ bw and ζ per , ↑ bw as the forward (or dual) and forward periodic Brownian Web trees .We denote their laws by Θ ↓↑ bw (d( ζ ↓ × ζ ↑ )) and Θ per , ↓↑ bw (d( ζ ↓ × ζ ↑ )) , with marginals Θ ↓ bw (d ζ ) , Θ ↑ bw (d ζ ) and Θ per , ↓ bw (d ζ ) , Θ per , ↑ bw (d ζ ) 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 doubleBrownian Web tree directly starting from a countable family of (independent) forward andbackward 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 3.2and Theorem 3.8.As a first consequence of the duality the Brownian Web tree enjoys we show that eachof the R -trees T ↑ bw and T ↓ bw has a unique open end with unbounded rays. This end shouldbe thought of as the point at ( ± ) ∞ where all the Brownian motions coalesce. As we willsee in Proposition 3.25, the periodic Brownian Web tree, instead, has (exactly) two openends with unbounded rays which are connected by a unique bi-infinite edge. Proposition 3.21 Let ζ ↑ bw and ζ ↓ bw be respectively the forward and backward Brownian Webtrees. Then, almost surely, the R -trees T ↑ bw and T ↓ bw have precisely one open end withunbounded rays, which we denote by † ↑ and † ↓ respectively.Proof. We prove the result for T ↑ bw , the other being analogous by duality. Notice that thestatement follows if we show that for every r > almost surely there exists a compacthe Brownian Web Tree and its dual K ⊂ T ↑ bw with T ↑ , ( r )bw ⊂ K , such that for all z , z (cid:48) ∈ K c the path connecting z and z (cid:48) does notintersect T ↑ , ( r )bw . Thanks to the double Brownian Web tree we are able to exhibit an explicitcompact set for which the latter claim holds. Let r > be fixed, (cid:68) be a countable dense setin R containing and recall that, with probability one, ζ ↓ bw = ζ ↓ ( (cid:68) ) .Using the same notations and conventions as in the proof of Proposition 3.2, let ˜ E NR,r be defined according to (3.14). Set τ def = τ ↓ ( π ↓ ˜ z + , π ↓ ˜ z − ) , X def = π ↓ ˜ z + ( τ n ) = π ↓ ˜ z − ( τ n ) and let ∆ N be the triangular region of R with vertices ˜ z ± and ( τ, X ) , base given by the segmentjoining ˜ z + and ˜ z − , and sides formed by the paths ( s, π ↓ ˜ z − ( s )) ˜ t − ≥ s ≥ τ , ( s, π ↓ z + n ( s )) t + n ≥ s ≥ τ . On ˜ E NR,r , ∆ N is compact and the properness of M ↑ bw guarantees that so is K N def = ( M ↑ bw ) − ( ∆ N ) .By point (ii) in Theorem 3.16 paths in the forward and backward Web trees do not cross,therefore T ↑ , ( r )bw ⊂ K N and the path connecting any two points in K cN cannot intersect T ↑ , ( r )bw . Hence, it remains to argue that there is an almost surely finite N for which therealisation of ζ ↓ bw belongs to ˜ E NR,r . This in turn is a direct consequence of (3.15) and astandard application of Borel–Cantelli.We are now interested in deriving properties of the inverse maps ( M • bw ) − and ( M per , • bw ) − ,for • ∈ {↑ , ↓} , and how these are related to the degrees of points in the R -trees T • bw and T per , • bw . We begin with the following proposition, which is a translation in the language ofthe present paper of [FINR06, Proposition 3.10]. Proposition 3.22 Let ζ ↓↑ bw = ( ζ ↑ bw , ζ ↓ bw ) and ζ ↓↑ , perbw = ( ζ ↑ , perbw , ζ ↓ , perbw ) be the double and doubleperiodic Brownian Web trees. Then, almost surely for every point z = ( t, x ) ∈ R | ( M ↑ bw ) − ( z ) | − | ( M ↓ bw ) − ( z ) | (cid:88) i =1 (deg( z ↓ i ) − ) (3.20) where { z ↓ i } i are the points in ( M ↓ bw ) − ( z ) and | ( M • bw ) − ( z ) | denotes the cardinality of ( M • bw ) − ( z ) . The relation (3.20) holds as well with the arrows ↑ , ↓ reversed and for theirperiodic counterpart.Proof. As usual we will focus on the non-periodic case, the other being analogous.We claim that for all z = ( t, z ) ∈ R , | ( M ↓ bw ) − ( z ) | = m b out ( z ) and the right-hand sideof (3.20) coincides with m b in ( z ) , where m b out ( z ) and m b in ( z ) are defined according to [FINR06,(3.11) and (3.10)] and respectively represent the number of distinct paths “leaving” and“entering” the point z for the backward Brownian Web (by removing the superscript b andreverting the arrows the same holds for the forward by duality).Indeed, for every z ↓ ∈ ( M ↓ bw ) − ( z ) , denoting by (cid:37) ↓ the radial map associated to ζ ↓ bw , wehave that ( −∞ , t ] (cid:51) s (cid:55)→ M ↓ bw ,x ( (cid:37) ↓ ( z ↓ , s )) is a path from z . On the other hand, deg( z ↓ ) − he Brownian Web Tree and its dual corresponds to the number of rays in the tree which coalesce at or reach z . Notice that, sincealmost surely ζ ↓ bw satisfies ( t ), the image of the rays coalescing or reaching z as well as thatof the rays from points in ( M ↓ bw ) − ( z ) are distinct so that the claim follows.Now, by Theorem 3.16 ( K ( ζ ↓ bw ) , ˆ K ( ζ ↑ bw )) is distributed as the double Brownian Weband almost surely ζ ↓↑ bw ∈ C α sp ( t ) × ˆ C α sp ( t ) . Since moreover the restriction of K to C α sp ( t ) isbijective 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 R or in R × T based on the meaningthey have for the (periodic) Brownian Web tree (and its dual) as we constructed it. Definition 3.23 Let ζ ↓↑ bw = ( ζ ↑ bw , ζ ↓ bw ) be the double Brownian Web tree. For • ∈ {↑ , ↓} , thetype of a point z ∈ R for ζ • bw is ( i, j ) ∈ N , where i = | ( M • bw ) − ( z ) | (cid:88) i =1 (deg( z • i ) − ) and j = | ( M • bw ) − ( z ) | . Above, { z • i : i ∈ { , . . . , | ( M • bw ) − ( z ) |}} = ( M • bw ) − ( z ) . We define S ↑ i,j (resp. S ↓ i,j ) as thesubset of R containing all points of type ( i, j ) for the forward (resp. backward) BrownianWeb tree. For the periodic Brownian Web ζ per , ↓↑ bw = ( ζ per , ↑ bw , ζ per , ↓ bw ) , the definition is the sameas above and the set of all of points in R × T of type ( i, j ) for the backward (resp. forward)periodic Brownian Web tree, will be denoted by S per , ↓ i,j (resp. S per , ↑ i,j ). Theorem 3.24 For the backward and backward periodic Brownian Web trees ζ ↓ bw and ζ ↓ , perbw ,almost surely, every z ∈ R (resp. R × T ) is of one of the following types, all of which occur: ( , ) , ( , ) , ( , ) , ( , ) , ( , ) and ( , ) . Moreover, almost surely, for every t ∈ R - S ↓ , has full Lebesgue measure on R and S ↓ , ∩ { t } × R has full Lebesgue measurein { t } × R ,- S ↓ , and S ↓ , have Hausdorff dimension / while S ↓ , ∩ { t } × R and S ↓ , ∩ { t } × R are both countable and dense in { t } × R ,- S ↓ , has Hausdorff dimension , S ↑ , and S ↑ , are countable and dense while S ↓ , ∩ { t } × R , S ↓ , ∩ { t } × R and S ↓ , ∩ { t } × R have each cardinality at most 1.For deterministic times t , S ↓ , ∩ { t } × R , S ↓ , ∩ { t } × R and S ↓ , ∩ { t } × R are almostsurely empty. Upon reversing all arrows, the properties above hold for the forward andforward periodic Brownian Web trees.Proof. Arguing as in the proof of Proposition 3.22, the statement follows immediatelyby [FINR06, Theorems 3.11, 3.13 and 3.14].he Brownian Web Tree and its dual Thanks to the classification above, we can now prove one of the features that distinguishesthe Brownian Web tree and its periodic version. In the next proposition we show that theperiodic Brownian Web tree possesses a unique bi-infinite path connecting its two openends with unbounded rays. Proposition 3.25 For • ∈ {↓ , ↑} , let ζ per , • bw = ( T per , • bw , ∗ per , • bw , d per , • bw , M per , • bw ) be the periodicbackward and forward Brownian Web trees of Definition 3.19. Then, almost surely, each T per , ↓ bw and T per , ↑ bw has exactly two open ends with unbounded rays and a unique bi-infiniteedge connecting them.Proof. Since T per , ↓ bw and T per , ↑ bw are periodic characteristic trees, we already know they haveone open end with unbounded rays, and this is the one for which (2.20) holds (for theforward periodic Web see Remark 2.21). Denote them by † ↓ and † ↑ and let (cid:37) ↓ per and (cid:37) ↑ per bethe radial maps introduced in (2.23) and Remark 2.24, respectively. Similarly to (3.7), for t , t ∈ R , t < t , we introduce Ξ ↑ T ( t , t ) def = { (cid:37) ↑ per ( z , t ) : z ∈ T per , ↑ bw and M per , ↑ t, bw ( z ) ≤ t } Ξ ↓ T ( t , t ) def = { (cid:37) ↓ per ( z , t ) : z ∈ T per , ↓ bw and M per , ↓ t, bw ( z ) ≥ t } and set η ↑ T ( t , t ) and η ↓ T ( t , t ) to be the cardinality of Ξ ↑ T ( t , t ) and Ξ ↓ T ( t , t ) respectively.We inductively define the sequence of stopping times τ def = inf { t > η ↑ T ( , t ) = 1 } τ k def = inf { t > τ k − : η ↑ T ( τ k − , t ) = 1 } . These stopping times coincide (in distribution) with those in the proof of [CMT19, Theorem3.1], where it is further showed that almost surely lim k →∞ τ k = + ∞ .Now, by definition, for every k ≥ , there must exist a point z k − ∈ T × { τ k − } suchthat | ( M per , ↑ bw ) − ( z k − ) | ≥ and the distance of (at least) two elements in ( M per , ↑ bw ) − ( z k − ) is ( τ k − τ k − ) . By (3.20) and Theorem 3.24, it follows that there exists exactly one point ( M per , ↓ bw ) − ( z k − ) whose degree is greater or equal to . Denote it by z k . Then the map β ↓ : R → T per , ↓ bw given by β ↓ ( s ) def = (cid:40) (cid:37) ↓ per ( z k , s ) , for s ∈ ( τ k − , τ k ] (cid:37) ↓ per ( z , s ) for s < .is not only well-defined by Theorem 3.16(ii) but also uniquely defined since so is the choiceof the point z k . The map β ↓ shows that there are exactly two open ends with unboundedrays, and β ↓ ( R ) is the unique linear subtree of T per , ↓ bw satisfying the properties in [Chi01,Lemma 3.7(i)].he Discrete Web Tree and convergence In this section, we introduce the discrete web and its dual and show that the couple convergesto the Double Brownian Web Tree of Definition 3.19. We begin our analysis with the spatial tree representation of a family of coalescing backwardrandom walks and its dual. The construction below will directly provide a coupling betweenforward and backwards paths under which one is determined by the other and the two satisfythe non-crossing property of Theorem 3.16(ii).Let δ ∈ ( , ] and ( Ω , (cid:65) , P δ ) be a standard probability space supporting four Poissonrandom measures, µ Lγ , µ Rγ , ˆ µ Lγ and ˆ µ Rγ . The first two, µ Lγ and µ Rγ , live on D ↓ δ def = R × δ Z , areindependent and have both intensity γλ , where, for every k ∈ δ Z , λ (d t, { k } ) is a copy ofthe Lebesgue measure on R and throughout the section γ = γ ( δ ) def = 12 δ . (4.1)The others live on D ↑ δ def = R × δ ( Z + 1 / ) , and are obtained from the formers by setting, forevery measurable A ⊂ D ↑ δ ˆ µ Lγ ( A ) def = µ Rγ ( A − δ/ ) and ˆ µ Rγ ( A ) def = µ Lγ ( A + δ/ ) . (4.2)Here, A ± δ/ is the translate of A in the spatial direction, i.e. A ± δ/ def = { z ± ( , δ/ ) : z ∈ A } .From now on, we will adopt the convention of writing z ∈ µ • γ , • ∈ { R, L } , if z is anevent of the given realisation of µ • γ . We represent the Poisson points of µ Lγ , µ Rγ , ˆ µ Lγ and ˆ µ Rγ with arrows as follows. If z ∈ µ Lγ (resp. µ Rγ ) then we draw an arrow from z to z − δ (resp. z + δ ), and similarly for ˆ µ Lγ and ˆ µ Rγ , as shown in Figure 2. We also define µ Tγ = { z − δ : z ∈ µ Lγ } ∪ { z + δ : z ∈ µ Rγ } , (4.3)and similarly for ˆ µ Tγ . (Here, T stands for ‘tip’ since µ Tγ denotes the collection of all tips ofarrows.)Let us now introduce two families of random walks. We define { π ↓ ,δz ( s ) } s ≤ t , for z = ( t, y ) ∈ D ↓ δ , as the random walk going backwards in time, “following the arrows”determined by µ Lγ and µ Rγ , and, for z = ( t, y ) ∈ D ↑ δ , { π ↑ ,δz ( s ) } s ≥ t as the forward randomwalk which follows those of ˆ µ L and ˆ µ R , as shown in Figure 2. (By convention, if z is thestart of an arrow, then π ↓ ,δz and π ↑ ,δz start by going downwards / upwards.) These are almosthe Discrete Web Tree and convergence y ˆ y t Figure 2: Graphical representation of the realisation of the Poisson processes µ L and µ R ,and their dual ˆ µ L and ˆ µ R which respectively live on D ↓ δ and D ↑ δ . The red and blue linesillustrate the restrictions of the backward and forward paths π ↓ ,δ ( t,y ) and π ↑ ,δ ( , ˆ y ) to the interval [ , t ] .surely well-defined µ Lγ and µ Rγ are disjoint with probability one and, for all z ∈ D ↓ δ and ˆ z ∈ D ↑ δ , π ↓ ,δz is càglàd (or càdlàg if we run time backwards from + ∞ to −∞ ), while π ↑ ,δ ˆ z iscàdlàg. Moreover, { π ↓ ,δz } z and { π ↑ ,δ ˆ z } ˆ z are coalescing families of paths starting from everypoint in D ↓ δ and D ↑ δ respectively, which do not cross. Definition 4.1 Let δ ∈ ( , ] , γ as in (4.1), µ Lγ and µ Rγ be two independent Poisson randommeasures on D ↓ δ of intensity γλ , ˆ µ L and ˆ µ R be given as in (4.2) and { π ↓ ,δz } z ∈ D ↓ δ and { π ↑ ,δ ˆ z } ˆ z ∈ D ↑ δ be the families of coalescing random walks introduced above. We define the Double Discrete Web Tree as the couple ζ ↓↑ δ def = ( ζ ↓ δ , ζ ↑ δ ) , in which- ζ ↓ δ def = ( T ↓ δ , ∗ ↓ δ , d ↓ δ , M ↓ δ ) is given by setting T ↓ δ = D ↓ δ , ∗ ↓ δ = ( , ) , M ↓ δ the canonicalinclusion, and d ↓ δ ( z, ¯ z ) = t + t (cid:48) − sup { s ≤ t ∧ t (cid:48) : π ↓ ,δz ( s ) = π ↓ ,δ ¯ z ( s ) } . (4.4)- ζ ↑ δ def = ( T ↑ δ , ∗ ↑ δ , d ↑ δ , M ↑ δ ) is built similarly, but with ∗ ↑ δ = ( , δ/ ) and the supremum in(4.4) replaced by inf { s ≥ t ∨ t (cid:48) : π ↑ ,δz ( s ) = π ↑ ,δ ¯ z ( s ) } .Notice that neither the Discrete Web Tree ζ ↓ δ nor its dual are characteristic spatial R -trees.Indeed, even though they satisfy the conditions of Definition 2.19 and Remark 2.21 theevaluation maps are discontinuous.To circumvent this technical issue, we introduce two connected subsets of R , (cid:83) ↓ δ and (cid:83) ↑ δ , obtained by interpolating the Poisson points of µ • γ and ˆ µ • γ , • ∈ { L, R } , and which willrepresent the image of modified evaluation maps. Fix a realisation of µ • γ , • ∈ { L, R } ,he Discrete Web Tree and convergence and consider µ Tγ as in (4.3). Given z = ( t, x ) ∈ D ↓ δ , we then define z ↓ as follows. Let t ↓ = sup { s < t : ( s, x ) ∈ µ Rγ ∪ µ Lγ ∪ µ Tγ } and set z ↓ = ( t ↓ , x + cδ ) , where c = 1 if ( t ↓ , x ) ∈ µ R , c = − if ( t ↓ , x ) ∈ µ L , and c = 0 otherwise. We then define (cid:83) ↓ δ as the union ofall closed line segments joining z to z ↓ with z ∈ µ Rγ ∪ µ Lγ ∪ µ Tγ . Given z = ( t, x ) ∈ D ↓ δ andsetting z ↑ = ( t ↑ , x ) with t ↑ = inf { s ≥ t : ( s, x ) ∈ µ Rγ ∪ µ Lγ ∪ µ Tγ } , we write ˜ M ↓ δ ( z ) ∈ (cid:83) ↓ δ for the unique element on the line segment joining z ↑ to z ↓ with the same time coordinate as z . The set (cid:83) ↑ δ is defined similarly, but with time reversed. It is immediate to see that, almostsurely, the sets (cid:83) ↓ δ and (cid:83) ↑ δ are well-defined and connected. With the previous construction athand we are ready for the following definition. Definition 4.2 In the same setting as Definition 4.1, we define the Interpolated DoubleDiscrete Web Tree as the couple ˜ ζ ↓↑ δ def = ( ˜ ζ ↓ δ , ˜ ζ ↑ δ ) in which ˜ ζ • δ def = ( T • δ , ∗ • δ , d • δ , ˜ M • δ ) , • ∈ {↑ , ↓} ,and ( T • δ , ∗ • δ , d • δ ) coincides with that of ζ • δ , while the evaluation map ˜ M • δ is defined as justdescribed. Proposition 4.3 For any δ ∈ ( , ] and α ∈ ( , ) , almost surely the interpolated doubleDiscrete Web tree ˜ ζ ↓↑ δ in Definition 4.2 belongs to C α sp × ˆ C α sp and the evaluation maps ˜ M • δ , • ∈ {↑ , ↓} are bijective on (cid:83) • δ . Moreover, it satisfies the following two properties(i δ ) − ˜ ζ ↑ δ + δ/ law = ˜ ζ ↓ δ where − ˜ ζ ↑ δ + δ/ def = ( T ↑ δ , ∗ ↑ δ , d ↑ δ , − ˜ M ↑ δ + δ/ ) (ii δ ) almost surely, for every z ↓ ∈ T ↓ δ and z ↑ ∈ T ↑ δ there exists c ∈ { +1 , − } such thatfor all ˜ M ↑ δ,t ( z ↑ ) ≤ s < s ≤ ˜ M ↓ δ,t ( z ↓ ) (cid:89) i =1 , ( ˜ M ↑ δ,x ( (cid:37) ↑ ( z ↑ , s i )) − ˜ M ↓ δ,x ( (cid:37) ↓ ( z ↓ , s i )) + cδ ) ≥ At last, almost surely, for • ∈ {↑ , ↓} sup z ∈ T • δ (cid:107) ˜ M • δ ( z ) − M • δ ( z ) (cid:107) ≤ δ (4.5) where M • δ 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 ofPoisson random measures and the definition of the sets (cid:83) ↓ δ and (cid:83) ↑ δ .he Discrete Web Tree and convergence We are now ready to show that the family { ˜ ζ ↓↑ δ } δ is tight. Proposition 4.4 Let α ∈ ( , ) and, for δ ∈ ( , ] , let Θ ↓↑ δ be the law on C α sp × ˆ C α sp of theInterpolated Double Discrete Web Tree ˜ ζ ↓↑ δ = ( ˜ ζ ↓ δ , ˜ ζ ↑ δ ) of Definition 4.2 and denote by Θ • δ with • ∈ {↑ , ↓} the law of ˜ ζ • δ . Then, for any α ∈ ( , ) the family Θ ↓↑ δ is tight in C α sp × ˆ C α sp .Furthermore, for any θ > and r > , the following holds lim K ↑∞ liminf δ ↓ Θ ↓ δ (cid:16) ∀ ε ∈ ( , ] , (cid:78) d ( T ( r ) , ε ) ≤ Kε − θ (cid:17) = 1 . (4.6) Proof. Let us point out that since by Proposition 4.3(i δ ), − ˜ ζ ↑ δ + δ/ law = ˜ ζ ↓ δ , it suffices toshow that the family { Θ ↓ δ } δ is tight in C α sp .In view of Proposition 2.16 and Lemma 2.22 we need to prove that for every r > , thelimits (4.6), and lim ε ↓ liminf δ ↓ Θ ↓ δ (cid:16) sup {(cid:107) M ( z ) − M ( w ) (cid:107) : z , w ∈ T ( r ) , d ( z , w ) ≤ ε } ≤ ε α (cid:17) = 1 , (4.7) lim K ↑∞ liminf δ ↓ Θ ↓ δ ( b ζ ( r ) ≤ K ) = 1 , (4.8)hold. These can be shown by following the same strategy and estimates as in the proof ofProposition 3.2, so that below we will adopt the notations and conventions therein.Notice at first that, for any z = ( t, x ) in a countable dense set (cid:68) of R , if { z δ } δ is suchthat for all δ ∈ ( , ] , z δ ∈ D ↓ δ and { z δ } δ converges to z , then, by Donsker’s invarianceprinciple, the backward random walk π ↓ ,δz δ defined above converges in law to a backwardBrownian motion π ↓ z started at z .Let { z ± δ } δ ⊂ Q ± R ∩ ( D δ ) be sequences converging to z ± . Denoting by E δR , the event E R in (3.4), but in which z ± is replaced by z ± δ , we see that the previous observation implies liminf δ ↓ P δ ( E δR ) = P ( E R ) (4.9)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 > and a < b limsup δ ↓ E δ [ η R ( a, b )] ≤ E [ η R ( a, b )] (4.10)where η R ( a, b ) is the cardinality of Ξ R ( a, b ) given in (3.7) and E δ is the expectation withrespect to P δ . Thanks to (4.9) and (4.10), we can argue as in Lemma 3.3 and obtain thathe Discrete Web Tree and convergence there exists a constant C = C ( r ) > independent of δ such that for all K > limsup δ ↓ P δ ( (cid:78) d ( T ( r ) , ε ) > Kε − θ ) ≤ C √ K so that by Borel-Cantelli (4.6) follows.As in Proposition 3.2, the uniform local Hölder continuity of the evaluation maps M ↓ δ can be reduced to properties of the paths π ↓ ,δ . For fixed R and r , let Ψ δ ( ε ) def = sup {| π ↓ ,δz ( s ) − π ↓ ,δz ( t ) | : z ∈ D δ , M ↓ δ ( s, π ↓ ,δz ) ∈ Λ r,R , t ∈ [ s − ε, s ] } If Ψ δ ( ε ) ≤ ε α / for every ε ≥ δ , then for every ( s, π ↓ ,δz ) , ( t, π ↓ ,δz (cid:48) ) ∈ T ↓ , ( r ) δ such that d ↓ δ (( s, π ↓ ,δz ) , ( t, π ↓ ,δz (cid:48) )) ≤ ε , we have | M ↓ δ,x ( s, π ↓ ,δz ) − M ↓ δ,x ( t, π ↓ ,δz (cid:48) ) | ≤ δ + ε α ≤ ε α . where we exploited the triangle inequality and (4.5). Therefore, (4.7) follows at once if limsup ε → liminf δ → Θ ↓ δ (cid:0) Ψ δ ( ε ) ≤ ε α / (cid:1) = 1 . (4.11)This in turn follows from the same arguments as in the proof of Lemma 3.4, together withthe fact that if { z + ,δ } δ and { z − ,δ } δ are sequences of points in R ± z ∩ ( D ↓ δ ) converging to z +0 and z − ∈ (cid:68) respectively, then liminf δ ↓ P δ (cid:16) sup h ∈ [ , ε ] | π ↓ ,δz ± ,δ ( t − h ) − x | ≤ ε α / (cid:17) = P ( E εR,r ( z )) . Finally, (4.8) can be proved by proceeding as in Lemma 3.5 and adapting the definitionof the event ˜ E KR,r in (3.14) as done for E R above.In the following theorem we show that the Interpolated Double Discrete Web treeconverges in law to the Double Brownian Web Tree. Theorem 4.5 Let α ∈ ( , / ) and, for δ ∈ ( , ] , Θ ↓↑ δ be the law on C α sp × ˆ C α sp of theInterpolated Double Discrete Web Tree ˜ ζ ↓↑ δ in Definition 4.2. Then, as δ ↓ , Θ ↓↑ δ convergesto Θ ↓↑ bw weakly on C α sp × ˆ C α sp .Proof. Thanks to Proposition 4.4, the sequence { ζ ↓↑ δ = ( ζ ↓ δ , ζ ↑ δ ) } δ is tight in C α sp × ˆ C α sp .Moreover, Proposition 4.3 (i δ ) and (ii δ ) imply that any limit point ζ ↓↑ = ( ζ ↓ , ζ ↓ ) must besuch that − ζ ↑ law = ζ ↓ and the non-crossing property holds. In view of Theorem 3.16, thehe Discrete Web Tree and convergence statement then follows once we show that ζ ↓ δ → ζ ↓ bw in law as δ → . To do so, we willapply Theorem 3.14, for which we need to verify the validity of (I) and (II).Clearly, for any z , . . . , z k ∈ R , if { z iδ } δ is such that z iδ ∈ D δ and z iδ → z i as δ → ,then ( π ↓ ,δz iδ ( · )) i converges in law to a family of coalescing Brownian motions starting at z , . . . , z k . Since furthermore (4.5) holds, (I) follows.For (II), our construction implies that, for any t, x ∈ R , h, ε > , { (cid:37) ↓ δ ( w , t − h ) w ∈ ( ˜ M ↓ δ ) − ( I t,x,ε ) } law = ˆ η δ ( t, t + h ; x − ε, x + ε ) , where (cid:37) ↓ δ is the radial map of ζ δ and ˆ η δ wasdefined in [FINR04, Definition 2.1] . For the latter, the statement was shown in the proofof [FINR04, Theorem 6.1]. References [ADH13] R. Abraham, J.-F. Delmas, and P. Hoscheit. A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces. Electron. J.Probab. , (2013), no. 14, 21. doi:10.1214/EJP.v18-2116 .[Ald91a] D. Aldous. The continuum random tree. I. Ann. 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