The Brownian Castle

February 8, 2021

G. Cannizzaro1,2 and M. Hairer1

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

Abstract We introduce a 1 + 1-dimensional temperature-dependent model such that the classical ballistic deposition model is recovered as its zero-temperature limit. Its ∞-temperature version, which we refer to as the 0-Ballistic Deposition (0-BD) model, is a randomly evolving interface which, surprisingly enough, does not belong to either the Edwards– Wilkinson (EW) or the Kardar–Parisi–Zhang (KPZ) universality class. We show that 0-BD has a scaling limit, a new stochastic process that we call Brownian Castle (BC) which, although it is “free”, is distinct from EW and, like any other renormalisation fixed point, is scale-invariant, in this case under the 1 : 1 : 2 scaling (as opposed to 1 : 2 : 3 for KPZ and 1 : 2 : 4 for EW). In the present article, we not only derive its finite-dimensional distributions, but also provide a “global” construction of the Brownian Castle which has the advantage of highlighting the fact that it admits backward characteristics given by the (backward) Brownian Web (see [TW98, FINR04]). Among others, this characterisation enables us to establish fine pathwise properties of BC and to relate these to special points of the Web. We prove that the Brownian Castle is a (strong) Markov and Feller process on a suitable space of càdlàg functions and determine its long-time behaviour. At last, we give a glimpse to its universality by proving the convergence of 0-BD to BC in a rather strong sense. arXiv:2010.02766v2 [math.PR] 5 Feb 2021 Contents

1 Introduction 2 1.1 The 0-Ballistic Deposition and its scaling limit ...... 5 1.2 The Brownian Castle: a global construction and main results ...... 7 Introduction 2

1.3 The BC universality class and further remarks ...... 11 1.4 Outline of the paper ...... 12 2 The Brownian Web Tree and its topology 15 2.1 Characteristic R-trees in a nutshell ...... 15 2.2 The tree map ...... 19 2.3 The (Double) Brownian Web tree ...... 22 3 Branching Spatial Trees and the Brownian Castle measure 25 3.1 Branching Spatial R-trees ...... 25 3.2 Stochastic Processes on trees ...... 27 3.3 Probability measures on the space of branching spatial trees ...... 31 3.4 The Brownian Castle measure ...... 39 4 The Brownian Castle 40 4.1 Pathwise properties of the Brownian Castle ...... 40 4.2 The Brownian Castle as a Markov process ...... 47 4.3 Distributional properties ...... 51 5 Convergence of 0-Ballistic Deposition 56 5.1 The Double Discrete Web tree ...... 57 5.2 The 0-BD measure and convergence to BC measure ...... 58 5.3 The 0-BD model converges to the Brownian Castle ...... 61 A The smoothened Poisson process 67 B On the cardinality of the coalescing point set of the Brownian Web 68 C Exit law of Brownian motion from the Weyl chamber 69 D Trimming and path correspondence 70

1 Introduction

The starting point for the investigation presented in this article is the one-dimensional Ballistic Deposition (BD) model [Fam86] and the long-standing open question concerning its large-scale behaviour. Ballistic Deposition (whose precise definition will be given below) is an example of random interface in (1 + 1)-dimensions, i.e. a map h: R+ × A → R, A being a subset of R, whose evolution is driven by a stochastic forcing. In this context, two universal behaviours, so-called universality classes, have generally been considered, the Kardar–Parisi–Zhang (KPZ) class, to which BD is presumed to belong [BS95, Qua12], and the Edwards–Wilkinson (EW) class. Originally introduced in [KPZ86], the first is conjectured to capture the large-scale fluctuations of all those models exhibiting some smoothing mechanism, slope-dependent growth speed, and short range randomness. The Introduction 3

“strong KPZ universality conjecture” states that for height functions h in this loosely defined class, the limit as δ → 0 of δ1h(·/δ3, ·/δ2) − C/δ2, where C is a model dependent constant, exists (meaning in particular that the scaling exponents of the KPZ class are 1 : 2 : 3), and is given by hKPZ, a universal (model-independent) stochastic process referred to as the “KPZ fixed point” (see [MQR16] for the recent construction of this process as the scaling limit of TASEP). If an interface model satisfies these features but does not display any slope-dependence, then it is conjectured to belong to the EW universality class [EW82], whose scaling exponents are 1 : 2 : 4 and whose universal fluctuations are Gaussian, given by the solutions hEW to the (additive) stochastic heat equation

1 2 ∂th = ∂ h + ξ , (1.1) EW 2 x EW with ξ denoting space-time white noise1. That said, there is a paradigmatic model in the KPZ universality class which plays a distinguished role. This model is a singular stochastic PDE, the KPZ equation, which can be formally written as 1 2 1 2 ∂th = ∂ h + (∂xh) + ξ . (1.2) 2 x 4 (The proof that it does indeed converge to the KPZ ﬁxed point under the KPZ scaling was recently given in [QS20, Vir20].) The importance of (1.2) lies in the fact that its solution is expected to be universal itself in view of the so-called “weak KPZ universality conjecture” [BG97, HQ15] which, loosely speaking, can be stated as follows. Consider any (suitably parametrised) continuous one-parameter family ε 7→ hε of interface growth models with the following properties

- the model h0 belongs to the EW universality class, - for ε > 0, the model hε belongs to the KPZ universality class.

Then it is expected that there exists a choice of constants Cε such that

1/2 −2 −1 lim ε hε(ε t, ε x) − Cε = h , (1.3) ε→0 with h solving (1.2). One can turn this conjecture on its head and take a result of the form (1.3) as suggestive evidence that the models hε are indeed in the KPZ universality class for ﬁxed ε. What we originally intended to do with the Ballistic Deposition model was exactly this: introduce a one-parameter family of interface models to which BD belongs and prove a limit of the type (1.3). Introduction 4

Figure 1: Example of two ballistic deposition update events. The two arrows indicate the locations of the two clocks that ring and the right picture shows the evolution of the left picture after these two update events.

The BD model is a Markov process h taking values in ZZ and informally deﬁned as follows. Take a family of i.i.d. exponential clocks (with rate 1) indexed by x ∈ Z and, whenever the clock located at x rings, the value of h(x) is updated according to the rule

h(x) 7→ max{h(x − 1), h(x) + 1, h(x + 1)} . (1.4)

This update rule is usually interpreted as a “brick” falling down at site x and then either sticking to the topmost existing brick at sites x − 1 or x + 1, or coming to rest on top of the existing brick at site x. See Figure 1 for an example illustrating two steps of this dynamic. The result of a typical medium-scale simulation is shown in Figure 2, suggesting that x 7→ h(x) is locally Brownian. A natural one-parameter family containing ballistic deposition is given by interpreting the maximum appearing in (1.4) as a “zero-temperature” limit and, for β ≥ 0, to consider instead the update rule

βy¯ h(x) 7→ y ∈ {h(x − 1), h(x) + 1, h(x + 1)} , P(y =y ¯) ∝ e . (1.5) As β → ∞, this does indeed reduce to (1.4), while β = 0 corresponds to a natural uniform reference measure for ballistic deposition. It is then legitimate to ask whether (1.3) holds if we take for hε the process just described with a suitable choice of β = β(ε). Surprisingly, this is not the case. The reason however is not that ballistic deposition isn’t in the KPZ universality class, but that its β = 0 version (which we will refer to as 0-Ballistic Deposition model) does not belong to the Edwards-Wilkinson universality class. Indeed, Figure 3 shows what a typical large-scale simulation of this process looks like. It is clear from this picture that its large-scale behaviour is not Brownian; in fact, it does not

1The choice of constants 1/2 and 1 appearing in (1.1) is no loss of generality as it can be enforced by a simple ﬁxed rescaling. Introduction 5

Figure 2: Medium-scale simulation of the ballistic deposition process. even appear to be continuous! The aim of this article is to describe the scaling limit of this process, which we denote by hBC and call the “Brownian Castle” in view of the turrets and crannies apparent in the simulation.

1.1 The 0-Ballistic Deposition and its scaling limit In order to understand how to characterise the Brownian Castle, it is convenient to take a step back and examine more closely the 0-Ballistic Deposition model. In view of (1.5), the dynamics of 0-BD is driven by three independent Poisson point processes µL, µR and µ• on R × Z, whose intensity is λ/2 for µR and µL and λ for µ•, λ being such that for every k ∈ Z, λ(dt, k) is the Lebesgue measure on R.2 Each event of µL, µR and µ• is responsible L of one of the three possible updates h0-bd(x) 7→ y of the height function, namely µ yields R • y = h0-bd(x + 1), µ yields y = h0-bd(x − 1), and µ yields y = h0-bd(x) + 1. Given a realisation of these processes, we can graphically represent them as in Figure 4, i.e. events of µL and µR are drawn as left / right pointing arrows, while for those of µ• are drawn as dots on R × Z. Z Assuming that the conﬁguration of 0-BD at time 0 is h0 ∈ Z , it is easy to see that for any z = (t, y) ∈ R+ × Z, the value h0-bd(z) can be obtained by going backwards following ↓ the arrows along the unique path πz starting at z and ending at a point in {0} × Z, say (0, x) (the red line in Figure 4), and adding to h0(x) the number of dots that are met along the way

2This is actually a slightly diﬀerent model from that described in (1.5) where the three processes have the same intensity. The present choice is so that as many constants as possible take the value 1 in the limit, but other (symmetric) choices yield the same limit modulo a simple rescaling. Introduction 6

Figure 3: Large-scale simulation of β = 0 ballistic deposition.

(in Figure 4, h0-bd(t, y) = h0(x) + 4). In order to obtain order one large-scale fluctuations for h0-bd, we clearly need to rescale ↓ space and time diffusively to ensure convergence of the random walks πz to Brownian motions. The size of the fluctuations should equally be scaled in a diffusive relation with time in order to have a limit for the fluctuations of the Poisson processes obtained by “counting the number of dots”. In other words, the scaling exponents governing the large-scale fluctuations should indeed be 1 : 1 : 2. The previous considerations immediately enable us to deduce the finite-dimensional distributions of the scaling limit of 0-BD and consequently lead to the following definition of the Brownian Castle. 3 Definition 1.1 Given h0 ∈ D(R, R) , we define the Brownian Castle (BC) starting from h0 as the process hbc : R+ × R → R with finite-dimensional distributions at space-time points {(ti, xi)}i≤k with ti > 0 given as follows. Consider k coalescing Brownian motions Bi, running backwards in time and such that each Bi is defined on [0, ti] with terminal value Bi(ti) = xi. For each edge e of the resulting rooted forest, consider independent Gaussian

3Here D(R, R) denotes all càdlàg functions from R to R. Introduction 7

y w t

0 x Z

Figure 4: Graphical representation of 0-Ballistic Deposition. The red lines represent the ↓,1 ↓,1 coalescing paths π(t,y) and π(t,w).

x1 x2 x3 x4 x5

B1(0) B3(0)

Figure 5: Example of construction of the 5-point distribution of the Brownian Castle.

random variables ξe with variance equal to the length `e of the time interval corresponding e h (t , x ) = h (B (0)) + P ξ E to . We then set bc i i 0 i e∈Ei e, where i denotes the set of edges that are traversed when joining the ith leaf to the root of the corresponding coalescence tree. See Figure 5 for a graphical description.

The existence of such a process hbc is guaranteed by Kolmogorov’s extension theorem. One goal of the present article is to provide a ﬁner description of the Brownian Castle which allows to deduce some of its pathwise properties and to show that 0-BD converges to it in a topology that is signiﬁcantly stronger than just convergence of k-point distributions.

1.2 The Brownian Castle: a global construction and main results As Definition 1.1 and the above discussion suggest, any global construction of the Brownian Castle must comprise two components. On the one hand, since we want to define it at all points simultaneously, we need a family of backward coalescing trajectories starting from Introduction 8 every space-time point and each distributed as a Brownian motion. On the other, we need a stochastic process indexed by the points on these trajectories whose increment between, say, 2 z1, z2 ∈ R , is a Gaussian random variable with variance given by their ancestral distance, i.e. by the time it takes for the trajectories started from z1 and z2 to meet. The first of these components is the (backward) Brownian Web which was originally constructed in [TW98] and further studied in [FINR04]. In the present context, it turns out to be more convenient to work with the characterisation provided by [CH21]. The latter has the advantage of highlighting the tree structure of the Web which in turn determines the distribution of the increments of the Gaussian process indexed by it. def ↓ The starting point in our analysis is to construct a random couple χbc = (ζbw,Bbc) (and an ↓ ↓ ↓ ↓ ↓ appropriate Polish space in which it lives), whose first component ζbw = (Tbw, ∗bw, dbw,Mbw) is the Brownian Web Tree of [CH21]. The terms in χbc can heuristically be described as

↓ ↓ ↓ - (Tbw, ∗bw, dbw) is an pointed R-tree (see Deﬁnition 2.1) which should be thought of as the set of “placeholders” for the points in the trajectories, and whose elements are ↓ ↓ morally of the form (s, πz ), where πz is a backward path in the Brownian Web W from z = (t, x) ∈ R2 (there can be more than 1!) and s < t is a time, and in which ↓ the distance dbw is the ancestral distance given by

↓ ↓ ↓ def ↓ ↓ ↓ dbw((t, πz ), (s, πz0 )) = (t + s) − 2τt,s(πz , πz0 ) , (1.6)

↓ ↓ ↓ ↓ def ↓ where the coalescence time τt,s is given by τt,s(πz , πz0 ) = sup{r < t ∧ s : πz (r) = ↓ πz0 (r)}, ↓ ↓ - Mbw is the evaluation map which associates to the abstract placeholder in Tbw the 2 ↓ ↓ ↓ actual space-time point in R to which it corresponds, i.e. Mbw : (s, πz ) 7→ (s, πz (s)), ↓ - Bbc is the branching map, which corresponds to the Gaussian process indexed by Tbw and such that

↓ ↓ 2 ↓ ↓ ↓ E[(Bbc(t, πz ) − Bbc(s, πz0 )) ] = dbw((t, πz ), (s, πz0 )) .

With such a couple at hand, we would like to deﬁne the Brownian Castle starting at h0 by setting

def ↓ ↓ ↓ hbc(z) = h0(πz (0)) + Bbc(t, πz ) − Bbc(0, πz ) , for all z ∈ R+ × R. (1.7)

The above definition implicitly relies on the fact that we are assigning to every point z ∈ R2 ↓ ↓ a point in Tbw (and consequently a path πz ∈ W), but, as it turns out (see [CH21, Theorem 3.24]), in the Brownian Web Tree there are “special points” from which more than one path originates. Since anyway this number is finite (at most 3), we can always pick the Introduction 9 right-most trajectory and ensure well-posedness of (1.7) (see the definition of the tree map in Section 2.2 that makes this assignment rigorous). The special points of the Brownian Web Tree are particularly relevant for the Brownian Castle because they are the points at which the discontinuities generating the turrets and crannies in Figure 2 can be located. Thanks to (1.7), we will not only be able to detect these points but also track the space-time behaviour of the (dense) set of discontinuities of hBC (see Section 4.1). In the following theorem, we loosely state some of the main results concerning the Brownian Castle which can be obtained by virtue of the construction above.

Theorem 1.2 The Brownian Castle in Deﬁnition 1.1 admits a version hbc such that for all h0 ∈ D(R, R), t 7→ hbc(t, ·) is a right-continuous map with values in D(R, R) endowed with the Skorokhod topology dSk in (1.15), which is continuous except for a countable subset of R+, but admits no version which is càdlàg in both space and time. hbc is a time-homogeneous D(R, R)-valued Markov process, satisfying both the strong Markov and the Feller properties, which is invariant under the 1 : 1 : 2 scaling, i.e. if i hbc with i ∈ {1, 2} are two instances of the Brownian Castle with possibly diﬀerent initial conditions at time 0, then, for all λ > 0, one has the equality in law

1 law −1 2 2 hbc(t, x) = λ hbc(λ t, λx), t ≥ 0, x ∈ R, (1.8) (viewed as an equality between space-time processes) provided that (1.8) holds as an equality between spatial processes at time t = 0. Moreover, when quotiented by vertical translations, hbc(t, ·) converges in law, as t → ∞, to a stationary process whose increments are Cauchy but which is singular with respect to the Cauchy process. A more precise formulation of this theorem, together with its proof, is split in the various statements contained in Section 4.

Remark 1.3 When we say that “a process h admits a version having property P ”, we mean that there exists a standard probability space Ω endowed with a collection of random variables h(z) such that for any ﬁnite collection {z1, . . . , zk}, the laws of (h(zi))i≤k and −1 (h(zi))i≤k coincide and furthermore h (P ) ⊂ Ω is measurable and of full measure. The second task of the present article is to show that the 0-Ballistic Deposition model indeed converges to it. Thanks to the heuristics presented above, in order to expect any meaningful limit, we need to recentre and rescale the 0-BD height function h0-bd according to the 1 : 1 : 2 scaling, so we set

δ def t x t h (t, x) = δ h0 , − , for all z ∈ + × . (1.9) 0-bd -bd δ2 δ δ2 R R Introduction 10

δ Now, given the way in which Definition 1.1 was derived, convergence of h0-bd to hbc in the sense of finite-dimensional distributions should not come as a surprise since it is an almost immediate consequence of Donsker’s invariance principle. That said, we aim at investigating a stronger form of convergence which relates 0-BD and BC as space-time processes. The major obstacle here is that Theorem 1.2 explicitly asserts that the Brownian Castle hbc does not live in any “reasonable” space which is Polish and in which point evaluation is a measurable operation, so that a priori it is not even clear in what sense such a convergence should be stated. It is at this point that our construction, summarised by the expression in (1.7), comes once more into play. As we have seen above, the version of the Brownian Castle hbc given in (1.7), is fully def ↓ determined by the couple χbc = (ζbw,Bbc), which in turn was inspired by the graphical representation of the 0-Ballistic Deposition model illustrated in Figure 4. For any realisation of the Poisson random measures µL, µR and µ• suitably rescaled and (for the latter) δ def ↓ ↓ def ↓ ↓ ↓ ↓ compensated, it is possible to build χ0-bd = (ζδ ,Nδ), in which ζδ = (Tδ , ∗δ, dδ,Mδ ) is the Discrete Web Tree of [CH21, Definition 4.1] and encodes the family of coalescing backward random walks π↓,δ naturally associated to the random measures µL and µR, while ↓ • Nδ is the compensated Poisson point process indexed by Tδ and induced by µ (the precise δ δ construction of χ0-bd can be found in Section 5.1). Given any initial condition h0 ∈ D(R, R), we now set, analogously to (1.7),

δ def δ ↓,δ ↓,δ ↓,δ h0-bd(z) = h0(πz¯ (0)) + Nδ(t, πz¯ ) − Nδ(0, πz¯ ) , for all z ∈ R+ × R, (1.10)

δ where, for z = (t, x), z¯ = (t, δbx/δc) ∈ R+ × (δZ). Notice that h0-bd is a càdlàg (in both δ δ space and time) version of h0-bd started from h0, in the sense that all of their k-point (in space-time) marginals coincide. In force of the previous construction, we are able to state the following theorem whose proof can be found at the end of Section 5.

δ δ Theorem 1.4 Let {h0}δ, h0 ⊂ D(R, R) be such that dSk(h0, h0) → 0. Then, for every δ sequence δn → 0 there exists a version of h0-bdn and hbc for which, almost surely, there exists a countable set of times D such that for every T ∈ R+ \ D

δ lim dSk(h0 bd(T, ·), hbc(T, ·)) = 0 . (1.11) δ→0 -

δ δ Here, h0-bd starts from h0 and is deﬁned in (1.10) while hbc is the Brownian Castle started from h0. This theorem also provides information concerning the nature and evolution of the discontinuities of the Brownian Castle. Indeed, for (1.11) to hold, it cannot be the case that many small discontinuities of 0-BD add up and ultimately create a large discontinuity for Introduction 11

BC. Instead, the statement shows that the major discontinuities of the former converge to those of the latter. To see this phenomenon and prove Theorem 1.4, the main ingredient is the convergence δ of χ0-bd to χbc (and that of the dual of the Discrete Web Tree to the dual of the Brownian Web Tree as stated in [CH21, Theorem 4.5]). The topology in which such a convergence holds (see Section 2) is chosen in such a way that the convergence of the evaluation maps morally provides a control over the sup norm distance of discrete and continuous backward trajectories and is therefore similar in spirit to that in, e.g., [FINR04], while that of the trees guarantees that couples of distinct discrete and continuous paths which are close also coalesce approximately at the same time. This is a crucial point (which moreover distinguishes our work from the previous ones) since it is at the basis of the convergence of δ Nδ to Bbc and ultimately ensures that of h0-bd to hbc. 1.3 The BC universality class and further remarks Over the last two decades, the KPZ (and EW) universality class has been at the heart of an intense mathematical interest because of the challenges it posed and the numerous physical systems which abide its laws (see [QS15] for a review and [ACQ11, SS08, MQR16, Vir20, QS20] among many other recent results). This article and the results stated herein establish the existence of a new universality class, which we will refer to as the BC universality class, by characterising a novel scale-invariant stochastic process, the Brownian Castle, which encodes the fluctuations of models in this class and arises as the scaling limit of a microscopic random system, the 0-Ballistic Deposition model. It is natural to wonder what are the features a model should exhibit in order to belong to it. Given the analysis of the Brownian Castle outlined above, it is reasonable to expect that any interface model which displays both horizontal and vertical fluctuations but no smoothing is an element of the BC universality class. The first type of fluctuations is responsible for the (coalescing) Brownian characteristics in the limit, while the second determines the Brownian motion indexed by them. A model which possesses these features and is somewhat paradigmatic for the class is the random transport equation given by

∂th = η ∂xh + µ , (1.12) where η and µ are two space-time stationary random fields, the first being responsible for the horizontal / lateral fluctuations and the second for the vertical ones. We conjecture that, provided the noises are sufficiently mixing so that some form of functional central limit theorem applies, under the 1 : 1 : 2 scaling the solution of (1.12) converges (in a weak sense) to the Brownian Castle. We conclude this introduction by pointing out some aspects of the construction of the Brownian Castle and the description of the 0-Ballistic Deposition model, commenting on their relation with the existing literature. Introduction 12

The importance of the Brownian Web lies in its connection with many interesting physical and biological systems (population genetics models, drainage networks, random walks in random environment...) and a thorough account on the advances of the research behind it can be found in [SSS17]. One of the most notable generalisations of the Brownian Web is the Brownian Net [SS08] (and the stochastic flows therein [SSS14]), which arises as the scaling limit of a collection of coalescing random walks that in addition have a small probability to branch. It is then natural to wonder if a construction similar to that carried out here is still possible starting from the Brownian Net, and what the corresponding “Castle” would be in this context. We believe that such considerations allow to build crossover processes connecting the BC and EW universality classes, namely such that their small scale statistics are BC, while their large scale fluctuations are EW. From the perspective of discrete interacting systems, let us also mention that the graphical representation of the 0-Ballistic Deposition is in itself not new. A picture analogous to Figure 4, can be found in [GM95], where the authors introduce the so-called noisy voter model. The latter can be obtained by the usual voter model (see [Lig05] for the definition), whose graphical representation is the same as that in Figure 4 but without •, by adding spontaneous flipping, illustrated by the realisation of µ•. Let us remark that not only the meaning but also the limiting procedure involving µ• is different in the two cases. In the present setting the intensity of µ• is fixed, while it is sent to 0 at a suitable rate in [FINR06].

1.4 Outline of the paper In Section 2, we recall the main definitions related to R-trees and the topology and ↓ construction of the Brownian Web Tree ζbw given in [CH21]. α,β In Section 3, we build the couple χbc. We introduce, for α, β ∈ (0, 1), the space Tbsp α,β (and its “characteristic” subset Cbsp ) in which the couple χbc lives, and define the metric that makes it Polish (Section 3.1). We then provide conditions under which a stochastic process X indexed by a spatial tree ζ admits a Hölder continuous modification, construct both Gaussian and Poisson processes indexed by a generic spatial tree, and prove that the map ζ 7→ Law(ζ, X) is continuous. (Sections 3.2-3.3.) At last, combining this with the α,β results of Section 2 we construct the law of χbc on Cbsp in Section 3.4. Section 4 is devoted to the Brownian Castle and its periodic counterpart, and contains the proof of Theorem 1.2. In Section 4.1, we first define the version formally given in (1.7) and determine its continuity properties (among which the finite p-variation for p > 1), then we study the location and structure of its discontinuities and analyse their relation with the special points of the Brownian Web. Afterwards, in Section 4.2, we prove the Markov, strong Markov, Feller and strong Feller properties (the latter holds only in the periodic case) and study its long-time behaviour. In Section 4.3, we derive the distributional properties of the Brownian Castle (scale invariance and multipoint distributions) and show that although Introduction 13 its invariant measure has increments that are Cauchy distributed and has finite p-variation for any p > 1, it is singular with respect to the law of the Cauchy process. In Section 5, we turn our attention to the 0-Ballistic Deposition model. At first, we δ associate the triplet χ0-bd to it and show that the latter converges to χbc (Sections 5.1 and 5.2) and then (Section 5.3), we prove Theorem 1.4. Finally, the appendix collects a number of relatively straightforward technical results.

Notations d We will denote by | · |e the usual Euclidean norm on R , d ≥ 1, and adopt the short-hand def def 2 notations |x| = |x|e and kxk = |x|e for x ∈ R and R respectively. Let (T , d) be a metric space. We deﬁne the Hausdorﬀ distance dH between two non-empty subsets A, B of T as

def ε ε dH (A, B) = inf{ε: A ⊂ B and B ⊂ A} (1.13) where Aε is the ε-fattening of A, i.e. Aε = {z ∈ T : ∃ w ∈ A s.t. d(z, w) < ε}. Let (T , d, ∗) be a pointed metric space, i.e. (T , d) is as above and ∗ ∈ T , and let M : T → Rd be a map. For r > 0 and α ∈ (0, 1), we define the sup-norm and α-Hölder norm of M restricted to a ball of radius r as |M(z) − M(w)| (r) def z (r) def e kMk∞ = sup |M( )|e , kMkα = sup α . z∈Bd(∗,r] z,w∈Bd(∗,r] d(z, w) d(z,w)≤1 where Bd(∗, r] ⊂ T is the closed ball of radius r centred at ∗, and, for δ > 0, the modulus of continuity as (r) def ω (M, δ) = sup |M(z) − M(w)|e . (1.14) z,w∈Bd(∗,r] d(z,w)≤δ In case T is compact, in all the quantities above, the suprema are taken over the whole space T and the dependence on r of the notation will be suppressed. Moreover, we say that a −α (r) function M is (locally) little α-Hölder continuous if for all r > 0, limδ→0 δ ω (M, δ) = 0. Let I ⊆ R be an interval and (X, d) be a complete separable metric space. We denote the space of càdàg functions on I with values in X as D(I, X) and, for f ∈ D(I, X), the set of discontinuities of f by Disc(f). We will need two different metrics on D(I, X), corresponding to the so-called J1 (or Skorokhod) and M1 topologies. For the first, let Λ(I) be the space of strictly increasing continuous homeomorphisms on I such that def λ(t) − λ(s) γ(λ) = sup |λ(t) − t| ∨ sup log < ∞ . t∈I s,t∈I t − s s

I def Then, for λ ∈ Λ(I) and f, g ∈ D(I, X) we set dλ(f, g) = 1 ∨ sups∈I d(f(s), g(λ(s))), so that the Skorokhod metric is given by Z ∞ def I def −t [−t,t] dSk(f, g) = inf γ(λ)∨dλ(f, g) , dSk(f, g) = inf γ(λ)∨ e dλ (f, g) dt , (1.15) λ∈Λ(I) λ∈Λ 0 where in the ﬁrst case I is assumed to be bounded. For the M1 metric instead, we restrict def to the case of X = R+ = [0, ∞). Given f ∈ D(I, R+), denote by Γf its completed graph, i.e. the graph of f to which all the vertical segments joining the points of discontinuity are added, and order it by saying that (x1, t1) ≤ (x2, t2) if either t1 < t2 or t1 = t2 and − − |f(t1 ) − x1| ≤ |f(t1 ) − x2|. Let Pf be the set of all parametric representations of Γf , which is the set of all non-decreasing (with respect to the order on Γf ) functions σf : I → Γf . Then, if I is bounded, we set

ˆc def dM1(f, g) = 1 ∨ inf kσf − σgk σf ,σg

c and dM1(f, g) to be the topologically equivalent metric with respect to which D(I, R+) is complete (see [Whi02, Section 8] for more details). If instead I = [−1, ∞), we define Z ∞ def −t c (t) (t) dM1(f, g) = e (1 ∧ dM1(f , g )) dt (1.16) 0 where f (t) is the restriction of f to [−1, t]. For p > 0, we say that a function f : R → X, (X, d) being a metric space, has finite p-variation if for every bounded interval I ⊂ R 1/p def p kfkp-var,I = sup d(f(tk), f(tk−1)) < ∞ (1.17) tk∈DI where DI ranges over all finite partitions of I. The Wasserstein distance of two probability measures µ, ν on a complete separable metric space (T , d) is defined as

def W(µ, ν) = inf Eγ[d(X,Y )] (1.18) g∈Γ(µ,ν) where Γ(µ, ν) denotes the set of couplings of µ and ν, the expectation is taken with respect to g and X,Y are two T -valued random variables distributed according to µ and ν respectively. At last, we will write a . b if there exists a constant C > 0 such that a ≤ Cb and a ≈ b if a . b and b . a. The Brownian Web Tree and its topology 15

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

2 The Brownian Web Tree and its topology

In this section, we provide the basic deﬁnitions and notations on (characteristic) R-trees and outline the construction and main properties of the (Double) Brownian Web Tree derived in [CH21].

2.1 Characteristic R-trees in a nutshell We begin by recalling the notion of R-tree (see [DLG05, Deﬁnition 2.1]), degree of a point, segment, ray and end.

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

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

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

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

A pointed R-tree is a triple (T , ∗, d) such that (T , d) is an R-tree and ∗ ∈ T . Given z ∈ T , the number of connected components of T \{z} is the degree of z, deg(z) in short. A point of degree 1 is an endpoint, of degree 2, an edge point and if the degree is 3 or higher, a branch point.

Deﬁnition 2.2 Let (T , d) be an R-tree and, for any z1, z2 ∈ T , fz1,z2 the isometric map in

Deﬁnition 2.1. We call the range of fz1,z2 , segment joining z1 and z2 and denote it by z1, z2 . For z ∈ T , a segment having z as an endpoint is said to be a T -ray from z if it is maximalJ K for inclusion. The ends of T are the equivalence classes of T -rays with respect to the equivalence relation according to which diﬀerent T -rays are equivalent if their intersection is again a T -ray. An end is closed if it is an endpoint of T and open otherwise, and, for † The Brownian Web Tree and its topology 16 an open end, we indicate by z, †i the unique T -ray from z representing †. † is said to be an J open end with (un-)bounded rays if for every z ∈ T , the map ιz : z, †i → R+ given by J ιz(w) = d(z, w) , w ∈ z, †i (2.1) J is (un-)bounded.

Throughout the article, we will work with (subsets of) the space of spatial R-trees, consisting of R-trees embedded into R2 via a map, called the evaluation map.

Deﬁnition 2.3 Let α ∈ (0, 1) and consider the quadruplet 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 proper4 map from T to R2.

α The space of pointed α-spatial R-trees Tsp is the set of equivalence classes of quadruplets as above with respect to the equivalence relation that identiﬁes ζ and ζ0 if there exists a bĳective isometry ϕ : T → T 0 such that ϕ(∗) = ∗0 and M 0 ◦ ϕ ≡ M, in short (with a slight abuse of notation) ϕ(ζ) = ζ0. α Elements ζ = (T , ∗, d, M) ∈ Tsp are further said to be characteristic, and their space α denoted by Csp, if the evaluation map M satisﬁes

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

Mt(zs) = (Mt(z0) − s d(z0, z1)) ∨ (Mt(z1) − (1 − s) d(z0, z1)) . (2.2)

where zs is the unique element of z0, z1 with d(z0, zs) = s d(z0, z1), (2) (Monotonicity in space) for everyJs < tK, interval I = (a, b) and any four elements z0,¯z0, z1,¯z1 such that Mt(z0) = Mt(¯z0) = t, Mt(z1) = Mt(¯z1) = s, Mx(z0) < Mx(¯z0), and M( z0, z1 ),M( ¯z0,¯z1 ) ⊂ [s, t] × (a, b), we have J K J K Mx(zs) ≤ Mx(¯zs) (2.3)

for every s ∈ [0, 1], (3) for all z = (t, x) ∈ R2, M −1({t} × [x − 1, x + 1]) 6= ∅. The space of those characteristic trees for which, in (1), (2.2) holds with ∨ instead of ∧ will ˆ α be denoted instead by Csp 4 0 0 α Namely such that limε→0 supz∈K supd(z,z0)≤ε kM(z) − M(z )k/d(z, z ) = 0 for every compact K and the preimage of every compact set is compact. The Brownian Web Tree and its topology 17

def Remark 2.4 Let T = R/Z be the torus of size 1 endowed with the usual periodic metric d(x, y) = infk∈Z |x − y + k|. When M is R × T-valued, we will say that ζ is periodic and α α denote the space of periodic (characteristic) pointed α-spatial R-trees by Tsp,per (or Csp,per). As in [CH21, Deﬁnition 2.19], we point out that (2) makes sense also in this case provided we restrict to intervals (a, b) that do not wrap around the torus.

α α In order to introduce a metric on Tsp (and Csp), recall that a correspondence C between two metric spaces (T , d), (T 0, d0) is a subset of T × T 0 such that for all z ∈ T there exists at least one z0 ∈ T 0 for which (z, z0) ∈ C and vice versa. Its distortion is given by

dis C =def sup{|d(z, w) − d0(z0, w0)| : (z, z0), (w, w0) ∈ C} .

0 0 0 0 0 α 0 Let ζ = (T , ∗, d, M) and ζ = (T , ∗ , d ,M ) ∈ Tsp be such that both T and T are compact, and C be a correspondence between them. We set

c, C 0 def 1 0 0 ∆sp (ζ, ζ ) = dis C+ sup kM(z) − M (z )k 2 (z,z0)∈ C nα 0 (2.4) + sup 2 sup kδz,wM − δz0,w0 M k n∈N (z,z0),(w,w0)∈ C 0 0 0 d(z,w),d (z ,w )∈An

def −n −(n−1) def where An = (2 , 2 ] for n ∈ N, and δz,wM = M(z) − M(w). In the above, we adopt the convention that if there exists no pair of couples (z, z0), (w, w0) ∈ C such that d(z, w) ∈ An, then the increment of M is removed and the supremum is taken among all 0 0 0 0 0 z , w such that d (z , w ) ∈ An and vice versa5 We can now deﬁne

c 0 def c 0 ∆sp(ζ, ζ ) = ∆sp(ζ, ζ ) + dM1(bζ , bζ0 ) (2.5) where

- the ﬁrst term is c 0 def c, C 0 ∆sp(ζ, ζ ) = inf ∆sp (ζ, ζ ) , (2.6) C: (∗,∗0)∈ C

- the map bζ is the properness map, which is 0 for r < 0, while for r ≥ 0 is deﬁned as

def bζ (r) = sup d(∗, z) , (2.7) z : M(z)∈Λr

def 2 2 per def for Λr = [−r, r] ⊂ R and Λr = Λr = [−r, r] × T, in the periodic case, 5If 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. The Brownian Web Tree and its topology 18

- dM1 is the metric on the space of càdlàg functions given in (1.16). Let us point out that by [CH21, Lemma 2.9], the properness map is non-decreasing and càdlàg so that the second summand in (2.5) is meaningful. α Since the elements of Tsp are locally compact, by [CH21, Theorem 2.6(a)] we can c 0 α generalise the deﬁnition of ∆sp to the non-compact case. For ζ, ζ ∈ Tsp, we set

Z +∞ 0 def −r h c (r) 0 (r) i ∆sp(ζ, ζ ) = e 1 ∧ ∆sp(ζ , ζ ) dr + dM1(bζ , bζ0 ) 0 (2.8) 0 =: ∆sp(ζ, ζ ) + dM1(bζ , bζ0 ). where, for r > 0, ζ(r) =def (T (r), ∗, d, M) (2.9) (r) def T = Bd(∗, r] being the closed ball of radius r in T .

Proposition 2.5 For any α ∈ (0, 1],

α (i) the space (Tsp, ∆sp) is a complete, separable metric space, α (ii) a subset A = {ζa = (Ta, ∗a, da,Ma) : a ∈ A}, A being an index set, of Tsp is relatively compact if and only if for every r > 0 and ε > 0 there exist (a) a ﬁnite integer N(r; ε) such that

(r) sup Nda (Ta , ε) ≤ N(r; ε) (2.10) a

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

(r) −α (r) sup kMak∞ ≤ C and sup δ ω (Ma, δ) < ε , (2.11) a∈A a∈A

0 sup bζa (r) ≤ C , (2.12) a

α α (iii) Csp is closed in Tsp.

Proof. The ﬁrst two points were shown in [CH21, Theorem 2.13 and Proposition 2.16], while the last is a consequence of [CH21, Lemma 2.22]. The Brownian Web Tree and its topology 19

α An important feature satisﬁed by any characteristic tree ζ = (T , ∗, d, M) ∈ Csp and shown in [CH21, Proposition 2.23], is that T possesses a unique open end † with unbounded rays (see Deﬁnition 2.2) such that for every z ∈ T and every w ∈ z, †i, one has J Mt(w) = Mt(z) − d(z, w) . (2.13)

It is by virtue of the previous property that we can introduce the radial map which allows to move along rays in the R-tree.

α Deﬁnition 2.6 Let α ∈ (0, 1], ζ = (T , ∗, d, M) ∈ Csp and † the open end with unbounded rays such that (2.13) holds. The radial map % associated to ζ is deﬁned as

def −1 %(z, s) = ιz (Mt(z) − s) , for z ∈ T and s ∈ (−∞,Mt(z)] (2.14)

ˆ α def where ιz was given in (2.1). If instead ζ ∈ Csp, the radial map %ˆ satisﬁes %ˆ(z, s) = −1 ιz (s − Mt(z)), where this time s ∈ [Mt(z), +∞). 2.2 The tree map In this subsection, we introduce a map, the tree map, which serves as an inverse of the evaluation map. Since the evaluation map is not necessarily bĳective, we need to determine a way to assign to a point in R2, one in the tree so that certain continuities properties can be deduced from those of the evaluation map. We begin with the following deﬁnition.

α Deﬁnition 2.7 Let α ∈ (0, 1), ζ = (T , ∗, d, M) ∈ Csp and % be the radial map associated to ζ given as in Deﬁnition 2.6. For z = (t, x) ∈ R2, we say that z is a right-most point for z if Mt(z) = t and

Mx(%(z, s)) = sup{Mx(%(w, s)) : Mt(w) = t & Mx(w) ≤ x} , (2.15) for all s < t. Left-most points are deﬁned as in (2.15) but replacing sup with inf and Mx(w) ≤ x with Mx(w) ≥ x. If there is a unique right-most (resp. left-most) point we will denote it by zr (resp. zl).

ˆ α Remark 2.8 For ζ ∈ Csp, a point is said to be a right-most (or left-most) point if (2.15) holds for all s > t.

For z ∈ R2 and an arbitrary characteristic tree, right-most and left-most point are not necessarily uniquely deﬁned. As we will see below, for elements in the measurable subset α of Csp given in [CH21, Deﬁnition 2.29], this is indeed the case. The Brownian Web Tree and its topology 20

α Deﬁnition 2.9 Let α ∈ (0, 1). We say that ζ = (T , ∗, d, M) ∈ Csp satisﬁes the tree condition if

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

α α We denote by Csp(t), the subset of Csp whose elements satisfy (t).

α 2 Lemma 2.10 Let α ∈ (0, 1) and ζ = (T , ∗, d, M) ∈ Csp(t). Then, for all z ∈ R there exist unique left-most and right-most points.

Proof. Since left-most and right-most points are exchanged under Mx 7→ −Mx, we only need to consider right-most points. Let z = (t, x) ∈ R2, † be the unique open end such that (2.13) holds and % be ζ’s radial map given in (2.14). Note first that we can assume without loss of generality that z ∈ M(T ) since the right-most points for z equal those of z¯ = (t, x¯), where x¯ = sup{y ≤ x : (t, y) ∈ M(T )}. Since M is proper, it is closed and therefore z¯ ∈ M(T ). def −1 Let {sn}n ⊂ R be a sequence such that sn ↑ t and An = {%(w, sn) : w ∈ M (z)}. An is finite since M −1(z) is totally disconnected, thanks to (2.13), and is compact by properness of the evaluation map. In particular this implies that also the number of paths connecting points in An with those in An+1 is finite. We inductively construct a sequence {wn}n ∈ −1 M (z) as follows. Let w1 be one of the points for which Mx(%(w1, s)) ≥ Mx(%(w, s)) −1 for all s ∈ [s0, s1] and w ∈ M (z). Assume we picked wn. If Mx(%(wn, s)) ≥ −1 def Mx(%(w, s)) for all s ∈ [sn, sn+1] and all w ∈ M (z) then set wn+1 = wn. Otherwise choose any wn+1 so that Mx(%(wn+1, s)) coincides with the right hand side of (2.15) for all s ∈ [sn, sn+1]. Notice that in the first case d(wn, wn+1) = 0. In the other instead, there exists s¯ ∈ [sn, sn+1] such that Mx(%(wn, s¯) < Mx(%(wn+1, s¯) hence by monotonicity in space Mx(%(wn, s) ≤ Mx(%(wn+1, s) for any s ≤ s¯. Moreover, for s ∈ [sn−1, sn], Mx(%(wn, s)) ≥ Mx(%(wn+1, s)) by construction, and therefore Mx(%(wn, s)) = Mx(%(wn+1, s)) for s ∈ [sn−1, sn].(t) then implies that %(wn, sn) = %(wn+1, sn) and we conclude that d(wn, wn+1) ≤ 2(t − sn). Hence, the sequence {wn}n is Cauchy and −1 converges to a unique limit z ∈ M (z). Since, for any n, d(wn−1, z) ≤ 2(t − sn) we necessarily have %(z, s) = %(wn, s) for all s ∈ [sn−1, sn] which implies that z is a right-most point. Now, if there existed another one, say ¯z, then by definition, %(¯z, ·) ≡ %(z, ·) on any subinterval I ⊂ (−∞, t) therefore, by (t), d(z,¯z) = 0.

Thanks to the above lemma, we are ready for the following deﬁnition. The Brownian Web Tree and its topology 21

α ˆ α α α Deﬁnition 2.11 Let α ∈ (0, 1) and Csp(t) (resp. Csp(t)) be the subset of Csp (resp. Csp(t)) α whose elements satisfy (t). For ζ ∈ Csp(t), we deﬁne the tree map T associated to ζ as

def T(z) = zr , for all z ∈ M(T ) (2.16) where zr is the unique right-most point (see Deﬁnition 2.7).

Remark 2.12 In the previous deﬁnition we could have analogously picked the left-most point. The choice above was made so that, under suitable assumptions on the evaluation map (see Proposition 2.13), the tree map is càdlàg. The following proposition determines the continuity properties of the tree map.

α Proposition 2.13 Let α ∈ (0, 1) and ζ ∈ Csp(t). Then, for every t ∈ R, x 7→ T(t, x) is càdlàg. Before proving the previous, we state and show the following lemma which contains more precise information regarding the roles of left-most and right-most point in the continuity properties of the tree map.

α 2 Lemma 2.14 Let α ∈ (0, 1) and ζ ∈ Csp(t). Let z = (t, x) ∈ R ∩ M(T ) and assume 2 there exists a sequence {zn = (t, xn)}n ⊂ R ∩ M(T ) converging to it. If

1. xn ↓ x then limn T(zn) = T(z), 2. xn ↑ x then limn T(zn) exists and coincides with zl,

α Proof. Let ζ = (T , ∗, d, M) ∈ Csp. We will prove only 1. since the other can be shown similarly. Let z and {zn}n be as in the statement. Since M is proper and continuous, the n def sequence {zr = T(zn)}n converges along subsequences and any limit point is necessarily in M −1(z). But now, monotonicity in space and (2.13) imply that

n n d(z, zr ) = d(z, zr) ∨ d(zr, zr ) hence, by uniqueness of zr, the result follows.

α Proof of Proposition 2.13. Let ζ = (T , ∗, d, M) ∈ Csp and recall that by definition M is both proper and continuous. Assume first that z = (t, x) ∈/ M(T ). Then, there exists ε > 0 such that {t} × [x − ε, x + ε] ∩ M(T ) = ∅. Hence, all w ∈ {t} × [x − ε, x + ε] have the same right-most point so that T is constant there. If z = (t, x) ∈ M(T ) and, for some ε > 0, {t} × (x, x + ε) ∩ M(T ) = ∅ then T is constantly equal to T(z) on {t} × [x, x + ε) and therefore it is continuous from the right. If The Brownian Web Tree and its topology 22 z = (t, x) ∈ M(T ) and, for some ε > 0, {t}×(x−ε, x)∩M(T ) = ∅, since by property (3) of characteristic trees M −1({t} × [x − ε − 2, x − ε]) 6= ∅ and M −1({t} × [x − ε − 2, x − ε]) is closed, there exists z¯ =def sup{w ∈ {t} × [x − ε − 2, x − ε]) ∩ M(T )}. But then, T is constantly equal to T(z¯) on {t} × (x − ε, x) which implies that limy↑x T(t, y) exists. The case of z being an accumulation point in {t} × [x − 1, x + 1] ∩ M(T ) was covered in Lemma 2.14, so that the proof is concluded. Remark 2.15 Notice that in view of the proof of Proposition 2.13 and Lemma 2.14, for α ζ = (T , ∗, d, M) ∈ Csp, T is continuous only at those points z such that the cardinality of M −1(z) is less or equal to 1. 2.3 The (Double) Brownian Web tree As pointed out in the introduction, a major role in the definition of the Brownian Castle is played by the Brownian Web and, in particular its characterisation as a characteristic spatial R-tree. In this subsection, we recall the main statements in [CH21], where such a characterisation was derived, focusing on those results which are instrumental to the present paper. We begin with the following theorem (see [CH21, Theorem 3.8 and Remark 3.9]) which establishes the existence of the Brownian Web Tree and uniquely identifies its law in α Csp.

1 α ↓ ↓ ↓ ↓ ↓ Theorem 2.16 Let α < 2 . There exists a Csp-valued random variable ζbw = (Tbw, ∗bw, dbw,Mbw) with radial map %↓, whose law is uniquely characterised by the following properties

1. for any deterministic point w = (s, y) ∈ R2 there exists almost surely a unique point ↓ ↓ w ∈ Tbw such that Mbw(w) = w, 2 2. for any deterministic n ∈ N and w1 = (s1, y1), . . . , wn = (sn, yn) ∈ R , the joint ↓ ↓ distribution of (Mbw,x(% (wi, ·)))i=1,...,n, where w1,..., wn are the points determined in 1., is that of n coalescing backward Brownian motions starting at w1, . . . , wn, 3. for any deterministic countable dense set D such that 0 ∈ D, let w be the point determined in 1. associated to w ∈ D and ∗˜↓ that associated to 0. Deﬁne ˜↓ ˜↓ ↓ ↓ ˜ ↓,D ζ∞(D) = (T∞(D), ∗˜ , d , M∞ ) as ˜↓ def ↓ 0 T∞(D) = {% (w, t) : w = (s, y) ∈ D , t ≤ s} (2.17) ˜ ↓,D ↓ def ↓ M∞ (% (w, t)) = Mbw(% (w, t))

↓ ˜↓ ˜↓ and d to be the ancestral metric in (1.6). Let T (D) be the completion of T∞(D) ↓ ˜ ↓,D ˜ ↓,D under d , M be the unique little α-Hölder continuous extension of M∞ and ˜↓ def ˜↓ ↓ ↓ ˜ ↓,D ˜↓ law ↓ ζ (D) = (T (D), ∗ , d , M ). Then, ζ (D) = ζbw. The Brownian Web Tree and its topology 23

The same statement holds upon taking the periodic version of all objects and spaces above and replacing the properties 1.-3. with 1per., 2per. and 3per. obtained from the former by adding the word “periodic” before any instance of “Brownian motion”. Thanks to the previous result, we can deﬁne the (periodic) Brownian Web Tree.

1 Deﬁnition 2.17 Let α < 2 . We deﬁne backward Brownian Web Tree and periodic backward α α ↓ ↓ ↓ ↓ ↓ Brownian Web tree, the Csp and Csp,per random variables ζbw = (Tbw, ∗bw, dbw,Mbw) per,↓ per,↓ per,↓ per,↓ per,↓ and ζbw = (Tbw , ∗bw , dbw ,Mbw ) whose distributions is uniquely characterised by properties 1.-3. and 1per., 2per. and 3per. in Theorem 2.16. The following proposition states some important quantitative and qualitative properties of both the Brownian Web Tree and its periodic counterpart.

1 ↓ Proposition 2.18 Let α < 2 and ζbw be the Brownian Web Tree given as in Deﬁnition 2.17. 3 Then, almost surely, for any ﬁxed θ > 2 and all r > 0 there exists a constant c = c(r) > 0 depending only on r such that for all ε > 0

↓, (r) −θ N↓ (Tbw , ε) ≤ cε (2.18) dbw

↓, (r) ↓ where N↓ (T , ε) is deﬁned as in (2.10). Moreover, almost surely Mbw is surjective dbw ↓ and (t) holds for ζbw. All the properties above remain true in the periodic case.

Proof. See [CH21, Proposition 3.2 and Theorem 3.8].

A key aspect of the Brownian Web Tree is that it comes naturally associated with a dual, which consists of a spatial R-tree whose rays, when embedded into R2, are distributed as a family of coalescing forward Brownian motions. In the following theorem and the subsequent deﬁnition (see [CH21, Theorem 3.1, Remark 3.18 and Deﬁnition 3.19]), we introduce the (periodic) Double Brownian Web Tree, a random couple of characteristic R-trees made of the Brownian Web Tree and its dual, and clarify the relation between the two.

α ˆ α ↓↑ def Theorem 2.19 Let α < 1/2. There exists a Csp × Csp-valued random variable ζbw = ↓ ↑ • • • • • • (ζbw, ζbw), ζbw = (Tbw, ∗bw, dbw,Mbw), ∈ {↓, ↑}, whose law is uniquely characterised by the following properties

↑ def ↑ ↑ ↑ ↑ ↓ (i) Both −ζbw = (Tbw, ∗bw, dbw, −Mbw) and ζbw are distributed as the backward Brownian Web tree in Deﬁnition 2.17. The Brownian Web Tree and its topology 24

↓ ↓ ↑ ↑ ↓ ↓ ↓ (ii) Almost surely, for any z ∈ Tbw and z ∈ Tbw, the paths Mbw(% (z , ·)) and ↑ ↑ ↑ ↓ ↑ ↓ Mbw(% (z , ·)) do not cross, where % (resp. % ) is the radial map of ζbw (resp. ↑ ζbw). ↓↑ α ˆ α ↑ ↓ Moreover, almost surely ζbw ∈ Csp(t) × Csp(t) and ζbw is determined by ζbw and vice-versa, ↑ ↓ meaning that the conditional law of ζbw given ζbw is almost surely given by a Dirac mass. The above statement remains true in the periodic setting upon replacing every object and space with their periodic counterpart.

1 Definition 2.20 Let α < 2 . We define the double Brownian Web tree and double periodic α ˆ α α ˆ α ↓↑ def Brownian Web tree as the Csp × Csp and Csp,per × Csp,per-valued random variables ζbw = ↓ ↑ per,↓↑ def per,↓ per,↑ ↑ per,↑ (ζbw, ζbw) and ζbw = (ζbw , ζbw ) given by Theorem 2.19. We will refer to ζbw and ζbw as the forward (or dual) and forward periodic, Brownian Web trees. ↓↑ ↓ ↑ per,↓↑ ↓ ↑ ↓ We denote their laws by Θbw(d(ζ ×ζ )) and Θbw (d(ζ ×ζ )), with marginals Θbw(dζ), ↑ per,↓ per,↑ Θbw(dζ) and Θbw (dζ), Θbw (dζ) respectively. As we will see in the next section, the continuity properties of the Brownian Castle are • −1 crucially connected to the cardinality and the degree of points of the inverse maps (Mbw) per,• −1 • and (Mbw ) , for ∈ {↑, ↓}. Based on these two features, it is possible to classify all points of R2 or R × T as was shown in [CH21, Theorem 3.24] and we now summarise this classification.

↓↑ ↓ ↑ • Deﬁnition 2.21 Let ζbw = (ζbw, ζbw) be the double Brownian Web tree. For ∈ {↑, ↓}, the 2 • 2 type of a point z ∈ R for ζbw is (i, j) ∈ N , where

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

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

↓ ↓,per Theorem 2.22 For the backward and backward periodic Brownian Web trees ζbw and ζbw , almost surely, every z ∈ R2 (resp. R × T) is of one of the following types, all of which ↓ occur: (0, 1), (1, 1), (2, 1), (0, 2), (1, 2) and (0, 3). Moreover, almost surely, S0,1 has full ↓ ↓ Lebesgue measure, S2,1 and S0,3 are countable and dense and for every t ∈ R Branching Spatial Trees and the Brownian Castle measure 25

↓ - S0,1 ∩ {t} × R has full Lebesgue measure in {t} × R, ↓ ↓ - S1,1 ∩ {t} × R and S0,2 ∩ {t} × R are both countable and dense in {t} × R, ↓ ↓ ↓ - S2,1 ∩ {t} × R, S1,2 ∩ {t} × R and S0,3 ∩ {t} × R have each cardinality at most 1.

↓ ↓ ↓ At last, for every deterministic t, S2,1 ∩ {t} × R, S1,2 ∩ {t} × R and S0,3 ∩ {t} × R are almost surely empty. ↓ ↑ 0 0 Moreover, S(i,j) = S(i0,j0) for (i, j)/(i , j ) = (0, 1)/(0, 1), (1, 1)/(0.2), (2, 1)/(0, 3), (0, 2)/(1, 1), (1, 2)/(1, 2) and (0.3)/(0, 3), and the same holds in the periodic case.

Proof. The ﬁrst part of the statement corresponds to [CH21, Theorem 3.24], while the last is a consequence of [CH21, Proposition 3.22].

3 Branching Spatial Trees and the Brownian Castle measure

As mentioned in the introduction, the Brownian castle is not just given by an R-tree T realised on R2 by a map M, but furthermore comes with a stochastic process X indexed by T , whose distribution depends on the metric structure of T . (In our case, we want X to be a Brownian motion indexed by T .) How to do it in such such a way that X admits a (Hölder) continuous modiﬁcation is the topic of the second section but ﬁrst, we want to introduce the space in which such an object lives.

3.1 Branching Spatial R-trees The space of branching spatial R-trees corresponds to spatial R-trees endowed with an additional (Hölder) continuous map, from the tree to R, which, for us, will encode a realisation of a suitable stochastic process. The term branching is chosen because this extra map (read, process) should be thought of as branching at the points in which the branches of the tree coalesce.

Deﬁnition 3.1 Let α, β ∈ (0, 1). The space of (α, β)-branching spatial pointed R-trees α,β α Tbsp is the set of couples χ = (ζ, X) with ζ ∈ Tsp and X : T → R, the branching map, is α,β 0 0 0 a locally little β-Hölder continuous map. In Tbsp two couples χ = (ζ, X), χ = (ζ ,X ) are indistinguishable if there exists a bĳective isometry ϕ : T → T 0 such that ϕ(∗) = ∗0, M 0 ◦ ϕ ≡ M and X0 ◦ ϕ ≡ X, in short ϕ ◦ χ = χ0. α If furthermore ζ ∈ Csp, we say that χ = (ζ, X) is a characteristic (α, β)-branching α,β α,β spatial R-tree and denote the subset of Tbsp containing such R-trees by Cbsp . Branching Spatial Trees and the Brownian Castle measure 26

α,β Similar to what done in Section 2.1, we endow the space Tbsp with the metric Z +∞ 0 def −r c (r) 0(r) ∆bsp(χ, χ ) = e [1 ∧ ∆bsp(χ , χ )]dr + dM1(bζ , bζ0 ) 0 (3.1) 0 =: ∆bsp(χ, χ ) + dM1(bζ , bζ0 ) .

0 α,β (r) 0 (r) for χ, χ ∈ Tbsp . Above, given r > 0, χ , χ are deﬁned as in (2.9) and

c (r) 0 (r) def c, C (r) 0 (r) ∆bsp(χ , χ ) = inf ∆bsp (χ , χ ) , C:(∗,∗0)∈ C the inﬁmum being taken over all correspondences C ⊂ T (r) × T 0(r) such that (∗, ∗0) ∈ C, and for such a correspondence C c, C (r) 0 (r) def c, C (r) 0 (r) 0 0 ∆bsp (χ , χ ) =∆sp (ζ , ζ ) + sup |X(z) − X (z )| (z,z0)∈ C nβ 0 (3.2) + sup 2 sup |δz,wX − δz0,w0 X | . n∈N (z,z0),(w,w0)∈ C 0 0 0 d(z,w),d (z ,w )∈An α,β The following lemma determines the metric properties of Cbsp and gives a compactness criterion for its subsets. The proof, as well as the statement, are completely analogous to α those for Tsp given in [CH21, Theorem 2.13 and Proposition 2.16], so we refer the reader to the aforementioned reference for further details.

α,β α,β Lemma 3.2 For α, β ∈ (0, 1), (Tbsp , ∆bsp) is a complete separable metric space and Cbsp α,β is closed in it. Moreover, a subset Aof Tbsp is relatively compact if and only if α 1. the projection of Aonto Tsp is relatively compact and 2. for every r > 0 and ε > 0 there exist constants K = K(r) > 0 and δ = δ(r, ε) > 0 such that (r) −β (r) sup kXχk∞ ≤ K and sup δ ω (Xχ, δ) < ε . (3.3) χ∈A ζ∈A The following lemma will be needed in some of the proofs below. It gives a way to estimate the distance between a branching spatial R-tree and one of its subset, provided the Hausdorﬀ distance (see (1.13)) between the metric spaces is known.

α,β Lemma 3.3 Let α, β ∈ (0, 1), χ = (T , ∗, d, M, X) ∈ Tbsp and assume T is compact. Let δ > 0, T ⊂ T be such that ∗ ∈ T and the Hausdorﬀ distance between T and T is bounded above by δ and deﬁne χ¯ = (T, ∗, d, MT,XT ). Then c −α −β ∆bsp(χ, χ¯) . (2δ) ω(M, 2δ) + (2δ) ω(X, 2δ) (3.4) Proof. The proof follows the very same steps of [CH21, Lemma 2.12]. Branching Spatial Trees and the Brownian Castle measure 27

3.2 Stochastic Processes on trees We want to understand how to realise the branching map X as a Hölder continuous real- valued stochastic process indexed by a pointed locally compact complete R-tree (T , d, ∗), whose covariance structure is suitably related to the metric d. In this section, we will mostly consider the case of a fixed (but generic) R-tree, and provide conditions for the process X to admit a β-Hölder continuous modification, for some β ∈ (0, 1). We will always view the α law of the process X on a given T , (T , ∗, d, M) ∈ Tsp, as a probability measure on the α,β space branching spatial R-trees Tbsp . A function ϕ : R+ → R+ is said to be a Young function if it is convex, increasing and such that limt→∞ ϕ(t) = +∞ and ϕ(0) = 0. From now on, fix a standard probability space (Ω, A, P). Given p ≥ 0 and a Young function ϕ, the Orlicz space on Ω associated to ϕ is the set Lϕ of random variables Z :Ω → R such that

def kZkϕ = inf{c > 0 : E[ϕ(|Z|/c)] ≤ 1} < ∞ . (3.5)

Notice that if Z is a positive random variable such that kZkLϕ ≤ C for some Young function ϕ and some ﬁnite C > 0, then Markov’s inequality yields

P(Z > u) ≤ 1/ϕ(u/C) , ∀ u > 0 . (3.6)

q The following proposition shows that if ϕ is of exponential type ϕ(x) = ex − 1 and Nd(T , ε) grows at most polynomially as ε → 0, then be obtain a modulus of continuity of order d| log d|1/q.

def xq Proposition 3.4 Let q ≥ 1 and ϕq(x) = e − 1. Let (T , ∗, d) be a pointed complete proper metric space and {X(z) : z ∈ T } a stochastic process indexed by T . Assume there are ν ∈ (0, 1], θ > 0 and, for every r > 0, there exists a constant c > 0 such that

(r) −θ Nd(T , ε) ≤ c ε , for all ε ∈ (0, 1) (3.7) and 0 0 ν 0 (r) kX(z) − X(z )kϕq ≤ c d(z, z ) , for all z, z ∈ T , (3.8) where T (r) is deﬁned as in (2.9). Then, X admits a continuous version such that for every r > 0, there exists a random variable K = K(ω, r) such that

|X(z) − X(z0)| ≤ K d(z, z0)ν | log d(z, z0)|1/q , for all z, z0 ∈ T (r).

q Furthermore, one has the bound P(K ≥ u) ≤ C1 exp(−C2u ) for some constants Ci > 0 depending only r, ν, c and θ. Branching Spatial Trees and the Brownian Castle measure 28

Proof. We closely follow the argument and notations of [Tal05, Sec. 1.2]. We also note that, since dν is again a metric, it suﬃces to consider the case ν = 1, so we do this now. −n Set εn = 2 and Nn = Nd(T , εn). Noting that in our case d(πn(z), πn+1(z)) ≤ 2εn, (3.6) yields 1/q q P(|X(πn(z)) − X(πn+1(z))| ≥ un εn) . exp(−cu n) , 2θn for some constant c > 0. Note furthermore that in our case NnNn−1 . 2 by assumption. Proceeding similarly to [Tal05, Sec. 1.2], we consider the event Ωu on which |X(πn(z)) − 1/q X(πn+1(z))| ≤ un εn for every n ≥ 0, so that

c X 2θn q q P(Ωu) . 2 exp(−cu n) . exp(−cu ) . n≥0

Furthermore, on Ωu, one has

0 X X 0 0 |X(z) − X(z )| ≤ |X(πn(z)) − X(πn+1(z))| + |X(πn(z )) − X(πn+1(z ))|

n≥n0 n>n0 0 + |X(πn0 (z)) − X(πn0+1(z ))| X 1/q 0 0 1/q ≤ 4u n εn ≈ u d(z, z )| log d(z, z )| ,

n≥n0

0 where n0 is such that εn0+1 ≤ d(z, z ) ≤ εn0 . The claim then follows at once.

Condition (3.7) on the size of the ε-nets will always be met by the R-trees we will consider, that this is the case for the Brownian Web Tree is stated in Proposition 2.18. Therefore, we introduce a subset of the space of characteristic spatial trees whose elements satisfy it locally uniformly. Let α ∈ (0, 1), θ > 0 and let c: R+ → R+ be an increasing ˜ ˜ ˜ function. We deﬁne Eα(c, θ), E(θ) and Eα respectively as

˜ def α (r) −θ Eα(c, θ) = {ζ = (T , ∗, d, M) ∈ Tsp : ∀r, ε > 0 , Nd(T , ε) ≤ c(r)ε } , ˜ def [ ˜ ˜ def [ ˜ (3.9) Eα(θ) = Eα(c, θ) and Eα = Eα(θ) . c θ

def ˜ α and Eα(c, θ) = Eα(c, θ) ∩ Csp and Eα(θ), Eα accordingly. Thanks to [BBI01, Proposition ˜ 7.4.12], it is not diﬃcult to verify that for every given c and θ as above Eα(c, θ) is closed α ˜ ˜ in Tsp, and consequently Eα(θ) and Eα are measurable with respect to the Borel σ-algebra induced by the metric ∆sp (and so are Eα(θ) and Eα). In the next proposition we show how (3.8) can be used to prove tightness for the laws, on the space of branching spatial R-trees, of a family of stochastic processes indexed by ˜ diﬀerent spatial R-trees that uniformly belong to Eα(c, θ), for some θ and c. Branching Spatial Trees and the Brownian Castle measure 29

˜ Proposition 3.5 Let q ≥ 1, α ∈ (0, 1) and let K ⊂ Eα(c, θ) for some c, θ > 0 be relatively compact. For every ζ = (T , ∗, d, M) ∈ K, let Xζ be a stochastic process indexed by T and denote by Qζ the law of (ζ, Xζ ). Assume that there exists ν ∈ (0, 1) such that for all ε > 0 and r > 0, there are constants c∞ = c∞(ε) > 0 and cν = cν(r) > 0 such that

inf Qζ (|Xζ (∗)| ≤ c∞) ≥ 1 − ε (3.10) ζ∈K and

ν (r) kXζ (z) − Xζ (w)kϕq ≤ cνd(z, w) , for all z, w ∈ T and ζ ∈ K. (3.11)

α,β Then, the family of probability measures {Qζ }ζ∈K is tight in Tbsp for any β < ν.

Proof. By Lemma 3.2, we only need to focus on the maps Xζ and, more speciﬁcally, on their restriction to the r-neighbourhoods of ∗. Since {Xζ (∗)}ζ∈K is tight by (3.10), it remains to argue that for every ε > 0

−β lim sup Qζ δ sup |Xζ (z) − Xζ (w)| > ε = 0 . (3.12) δ→0 ζ∈K d(z,w)≤δ

This in turn is immediate from Proposition 3.4 and our assumption.

Our main example is that of a Brownian motion indexed by a pointed R-tree. Let (T , ∗, d) be a pointed locally compact complete R-tree, and let {B(z) : z ∈ T } be the centred Gaussian process such that B(∗) =def 0 and such that

0 2 0 E[(B(z) − B(z )) ] = d(z, z ) , (3.13) for all z, z0 ∈ T . We call B the Brownian motion on T .

Remark 3.6 The existence of a Gaussian process whose covariance matrix is as above is guaranteed by the fact that any R-tree (T , d) is of strictly negative type, see [HLMT98, Cor. 7.2]. ˜ Remark 3.7 If ζ = (T , d, ∗,M) ∈ Eα, then it follows from Proposition 3.4 that, for B a α,β def Brownian motion on T , one has (ζ, B) ∈ Tbsp almost surely, for every β < βGau = 1/2. Gau α,β We will denote by Qζ its law on Tbsp . In the study of 0-Ballistic Deposition, we will also consider Poisson processes indexed by an R-tree. The Poisson process is clearly not continuous so, in order to ﬁt it in our framework, we introduce a smoothened version of it. First, recall that for any locally Branching Spatial Trees and the Brownian Castle measure 30 compact complete R-tree T , the skeleton of T , T o, is deﬁned as the subset of T obtained by removing all its endpoints, i.e.

def [ T o = ∗, z . (3.14) z∈T J J

For any T as above, there exists a unique σ-ﬁnite measure ` = `T , called the length measure such that `(T \ T o) = 0 and `( z, z0 ) = d(z, z0) , (3.15) J K for all z, z0 ∈ T .

Remark 3.8 One important property of the Brownian motion B is that, given any four points zi, one has

E[(B(z1) − B(z2))(B(z3) − B(z4))] = `( z1, z2 ∩ z3, z4 ) , J K J K where the equality follows immediately by polarisation and the tree structure of T . In particular, increments of B are independent on any two disjoint subtrees of T . α Let α ∈ (0, 1), ζ = (T , ∗, d, M) ∈ Csp a characteristic tree and † the open end for which (2.13) holds. For γ > 0, let µγ be the Poisson random measure on T with intensity γ` and, for a > 0, let ψa be a smooth non-negative real-valued function on R, compactly R supported in [0, a] and such that ψa(x) dx = 1. We define the smoothened Poisson R random measure on T as Z a def µγ(w) = ψa(d(w,¯z))µγ(d¯z) , w ∈ T . (3.16) w,†i J In other words, we are smoothening the Poisson random measure µγ by fattening its points along the rays from the endpoints to the open end †, so that the value at a given point w depends only on Poisson points in w, w(a) , where w(a) is the unique point on the ray w, †i such that d(w, w(a)) = a. J K J α Definition 3.9 Let α ∈ (0, 1) and ζ = (T , ∗, d, M) ∈ Csp with length measure `. Let γ > 0 and µγ be the Poisson random measure on T with intensity measure γ`. For a > 0, let ψa be a smooth non-negative real-valued function on R, compactly supported in [0, a] R and such that ψa(x) dx = 1. We define the rescaled compensated smoothened (RCS in R short) Poisson process on T as Z a def 1 a Nγ (z) = √ (µγ(w) − γ)`(dw) , for z ∈ T , (3.17) γ ∗,z J K a where µγ is the smoothened Poisson random measure given in (3.16). Branching Spatial Trees and the Brownian Castle measure 31

In case the R-tree has (locally) finitely many endpoints and γ and a are fixed, it is easy to see that the smoothened Poisson process defined above is Lipschitz. That said, we want to obtain more quantitative information about its regularity and how the latter relates to the parameters γ and a, in order to be able to identify a regime in which a family of RCS Poisson processes on a given tree converges weakly. In the following Lemma (and the rest of the paper), for ζ = (T , ∗, d, M) ∈ Eα and X a stochastic process indexed by T , which admits a β-Hölder continuous modification, we will denote by Qζ (dX) the law of (ζ, X) in the space of (α, β)-characteristic branching α,β α,β spatial R-trees and by M(Cbsp ) the space of probability measures on Cbsp endowed with the topology of weak convergence.

a Lemma 3.10 Let α ∈ (0, 1), ζ = (T , ∗, d, M) ∈ Eα, Nγ be as in Deﬁnition 3.9 and set def 1 −p a α,β βPoi = 2p . If a = γ for some p > 1, then for any β < βPoi, (ζ, Nγ ) ∈ Cbsp almost surely. Poiγ Furthermore, denoting its law by Qζ , these are tight over γ ≥ 1.

Proof. Thanks to Propositions 3.4 and 3.5, it suﬃces to show that there exists a constant C depending only on r such that

1 a a 2p kNγ (z) − Nγ (w)kϕ1 ≤ Cd(z, w) , (3.18) for all z, w ∈ T (r), which in turn is essentially a consequence of Lemma A.1 in the appendix. ψa ψa Indeed, if the points z, w belong to the same ray, then the increment Nγ (z) − Nγ (w) a coincides in distribution with that of Pγ (d(z, w)) in (A.1), so that (3.18) follows from (A.2). If z and w lie on diﬀerent branches, let z† be the unique point for which z, †i ∩ w, †i = z†, †i. Then, the triangle inequality for Orlicz norms yields J J J a a a a a a kNγ (z) − Nγ (w)kϕ1 ≤ kNγ (z) − Nγ (z†)kϕ1 + kNγ (z†) − Nγ (w)kϕ1 , and the required bound follows from (3.18).

3.3 Probability measures on the space of branching spatial trees The results in the previous section identify suitable conditions on a spatial R-tree ζ = (T , ∗, d, M) and the distribution of the increments of a stochastic process X indexed by T , under which the couple (ζ, X) is (almost surely) a branching spatial R-tree. We now want to let ζ vary and understand the behaviour of the map ζ 7→ Qζ = Law(ζ, X). Since in rest of the paper we will only deal with characteristic R-trees, we will directly work with these, even though some of our statements remain true for general spatial R-trees. We now write γ,p a −p Gau Qζ = Law(ζ, X) for X = Nγ as in (3.17) with the choice a = γ , and Qζ for X = B as in (3.13). We then have the following continuity property. Branching Spatial Trees and the Brownian Castle measure 32

Gau γ,p Proposition 3.11 Let α ∈ (0, 1), c, θ > 0, and Qζ = Qζ , respectively Qζ = Qζ for some γ > 0 and p > 1. Then, the map

α,β Eα(c, θ) 3 ζ 7→ Qζ ∈ M(Cbsp ) (3.19)

1 is continuous, provided that β < βGau = 2 , respectively β < βPoi as in Lemma 3.10. In view of Lemma 3.10, Proposition 3.5, and the central limit theorem, it is clear that, γ,p Gau for a ﬁxed tree ζ, Qζ converges weakly to Qζ as γ ↑ ∞, for any ﬁxed p. In the next statement, we show that such a convergence is locally uniform in ζ.

Proposition 3.12 Let α ∈ (0, 1), p > 1, and c, θ > 0. Then, for any β < βPoi, γ,p Gau α,β limγ→∞ Qζ = Qζ in M(Cbsp ), uniformly over compact subsets of Eα(c, θ). The remainder of this subsection is devoted to the proof of the previous two statements. We will first focus on Proposition 3.11 since some tools from its proof will be needed in that of Proposition 3.12. Before delving into the details we need to make some preliminary considerations which apply to both. Let c, θ > 0 be fixed and K be a compact subset of Eα(c, θ). Since the constant C in (3.18) is independent of both γ and the specific features of the tree, Proposition 3.5 γ,p Gau α,β implies that the families {Qζ : γ > 0, ζ ∈ K} and {Qζ : ζ ∈ K} are tight in Cbsp for any β < βPoi and β < βGau respectively, and jointly, for β < βPoi ∧ βGau. Then, the proof of both Propositions 3.11 and 3.12 boils down to show that if {ζn}n ⊂ K is a sequence converging to ζ with respect to ∆sp then there exists a coupling between (ζn,Xn) and (ζ, X) such that ∆bsp((ζn,Xn), (ζ, X)) converges to 0 locally uniformly over ζ ∈ K and the Hölder norms of Xn and X. If we denote by Qn and Qζ the laws of (ζn,Xn) ζ, X γ,p γ,p and ( ) then for the first statement, we need to pick Qn and Qζ to be either Qζn and Qζ , γ > 0 Gau Gau ζ = ζ n = γn,p for fixed, or Qζn and Qζ , while for the second, n for all , Qn Qζ , with Gau γn → ∞ and Qζ = Qζ . The problem in the first case is that, since X and Xn are indexed by different spaces, it is not a priori clear how to build these couplings. In the next subsection, which represents the core of the proof, we construct one.

3.3.1 Coupling processes on diﬀerent trees Fix α ∈ (0, 1) and η > 0, and consider characteristic trees ζ = (T , ∗, d, M), ζ0 = 0 0 0 0 α 0 (T , ∗ , d ,M ) ∈ Csp such that T and T are compact. As a shorthand, we set δ = c (r) 0 (r) (r) 0 (r) ∆sp(ζ , ζ ) and we ﬁx a correspondence C between T and T such that

c, C (r) 0 (r) ∆sp (ζ , ζ ) ≤ 2δ . (3.20) Branching Spatial Trees and the Brownian Castle measure 33

We will always assume that our two trees are sufficiently close so that δ ≤ η. (We typically think of the case δ η.) Let then B be the Gaussian process on T such that (3.13) holds a and, for γ > 0, let µγ be the Poisson random measure on T with intensity γ` and Nγ be the RCS Poisson process of Definition 3.9 and Lemma 3.10. The aim of this subsection is to first inductively construct subtrees T and T 0 of T and T 0 respectively that are close to each other and whose distance from the original trees is easily quantifiable. Simultaneously, we build a bĳection ϕ: T → T 0 which preserves the length measure and has small distortion. This provides a natural coupling between µγ and a 0 0 0 ∗ def −1 Poisson random measure µγ on T by µγ(A) = ϕ µγ(A) = µγ(ϕ (A)), and similarly for the white noise underlying B. To start our inductive construction, we simply set

0 0 0 T0 = {∗} , T0 = {∗ } , ϕ(∗) = ∗ .

0 Assume now that, for some m ∈ N, we are given subtrees Tm−1 and Tm−1 as well as a 0 length-preserving measure ϕ: Tm−1 → Tm−1. Let then v ∈ T \ Tm−1 be a point whose distance from Tm−1 is maximal and denote by bm the projection of v onto Tm−1. We also set def Cm−1 = C∪ ϕm−1 = C∪ {(z, ϕ(z)) : z ∈ Tm−1} , where we have identiﬁed the bĳection ϕm−1 = ϕTm−1 with the natural correspondence 0 induced by it. If d(v, bm) ≤ 2(η ∨ dis Cm−1), we terminate our construction and set

def 0 def 0 def 0 def 0 0 0 0 0 T = Tm−1 , T = Tm−1 , Z = (T, ∗, d, MT ) , Z = (T , ∗ , d ,M T ) . (3.21a)

0 0 0 0 0 Otherwise, let v ∈ T be such that (v, v ) ∈ C and bm be its projection onto Tm−1. If 0 0 0 0 0 d(v, bm) ≥ d (v , bm) then we set vm = v and denote by vm ∈ bm, v the unique point such 0 0 0 J K 0 that d(vm, bm) = d (vm, bm). Otherwise, we set vm = v and deﬁne vm correspondingly. We then set def 0 def 0 0 0 Tm = Tm−1 ∪ bm, vm ,Tm = Tm−1 ∪ bm, vm , (3.21b) J K J K 0 and we extend ϕ to bm, vm \{bm} to be the unique isometry such that ϕ(vm) = vm. J K 0 0 0 We also write `m = d(bm, vm) = d (bm, vm). The following shows that this construction terminates after ﬁnitely many steps.

Lemma 3.13 Let N be the minimal number of balls of radius η/8 required to cover T . Then, the construction described above terminates after at most N steps and, until it does, 0 0 one has `m ≥ η/2 so that in particular vm ∈/ Tm−1. Branching Spatial Trees and the Brownian Castle measure 34

Proof. We start by showing the second claim. Assuming the construction has not terminated 0 0 0 0 0 yet, we only need to consider the case d (v , bm) < d(v, bm) so that vm = v . Take j < m 0 0 0 such that bm ∈ bj, vj , then J K 1 ` = d0(v0 , b0 ) = (d0(v0 , v0 ) + d0(v0 , b0 ) − d0(v0 , b0 )) m m m 2 m j m j j j 1 3 0 ≥ (d(v, vj) + d(v, bj) − d(vj, bj)) − dis C (3.22) 2 2 m−1 3 η ≥ d(v, b ) − dis C0 ≥ . m 2 m−1 2

0 0 0 The passage from the ﬁrst to the second line is a consequence of the fact that (v, vm), (vj, vj), (bj, bj) ∈ 3 η Cm−1, and the last bound follows from the fact that d(v, bm) ≥ 2 dis Cm−1+ 2 by assumption. It remains to note that since `m > η/2, the points vj are all at distance at least η/2 from each other, so there can only be at most N of them.

Thanks to Lemma 3.13, we can also deﬁne

0 0 0 0 def −1 0 0 0 0 B (z ) − B (bm) = B(ϕm (z )) − B(bm) , for z ∈ bm, vm (3.23) 0 0 0 def ∗ J K µγ bm, vm = ϕmµγ , (3.24) J K 0 a 0 and Nγ accordingly. In the following lemma, we denote by X and X the processes on T 0 0 a 0 a and T , which correspond to either B and B or to Nγ and Nγ .

Lemma 3.14 In the setting above, let Z,Z0 be as in (3.21a) and X and X0 be constructed 0 0 α,β inductively via (3.23) or (3.24). If (3.20) holds, then (Z,X) and (Z ,X ) belong to Cbsp and c,ϕ 0 0 N −α N Nκ κ ∆bsp((Z,X), (Z ,X )) . (5 δ) ω(M, 5 δ) + N5 kXkβ+κδ . (3.25) for any κ ∈ (0, 1) suﬃciently small, where N is as in Lemma 3.13. Moreover, the Hausdorﬀ distance between both T and T , and T 0 and T 0, is bounded above by 2(5N δ) + η.

Proof. We will ﬁrst bound the distortion dis ϕ between T and T 0 by induction on m. We 0 0 7 begin by showing that dis( Cm−1 ∪ {(vm, vm), (bm, bm)}) ≤ 2 dis Cm−1. Assume without 0 0 0 0 loss of generality that d(v, bm) > d (v , bm) and let (w, w ) ∈ Cm−1. By the triangle 0 inequality and the fact that, by construction, (v, vm), (bm, ϕm−1(bm)) ∈ Cm−1 we have

0 0 0 |d(vm, w) − d (vm, w )| ≤ d(v, vm) + dis Cm−1 , 0 0 0 0 0 |d(bm, w) − d (bm, w )| ≤ dis Cm−1 + d (ϕm−1(bm), bm) , Branching Spatial Trees and the Brownian Castle measure 35

0 0 where we added and subtracted d(vm, w) in the ﬁrst and d (ϕm−1(bm), w ) in the second. Now, as a consequence of (3.22)

0 0 0 3 d(v, vm) = d(v, bm) − d(vm, bm) = d(v, bm) − d (v , bm) < 2 dis Cm−1 while

0 0 0 0 0 0 0 d (ϕm−1(bm), bm) = d (vm, ϕm−1(bm)) − d (vm, bm) 5 ≤ dis Cm−1 + d(v, bm) − d(vm, bm) = dis Cm−1 + d(v, vm) ≤ 2 dis Cm−1

0 0 0 and both hold since, by construction, d(vm, bm) = d (vm, bm). Therefore the claim 0 0 7 dis( Cm−1 ∪ {(vm, vm), (bm, bm)}) ≤ 2 dis Cm−1, follows at once. Now, for any z ∈ Tm \ Tm−1, let ˜z ∈ T be such that (˜z, ϕm(z)) ∈ Cm−1. We clearly have

0 0 |d(z, w) − d (ϕm(z), w )| ≤ d(z,˜z) + dis Cm−1 . ˜ Denote by b ∈ Tm the projection of ˜z onto Tm. In order to bound d(z,˜z), it suﬃces to exploit ˜ ˜ the fact that if b ∈ z, vm , then d(z,˜z) = d(bm,˜z) − d(bm, z), while if b ∈ Tm \ z, vm , J K 0 0 0 J K then d(z,˜z) = d(vm,˜z) − d(vm, z). Let (y, y ) be either (vm, vm) or (bm, bm), so that

0 0 0 0 d(z,˜z) = d(y,˜z) − d(y, z) ≤ d (y , ϕm(z)) + dis( Cm−1 ∪ {(vm, vm), (bm, bm)}) − d(y, z) 0 0 7 = dis( Cm−1 ∪ {(vm, vm), (bm, bm)}) ≤ 2 dis Cm−1

0 0 where we used that, by construction, d (y , ϕm(z)) = d(y, z). Hence, dis ϕm ≤ dis Cm ≤ 9 9 m 2 dis Cm−1 and since dis C0 = dis C, we conclude that dis ϕm ≤ dis Cm ≤ ( 2 ) dis C and N therefore dis ϕ ≤ dis( C∪ ϕ) . 5 δ. Concerning the evaluation maps, take z1, z2 ∈ T and choose w1, w2 ∈ T such that (wi, ϕ(zi)) ∈ C. One has

0 N d(zi, wi) = |d(zi, wi) − d (ϕ(zi), ϕ(zi))| ≤ dis(ϕ ∪ C) . 5 δ , (3.26) so that 0 0 kM(z1) − M (ϕ(z1))k ≤ kδz ,w Mk + kM(w1) − M (ϕ(z1))k 1 1 (3.27) ≤ ω(M, 5N δ) + 2δ , where we used the little Hölder continuity of M and (3.20). For the Hölder part of the distance instead, let n ∈ N and assume further that d(z1, z2), d(ϕ(z1), ϕ(z2)) ∈ An. If there exist no w1, w2 ∈ T such that (wi, ϕ(zi)) ∈ C and also d(w1, w2) ∈ An, then for n ∈ N such that 2−n > 5N δ, we exploit (3.27) to obtain

0 −nα N −α N kδz1,z2 M − δϕ(z1),ϕ(z2)M k . 2 ((5 δ) ω(M, 5 δ)) Branching Spatial Trees and the Brownian Castle measure 36

−n N c while for n such that 2 ≤ 5 δ, by deﬁnition of ∆sp (see below (2.4)), we get

0 0 −nα nα −n kδz1,z2 M − δϕ(z1),ϕ(z2)M k ≤ kδz1,z2 Mk + kδϕ(z1),ϕ(z2)M k . 2 (2 ω(M, 2 ) + δ)

−nα N −α N which in turn is bounded by 2 ((5 δ) ω(M, 5 δ)). In case instead d(w1, w2) ∈ An, then we simply apply the triangle inequality to write

0 0 kδz1,z2 M − δϕ(z1),ϕ(z2)M k ≤ kδz1,z2 M − δw1,w2 Mk + kδw1,w2 M − δϕ(z1),ϕ(z2)M k

Thanks to the estimate on the distortion of C ∪ ϕ and (3.27), the ﬁrst summand can be controlled as in the proof of [CH21, Lemma 2.12], while the second is bounded by 2δ thanks to (3.20). We focus now on the branching maps, for which we proceed once again by induction on the iteration step m. Notice that for m = 0, there is nothing to prove. We now assume that for some m < N, there exists K,K0 > 0 such that

sup |X(z) − X0(z0)| ≤ Kδ% (3.28) 0 (z,z )∈ϕm−1 0 0 0 kX Tm−1k% ≤ K kXk% (3.29)

0 for % < β, but arbitrarily close to it. Let (z, z ) ∈ ϕm \ ϕm−1 be such that z ∈ bm, vm and 0 J K w be the point in Tm−1 for which (w, bm) ∈ Cm−1. Then,

0 0 0 0 0 0 |X(z) − X (z )| = |X(bm) − X (bm)| ≤ |δbm,wX| + |X(w) − X (bm)| % % N% % ≤ kXk%d(bm, w) + Kδ ≤ (5 kXk% + K)δ where the passage from the ﬁrst to the second line is a consequence of the Hölder regularity 0 of X and (3.28) while in the last inequality we exploited the fact that both (bm, bm) and 0 0 (w, bm) ∈ ϕm and the same bound as in (3.26). Concerning the Hölder norm of X , let 0 0 0 0 0 z ∈ Tm \ Tm−1 and w ∈ Tm−1, then by triangle inequality we have

0 0 0 0 0 0 0 % 0 0 0 0 0 0 |δz ,w X | ≤ |δz ,bm X | + |δbm,w X | ≤ kXk%(1 + K )d (z , w )

0 0 where we used (3.29). Hence, the %-Hölder norm of X on T is bounded above by NkXk%. For the second summand in (3.2), let (z, z0), (w, w0) ∈ ϕ be such that d(z, w), d0(z0, w0) ∈ An. Then, we have

0 −n% 0 −n% |δz,wX − δz0,w0 X | . 2 (kXk% + kX k%) . 2 NkXk% as well as 0 N% % |δz,wX − δz0,w0 X | . N5 kXk%δ . Branching Spatial Trees and the Brownian Castle measure 37 hence, by geometric interpolation, (3.25) follows. For the last part of the statement, notice that the Hausdorff distance between T and T 0 N is bounded above by 2(η ∨ dis C ) . η + 5 δ by the definition of our halting condition. Concerning the Hausdorff distance between T 0 and T 0, let z0 ∈ T 0. Take z ∈ T such that 0 N N (z, z ) ∈ C, w ∈ T such that d(z, w) . η + 5 δ, which exists since dH (T ,T ) . η + 5 δ, and w0 ∈ T 0 such that (w, w0) ∈ ϕ. Then

0 0 0 0 0 0 N d (z , w ) ≤ |d (z , w ) − d(z, w)| + d(z, w) . dis( C∪ ϕ) + η + 5 δ from which the claim follows at once.

We are now ready for the proof of Propositions 3.11 and 3.12.

Proof of Proposition 3.11. Let {ζn}n , ζ ⊂ Eα(c, θ) be such that ∆sp(ζn, ζ) converges to 0. α,β Let ε > 0 be ﬁxed. Since the family {Qn}n is tight, there exists Kε ⊂ Cbsp compact such that inf Qn(Kε) ≥ 1 − ε . (3.30) n∈N

We want to bound the Wasserstein distance between Qn and Q. By the previous we have

W(Qn, Qζ ) = inf Eg[∆bsp((ζn,Xn), (ζ, X))] g∈Γ(Qζn ,Qζ ) 1 ≤ inf Eg[∆bsp((ζn,Xn), (ζ, X)) Kε ] + ε g∈Γ(Qζn ,Qζ ) (3.31) c (r) (r) (r) (r) 1 ≤ inf Eg ∆bsp((ζn ,Xn ), (ζ ,X )) Kε + 2ε , g∈Γ(Qζn ,Qζ ) where in the last passage we used the deﬁnition of metric ∆bsp in (3.1) and chose r so that e−r < ε. We now apply the construction (3.21) with ζ replaced by ζ(r) and ζ0 replaced by (r) ζn . The triangle inequality then yields

c (r) (r) (r) (r) c (r) (r) ∆bsp((ζn ,Xn ), (ζ ,X )) ≤ ∆bsp((Zn,Xn ), (Z,X )) (3.32) c (r) (r) (r) c (r) (r) (r) + ∆bsp((ζn ,Xn ), (Zn,Xn )) + ∆bsp((ζ ,X ), (Z,X )) . (3.33) Since the summands in (3.33) only depend on one of the two probability measures, their coupling is irrelevant. By the last point of Lemma 3.14, the Hausdorff distance of both (r) (r) 0 N Tn and T from T and T is at most of order η + 5 δ so that we can apply Lemma 3.3. If we now choose first η ≈ ε small enough, then we can guarantee that each of the two terms in (3.33) is less than ε, provided that n is sufficiently large (and therefore δ sufficiently small). Note here that even though N depends (badly) on η, it is independent of n. At last, upon choosing at most an even smaller δ and exploiting the coupling of the previous section, we can use (3.25) to control (3.32) by ε, so that the proof is concluded. Branching Spatial Trees and the Brownian Castle measure 38

γ γ,p Gau Proof of Proposition 3.12. Throughout the proof we will write Qζ for Qζ and Qζ for Qζ and we fix a compact set K ⊂ Eα(c, θ). γ α,β Let ε > 0 be fixed. Since the family {Qζ , Qζ : γ > 0, ζ ∈ K} is tight in Cbsp for any α,β β < βPoi, there exists Kε ⊂ Cbsp compact such that (3.30) holds with the infimum taken over ζ ∈ K and γ > 0. Then, proceeding as in (3.31) and following the strategy used to control (3.33) in the previous proof, we see that we can choose r, η > 0 in such a way that sup W(Qγ, Q ) ≤ sup inf ∆c ((Z,X(r)), (Z,X(r)))1 + 4ε ζ ζ γ Eg bsp γ Kε , (3.34) ζ∈K ζ∈K g∈Γ(Qζ ,Qζ ) where Z = Zζ is again constructed from ζ as in (3.21). We are left to determine a coupling under which the first term is small. Let W be a standard Brownian motion and Pγ a rescaled compensated Poisson process of intensity γ on R+, coupled in such a way that, with probability at least 1 − ε, one has −1/5 def sup |W (t) − Pγ(t)| ≤ (1 + L)γ = dγ , (3.35) t∈[0,L]

def provided that γ is sufficiently small. Here L = supζ∈K `ζ (T ) with `ζ the length measure on T is finite by Lemma 3.13 and the existence of such a coupling (with 1/5 replaced by any exponent less than 1/4) is guaranteed by the quantitative form of Donsker’s invariance principle. Similarly, we have supζ∈K #{z ∈ Tζ : deg(z) = 1} < ∞ and we will denote its value by N. For ζ ∈ K, we order the endpoints and the respective branching points of T according to the procedure of Section 3.3.1 and recursively define the subtrees Tj, j ≤ N, of T as in (3.21a). For every j ≤ N, denote by ϕj the unique bĳective isometry from bj, vj to [`ζ (Tj), `ζ (Tj) + d(bj, vj)] with ϕj(bj) = `ζ (Tj). GivenJ anyK function Y : [0,L] → R, we then define Y˜ : T → R to be the unique function such that Y˜ (∗) = 0 and ˜ ˜ Y (z) = Y (ϕj(z)) − Y (ϕj(bj)) + Y (bj) , for z ∈ Tj \ Tj−1. (3.36) ˜ ˜ This then allows us to construct the desired coupling by setting B = W and Nγ = Pγ, a as well as Nγ to be the smoothened version of Nγ. It follows from (3.36) that, setting δ = |B z − N z | δ ≤ δ + 2d j supz∈Tj ( ) γ( ) , we have j+1 j γ on the event (3.35). We now remark that, for any integer k > 0, we can guarantee that 1 a − 2 p+k(1−p) P sup |Nγ(z) − Nγ (z)| ≥ kγ ≤ LN(2γ) , (3.37) z∈T so that, choosing k sufficiently large so that k(p − 1) > p and then choosing γ sufficiently large, one has 1 a − 2 P sup |B(z) − Nγ (z)| > 2Ndγ + kγ ≤ 2ε . (3.38) z∈T The claim now follows at once by combining this with Lemma 3.10. Branching Spatial Trees and the Brownian Castle measure 39

3.4 The Brownian Castle measure In this section, we collect the results obtained so far and show how to deﬁne a measure, which we will refer to as the Brownian Castle measure, on the space of branching spatial trees which encodes the inner structure of the Brownian Castle. We begin with the following proposition, which determines the existence and the continuity properties of the Gaussian process deﬁned via 3.13 on the Brownian Web Tree of Section 2.3.

↓ per,↓ Proposition 3.15 Let ζbw and ζbw be the backward and backward periodic Brownian Web per ↓ trees in Deﬁnition 2.20. There exist Gaussian processes Bbc and Bbc indexed by Tbw, per,↓ Tbw that satisfy (3.13). Moreover, they admit a version whose realisations are locally little β-Hölder continuous for any β < 1/2.

↓ per,↓ Proof. The existence part of the statement is due to the fact that Tbw and Tbw are almost ↓ per,↓ surely R-trees (see Remark 3.6), while, since by Proposition 2.18 ζbw and ζbw belong to Eα almost surely, the Hölder regularity is a direct consequence of Proposition 3.4.

The previous proposition represents the last ingredient to deﬁne the Brownian Castle def ↓ measure. This is given by the law of the couple χbc = (ζbw,Bbc) in the space of characteristic branching spatial trees.

↓ per,↓ Theorem 3.16 Let ζbw and ζbw be the backward and backward periodic Brownian Web per trees in Deﬁnition 2.20, and Bbc and Bbc be the Gaussian processes built in Proposition 3.15. def ↓ Then, almost surely the couple χbc = (ζbw,Bbc) is a characteristic (α, β)-branching spatial pointed R-tree according to Deﬁnition 3.1, for any α, β < 1/2. We call its law on α,β Cbsp the Brownian Castle Measure. The latter can be written as Z def Gau ↓ Pbc(dχ) = Qζ (dX) Θbw(dζ) (3.39)

Gau α,β where Qζ (dX) denotes the law of the Gaussian process Bbc on Cbsp . Analogously, for per def per,↓ per α,β the same range of the parameters α, β, χbc = (ζbw ,Bbc ) almost surely belongs to Cbsp,per per and we deﬁne its law on it, Pbc (dχ), as in (3.39).

α,β Proof. That χbc almost surely belongs to Cbsp is a direct consequence of Theorem 2.16 and Propositions 3.15. The needed measurability properties that allow to deﬁne (3.39) follow by Proposition 3.11. The Brownian Castle 40

4 The Brownian Castle

The aim of this section is to rigorously deﬁne the “nice” version of the Brownian Castle (and its periodic version, see Remark 4.1 for its deﬁnition) sketched in (1.7), and prove Theorem 1.2 along with other properties.

per Remark 4.1 The periodic Brownian Castle hbc is defined similarly to hbc. We require it to start from a periodic càdlàg function in D(T, R) and its finite-dimensional distributions to be characterised as in Definition 1.1, but z1, . . . , zn ∈ (0, +∞) × T and the coalescing backward Brownian motions Bk’z are periodic. 4.1 Pathwise properties of the Brownian Castle 1 Let α, β < 2 , χ = (ζ, X), for ζ = (T , ∗, d, M), be a (periodic) (α, β)-branching characteristic spatial R-tree according to Definition 3.1 satisfying (t) (see Definition 2.9), and % be its radial map as in (2.14). We introduce the following maps

s def 2 hχ(z) = X(T(z)) z ∈ R (4.1) and, for h0 ∈ D(R, R) (or in D(T, R)),

h0 def s hχ (z) = h0(Mx(%(T(z), 0))) + h (z) − X(%(T(z), 0)) (4.2) for z = (t, x) ∈ R+ × R (resp. R+ × T). We are now ready for the following theorem and the consequent deﬁnition, which identify the version of the Brownian Castle we will be using throughout the rest of the paper.

α,β Theorem 4.2 Let χbc be the Cbsp -valued random variable introduced in Theorem 3.16 and Pbc be its law given by (3.39). Then, for every càdlàg function h0 ∈ D(R, R), the map hh0 in is -almost surely well-defined and is a version of the Brownian Castle h , χbc (4.2) Pbc bc i.e. it starts at h0 at time 0 and its finite-dimensional distributions are as in Definition 1.1. In the periodic setting, the same statement holds, i.e. for every periodic càdlàg function h0 per h0 ∈ D(T, R) h per is Pbc -almost surely well-defined, starts at h0 at time 0 and has the χbc per same finite-dimensional distributions as hbc in Remark 4.1. Proof. The proof is a direct consequence of our construction in Sections 2.3 and 3.4 and follows by Proposition 2.18, Lemma 2.10 and Theorems 2.16 and 3.16.

s per,s Definition 4.3 We define the stationary (periodic) Brownian Castle, hbc (resp. hbc ), as s 2 s the field hχ on R (resp. h per on R × T) given by (4.1), while, for h0 ∈ D(R, R) (resp. bc χbc per h0 ∈ D(T, R)), we define the (periodic) Brownian Castle starting at h0, hbc (resp. hbc ), h0 h0 h h per as the map χbc (resp. ) in (4.2). χbc The Brownian Castle 41

s Remark 4.4 Since we require the Gaussian process Bbc to start from 0 at ∗, hbc is not, ˜s strictly speaking, stationary but its increments are. As a consequence, writing hbc for the s projection of hbc onto a space of functions in which two elements are identified if they differ ˜s by a fixed constant, we see that hbc is truly stationary in time.

The previous theorem guarantees that thanks to χbc it is possible to provide a construction of the Brownian Castle which highlights its inner structure. We will now see how to exploit this construction in order to prove certain continuity properties that the Brownian Castle hbc per and its periodic counterpart hbc enjoy.

Proposition 4.5 Pbc-almost surely, for every initial condition h0 ∈ D(R, R), the map t 7→ hbc(t, ·) takes values in D(R, R), and is continuous from above, i.e. for every t ∈ R+ lims↓t dSk(hbc(s, ·), hbc(t, ·)) = 0. Moreover, Pbc-almost surely, for every t ∈ R+ such ↓ that there is no x ∈ R for which (t, x) ∈ S(0,3), t 7→ hbc(t, ·) is continuous at t, i.e. per lims→t dSk(hbc(s, ·), hbc(t, ·)) = 0. The same holds in the periodic setting Pbc -almost surely.

Proof. The definition of hbc, together with Proposition 2.13 and the continuity of Bbc immediately imply that almost surely for every t ∈ R, R 3 x 7→ hbc(t, x) is càdlàg and therefore belongs to D(R, R). In order to prove the second part of the statement, fix t > 0 and let s > t. By definition of the Skorokhod distance, it suffices to exhibit a λs ∈ Λ such that γ(λs) < ε for s sufficiently small and supx∈[−R,R] |hbc(s, λs(x)) − hbc(t, x)| < ε for R big enough. Since hbc(t, ·) ∈ D(R, R),[Bil99, Lemma 1.12.3] implies that there exist −R = x1 < ··· < xn = R such that for all i = 1, . . . , n

sup |hbc(t, x) − hbc(t, y)| < ε/2 . (4.3) x,y∈[xi,xi+1)

↓ We assume, without loss of generality, that for every x ∈ [xi, xi+1), % (Tbw(t, x), 0) coincide. Indeed, if this is not the case, it suffices to add a finite number of points xi and (4.3) would i ↑ still hold. Now, for each of the zi = (xi, t), we consider zl ∈ Tbw, i.e. the left-most point (see Remark 2.8) in the preimage of zi for the forward Brownian Web tree (which, by ↓ Theorem 2.19 is deterministically fixed by ζbw). Now, let κ > 0 be small enough so that for s ∈ (t, t + κ) ↑ ↑ 1 ↑ ↑ n Mbw,x(% (zl , s)) < ··· < Mbw,x(% (zl , s)) , set def ↑ ↑ i λs(xi) = Mbw,x(% (zl , s)) and define λs(x) for x 6= xi by linear interpolation. Clearly, γ(λs) converges to 0 as s ↓ t, so that we can choose s˜ sufficiently close to t for which γ(λs) < ε for The Brownian Castle 42 all s ∈ (t, s˜). Now, by the non-crossing property (see point (ii) in Theorem 2.19), def ↓ ↓ y = Mbw,x(% (Tbw(λs(x), s), t)) ∈ [xi, xi+1) for s ∈ (t, s˜) and x ∈ [xi, xi+1) and clearly ↓ ↓ dbw(Tbw(λs(x), s),% (Tbw(λs(x)), t)) = s−t. Recall that Bbc is locally β-Hölder continuous so that upon taking s¯ =def s ˜ ∧ (t + ε1/β/2), we obtain

|hbc(s, λs(x)) − hbc(t, x)| ≤ |hbc(s, λs(x)) − hbc(t, y)| + |hbc(t, y) − hbc(t, x)| ε < |B (T (s, λ (x))) − B (T (t, y))| + < ε bc bw s bc bw 2 for all x ∈ [−R,R) and s ≤ s¯, and from this the result follows. ↓ It remains to prove the last part of the statement. Let t ∈ R+ be such that {t}×R∩S(0,3) = 1/β ˜↓ ∅ and ε > 0. We now consider a ﬁnite subset of {t − ε } × R, Ξ[−R,R] (which is the ↓ image via Mbw of ΞR in (B.1)), given by ˜↓ 1/β def ↓ ↓ 1/β ↓ Ξ[−R,R](t, t − ε ) = {Mbw,x(% (z, t − ε )) : Mbw(z) ∈ {t} × [−R,R]} . (4.4) ˜↓ 1/β Order the elements in the previous set in increasing order, i.e. Ξ[−R,R](t, t − ε ) = {xi : def ˜↓ 1/β ˜↓ 1/β i = 1,...,N} and x1 = min Ξ[−R,R](t, t − ε ). Now, for any xi ∈ Ξ[−R,R](t, t − ε ), let

def ↓ 1/β yi = inf{y ∈ R : % (Tbw(t, y), t − ε ) = xi} , i = 1,...,N and (4.5) def ↓ 1/β yN+1 = sup{y ∈ R : % (Tbw(t, y), t − ε ) = xN } then, since by Theorem 2.19 (ii) forward and backward paths do not cross, we know that {yi} Ξ˜↑ (t − ε1/β, t) Ξ˜↓ coincides with [x1,xN ] (deﬁned as but with all arrows reversed). By duality ↓ ↑ ↑ ˜ 1/β S(0,3) = S(2,1), hence {(yi, t)} ∩ S(2,1) = ∅. Therefore, there exists a time t ∈ (t − ε , t) 1/β such that no pair of forward paths started before t−ε and passing through [x1, xN ] at time t − ε1/β s ∈ (t,˜ t] Ξ˜↑ (t − ε1/β, t) , coalesces at a time . In other words, the cardinality of [x1,xN ] Ξ˜↑ (t − ε1/β, s) s ∈ (t,˜ t] coincides with that of [x1,xN ] for any . ↑ ↑ For i ≤ N, let zi be the unique point in Tbw, such that for all z ∈ Tbw for which ↑ 1/β ↑ ˜ ˜ Mbw,x(z) ∈ {t − ε } × [xi, xi+1], % (z, t) = zi. We deﬁne the map λs, s ∈ (t, t), as

def ↑ λs(xi) = Mbw,x(zi) and for x 6= xi we extend it by linear interpolation. Clearly, γ(λs) converges to 0, so that we can choose s˜ > t˜suﬃciently close to t so that γ(λs) < ε. Now, notice that, by construction (and Theorem 2.19 (ii)), for all x ∈ [−R,R] and s ∈ (s,˜ t), Tbw(t, x) and Tbw(s, λs(x)) must ↓ 1/β be such that d (Tbw(t, x), Tbw(s, λx(x))) < ε which, by the (local) β-Hölder continuity of Bbc, guarantees that

|hbc(t, x) − hbc(s, λs(x))| = |Bbc(Tbw(t, x)) − Bbc(Tbw(s, λx(x))| . ε The Brownian Castle 43 which concludes the proof in the non-periodic setting. The periodic case follows the same steps but, as spatial interval, one can directly take the whole of T instead of [−R,R].

Remark 4.6 By Proposition 2.13, the fact that hbc(t, •) is càdlàg simply follows from its α,β description in terms of an element of Cbsp . The fact that it is right-continuous as a function of time however uses specific properties of the Brownian Castle itself and wouldn’t be true α,β for hbc built from an arbitrary element of Cbsp . In the next proposition, we show that it is possible to obtain a finer control over the fluctuations of the Brownian Castle.

Proposition 4.7 Pbc-almost surely, for every t > 0, hbc(t, ·) has ﬁnite p-variation for every p > 1, locally on any bounded interval of R.

Proof. Let t, R > 0 and consider hbc(t, ·) restricted to the interval [−R,R]. At first, we will approximate hbc(t, ·) by piecewise constant functions. ˜ −n def For any n ≥ 0, let Ξ(t, t − 2 ) be the set defined according to (4.4) and let Nn = −n ςn ηR(t, 2 ) be its cardinality, which we recall satisfies the bound Nn . 2 (for some random proportionality constant independent of n) given in (B.3). We order the points of Ξ˜ t, t − 2−n x(n) < ··· < x(n) ( ) as in the proof of Proposition 4.5, denote them by 1 Nn , and set x(n) =def −R x(n) =def R hn t, · 0 and Nn+1 . We define the piecewise constant function bc( ) by

n (n) (n) (n) hbc(t, x) = hbc(t, xi ) for x ∈ [xi , xi+1).

We then note that, for any x ∈ [−R,R] we have the identity

0 X n+1 n hbc(t, x) − hbc(t, x) = (hbc (t, x) − hbc(t, x)) n≥0 so that in particular, for any p ≥ 1,

0 X n+1 n khbc(t, ·) − hbc(t, ·)kp-var ≤ khbc (t, ·) − hbc(t, ·)kp-var , (4.6) n≥0 the p-variation norm k · kp-var being deﬁned as in (1.17). Thanks to the β-Hölder continuity of Bbc, we then have

n+1 n −βn 1/p khbc (t, ·) − hbc(t, ·)kp-var . 2 Nn+1 ,

n+1 n since hbc (t, ·) − hbc(t, ·) is a piecewise constant function with sup-norm over [−R,R] −βn bounded by C2 (for some random C independent of n) and at most Nn+1 jumps. The Brownian Castle 44

Inserting this bound into (4.6) and exploiting the bound on Nn+1 provided by Lemma B.1, we obtain 0 X (ς/p−β)n khbc(t, ·) − hbc(t, ·)kp-var . 2 , n≥0 which is finite for any p > ς/β. Since both ς and β can be chosen arbitrarily close to 1/2, the statement follows. per Combining the (Hölder) continuity of the map Bbc (or of Bbc ) with Proposition 2.13, ↓ ↓ ↓ we conclude that the set of discontinuities of hbc is contained in S0,2 ∪ S1,2 ∪ S0,3 (see ↑ Definition 2.21) or, by duality (see Theorem 2.22), in the image through Mbw of the skeleton o,↑6 per of the forward Brownian Web Tbw given by (3.14), and the same holds for hbc . This means that we can identify specific events in the spatio-temporal evolution of the (periodic) Brownian Castle with special points of the (periodic) Brownian Web. Let us define the basin of attraction for the shock at z = (t, x) ∈ R+ × R as def 0 0 0 2 0 Az = {z = (t , x ) ∈ R : t < t and there exists 0 ↑ ↑ 0 0 ↑ ↑ 0 (4.7) z ∈ Tbw s.t. Mbw(z ) = z and Mbw(% (z , t)) = z} and the age of the shock as 0 0 az = t − sup{t < t : {t } × R ∩ Az = ∅} . (4.8) per per and define mutatis mutandis Az and az as the basin of attraction of a shock at z ∈ R+ × T and its age, in the periodic setting. In the following proposition, we show properties of the age of a point z and characterise its basin of attraction. Proposition 4.8 1. In both the periodic and non-periodic case, the set of points with strictly positive age coincides with the union of points of type (i, j) for j > 1. 2. In the non-periodic case, almost surely, for every z, az < ∞ and there exists a unique 0 0 0 0 z = (t , x ) ∈ Az that realises the supremum in (4.8), i.e. such that az = t − t . In the periodic case, for every t ∈ R, the previous holds for all z ∈ {t} × T except for

exactly one value zper = (t, xper), which is such that azper = ∞. 0 3. In the non-periodic case, if z = (t, x) is such that az > 0, then the unique z = 0 0 0 ↑ (t , x ) ∈ Az (determined in the previous point) such that az = t − t , belongs to S0,3. ↓ −1 Moreover, the left-most and right-most points at z, zl, zr ∈ (Mbw) (z) are such that zl 6= zr and ˚ [ ↓ ↓ ↓ ↓ Az = (Mbw(% (zl, s)),Mbw(% (zr, s))) . (4.9) s

6 ↑ per,↑ In [CH21], it was shown that the skeleton is given by T∞(D) (resp. T∞ (D)), D being any countable dense of R2 (resp. T × R), but with the endpoints removed The Brownian Castle 45

˚ where Az denotes the interior of Az and Az is compact.

Proof. Point 1. is an immediate consequence of Theorem 2.22. Indeed, if z is such that ↑ −1 az > 0, then there exists a point in (Mbw) (z) whose degree is strictly greater than 1, which ↓ −1 ↓ implies that |(Mbw) (z)| ≥ 2 so that z belongs to the union of Si,j for j > 1. Vice-versa, ↓ ↑ if z belongs to one of the Si,j for j > 1 then, by duality, it belongs to one of Si,j for 0 ↑ ↑ 0 (i, j) = (1, 1), (2, 1) or (1, 2) so that there exists at least one z ∈ Tbw such that Mbw,t(z ) < t ↑ ↑ 0 ↑ 0 and Mbw(αbw(z , t)) = z. Hence az ≥ t − Mbw,t(z ) > 0. For point 2., we first show that if az is finite then the point realising the supremum is unique, the proof being the same in the periodic and non-periodic setting. Assume 0 0 0 00 0 00 there exist z = (t , x ), z = (t , x ) ∈ Az realising the supremum in (4.8). Then, by the 0 0 00 0 0 00 coalescing property, every point in {t } × [x , x ] (or {t } × (T \ [x , x ])) belongs to Az. ↑ But, according to Theorem 2.22 almost surely for every s ∈ R, S1,1 ∩ {s} × R is dense in ↑ 0 0 00 ↑ ↑ 0 {s} × R, hence, there is z˜ ∈ S1,1 ∩ {t } × [x , x ] and z ∈ Tbw such that Mbw,t(z) < t and ↑ ↑ 0 ↑ 0 Mbw,t(% (z, t )) =z ˜. But then az ≥ t − Mbw,t(z) > t − t , which is a contradiction. ↑ Since, by [CH21, Proposition 3.21], Tbw has a unique open end with unbounded rays, per,↑ for every z, az < ∞. This is not true anymore for Tbw which has exactly two open ends with unbounded rays, but since there exists a unique bi-infinite edge connecting them (see [CH21, Proposition 3.25]), it follows that for every t there is a unique xper ∈ T such that (t, xper) has infinite age. Let us now focus on 3. Let z = (t, x) be such that az > 0. From 2., there exists a unique 0 2 0 ↑ ↑ 0 0 ↑ ↑ 0 point z ∈ R and a point z ∈ Tbw such that Mbw(z ) = z and Mbw(% (z , t)) = z, hence z ∈ ↑ ↑ ↑ ↓ −1 S1,1 ∪S2,1 ∪S1,2. Thanks to Theorem 2.22, the right-most and left-most points in (Mbw) (z), ↓ zr, zl ∈ Tbw must be distinct. By Theorem 2.19 (ii), forward and backward trajectories cannot 0 ↓ ↓ ↑ ↑ 0 ↓ ↓ cross, therefore for every s ∈ (t , t), Mbw(% (zl, s)) < Mbw(% (z , s)) < Mbw(% (zr, s)). In particular, the backward paths starting from z cannot coalesce before t0. They cannot coalesce after t0 either since, if this were to be the case, then for the same reasons as above ↓ ↓ ↓ ↓ the path in the forward web starting from any point in {s} × (Mbw(% (zl, s),Mbw(% (zr, s))), 0 s < t would be contained in Az, contradicting point 2. It follows that the point at which the 0 0 ↓ ↑ two backward paths coalesce is exactly z , which implies that z ∈ S2,1 = S0,3. Moreover the previous argument also shows that (4.9) holds (with z1 = zr and z2 = zl).

Remark 4.9 Proposition 4.8 and its proof underline one of the main visible diﬀerences per,↑ between the Brownian Castle and its periodic counterpart. Indeed, only Tbw possesses a ↑ per bi-inﬁnite edge β , which implies that hbc exhibits a “master shock” starting back at −∞ per,↑ ↑ and running along Mbw (β (·)). Indeed, as we have seen above, for every s ∈ R there exist per,↑ ↑ two backward paths starting in or passing through Mbw (β (s)) that before meeting need to The Brownian Castle 46

transverse the whole torus. On the other hand, all the discontinuities of hbc have a ﬁnite origin that can be tracked with the methods shown in Proposition 4.8. The following proposition collects the most important connections between certain events we witness on the Brownian Castle and special points in the Web.

per Proposition 4.10 1. Shocks for hbc and hbc correspond to the trajectories of the forward and periodic forward Brownian Web trees respectively, i.e. they are points of type ↑ per,↑ (1, 1) or (1, 2) for ζbw (resp. ζbw ). 2. If two shocks merge at z, then z is of type (0, 3) for the backward (periodic) Brownian Web.

Proof. The result follows by the fact that, by construction, the paths of backward Brownian Web tree represent the backward characteristics of the Brownian Castle, and by duality.

The previous proposition provides the reason why there is no chance for the Brownian Castle hbc(s, ·) to admit a limit as s ↑ t for all t ∈ R+, in the Skorokhod topology (or any of the M1, J2 and M2-topologies on this space, see [Whi02, Section 12]), independently of the specifics of the proof of Proposition 4.5 or our construction. Indeed the Skorokhod topology allows for discontinuities to evolve continuously and to merge only if their difference continuously converges to 0. This is not necessarily the case here. Indeed, if ↑ z = (t, x) ∈ S2,1, then there are two paths in the forward Web that coalesce at z, i.e. two discontinuities merging there. According to Proposition 2.13, for s sufficiently close to t these discontinuities evolve continuously up to the time at which they merge but there is no reason for their difference to vanish. The pointwise limit of hbc(s, ·) as s ↑ t would then need to encode three different values at the point z, but the resulting object is not an ↑ element of D(R, R). Furthermore, according to Theorem 2.22 S2,1 is a countable yet dense subset of R2 so that points at which càdlàg continuity fails are very common! In the following proposition, whose proof is based on the above heuristics, we show that for any choice of initial condition, there is no version of the Brownian Castle hbc (defined by simply specifying its finite dimensional distributions) which is càdlàg in time and space.

Proposition 4.11 Given any initial condition h0 ∈ D(R, R) and T > 0, the Brownian Castle starting at h0 does not admit a version in D([0,T ],D(R, R)). The same is true for the periodic Brownian Castle.

Proof. Since a right-continuous function with values in D(R, R) is uniquely determined by its values at space-time points with rational coordinates (for example), it suﬃces to show that the exists a (random) time for which hbc admits no left limit in D(R, R). For this, it The Brownian Castle 47

(i) suffices to find a point (t, x) and three sequences (tk, xk )k≥0 (here i ∈ {1, 2, 3}) with tk ↑ t, (i) (i) xk → x, and limk→∞ hbc(tk, xk ) = Li with all three limits Li different from each other. ↑ ↑ Now, notice that, almost surely, one can find two elements x0, x1 ∈ Tbw with Mbw(xi) = ↑ (xi, 0) and x0, x1 in [−1, 1] such that, for the forward Brownian Web tree, one has % (x0,T ) = ↑ ↑ ↑ ↑ ↑ % (x1,T ). Writing t = inf{s > 0 : % (x0, s) = % (x1, s)} and x = Mbw(% (x0, t)), we then ↓ necessarily have (t, x) ∈ S0,3 by duality and, furthermore, the three trajectories emanating from (t, x) in the backwards Brownian Web cannot coalesce before time 0 by the non- crossing property. Since further, the Gaussian process Bbc is locally Hölder continuous and, with high probability, t ≥ c > 0 for some positive constant c, the claim then follows by (i) taking for (tk, xk ), sequences accumulating at (t, x) and belonging to these three trajectories.

4.2 The Brownian Castle as a Markov process We are now interested in studying the properties of the (periodic) Brownian Castle as a random interface evolving in time, i.e. as a stochastic process with values in D(R, R) (resp. D(T, R)). To do so, we need to introduce a suitable filtration on the probability space per 7 (Ω, F, Pbc) (resp. (Ωper, Fper, Pbc )) on which the Castle is defined . From now on, we assume that all sub-σ-algebras of F(resp. Fper) that we consider contain all null events. We will make use of the following construction. Given a metric space X, we write X2c ⊂ X2 for the set of all pairs (x, y) such that x and y are in the same path component of X. Given B : X → R (or into any abelian group), we then write δB : X2c → R for the map given by δB(x, y) = B(y) − B(x). α,β Let χ = (T , ∗, d, M, B) ∈ Ω = Cbsp (or Ωper) and, for −∞ ≤ s ≤ t < +∞, define def −1 def −1 Ts,t = M ((s, t] × R) (or Ts,t = M ((s, t] × T)). Let evals,t be the map given by

def evals,t(χ) = (ζs,t, δ(BTs,t)) (4.10)

def where ζs,t = (Ts,t, d, MTs,t). We use the notations

def def Fs,t = σ(evals,t) , Ft = σ(eval−∞,t) , (4.11) for the σ-algebras that they generate. The following property is crucial.

Lemma 4.12 If the intervals (s, t] and (u, v] are disjoint, then Fs,t and Fu,v are independent.

Proof. The fact that Ts,t and Tu,v are independent under the law of the Brownian Web tree was shown for example in [HW09, Prop. 2] (this is for a slightly diﬀerent representation of 7 α,β For example Ω can be taken to be Cbsp and Fthe Borel σ-algebra induced by the metric in (3.1) The Brownian Castle 48 the Brownian web, but the topological space T can be recovered from it in a measurable way). It remains to note that, conditionally on Ts,t and Tu,v, the joint law of δ(BTs,t) and δ(BTu,v) is of product form with the two factors being Ts,t and Tu,v-measurable respectively. This follows immediately from the independence properties of Brownian increments as formulated in Remark 3.8.

per One almost immediate consequence is that both hbc and hbc are time-homogeneous strong Markov processes satisfying the Feller property.

per Proposition 4.13 The (periodic) Brownian Castle hbc (resp. hbc ) is a time-homogeneous D(R, R) (resp. D(T, R))-valued Markov process on the complete probability space per (Ω, F, Pbc) (resp. (Ωper, Fper, Pbc )), with respect to the ﬁltration {Ft}t≥0 introduced per • • in (4.11). Moreover, both {hbc(t, )}t≥0 and {hbc (t, )}t≥0 are strong Markov and Feller.

Proof. The proof works mutatis mutandis for both the periodic and non-periodic case so we will focus on the latter. We have already shown that Pbc-almost surely for every h0 and t ≥ 0, hbc(t, •) ∈ D(R, R) (see Proposition 4.5). Moreover, by construction, hbc(t, •) only depends on eval−∞,t(χbc) so that it is clearly Ft-measurable. Notice that, by deﬁnition, for every 0 ≤ s < t and x ∈ R, we can write

↓ ↓ ↓ hbc(t, x) = hbc(s, Mbw(% (Tbw(t, x), s))) + Bbc(Tbw(t, x)) − Bbc(% (Tbw(t, x), s)) , so that hbc(t, •) is hbc(s, •) ∨ Fs,t-measurable. Since Fs,t and Fs are independent by Lemma 4.12, the Markov property follows, while the time homogeneity is an immediate ↓ ↓ consequence of the stationarity of (Tbw,Mbw,Bbc). Stochastic continuity was already shown in Proposition 4.5, so, if we show that the law of hbc(t, •) depends continuously (in the topology of weak convergence) on h0, then the Feller property holds. By the deﬁnition of the Skorokhod topology, it is suﬃcient to n show that, if {h0 }n∈N ⊂ D(R, R) is a sequence converging to h0 in D(R, R) then, for every n n R > 0, one has sup|x|≤R |hbc(t, x) − hbc(t, x)| → 0 in probability, where we write hbc for n the Brownian Castle with initial condition h0 . Note that

n n sup |hbc(t, x) − hbc(t, x)| ≤ sup |h0 (y(x)) − h0(y(x))| (4.12) |x|≤R |x|≤R

def ↓ ↓ where y(x) = Mbw(% (Tbw(t, x), 0)). With probability one, the set {y(x) : x ∈ [−R,R]} is ﬁnite and has empty intersection with the set of discontinuities of h0. Hence, the right-hand The Brownian Castle 49 side of (4.12) converges to 0 (almost surely and therefore also in probability) by [EK86, Prop. 3.5.2]. Since the Brownian Castle almost surely admits right continuous trajectories and is Feller, the same proof as in [RY91, Theorem III.3.1] guarantees that it is strong Markov (even though its state space is not locally compact).

per The periodic Brownian Castle hbc , is not only Feller, but also strong Feller, namely its Markov semigroup maps bounded functions to continuous functions. It will be convenient to write T [t] = M −1({t} × T) for the time-t “slice” of a spatial R-tree. Proposition 4.14 The periodic Brownian Castle satisﬁes the strong Feller property.

Proof. Let Φ ∈ Bb(D(T, R)) bounded by 1, h0 ∈ D(T, R) and t > 0. We aim to show that, ¯ for every ε > 0 there exists δ > 0 such that whenever dSk(h0, h0) < δ per per ¯ |Ebc[Φ(hbc (t, ·))|h0] − Ebc[Φ(hbc (t, ·))|h0]| < ε . (4.13)

Fix ε > 0. Let ν ∈ (0, t) suﬃciently small and N¯ big enough so that the probability of the events

def n ↓ per,↓ ↓ per,↓ o A1 = #{% (z, ν) : z ∈ Tbw [t]} = #{% (z, 0) : z ∈ Tbw [t]} n o def ↓ per,↓ ¯ A2 = #{% (z, 0) : z ∈ Tbw [t]} ≤ N

ε ¯ is each at least 1 − 3 . This is certainly possible since as ν goes to 0 and N tends to ∞ the probability of both A1 and A2 goes to 1. On A1 ∩ A2, let z1,..., zN be the list of all ↓ per,↓ ¯ per,↓ distinct points of {% (z, ν) : z ∈ Tbw [t]} (clearly, N ≤ N) and set yi = Mbw,x (yi) where ↓ yi = % (zi, 0). As before, the probability that one of the yi’s is a discontinuity point for h0 is 0. Hence, by [EK86, Prop. 3.5.2], for every εˆ > 0 it is possible to choose δ > 0 small ¯ def ¯ enough so that whenever dSk(h0, h0) < δ then δi = h0(yi) − h0(yi) satisﬁes |δi| < εˆ for all i. per,↓ Write simply B instead of Bbc as a shorthand. Note now that for every x ∈ T there exists i ≤ N such that one can write

per hbc (t, x) = h0(yi) + δB(yi, zi) + δB(zi, Tbw(t, x)) , (4.14)

per,↓ ↓ and, conditional on Tbw , the collection of random variables {δB(% (z, ν), z) : z ∈ per,↓ per,↓ Tbw [t]} is independent of the collection {δB(yi, zi)}i≤N . Conditional on Tbw and ¯ restricted to A1 ∩ A2, the law of the latter is N(0, νIdN ) for some N ≤ N. We now choose ¯ εˆ small enough so that kN(0, νIdN ) − N(h, νIdN )kTV ≤ ε/3, uniformly over all N ≤ N N and all h ∈ R with |hi| ≤ εˆ. The Brownian Castle 50

¯per ¯ Writing hbc for the Brownian Castle with initial condition h0, it immediately follows ¯per per from the properties of the total variation distance that we can couple hbc and hbc in such a per per ¯ • • ¯ ¯ way that P(hbc (t, ) = hbc (t, )) ≥ 1 − ε, uniformly over h0 with dSk(h0, h0) < δ, and (4.13) follows. We now want to study the large time behaviour of the Brownian Castle and its periodic counterpart. Notice at first that for any sublinearly growing initial condition h0, the variance ↓ of hbc(t, 0) grows like t since, by Definition 1.1, hbc(t, 0) conditioned on Tbw is Gaussian with variance t and mean given by h0, evaluated at the point where the backward Brownian motion starting at (t, 0) hits {0} × R. On the other hand, it is immediate that the Brownian Castle is equivariant under the action of R by vertical translations in the sense that one h0+a h0 ˜ has hbc = hbc + a. As a consequence, writing D(R, R) = D(R, R)/R for the quotient space, the canonical projection of the Brownian Castle onto D˜ is still a Markov process. ˜ ˜per We henceforth write hbc (respectively hbc ) for this Markov process. s Recall the stationary Brownian Castle hbc given in Definition 4.3. As above, we write ˜s ˜ hbc for its canonical projection to D and similarly for its periodic version, which, according to Remark 4.4, are truly stationary. With these notations, we then have the following result. Proposition 4.15 There exists a stopping time τ with exponential tails such that, for t ≥ τ, per per,s ˜ • ˜ • one has hbc (t, ) = hbc (t, ) independently of the initial condition h0. Proof. It suffices to take for τ the first time such that all the backward paths starting from {t} × T coalesce before hitting time 0, namely

def n ↓ 0 0 per,↓ o τ = inf t ≥ 0 : dbw(z, z ) ≤ 2t , ∀ z, z ∈ Tbw [t] .

Notice that τ coincides in distribution with T ↑(0) introduced in [CMT19, Sec. 3.1]. (This is by duality: the non-crossing property guarantees that all backwards trajectories starting from t coalesce before time 0 precisely when all forward trajectories starting from 0 have per ˜ • coalesced.) It follows immediately from the deﬁnitions that, for all t ≥ τ, hbc (t, ) is per,s ˜ • independent of h0 and therefore equal to hbc (t, ). Exponential integrability of τ then follows from [CMT19, Prop. 3.11 (ii)]. In the non-periodic case, one cannot expect such a strong statement of course, but the following bound still holds. Proposition 4.16 The bound

˜ ˜s log t Ebc[dSk(hbc(t, ·), hbc(t, ·))] . √ , t holds independently of the initial condition h0. The Brownian Castle 51

˜ ˜s Proof. The deﬁnition of dSk implies that if hbc(t, x) = hbc(t, x) for all x with |x| ≤ R, then ˜ ˜s −R dSk(hbc(t, ·), hbc(t, ·)) ≤ e and that, in any case, dSk is bounded by 1. It follows that ˜ ˜s c −R EdSk(hbc(t, ·), hbc(t, ·)) ≤ P(AR) + e P(AR) , ˜ ˜s for any R > 0 and any event AR implying that hbc(t, ·) and hbc(t, ·) agree on [−R,R]. It suﬃces to take for AR the event that the two√ backwards trajectories starting at (t, ±R) c coalesce before time 0. Since P(AR) . R/ t, choosing R = log t yields the claim. 4.3 Distributional properties In this section, we will focus on the distributional properties of the stationary Brownian Castle which, as showed in the previous section, describes the long-time behaviour of hbc per (resp. hbc ), at least modulo vertical translations. We begin with the following proposition which shows that hbc is indeed invariant with respect to the 1:1:2 scaling, i.e. its scaling exponents are indeed those characterising its own universality class.

s Proposition 4.17 Let hbc be the stationary Brownian Castle deﬁned according to (4.1). Then, for any λ > 0, 2 law s • • s • • λhbc( /λ , /λ) = hbc( , ) . (4.15) Proof. The claim clearly holds for all ﬁnite-dimensional distributions from the scaling s properties of Brownian motion. Since these characterise the law of hbc, (4.15) follows at once.

Remark 4.18 Note that (4.15) holds without having to quotient by vertical shifts, while this is necessary to have space-time translation invariance. Although Definition 1.1 provides a graphic description of the finite dimensional distributions of the Brownian Castle, we would like to obtain more explicit formulas characterising them. In the next proposition, we begin our analysis by studying the distribution of the increments at fixed time and as time goes to +∞.

Proposition 4.19 Let h0 ∈ D(R, R), t > 0 and hbc be the Brownian Castle with initial condition h0. Then, as |x − y| converges to 0

h (t, x) − h (t, y) law bc bc −→ Cauchy(0, 2) (4.16) x − y where, for a ∈ R and γ > 0, Cauchy(a, γ) denotes a Cauchy random variable with location parameter a and scale parameter γ. Moreover, for any x, y ∈ R, law hbc(t, x) − hbc(t, y) −→ Cauchy(0, 2|x − y|) (4.17) The Brownian Castle 52

s as t ↑ +∞. In particular, for any x, y ∈ R the stationary Brownian Castle hbc satisﬁes s s law hbc(t, x) − hbc(t, y) = Cauchy(0, 2|x − y|) for any t ≥ 0.

s Proof. The claim for hbc is clearly true since from its deﬁnition we have

s s law hbc(t, x) − hbc(t, y) = N(0, 2τy−x) , where τr is the law of the ﬁrst hitting time of r for a standard Brownian motion starting at 0. law 2 law 2 Now, τr = Levy(0, r ) and Cauchy(t) = N(Levy(0, t /2)), which implies the result. (4.17) follows from Proposition 4.16, while (4.16) can be reduced to (4.17) by Proposition 4.17.

We now turn our attention to the n-point distribution for n ≥ 3. Given x1, . . . , xn ∈ R, we aim at deriving an expression for the characteristic function of (Hbc(x1),..., Hbc(xn)), s • where we use the shorthand Hbc = hbc(0, ). By the definition of Hbc (and Definition 1.1), once the full ancestral structure of n independent coalescing (backward) Brownian motions starting from x1, . . . , xn respectively, is known, the conditional joint distribution of (Hbc(x1),..., Hbc(xn)) (modulo vertical shifts) is Gaussian and therefore easily accessible. In order to get our hands on the aforementioned ancestral structure we will proceed inductively using the strong Markov property of a finite family of coalescing Brownian motions. This will be possible if we are able to simultaneously keep track of the first time at which any two Brownian motions meet, which are the Brownian motions meeting and the position of all of them at that time. Let x1 < ··· < xn and let (yi)i≤n be independent standard Brownian motions starting at n−1 n−1 xi. Denote by Z = (τ, ι, Π ) the R+ × [n − 1] × R -valued random variable in which τ, ι and Πn−1 are defined by

def τ = inf {t > 0 : ∃ ι ∈ [n − 1] such that yι(t) = yι+1(t)} , (4.18) n−1 def Π = (y1(τ), . . . , yι−1(τ), yι+1(τ), . . . , yn(τ)) , where ι is implicitly deﬁned as the (almost surely unique) value appearing in the deﬁnition of τ. The random variable Z admits a density with respect to the product of the one-dimensional Lebesgue measure on R+, the counting measure on [n − 1] and the (n − 1)-dimensional Lebesgue measure, as the following variant of the Karlin–McGregor formula [KM59] shows.

n−1 Lemma 4.20 Let Z be the R+ × [n − 1] × R -valued random variable deﬁned in (4.18). Then, with the usual abuse of notation,

n−1 j Px τ ∈ dt, ι = j, Π ∈ dy = det Mt (x, y) dy dt (4.19) The Brownian Castle 53 where the n × n matrix M j is deﬁned by  p (y − x ) , for k < j,  t k i j def 0 Mt (x, y)i,k = pt(yj − xi) , for k = j, (4.20)  pt(yk−1 − xi) , for k > j, p is the heat kernel and p0 its spatial derivative.

Proof. This is an immediate corollary of Theorem C.1, combined with the Karlin–McGregor formula.

In the following proposition, we derive a recursive formula for the characteristic function of the n-point distribution of Hbc.

Proposition 4.21 Let Hbc be as above. For n ∈ N, α = (α1, . . . , αn), x = (x1, . . . , xn) ∈ n P R such that j αj = 0 and x1 < ··· < xn, let Fn(α, x) be the characteristic function of (Hbc(x1),..., Hbc(xn)) evaluated at α. Then, Fn satisﬁes the recursion n−1 Z Z X − 1 |α|2t j Fn(α, x) = e 2 Fn−1(cjα, y) det Mt (x, y) dy dt (4.21) j=1 R+ y1<···

j n−1 where the n × n matrix M was given in (4.20) and cjα ∈ R is the vector deﬁned by  αl , l < j def  (cjα)l = αj + αj+1 , l = j  αl+1 , l > j .

n Proof. Fix x1 < ··· < xn and consider the stochastic process (yi,Bij)i,j=1 where the yi are coalescing Brownian motions with initial conditions yi(0) = xi and the Bij are given by 1 dBij(t) = yi6=yj (dWi(t) − dWj(t)) , Bij(0) = 0 , (4.22) the Wi being i.i.d. standard Wiener processes independent of the yi’s. This process is strong Markov since so is the family of yi’s (see [TW98]). Furthermore, since the yi’s all coalesce at some time, the limit Bij(∞) = limt→∞ Bij(t) is well-defined and, by construction, one has n law n (Hbc(xi) − Hbc(xj))i,j=1 = (Bij(∞))i,j=1 We now combine the strong Markov property with the fact that B, as defined by (4.22), depends on its initial condition in an affine way with unit slope. This implies that, writing The Brownian Castle 54

˜ Hbc for a copy of Hbc that is independent of the process (y, B) and τ for any stopping time, one has the identity in law law ˜ ˜ (Bij(∞))i,j = (Bij(τ) + Hbc(yi(τ)) − Hbc(yj(τ)))i,j .

Write now (Ft) for the ﬁltration generated by the yi and the Wi and τ for the ﬁrst time at which any two of the y’s coalesce. Using furthermore the shorthand Hbc(x) for (Hbc(x1),..., Hbc(xn)) we have

h h ihα,H (x)i ii h ihα,H˜ (y(τ))i− 1 |α|2τ i bc bc 2 Fn(α, x) = E E e Fτ = E e n−1 ∞ Z Z |α|2 X − t ihα,Hbc(y)i n−1 = e 2 E e Px(τ ∈ dt, ι = l, Π ∈ dy) l=1 0 y1<···

Thanks to the results in Sections 4.1 and 4.2, and Proposition 4.19, we know that Hbc has increments which are stationary and distributed according to a Cauchy random variable with parameter given by their lengths, and admits a version whose trajectories are càdlàg and have (locally) ﬁnite p-variation for any p > 1 (see Proposition 4.7). If moreover we knew that the increments were independent, we could conclude that Hbc is nothing but a Cauchy process. The lack of independence is already evident by formula (4.21) in Proposition 4.21, therefore the k-point distributions of Hbc with k > 2 are diﬀerent from those of the Cauchy process. In the next proposition, we actually show more, namely that the law of Hbc is a genuinely new measure on D(R, R) since it is singular with respect to that of the Cauchy process.

Proposition 4.22 Let Hbc be as above and C be the standard Cauchy process on R+. Then, when restricted to [0, 1], the laws of Hbc and C are mutually singular.

Proof. We want to exhibit an almost sure property that distinguishes the laws of Hbc and 1 m C on D([0, 1], R). Let 1 ≥ x0 > ··· > xm ≥ 2 , ϕ : R → R be a bounded function and {λn}n an increasing sequence in [1, ∞) such that λn+1 ≥ 4λn for all n ∈ N. For n ≥ 1, deﬁne the functional Φn : D([0, 1], R) → R by n def 1 X def Φn(h) = ϕ(Ik(h)) , where (Ik(h))i = λk(h(xi/λk) − h(x0/λk)) n k=1 The Brownian Castle 55 for h ∈ D([0, 1], R). By scaling invariance and the independence properties of the Cauchy process {Ik(C)}k is i.i.d. while, by Proposition 4.17, {Ik(Hbc)}k is a sequence of identically distributed (but not independent!) random variables. Since furthermore ϕ is bounded, the classical strong law of large numbers holds for {ϕ(Ik(C))}k, which implies that almost surely lim Φn(C) = [Φ1(C)] . (4.23) n→∞ E

We claim that, provided that we choose a sequence {λn}n that increases sufficiently fast, (4.23) holds also for Hbc. Before proving the claim, notice that, assuming it holds, we are done. Indeed, it suffices to take m ≥ 2, and determine a function ϕ such that E[Φ1(C)] 6= Ebc[Φ1(Hbc)]. Such a function clearly exists since by Proposition 4.21 C and Hbc have different n-point distributions for n ≥ 3. ˆ We now turn to the proof of the claim. We will construct two sequences {Jk}k and m {Jk}k of R -valued random variables such that ˆ law J1 = J1 , {Jk}k∈N = {Ik(Hbc)}k∈N , ˆ ˆ and the sequence Jk is i.i.d. Arguing as for the Cauchy process, (4.23) holds for {ϕ(Jk)}k ˆ hence the claim follows if we can build {Jk}k and {Jk}k in such a way that, almost surely, ˆ Jk = Jk for all but finitely many values of k. Let {Wk,i : k ∈ N, i ∈ {0, . . . , m}} be a collection of i.i.d. standard Wiener processes and zk,i = (0, xk,i) be points with xk,i = xi/λk. We use them in two different ways. First, we apply the construction given at the start of Section 2.3 to each of the groups {(Wk,i, zk,i)}i≤m separately, which yields a collection {ζk = (Tk, ∗k, dk,Mk)}k∈N of characteristic spatial R-trees with each ζk representing coalescing Brownian motions starting from {zk,i}i≤m and Mk(∗k) = z0,k. We then apply the construction to the whole collection at once, taken in lexicographical order, so to obtain one “big” spatial R-tree ζ = (T , d, ∗,M). Let ˆzk,i ∈ Tk and zk,i ∈ T be the unique points such that Mk(ˆzk,i) = zk,i and M(zk,i) = zk,i, respectively. Write furthermore

τk = sup{t < 0 : %(zk,0, t) = %(zk−1,m, t)} .

Since both {ζk}k and ζ are built via the same Brownian motions, we clearly have ¯ Mk(%(ˆzk,i, t)) = M(%(zk,i, t)) for all i ≤ m and all t ∈ [τk, 0]. Denote by T ⊂ T the subspace given by ¯ T = {%(zk,i, t) : t ∈ [τk, 0], i ≤ m, k ∈ Z} , ¯ ⊂ ι: S ¯ → ¯ and similarly for Tk Tk, so that there is a canonical bĳection k∈N Tk T . We now turn to the branching maps. Fix independent Brownian motions Bk on each of B˜ : S ¯ → B the Tk and write k∈N Tk R for the map that restricts to k on each Tk. We then construct a Brownian motion B on T such that Convergence of 0-Ballistic Deposition 56

• Writing B¯ for the restriction of B to T¯, one has δB¯(z, z0) = δB˜(ιz, ιz0). • Conditionally on the W ’s, all the increments B(z) − B(z0) are independent of all the 0 ¯ Bk’s for any z, z ∈ T \ T . The independence properties of Brownian motion mentioned in Remark 3.7 guarantee that such a construction is possible and uniquely determines the law of B, conditional on the Wk,i’s and the Bk’s. We claim that setting ˆ Jk,i = λk(B(zk,i) − B(zk,0)) , Jk,i = λk(Bk(ˆzk,i) − Bk(ˆzk,0)) , ˆ the sequences {Jk}k and {Jk}k satisfy all the desired properties mentioned above. ˆ Clearly the Jk are independent and they are identically distributed by Brownian scaling. The fact that the Jk,i are distributed like Ik(Hbc) is also immediate from the construction, so ˆ it remains to show that Jk = Jk for all but ﬁnitely many values of k. Writing

τˆk = sup{t < 0 : %(ˆzk,0, t) = %(ˆzk,m, t)} , ˆ we have that Jk = Jk as soon as |τˆk| ≤ |τk|. For k > 0,

2 law xm x0 −2 |τk| = − Levy(1) ≥ (4λk−1) Levy(1) , λk−1 λk 2 law x0 − xm −2 |τˆk| = Levy(1) ≤ λk Levy(1) , λk where we use the fact that λk ≥ 4λk−1 in the ﬁrst inequality. If we choose 0 < ck < Ck < ∞ such that −2 −2 P(Levy(1) < ck) ≤ k , P(Levy(1) > Ck) ≤ k , −2 we can conclude that P(|τˆk| ≥ |τk|) ≤ 2k provided that we choose the λk’s in such a way 2 2 ˆ that ckλk ≥ 16Ckλk−1. Hence, by Borel-Cantelli we have Jk = Jk for all but ﬁnitely many k’s and the proof is concluded.

5 Convergence of 0-Ballistic Deposition

In this last section, we show that the 0-Ballistic Deposition model does indeed converge to the Brownian Castle. In order to prove Theorem 1.4, we will begin by associating to 0-BD a branching spatial R-tree and prove that, when suitably rescaled, the law of the latter converges to Pbc deﬁned in (3.39). Convergence of 0-Ballistic Deposition 57

5.1 The Double Discrete Web tree We begin our analysis by recalling the construction and the results obtained in [CH21, Section 4], concerning the spatial tree representation of a family of coalescing backward random walks and its dual. Let δ ∈ (0, 1] and (Ω, A, Pδ) be a standard probability space supporting four Poisson L R L R L R ↓ def random measures, µγ , µγ , µˆγ and µˆγ . The ﬁrst two, µγ and µγ , live on Dδ = R × δZ, are independent and have both intensity γλ, where, for every k ∈ δZ, λ(dt, {k}) is a copy of the Lebesgue measure on R and throughout the section 1 γ = γ(δ) =def . (5.1) 2δ2

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

L def R R def L µˆγ (A) = µγ (A − δ/2) and µˆγ (A) = µγ (A + δ/2) . (5.2)

Here, A ± δ/2 =def {z ± (0, δ/2) : z ∈ A}. Representing the previous Poisson processes via arrows as in Figure 4, it is not hard to ↓,δ ↑,δ define two families of coalescing random walks {π } ↓ and {π } ↑ , the first running z∈Dδ z∈Dδ backward in time and the second, forward, in such a way they never cross (see [CH21, Section 4.1] and in particular Figure 1 therein). Thanks to these, we can state the following definition, which is taken from [CH21, Definition 4.1].

L R Deﬁnition 5.1 Let δ ∈ (0, 1], γ as in (5.1), µγ and µγ be two independent Poisson random ↓ L R ↓,δ measures on δ of intensity γλ, µˆ and µˆ be given as in (5.2) and {πz } ↓ and D z∈Dδ ↑,δ {πzˆ } ↑ be the families of coalescing random walks introduced above. We deﬁne the zˆ∈Dδ ↓↑ def ↓ ↑ Double Discrete Web Tree as the couple ζδ = (ζδ , ζδ ), in which

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

↓ 0 0 ↓,δ ↓,δ dδ(z, z¯) = t + t − 2 sup{s ≤ t ∧ t : πz (s) = πz¯ (s)} . (5.3)

↑ def ↑ ↑ ↑ ↑ ↑ - ζδ = (Tδ , ∗δ, dδ,Mδ ) is built similarly, but with ∗δ = (0, δ/2) and the supremum in 0 ↑,δ ↑,δ (5.3) replaced by inf{s ≥ t ∨ t : πz (s) = πz¯ (s)}. As was pointed out in [CH21], the Discrete Web Tree and its dual are not spatial trees since the random walks π↓,δ, π↑,δ are not continuous. To overcome the issue, in [CH21, Convergence of 0-Ballistic Deposition 58

˜↓↑ ↓↑ Section 4.1], it was introduced a suitable modification ζδ of ζδ , called the Interpolated Double Discrete Tree (see [CH21, Definition 4.2]). Simply speaking, the latter was obtained • • • • by keeping the same pointed R-trees (Tδ , ∗δ, dδ), ∈ {↑, ↓}, as those of the Double Discrete ˜ ↓ ˜ ↑ Tree and defining new evaluation maps Mδ and Mδ by interpolating the discontinuities of ↓ ↑ Mδ and Mδ in such a way that the properties (1), (2) (and (3)) of Definition 2.3 were kept and continuity was restored. The following result is a consequence of Propositions 4.3 and 4.4, and Theorem 4.5 in [CH21].

Theorem 5.2 For any δ ∈ (0, 1) and α ∈ (0, 1/2), the Interpolated Double Discrete ˜↓↑ α ˆ α Web Tree ζδ introduced above almost surely belongs to Csp × Csp and satisfies (t) of Definition 2.9. ↓↑ ˜↓↑ α ˆ α ↓ ↑ ↓ Let Θδ be the law of ζδ on Csp × Csp, with marginals Θδ and Θδ. Then, Θδ is tight in α 3 ↓↑ E (θ) (see (3.9)) for any θ > 2 and, as δ ↓ 0, Θδ converges to the law of the Double Web ↓↑ α ˆ α Tree Θbw of Definition 5.1 weakly on Csp × Csp. At last, almost surely, for • ∈ {↑, ↓}

˜ • • sup kMδ(z) − Mδ(z)k ≤ δ (5.4) • z∈Tδ

• where Mδ are the evaluation maps of the double Discrete Web Tree in Deﬁnition 5.1. 5.2 The 0-BD measure and convergence to BC measure We are now ready to introduce the missing ingredient in the construction of the 0-BD tree, namely the Poisson random measure responsible of increasing the height function by 1. L R • Let δ ∈ (0, 1], γ as in (5.1), µγ and µγ be as in the previous section and µγ be a Poisson ↓ L R random measure on Dδ of intensity 2γλ and independent of both µγ and µγ . L R ↓ For a typical realisation of µγ and µγ , consider the Discrete Web Tree ζ = ↓ ↓ ↓ ↓ • (Tδ , ∗δ, dδ,Mδ ) in Deﬁnition 5.1. Let µγ be the measure on Tδ induced by µγ via

def • ↓ µγ(A) = µγ(Mδ (A)) (5.5)

↓ ↓ ↓ for any A Borel subset of Tδ . Notice that Mδ is bĳective on Dδ so that µγ is well-deﬁned • and, since µγ is independent over disjoint sets, µγ is distributed according to a Poisson ↓ ↓ random measure on Tδ of intensity 2γ`, ` being the length measure on Tδ (see (3.15).

L R • Deﬁnition 5.3 Let δ ∈ (0, 1), γ as in (5.1), µγ , µγ and µγ be three independent Poisson random measures on Dδ of respective intensities γλ, γλ, and 2γλ. We deﬁne the 0-BD Tree Convergence of 0-Ballistic Deposition 59

δ def ↓ ↓ as the couple χ0-bd = (ζδ ,Nγ) in which ζδ is as in Deﬁnition 5.1 while Nγ is the rescaled compensated Poisson process given by

def ↓ Nγ(z) = δ(µγ( ∗, z ) − 2γdδ(z, ∗)) . (5.6) J K and µγ is the measure given in (5.5). As before, the 0-BD tree also fails to be a branching spatial tree since neither the evaluation nor the branching map are continuous. The remedy here was already presented in Section 3.2 where we introduced the smoothened RC Poisson process.

L R • Remark 5.4 We can view the triple (µγ , µγ , µγ) as an element of the space of locally finite integer-valued measures endowed with the topology of vague convergence. All functions of δ χ0-bd mentioned later on are Borel measurable with respect to this. Definition 5.5 In the setting of Definition 5.3 and for p > 2 and a = (2γ)−p, we define δ def ˜↓ a ˜↓ the smoothened 0-BD Tree as the couple χ˜0-bd = (ζδ ,Nγ ) in which ζδ is the Interpolated a Discrete Web Tree of Section 5.1 while Nγ is the RCS Poisson process of Definition 3.9 associated to the Poisson random measure µγ given in (5.5).

δ Proposition 5.6 For any δ ∈ (0, 1] and α, β ∈ (0, 1) the smoothened 0-BD Tree χ˜0-bd = ˜↓ a (ζδ ,Nγ ) in Deﬁnition 5.5 is almost surely a characteristic (α, β)-branching spatial pointed δ α,β R-tree. Its law P0-bd on Cbsp , which we call the 0-BD measure, can be written as Z δ def Poiγ ↓ P0-bd(dχ) = Qζ (dχ)Θδ(dζ) (5.7)

↓ ˜↓ α Poiγ where Θδ denotes the law of ζδ in Theorem 5.2 on Csp and Qζ that of the RCS Poisson a α,β process Nγ on Cbsp . Moreover, almost surely (5.4) holds and for every r > 0 there exists a constant C = C(r) > 0 such that for all δ small enough and k > p/(p − 2)

a Pδ sup |Nγ (z) − Nγ(z)| > kδ ≤ Cδ , (5.8) ↓ (r) z∈(Tδ )

δ where Nγ is the branching map of χ˜0-bd in Definition 5.3. Proof. The first part of the statement is a direct consequence of the fact that almost surely ˜↓ ↓ ζδ ∈ Eα since Tδ is almost surely locally finite, so that Lemma 3.10 applies, while the measurability conditions required to make sense of (5.7) are implied by Proposition 3.11. Moreover, (5.4) follows by Theorem 5.2 so that we are left to show (5.8). Convergence of 0-Ballistic Deposition 60

δ Let us recall the deﬁnition of the event ER given in [CH21, Proposition 4.4] (see also ± Proposition 3.2 Eq. (3.4) in the same reference). For r ≥ 1 and R > r, let QR be two squares of side 1 centred at (r + 1, ±(2R + 1)), z± = (t±, x±) be two points in the interior ± ± ± ± of QR and {zδ }δ ⊂ QR ∩ (Dδ) be sequences converging to z . We set

Eδ =def { sup |π↓,δ(s)| ≤ R, sup |π↓,δ(s) − x±| ≤ R} R 0 z± δ 0≥s≥−r ± δ tδ ≥s≥−r

Notice that, as was shown in [CH21, Eq. (4.9) and (3.5)], √ r R2 δ − 2r liminf Pδ(ER) ≥ 1 − e . (5.9) δ↓0 R

δ ˜ ↓ ↓ (r) δ def ↓ Moreover, on ER, Mδ ((Tδ ) ) ⊂ Λr,R = ([−r, r] × [−3R − 1, 3R + 1]) ∩ Dδ. a −1/2 Now, for any positive integer k, supz |Nγ (z) − Nγ(z)| > k(2γ) only if there exists ↓ (r) −p z ∈ (Tδ ) and a neighbourhood of z of size a = (2γ) which contains more than k µγ- −1 points. By the deﬁnition of µγ in (5.5), this implies that there must exist i = 0,..., d2ra e δ def • such that the rectangle ([ti+1, ti] × R) ∩ Λr,R, ti = r − 2ia, contains at least k µγ-points. These considerations together with (5.9), lead to the bound √ 1 2 a − r − R −2p 2p−3 k 2 2r P sup |Nγ (z) − Nγ(z)| > kγ . e + 2rδ (4δ R) . ↓ (r) R z∈(Tδ )

Therefore, taking R = δ−1, choosing k suﬃciently large so that k > p/(p − 2), and then δ suﬃciently small (5.8) follows at once.

We are now ready to show that the law of smoothened 0-BD tree converges to the Brownian Castle measure Pbc of Theorem 3.16.

−p δ Theorem 5.7 Let α ∈ (0, 1/2) and, for δ ∈ (0, 1] and p > 1, a = (2γ) and P0-bd be the δ law of the smoothened 0-BD tree given in (5.7). Then, as δ ↓ 0, P0-bd converges to Pbc α,β 1 weakly on Cbsp , for any β < βPoi = 2p .

α,β Proof. Since Cbsp is a metric space, it suﬃces to show convergence when testing against any R δ Lipschitz continuous bounded function F . By (3.39) and (5.7), we see that | F (χ)(P0-bd − Pbc)(dχ)| ≤ I1 + I2 with ZZ I =def F χ Poiγ χ − Gau χ Θ↓ ζ 1 ( ) Qζ (d ) Qζ (d ) δ(d ) Convergence of 0-Ballistic Deposition 61

Z Z I =def F χ Gau χ Θ↓ ζ − Θ↓ ζ . 2 ( )Qζ (d ) δ(d ) bw(d )

↓ α Since Θδ converges by Theorem 5.2, for every ε > 0 we can ﬁnd a compact subset Kε ⊂ Csp ↓ with supδ>0 Θδ(Kε) ≥ 1 − ε. Hence, Z Z I ≤ F χ Poiγ χ − Gau χ Θ↓ ζ + 2kF k ε 1 ( ) Qζ (d ) Qζ (d ) δ(d ) ∞ Kε Z ≤ F χ Poiγ χ − Gau χ + 2kF k ε . sup ( ) Qζ (d ) Qζ (d ) ∞ ζ∈Kε As δ → 0 the ﬁrst term converges to 0 by Proposition 3.12 and, since the left hand side is independent of ε, we conclude that I1 → 0. Finally, Proposition 3.11 and the Lipschitz R Gau continuity of F imply that the map ζ 7→ F (χ)Qζ (dχ) is continuous so that I2 → 0 by Theorem 5.2.

5.3 The 0-BD model converges to the Brownian Castle In order to establish the convergence of the 0-Ballistic Deposition model to the Brownian δ δ Castle, let δ > 0 and χ0-bd the 0-BD Tree given in Deﬁnition 5.3. Let h0 ∈ D(R, R) and, as in (4.2), set

δ def δ ↓ ↓ ↓ h0-bd(z) = h0(Mδ,x(%δ(Tδ(z), 0))) + Nγ(Tδ(z)) − Nγ(%δ(Tδ(z), 0)) (5.10)

↓ for all z ∈ R+ × R, where Tδ is the tree map associated to ζδ of Deﬁnition 2.11. Even δ though χ0-bd is not a characteristic branching spatial tree, (5.10) still makes sense and provides a version (say, in D(R+,D(R, R))) of the rescaled and centred 0-BD in the sense δ that its k-point distributions agree with those of h0-bd in (1.9). Before proving Theorem 1.4, let us state the following lemma which will be needed in the proof.

↓ Lemma 5.8 Let ζbw be the backward Brownian Web tree of Deﬁnition 2.17 and A ⊂ R be a ﬁxed subset of measure 0. Then, with probability 1

↓ ↓ ↓ {Mbw,x(% (z, 0)): Mbw,t(z) > 0} ∩ A = ∅ .

↓ law ˜↓ 2 Proof. It suﬃces to note that, by Theorem 2.16, ζbw = ζ (Q ) and that, for a Brownian motion B, one has P(Bt ∈ A) = 0 for any ﬁxed t > 0.

Proof of Theorem 1.4. By Theorem 5.7 and Skorokhod’s representation theorem, there χδn χ˜δn ζ˜↑ χ ζ↑ exists a probability space supporting the random variables 0-bd, 0-bd, δn , bc and bw in such a way that the following properties hold. Convergence of 0-Ballistic Deposition 62

χ˜δn χδn ζ˜↑ 1. The random variables 0-bd, 0-bd and δn are related by the constructions in Section 5.1 and Definitions 5.3 and 5.5. ↑ 2. Similarly, the random variables χbc and ζbw are related by the construction of Definition 2.20. χ˜δn → χ ζ˜↑ → ζ↑ α,β α 3. One has 0-bd bc and δn bw almost surely in Cbsp and Csp respectively, for all α ∈ (0, 1/2) and β < βPoi. We consider this choice of random variables fixed from now on and, in order to shorten notations, we will henceforth replace δn by δ with the understanding that we only ever consider values of δ belonging to the fixed sequence. We now define the countable set D ⊂ R appearing in the statement of the theorem as ↓ the set of times t ∈ R+ for which there is x ∈ R with (t, x) ∈ S(0,3) (see Definition 2.21). Our goal is then to exhibit a set of full measure such that, for every T/∈ D, every R > 0 and every ε > 0, there exists δ > 0 and λ ∈ Λ([−R,R]) for which γ(λ) ∨ [−R,R] δ dλ (hbc(T, ·), h0-bd(T, ·)) < ε. The proof will be divided into four steps, but before delving into the details we will need some preliminary considerations. We henceforth consider a sample of the random variables mentioned above as given δ and we fix some arbitrary T/∈ D and R, ε > 0. Since the sets {χbc, χ0-bd}δ and ˜↑ ˜↑ {ζbw, ζδ }δ are compact, point 3. of Proposition 2.5 and (5.4) imply that if we choose r > 2 supδ b˜↓ (2(R ∨ T )) ∨ b˜↑ (2(R ∨ T ))then, for all δ and • ∈ {↓, ↑}, ζδ ζδ

• −1 •, (r) (Mδ) ({T } × [−R,R]) ⊂ Tδ .

• where Mδ are the evaluation maps in Deﬁnition 5.1 (for the non-interpolated trees). Invoking once more Proposition 2.5, we also know that the constant Cr > 0 given by

def • (r) ˜ • (r) (r) a (r) Cr = sup{kMbwkα , kMδkα , kBbckβ , kNγ kβ : δ ∈ (0, 1]} ∨ 1 is finite. Step 1. As a first step in our analysis, we want to determine a set of distinct points y1 < ··· < yN+1 for which the modulus of continuity of hbc on {T } × [yi, yi+1) can be easily controlled. ˜↓ Let 0 < η1 < ε be sufficiently small and Ξ[−R,R](T,T − η1) be defined according ˜↓ to (4.4). We order its elements in increasing order, i.e. Ξ[−R,R](T,T − η1) = {x1, . . . , xN } def ˜↓ with x1 = min Ξ[−R,R](T,T − η1) and let {yi : i = 1,...,N + 1} be as in (4.5). ↓ ↑ Since T/∈ S(0,3) = S(2,1), arguing as in the proof of Proposition 4.5, there exists tc ∈ (T − η1,T ) such that no pair of forward paths starting before T − η1 and passing through + − {T − η1} × [x1, xN ] coalesces at a time s ∈ (tc,T ]. For each i = 1,...,N, let xi , xi be Convergence of 0-Ballistic Deposition 63

↑ −1 the points in (Mbw) (T − η1, xi) from which the right-most and left-most forward paths ↑ ↑ − ↑ ↑ + from (T − η1, xi) depart and such that Mbw(% (xi ,T )) = yi and Mbw(% (xi ,T )) = yi+1. Notice that these coincide with the right-most and left-most point from (T, xi) defined in ↑ Remark 2.8 unless (T, xi) ∈ S(0,3). Step 2. As a second step, we would like to determine a sufficiently small δ and points δ δ δ y1 < ··· < yN+1 which play the same role as the yi’s, but for h0-bd, and are close to them. 1 ↑,(r),η Let η < 2 η1 and M ≥ 1 the number of endpoints of Tbw , which is finite by points 2. and 3. of Lemma D.1. Let η2 > 0 be such that

α |T − tc| 12Crη < min{|yi − yi+1|: i = 1,...N} ∧ . (5.11) 2 10M δ ˜↑ ↑ Thanks to the fact that ∆bsp(χbc, χ˜0-bd) ∨ ∆sp(ζδ , ζbc) → 0 and Lemma D.1, there exists ↓ ↓,(r) ↓,(r) ↑ δ = δ(η2) > 0 and correspondences C , between Tbw and Tδ , and C , between ↑,(r),η ↑,(r),η Tbw and Tδ (see (D.3) below), such that c, C↓ (r) δ,(r) c, C↑ ↑,(r),η ˜↑,(r),η ∆bsp (χbc , χ˜0-bd) ∨ ∆sp (ζbw , ζδ ) < η2 . (5.12) ↑ ↑ Let us define the subtrees Tbw and Tδ according to (D.5) and the corresponding spatial trees ↑ ˜↑ Zbw and Zδ as in (D.6). • def ↑ • For 1 ≤ i ≤ N and • ∈ {+, −}, define wi = % (xi,T − η1 + η + η2). Applying • ↑ • ↑ Lemma D.2, it follows from (D.9) that wi,% (wi,T ) ⊂ Tbw and, by the definition of ↑ ↑ J ↑ K •,δ ↑ Tbw, Tδ and of the path correspondence Cp of (D.7), there exists wi ∈ Tδ such that • •,δ ↑ δ (wi, wi ) ∈ Cp . In the following lemma, we determine the yi ’s and complete the second step of the proof.

def ↑ ↑ •,δ • Lemma 5.9 For η2 as in (5.11), the set Yδ = {Mδ (%δ(wi ,T )) : 1 ≤ i ≤ N, ∈ {+, −}} δ δ δ δ 1 contains exactly N + 1 points y1 < ··· < yN+1 which satisfy |yi+1 − yi | ≥ 3 mini{|yi − ↑ ↑ y1+1|}. Moreover, there exists no point zδ ∈ Tδ such that Mδ,t(zδ) < T − η1 − 5Mη2 and, δ ↓ ↑ δ for some i, yi < Mδ,x(%δ(zδ,T )) < yi+1.

Proof. In order to verify that Yδ has at most N + 1 points, it suﬃces to show that, for +,δ −,δ all i ∈ {1,...,N − 1}, the rays starting at wi and wi+1 coalesce before time T . By +,δ −,δ Lemma D.2, the distance between wi and wi+1 is bounded by

↑ +,δ −,δ ↑ + − dδ(wi , wi+1) ≤ dbw(wi , wi+1) + 4Mη2 ≤ 2(tc − (T − η1 + η + η2) + 2Mη2) ↑ +,δ ↑ −,δ so that if s¯ is the ﬁrst time at which %δ(wi , s¯) = %δ(wi+1, s¯) then, by (5.11),

1 ↑ +,δ −,δ s¯ = T − η1 + η + η2 + 2 dδ(wi , wi+1) ≤ tc + (4M + 1)η2 < T . Convergence of 0-Ballistic Deposition 64

Hence, the cardinality of Yδ is not bigger than N + 1 and we can order its elements as δ δ y1 ≤ · · · ≤ yN+1. To show that the inequalities are strict, notice that, again by Lemma D.2 and (5.4), we have

δ ↑ ↑ − ↑ ↑ −,δ |yi − yi | = |Mbw(% (wi ,T )) − Mδ,x(%δ(wi ,T ))| ↑ ↑ − ˜ ↑ ↑ −,δ ≤|Mbw(% (wi ,T )) − Mδ,x(%δ(wi ,T ))| (5.13) ˜ ↑ ↑ −,δ ↑ ↑ −,δ α 1 + |Mδ,x(%δ(wi ,T ))| − Mδ,x(%δ(wi ,T ))| . Crη2 + δ ≤ min{|yi − yi+1|} . 6 i

δ δ The lower bound on |yi − yi+1| follows at once. ↑ For the second part of the statement, we argue by contradiction and assume zδ ∈ Tδ ↑ δ ↓ ↑ δ is such that Mδ,t(zδ) < T − η1 − 5Mη2 and yi < Mδ,x(%δ(zδ,T )) < yi+1. Note that def ↑ ↑ ↑ Izδ = %δ(zδ,T − η1 + η),%δ(zδ,T ) ⊂ Tδ since all the points in the segment are at J K ↑,(r) ↑ distance at least η + 5Mη2 from zδ and, by (D.9), Rη+5Mη2 (Tδ ) ⊂ Tδ . Hence, there ↑ ↑ ↑ ↑ exists w ∈ Tbw such that for all s ∈ [T − η1 + η, T ], (% (w, s),%δ(zδ, s)) ∈ Cp . Now, ↑ ↑,(r),η w ∈ Tbw and the latter is contained in Tbw by (D.9), therefore there must be a point ↑,(r) ↑ ↑ w¯ ∈ Tbw such that Mbw,t(w¯ ) ≤ T −η1 and % (w¯ ,T −η1 +η) = w. Since, by construction, ↑ all the rays in Tbw starting before T − η1 must coalesce before time tc and the tree is ↑ ↑ ↑ + characteristic, w must be such that either Mbw,x(% (w,T − η1 + η + η2)) ≥ Mbw,x(wi ) or ↑ ↑ ↑ − Mbw,x(% (w,T −η1 +η +η2)) ≤ Mbw,x(wi ). Assume the ﬁrst (the other case is analogous), ↑ ↑ + then, by the coalescing property, for all s ≥ tc, % (w, s) = % (wi , s), which means that ↑ + ↑ ↑ (% (wi , s),%δ(zδ, s)) ∈ Cp . Therefore,

↑ ↑ i ↑ +,δ ↑ ↑ ↑ +,δ dδ(%δ(zδ,T − tc),%δ(wi ,T − tc)) = |dδ(%δ(zδ,T − tc),%δ(wi ,T − tc)) ↑ ↑ + ↑ + − d (% (wi ,T − tc),% (wi ,T − tc))| < 4Mη2 ≤ T − tc .

−,δ ↑ −,δ +,δ ↑ +,δ However, the segment Izδ cannot intersect either wi ,%δ(wi ,T ) or wi ,%δ(wi ,T ) , ↓ ↑ δ δ J ↑ ↑K i J ↑ +,δ K since otherwise Mδ,x(%δ(zδ,T )) = yi or yi+1. This implies that dδ(%δ(zδ,T −tc),%δ(wi ,T − tc)) > T − tc, which is a contradiction thus completing the proof.

Before proceeding, let us introduce, for all i = 1,...,N, the following trapezoidal δ 2 regions ∆i and ∆i in R

def [ ↑ ↑ − ↑ ↑ + ∆i = [Mbw,x(% (wi , s),Mbw,x(% (wi , s))] s∈[T −η +η+η ,T ] 1 2 (5.14) δ def [ ↑ ↑ −,δ ↑ ↑ +,δ ∆i = [Mδ,x(%δ(wi , s),Mδ,x(%δ(wi , s))] . s∈[T −η1+η+η2,T ] Convergence of 0-Ballistic Deposition 65

By Lemma 5.9 and the non-crossing property of forward and backward trajectories (see Theorem 2.19(ii) and the construction of the double Discrete Web Tree in Definition 5.1), δ δ δ ↓ every couple of points z1, z2 ∈ Tbw(∆i) and z1, z2 ∈ Tδ(∆i ) satisfies dbw(z1, z2) ≤ 2η1 ↓ δ δ δ δ δ and dδ(z1, z2) ≤ 2(η1 + 5Mη2). Indeed, if there existed points z1, z2 ∈ Tδ(∆i ), for ↓ δ δ ↓ ↓ δ which dδ(z1, z2) > 2(η1 + 5Mη2), then the paths Mδ (%δ(zi , ·)) would coalesce before T − η1 − 5Mη2. This in turn would imply the existence of a forward path starting before T − η1 − 5Mη2 at a position x lying in between the two trajectories, which, because of δ δ the non-crossing property, at time T would be located strictly between yi and yi+1, thus contradicting the above lemma. i i ↓ Step 3. In this third step, we want to show that for every i we can find a couple (z , zδ) ∈ C i δ ↓ i such that zδ ∈ Tδ(∆i ) and % (z , s¯) ∈ T(∆i) for some a s¯ sufficiently close to T . i ↓ ↓ i δ Let i ∈ {1,...,N} and zδ ∈ Tδ be such that Mδ (zδ) = (T, x) and x ∈ (yi + α δ α 6Crη2 , yi+1 − 6Crη2 ), which exists thanks to Lemma 5.9 if we choose η2 as in (5.11). i δ Clearly, zδ ∈ Tδ(∆i ). i ↓ i i ↓ ↓ i ↓ i ↓ i Let z ∈ Tbw be such that (z , zδ) ∈ C . If Mbw,t(z ) > T , since dbw(z ,% (z ,T )) = ↓ i ↓ i |Mbw,t(z ) − Mδ,t(zδ)| < η2 (the last inequality being a consequence of (5.12)) we have ↓ i ↓ ↓ i α |Mbw,x(z ) − Mbw,x(% (z ,T ))| ≤ Crη2 . Hence

↑ ↑ − ↑ ↑ + α sup |Mbw(% (wi , s)) − yi| ∨ |Mbw(% (wi , s)) − yi+1| ≤ Crη2 . s∈[T −η2,T ]

↓ ↑ ↑ − ↓ so that we can argue as above and show |Mbw,x(z) − Mbw(% (wi , t))| ∧ |Mbw,x(z) − ↑ ↑ + Mbw(% (wi , t))| > 0. As a consequence of the coalescing property and the previous bounds, for all points δ w ∈ Tbw(∆i) and wδ ∈ Tδ(∆i ) we have

↓ i ↓ i dbw(w, z ) ≤ η2 + 2η1 , dδ(wδ, zδ) ≤ 2(η1 + 5Mη2) . (5.15)

Step 4. We can now exploit what we obtained so far, go back to the height functions hbc δ and h0-bd, and complete the proof. First, let λ : R → R be the continuous function such δ 0 that λ(yi) = yi for all i, interpolating linearly between these points, and λ (x) = 1 for Convergence of 0-Ballistic Deposition 66

δ δ x 6∈ [y1, yN+1]. In particular, one has λ(x) ∈ [yi , yi+1) if x ∈ [yi, yi+1). Note that as a consequence of (5.13), we can choose η1 and δ suﬃciently small so that γ(λ) ≤ ε. Let x ∈ [yi, yi+1). Then,

δ ↓ ↓ i ↓,δ ↓ ↓ i |hbc(T, x) − h0-bd(T, λ(x))| ≤ |h0(Mbw,x(% (z , 0))) − h0 (Mδ,x(%δ(zδ, 0)))| (5.16) ↓ i ↓ i + |Bbc(T(T, x)) − Nγ(Tδ(T, λ(x)))| + |Bbc(% (z , 0)) − Nγ(%δ(zδ, 0))| , where we chose η1 and η2 suﬃciently small, so that T − η1 − 5Mη2 > 0 and consequently, ↓ ↓ i ↓ ↓ i by (5.15), % (T(T, x), 0) = % (z , 0) and %δ(Tδ(T, λ(x)), 0) = %δ(zδ, 0). For the second term a in (5.16) we exploit the Hölder continuity of Bbc and Nγ , (5.8), (5.15) and (5.12) which give

For the last term in (5.16), arguing as in the proof of [CH21, Lemma 2.28] (replacing M1 a and M2 by B and Nγ in the statement) and using (5.8), we have

↓ ↓ ↓ {Mbw,x(% (z, 0)): Mbw,t(z) > 0} ∩ Disc(h0) = ∅ .

↓ ↓ i In particular, for all i, Mbw,x(% (z , 0)) is a continuity point of h0 and by assumption δ dSk(h0, h0) → 0. By choosing η1, η2 and δ suﬃciently small, we can guarantee on the one hand that each of (5.17) and (5.18) is smaller than ε/3, while on the other that the distance between ↓ ↓ i ↓ ↓ i Mbw,x(% (z , 0)) and Mδ,x(%δ(zδ, 0)) is arbitrarily small. This, together with the fact that by δ ↓ ↓ i assumption dSk(h0, h0) → 0 and that Mbw,x(% (z , 0)) is a continuity point for h0, implies that also the third term in (5.16) can be made smaller than ε/3. We conclude that [−R,R] δ γ(λ) ∨ dλ (hbc(T, ·), h0-bd(T, ·)) < ε as required to complete the proof. The smoothened Poisson process 67

Appendix A The smoothened Poisson process

In this appendix, we derive bounds on the Orlicz norm of the increment of a smoothened version of the Poisson process. Let a > 0 and ψa be a smooth non negative function R supported in [−a, 0] or [0, a] such that ψa(x)dx = 1. For λ > 0 let µλ be a Poisson random measure on R+ with intensity measure λ`, where ` is the Lebesgue measure on a R+, and deﬁne the rescaled compensated smoothened Poisson process Pλ and the rescaled compensated Poisson process Pλ respectively by Z t a 1 def 1 Pλ = √ ψa ∗ µλ(s)ds − λt , Pλ(t) = √ (µλ([0, t]) − λt) . (A.1) λ 0 λ Then, the following lemma holds.

a Lemma A.1 In the setting above, let Pλ be the rescaled compensated smoothened Poisson process on [0,T ] deﬁned in (A.1). Let p > 1 and assume aλp = 1. Then, there exists a positive constant C depending only on T such that for every 0 ≤ s < t ≤ T , we have

1 a a 2p kPλ (t) − Pλ (s)kϕ1 ≤ C(t − s) (A.2) where the norm appearing on the left hand side is the Orlicz norm deﬁned in (3.5) with def x ϕ1(x) = e − 1.

Proof. We prove the result for ψa supported in [−a, 0]. Also, writing P (t) = P1(t), we have EeP (t)/c = exp(t(e1/c − 1 − 1/c)), so that √ kP (t)kϕ1 . 1 + t . (A.3) Fix 0 ≤ s < t ≤ T and consider ﬁrst the case t − s ≥ a. Notice that we have Z a a 1 Pλ (t) − Pλ (s) = √ (ψa(t − u) − ψa(s − u))µλ([0, u])du − λ(t − s) λ R √ law 1 √ ≤ Pλ(t + a) − Pλ(s) + λa = √ P (λ(a + t − s)) + λa . λ It follows from (A.3) and the triangle inequality that

√ 1 √ √ 1 1 a a 2p kPλ (t) − Pλ (s)kϕ1 . t − s + a + √ + λa . t − s + √ . (t − s) . λ λ a For t − s < a, we bound the increment of Pλ by Z t Z u+a a a 1 Pλ (t) − Pλ (s) = √ ψa(u − r)µλ(dr)du − λ(t − s) λ s u On the cardinality of the coalescing point set of the Brownian Web 68

Z t 1 1 law t − s ≤ √ µλ([u, u + a])du − λ(t − s) = √ P (λa) . λ a s λa

1 λa ≤ 1 kP a(t) − P a(s)k √t−s (t − s) 2p Since , it follows that in this case λ λ ϕ1 . λa . as claimed.

Appendix B On the cardinality of the coalescing point set of the Brow- nian Web

The aim of this appendix is to derive a uniform bound on the cardinality of the coalescing ↓ point set of the Brownian Web Tree ζbw of Theorem 2.16. Let R, ε > 0 and t ∈ R, set

def ↓ ↓ ΞR(t, t − ε) = {% (z, t − ε): Mbw(z) ∈ [t, ∞) × [−R,R]} (B.1) def ηR(t, ε) = #ΞR(t, t − ε) (B.2) where, for a set A, #A denotes its cardinality. Then, the following lemma holds.

Lemma B.1 Almost surely, for any ς > 1/2 and all r, R > 0 there exists a constant C = C(r, R) such that −ς ηR(t, ε) ≤ Cε , (B.3) for all t ∈ [−r, r] and ε ∈ (0, 1]

Proof. Notice ﬁrst that, by a simple duality argument, ηR(t, ε) is equidistributed with ηˆ(t, t + ε; −R,R) of [FINR04, Deﬁnition 2.1]. According to [GSW16, Lemma C.2], for R, t, ε η t, ε 2√R λ any , R( ) is a negatively correlated point process with intensity measure t , where λ is the Lebesgue measure on R. In particular, [GSW16, Lemma C.5] implies that, for any p > 1 p p 2R [ηR(t, ε) ] p √ , (B.4) E . ε where the hidden constant depends only on p. Moreover, the random variables ηR(t, ε) are monotone in the following sense, for any R, t, ε we have

0 0 ηR(t, ε) ≤ ηR0 (t , ε ) (B.5) for all R0 ≥ R, t0 ∈ (t − ε, t] and ε0 ∈ (0, ε − (t − t0)]. m def −m Let R, r > 0, for k = 0,..., 2dre2 set tk,m = −r + k2 and consider the event

def m −n m+1 nς Em = {∀k = 0,..., 2dre2 , n ∈ N, ηR(tk,m, 2 ) ≤ 2 R2 } . Exit law of Brownian motion from the Weyl chamber 69

By Markov’s inequality and (B.4), we have

2dre2m c X X −n m+1 nς m(1−p) X −np(ς−1/2) P(Em) ≤ P(ηR(tk,m, 2 ) ≤ 2 R2 ) .p dre2 2 k=0 n n

m(1−p) which, for ς > 1/2, is ﬁnite and Or(2 ). Therefore, upon choosing p > 1 and applying Borel-Cantelli and (B.5), the statement follows at once.

Appendix C Exit law of Brownian motion from the Weyl chamber

For n ≥ 2, we deﬁne the Weyl chamber Wn as

n Wn = {x ∈ R : x1 < ··· < xn} .

x Let (Bt )t≥0 be a standard n-dimensional Brownian motion and, given a suﬃciently regular n x domain W ⊂ R , let τW = inf{t > 0 : Bt ∈ ∂W } and

W def x Pt (x, y) dy = P(τW > t, Bt ∈ dy) , x, y ∈ W. We then have the following result.

x Theorem C.1 Let (Bt )t≥0 with x ∈ Wn be as above and let τ = τWn . Then,

x Wn def P(τ ∈ dt, Bτ ∈ dy) = ∂ny Pt (x, y) σWn (dy) dt = νx(dt, dy) , (C.1) where ∂ny is the derivative in the inward normal direction at y ∈ ∂Wn and σWn is the surface measure on ∂Wn.

Proof. For smooth cones, the claim was shown for example in [BD06, Thm 1.3], so it (ε) remains to perform an approximation argument. Choose a sequence of smooth cones Wn (ε) (ε) c c such that, for every ε > 0, one has Wn ⊂ Wn and furthermore Wn ∩ Cε = Wn ∩ Cε , where Cε denotes those “corner” conﬁgurations where at least two distinct pairs of points ε τ = τ (ε) are at distance less than from each other. Writing ε Wn , it follows immediately from (ε) Wn Wn (ε) these two properties that Pt (x, y) ≤ Pt (x, y) for all t ≥ 0 and x, y ∈ Wn , so that in particular, [BD06, Thm 1.3] implies

(ε) (ε) def x Wn νx (dt, dy) = P τε ∈ dt, Bτε ∈ dy = ∂ny Pt (x, y) σWn (dy) dt ≤ νx(dt, dy) ,

(ε) c Wn Wn for all y ∈ ∂Wn ∩ Cε . Here, we also used the fact that Pt and Pt both vanish on c ∂Wn ∩ Cε . We also note that νx is a probability measure, as can be seen by combining the Trimming and path correspondence 70

Wn divergence theorem (on the space-time domain R+ × Wn) with the fact that Pt solves the heat equation on Wn with Dirichlet boundary conditions and initial condition δx. (ε) On the other hand, writing µx for the (positive) measure such that

x (ε) c (ε) P(τ ∈ dt, Bτ ∈ dy) = νx (dt, dy ∩ Cε ) + µx (dt, dy) , one has the bound

def (ε) x cε = µx (R+ × ∂Wn) ≤ P(τˆε ≤ τ) , τˆε = inf{t : Bt ∈ Cε} . Since τ < ∞ almost surely and Brownian motion does not hit subspaces of codimension 2, we have limε→0 cε = 0. For any two measurable sets I ⊂ R+ and A ⊂ ∂Wn such that A ∩ Cδ = ∅ for some δ > 0, we then have

x (ε) c P(τ ∈ I,Bτ ∈ A) ≤ νx (I,A ∩ Cε ) + cε ≤ νx(I,A) + cε , for all ε ≤ δ, so that

x P(τ ∈ dt, Bτ ∈ dy) ≤ νx(dt, dy) +µ ˆ(dt, dy) , (C.2) T where µˆ is supported on R+ × ε>0 Cε. As before, one must have µˆ = 0 since Brownian motion does not hit subspaces of codimension 2, so that the desired identity follows from the fact that both νx and the left-hand side of (C.2) are probability measures.

Appendix D Trimming and path correspondence

In this appendix we introduce some further tools in the context of spatial trees, which plays a major role in the proof of Theorem 1.4. Let (T , ∗, d) be a pointed locally compact complete R-tree and ﬁx η > 0. We deﬁne the η-trimming of T as

def Rη(T ) = {z ∈ T : ∃ w ∈ T such that z ∈ ∗, w and d(z, w) ≥ η} ∪ {∗} . (D.1) J K (The explicit inclusion of ∗ is only there to guarantee that Rη(T ) is non-empty if T is of diameter less than η.) Rη(T ) is clearly closed in T and furthermore, it is a locally ﬁnite R-tree. With a slight abuse of notation, we denote again by Rη the trimming of α α α a spatial R-tree, i.e. the map Rη : Tsp → Tsp deﬁned on ζ = (T , ∗, d, M) ∈ Tsp as Rη(ζ) = (Rη(T ), ∗, d, M). In the following lemma we summarise further properties of the trimming map.

α Lemma D.1 Let α ∈ (0, 1). For all η > 0, Rη is continuous on Tsp. Moreover, if {(Ta, ∗a, da)}a∈A, A being an index set, is a family of compact pointed R-trees then Trimming and path correspondence 71

1. the Hausdorﬀ distance between Rη(Ta) and Ta is bounded above by η, −1 2. the number of endpoints of Rη(Ta) is bounded above by (cη) `a(Rcη(Ta)) < ∞, for any c ∈ (0, 1),

3. the family is relatively compact if and only if supa∈A `a(Rη(Ta)) < ∞ for all η > 0.

Proof. The continuity of the trimming map is an easy consequence of the deﬁnition of the metric ∆sp in (2.8),[EPW06, Lemma 2.6(ii)] and [CH21, Lemma 2.17]. Points 1., 2. and 3. were respectively shown in [EPW06, Lemma 2.6(iv)], the proof of [EW06, Lemma 2.7] and [EW06, Lemma 2.6].

ˆ α Let α ∈ (0, 1) and ζ1, ζ2 ∈ Csp (see Deﬁnition 2.3) be such that

M1,t(∗1) = 0 = M2,t(∗2) . (D.2)

For j = 1, 2, denote by %j the radial map of ζj. For r, η > 0, set

(r), η def (r) Tj = Rη(Tj ∪ ∗j,%j(∗j, r + η) ) (D.3) J K (r), η def (r), η (r), η and ζj = (Tj , ∗j, dj,Mj). Assume there exists a correspondence C between ζ1 (r), η and ζ2 for which c, C (r), η (r), η ∆sp (ζ1 , ζ2 ) < ε , (D.4) (r), η for some ε > 0. Let N be the number of endpoints of ζ1 , which is ﬁnite by Lemma D.1 (r), η 1 points 2. and 3. We now number the endpoints of T1 and denote them by {v˜i : i = 1 def 1 2 def 2 (r), η 0,...,Nη}, where v˜0 = v0 = ∗1. Let v0 = ∗2 and for every i = 1,...,Nη, let v˜i ∈ T2 1 2 1 2 1 def 1 2 def 2 1 be such that (v˜i , v˜i ) ∈ C. If M1,t(v˜i ) ≥ M2,t(v˜i ), set vi = v˜i and vi = %2(v˜i ,M1,t(vi )), 2 def 2 1 def 1 2 otherwise set vi = v˜i and vi = %1(v˜i ,M2,t(vi )). (r) (r), η Setting ∗j = %j(∗j, r), we deﬁne the subtree Tj ⊂ Tj by

def [ j (r) Tj = vi , ∗j . (D.5) i≤NηJ K

We also write Zj for the corresponding spatial tree

def Zj = (Tj, ∗j, dj,Mj) . (D.6)

Finally, we deﬁne the path correspondence between T1 and T2 by

Nη def [ 1 2 1 Cp = {(%1(vi , t),%2(vi , t)): M1,t(vi ) ≤ t ≤ r} . (D.7) i=0 Trimming and path correspondence 72

ˆ α Lemma D.2 Let α ∈ (0, 1) and ζ1, ζ2 ∈ Csp, r, η, ε > 0 be such that (D.2) and (D.4) hold. Then, (r) α dis Cp + sup kM1(z1) − M2(z2)k . 4Nηε + kM1kα ε . (D.8) (z1,z2)∈ Cp (r), η Moreover, the Hausdorﬀ distance between T1 and T1 is bounded above by ε, while that (r), η between T2 and T2 is bounded by 5Nηε and the following inclusions hold

(r) (r), η (r) (r), η Rη+ε(T1 ) ⊂ T1 ⊂ T1 ,Rη+5Nηε(T2 ) ⊂ T2 ⊂ T2 . (D.9) Proof. The statement follows by applying iteratively [CH21, Lemma 2.28]. Indeed, for ˜ def m ≤ Nη, let Cm = C ∪ Cm−1, where Cm−1 is defined as the right hand side of (D.7) but the union runs from 0 to m − 1 (so that in particular CNη = Cp). Then, for all m, ˜ = C Cm C˜m−1 , the right hand side being defined according to [CH21, Eq. (2.29)]. Hence, ˜ we immediately see that dis Cp ≤ dis CNη . 4Nηε and the bound on the evaluation map can be similarly shown. For the last part of the statement, notice that the Hausdorff distance between T1 and (r),η T1 is bounded by ε by construction, while, arguing as in the proof of Lemma 3.14, it (r),η is immediate to show that the Hausdorff distance between T2 and T2 is controlled by 4Nηε + ε. These bounds together with the definition of the trimming map, guarantee that, (r) (r) for any a > 0, all the endpoints of Rη+ε(T1 ) and Rη+5Nηε+a(T2 ) must belong to T1 and T2 respectively, which in turn implies

(r) (r) Rη+ε+a(T1 ) ⊂ T1 ,Rη+5Nηε+a(T2 ) ⊂ T2 . By letting a → 0 using Lemma D.1 point 1., the conclusion follows.

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