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

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The Brownian Web As a Random $\Mathbb R $-Tree The Brownian Web as a random R-tree February 9, 2021 G. Cannizzaro1;2 and M. Hairer1 1 Imperial College London, SW7 2AZ, UK 2 University of Warwick, CV4 7AL, UK Email: [email protected], [email protected] Abstract Motivated by [CH21], we provide an alternative characterisation of the Brownian Web [TW98, FINR04], i.e. a family of coalescing Brownian motions starting from 2 every point in R simultaneously, and fit it into the wider framework of random (spatial) R-trees. We determine some of its properties (e.g. its box-counting dimension) and recover some which were determined in earlier works, such as duality, special points and convergence of the graphical representation of coalescing random walks. Along the way, we introduce a modification of the topology of spatial R-trees in [DLG05, BCK17] which makes it Polish and could be of independent interest. Contents 1 Introduction 2 1.1 Outline of the paper . .5 2 Preliminaries 7 2.1 R-trees in a nutshell . .7 2.2 Spatial R-trees . .9 arXiv:2102.04068v1 [math.PR] 8 Feb 2021 2.3 Characteristic R-trees and the radial map . 18 2.4 Alternative topologies . 21 3 The Brownian Web Tree and its dual 26 3.1 An alternative characterisation of the Brownian Web . 26 3.2 A convergence criterion to the Brownian Web tree . 36 3.3 The double Brownian Web tree and special points . 38 Introduction 2 4 The Discrete Web Tree and convergence 43 4.1 The Double Discrete Web Tree . 43 4.2 Tightness and convergence . 46 1 Introduction The Brownian Web is a random object that can be heuristically described as a collection of coalescing Brownian motions starting from every space-time point in R2, a typical realisation of which is displayed in Figure 1. Its study originated in the PhD thesis of Arratia [Arr79], who was interested in the Voter model [Lig05], its dual, given by a family of (backward) coalescing random walks, and their diffusive scaling limit. Rediscovered by Tóth and Werner in [TW98], the authors provided the first thorough construction, determined its main properties and used it to introduce the so-called true self-repelling motion. A different characterisation was subsequently given in [FINR04] where, by means of a new topology, a sufficient condition for the convergence of families of coalescing random walks was derived. Later on, further generalisations via alternative approaches appeared, e.g. in [NT15], motivated by the connection with Hastings–Levitov planar aggregation models, and in [BGS15], where the optimal convergence condition was obtained and a family of coalescing Brownian motions on the Sierpinski gasket were built. For an account of further developments of the Brownian Web and the diverse contexts in which it emerged, we refer to the review paper [SSS17]. In most (if not all) of these works, the Brownian Web is viewed as a random (compact) collection of paths Win a suitable space. The present paper aims at highlighting yet another of its characterising features, namely its coalescence or tree structure, clearly apparent in Figure 1. The main motivation comes from the companion paper [CH21] in which such a structure is used to construct and study the Brownian Castle, a stochastic process whose value at a given point equals that of a Brownian motion indexed by a Brownian web. Since the characteristics of the Brownian Castle are given by backward (coalescing) Brownian motions, in what follows we will (mainly) consider the case in which paths in W run backward in time (the so-called backward Brownian Web [FINR04]). To carry out this programme, we would like to view the set of points in the trajectories of paths in W as a metric space with metric given by the intrinsic distance, namely the # distance between (ti; πi ) 2 R × W, i = 1; 2, is the minimal time it takes to go from # # # # (t1; π1(t1)) to (t2; π2(t2)) moving along the trajectories of π1 and π2 at unit speed. Instead of working directly on R2, it turns out to be more convenient to first encode the points of the trajectories in an abstract space and then suitably embed the latter into R2. More precisely, # def # # # # we will construct a (random) quadruplet ζbw = (Tbw; ∗bw; dbw;Mbw) whose elements we # # now describe. The first three form a pointed R-tree, which means that (Tbw; dbw) is a Introduction 3 Figure 1: A typical realisation of the Brownian web: coalescing Brownian trajectories emanate from every point of the plane simultaneously. Trajectories are coloured according to their creation time / age. Introduction 4 # # connected metric space with no loops and ∗bw is an element of Tbw (see Definition 2.1). # # # Morally, points in Tbw are of the form (t; π ) for π 2 W and t ≤ σπ# , where σπ# is the # # starting time of π , and dbw is the ancestral distance defined as # # # def # # # # # # dbw((t; π ); (s; π~ )) = (t + s) − 2τt;s(π ; π~ ) , for all (t; π ); (s; π~ ) 2 Tbw, # # # # # # def τt;s being the first time at which π and π~ meet, i.e. τt;s(π ; π~ ) = supfr < t ^ s : # # # # 2 π (r) =π ~ (r)g. Mbw is the evaluation map which embeds Tbw into R , and is given by # # # 2 Tbw 3 (t; π ) 7! (t; π (t)) 2 R . The main task of the present paper is to identify a “good” space in which the quadruplet # ζbw lives, which is Polish and allows for a manageable characterisation of its compact subsets. Elements of the form ζ = (T ; ∗; d; M) are said to be spatial R-trees and have already been considered in the literature. Similar to [DLG05, BCK17], we endow the space of α spatial R-trees Tsp, α 2 (0; 1), with a Gromov–Hausdorff-type topology (for an introduction in the case of general metric and length spaces we refer to the monograph [BBI01], and to [Eva08] for the specific case of R-trees) with an important caveat, namely our metric (see (2.10)) takes into account two extra conditions imposed at the level of the evaluation map M. More precisely, M is required to be both locally little α-Hölder continuous, i.e. 0 0 α lim"!0 supz2K supd(z;z0)≤" kM(z) − M(z )k=d(z; z ) = 0 for every compact K, and proper, i.e. preimages of compacts are compact. As pointed out in [BCK17, Remark 3.2], without α the first property the space Tsp would not be Polish (the space Tsp of [DLG05, BCK17] lacks completeness). The second property prevents the existence of sequences of points that are order 1 distance apart in T but whose M-image is arbitrarily close in R2. In the weighted metric ∆sp in (2.10), necessary to consider the case of unbounded R-trees (as is # Tbw) and which is in essence that of [BCK17], this is allowed. The topology introduced in Section 2.2 and briefly described above guarantees that a α sequence fζn = (Tn; ∗n; dn;Mn)gn converges to ζ = (T ; ∗; d; M) in Tsp provided that, morally, the metrics dn converge to d, which in the present context means that couples of distinct paths which are close also coalesce approximately at the same time, and the evaluation maps Mn converge to M in Hölder sense, which in turn ensures control over the sup-norm distance of paths and is somewhat similar in spirit to that of [FINR04]. Notice that it is not always possible to assign to a family of paths an R-tree (trivially, consider the case of paths which are not coalescing) and, conversely, there is no canonical way to associate a collection of paths to a generic spatial R-tree. However, we identify a subset α of Tsp for which this is the case and prove that, as suggested by the heuristic description above, our topology is strictly finer than that in [FINR04] (see Proposition 2.25). While this ensures that many of the results obtained for the Brownian Web (existence of a dual, its properties, special points) can be translated to the present setting (see Section 3.3), Introduction 5 α convergence statements in Tsp do not follow from those previously established. This is # remedied in Section 3.2, where a convergence criterion to ζbw is derived. As shown in [CH21], there are two main advantages of the characterisation of the Brownian Web outlined above. First, it allows to preserve information on the intrinsic metric on the set of trajectories, which in turn is at the basis of the properties and the proof of the universality statement for the Brownian Castle in [CH21, Theorem 1.4]. Moreover, # the R-tree structure automatically endows Tbw with a σ-finite length measure (see [Eva08, Section 4.5.3]) that can be useful in many contexts and, for example, could provide a more direct construction of the marked Brownian Web of [FINR06]. At last, the present paper fits the Brownian Web into the wider framework of random R-trees. Many fascinating objects belong to this realm and display interesting relations to important statistical mechanics models, such as Aldous’s CRT in [Ald91a, Ald91b, Ald93], the Lévy and Stable trees of Le Gall and Duquesne and their connection to superprocesses [DLG05], the Brownian Map and random plane quadrangulations [LG13, Mie13], the scaling limit of the Uniform Spanning Tree and SLE [Sch00, BCK17], just to mention a few. As expected, the law of the Brownian Web as a random R-tree is different from those alluded to above (see Corollary 3.11 and Remark 3.12) but it would be interesting to explore further this new interpretation in light of the aforementioned works to see if extra properties of the Brownian Web itself or the Brownian Castle of [CH21] can be derived.
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