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The Brownian Web The Annals of Probability 2008, Vol. 36, No. 3, 1153–1208 DOI: 10.1214/07-AOP357 c Institute of Mathematical Statistics, 2008 THE BROWNIAN NET By Rongfeng Sun and Jan M. Swart1 TU Berlin and UTIA´ Prague The (standard) Brownian web is a collection of coalescing one- dimensional Brownian motions, starting from each point in space and time. It arises as the diffusive scaling limit of a collection of coalescing random walks. We show that it is possible to obtain a nontrivial limiting object if the random walks in addition branch with a small probability. We call the limiting object the Brownian net, and study some of its elementary properties. Contents 1. Introduction and main results....................... .....................1154 1.1. Arrow configurations and branching-coalescing random walks.......1154 1.2. Topology and convergence......................... .................1155 1.3. The Brownian web................................. ................1158 1.4. Characterization of the Brownian net using hopping . .............1159 1.5. The left-right Brownian web....................... .................1160 1.6. Characterization of the Brownian net using meshes . .............1161 1.7. The dual Brownian web............................. ...............1162 1.8. Dual characterization of the Brownian net........... ...............1164 1.9. The branching-coalescing point set ................ ................. 1164 1.10. The backbone................................... ...................1166 1.11. Discussion, applications and open problems . .................1167 1.12. Outline....................................... ......................1169 2. The left-right Brownian web......................... .....................1170 2.1. The left-right SDE............................... ..................1170 2.2. Left-right coalescing Brownian motions............ .................1173 2.3. The left-right Brownian web....................... .................1174 arXiv:math/0610625v5 [math.PR] 13 Jun 2008 3. Properties of the left-right Brownian web............. ....................1175 3.1. Properties of the left-right SDE................... ..................1175 Received October 2006; revised June 2007. 1Supported by the DFG and by GACRˇ grants 201/06/1323 and 201/07/0237. AMS 2000 subject classifications. Primary 82C21; secondary 60K35, 60F17, 60D05. Key words and phrases. Brownian net, Brownian web, branching-coalescing random walks, branching-coalescing point set. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Probability, 2008, Vol. 36, No. 3, 1153–1208. This reprint differs from the original in pagination and typographic detail. 1 2 R. SUN AND J. M. SWART 3.2. Properties of the Brownian web ..................... ............... 1176 3.3. Properties of the left-right Brownian web........... ................1178 4. The Brownian net................................... .....................1179 4.1. hop wedge ................................................... ...1179 N ⊂ N 4.2. wedge hop ................................................... ...1181 4.3. CharacterizationsN ⊂ N with hopping and wedges. ...............1184 5. Convergence...................................... .......................1184 5.1. Convergence to the left-right SDE.................. ................1184 5.2. Convergence to the left-right Brownian web.......... ...............1187 5.3. Convergence to the Brownian net.................... ...............1188 6. Density calculations.............................. ........................1190 6.1. The density of the branching-coalescing point set . ...............1191 6.2. The density on the left of a left-most path............ ..............1193 7. Characterization with meshes ....................... .....................1198 8. The branching-coalescing point set .................. .....................1199 9. The backbone...................................... ......................1203 9.1. The backbone of branching-coalescing random walks . .............1203 9.2. The backbone of the branching-coalescing point set . ..............1205 Appendix: Definitions of path space..................... ....................1206 Acknowledgments.................................... .......................1207 References......................................... .........................1207 1. Introduction and main results. 1.1. Arrow configurations and branching-coalescing random walks. The Brownian web originated from the work of Arratia [1, 2], and has since been studied by T´oth and Werner [17], and Fontes, Isopi, Newman and Ravishankar [7, 8, 9]. It arises as the diffusive scaling limit of a collection of coalescing random walks. In this paper we show that it is possible to obtain a nontrivial limiting object if the random walks in addition branch with a small probability. 2 2 Let Zeven := (x,t) : x,t Z, x + t is even be the even sublattice of Z . We interpret the{ first coordinate∈ x as space} and the second coordinate t as time, which is plotted vertically in figures. Fix a branching probability 2 1 β β [0, 1]. Independently for each (x,t) Zeven, with probability −2 , draw ∈ ∈ 1 β an arrow from (x,t) to (x 1,t + 1), with probability −2 , draw an arrow from (x,t) to (x + 1,t + 1),− and with the remaining probability β, draw two arrows starting at (x,t), one ending at (x 1,t + 1) and the other at (x + 1,t + 1). (See Figure 1.) We denote the random− configuration of all arrows by 2 2 (1.1) := (z, z′) Z Z : there is an arrow from z to z . ℵβ { ∈ even × even ′} THE BROWNIAN NET 3 Fig. 1. An arrow configuration. By definition, a path along arrows in β, in short an β-path, is the graph of a function π : [σ , ] R ,ℵ with σ Z ℵ , such that π ∞ → ∪ {∗} π ∈ ∪ {±∞} ((π(t),t), (π(t + 1),t + 1)) β and π is linear on the interval [t,t + 1] for all t [σ , ] Z, while π( ∈ℵ)= whenever [σ , ]. We call σ the ∈ π ∞ ∩ ±∞ ∗ ±∞ ∈ π ∞ π starting time, π(σπ) the starting position and zπ := (π(σπ),σπ) the starting point of the β-path π. For any Aℵ Z2 ( , ) , we let (A) denote the collection of ⊂ even ∪ { ∗ ±∞ } Uβ all β-paths with starting points in the set A, and we use the shorthands (ℵz) := ( z ) and := (Z2 ( , ) ) for the collections of all Uβ Uβ { } Uβ Uβ even ∪ { ∗ ±∞ } β-paths starting from a single point z and from any point in space-time, respectively.ℵ An arrow configuration β is in fact the graphical representation for a system of discrete time branching-coalescingℵ random walks. Indeed, if we set (1.2) ηA := π(t) : π (A) (t Z, A Z2 ( , ) ), t { ∈Uβ } ∈ ⊂ even ∪ { ∗ ±∞ } A and we interpret the points in ηt as being occupied by a particle at time t, A then (ηt )t Z is a collection of random walks, which are introduced into the system at∈ space-time points in A. At each time t Z, independently each 1 β ∈ particle with probability −2 jumps one step to the left (resp. right), and with probability β branches into two particles, one jumping one step to the left and the other one step to the right. Two walks coalesce instantly when they jump to the same lattice site. Note that the case β = 0 corresponds to coalescing random walks without branching. We are interested in the limit of β under diffusive rescaling, letting at the same time β 0. Thus, we rescaleU space by a factor ε, time by a factor ε2, and let ε 0→ and β 0 at the same time. For the case β = 0, it has been → → shown in [8] that 0 diffusively rescaled converges weakly in law, with respect to an appropriateU topology, to a random object , called the Brownian web. We will show that if β/ε b for some b 0, thenW in (essentially) the same topology as in [8], diffusively→ rescaled converges≥ in law to a random object Uβ 4 R. SUN AND J. M. SWART 2 2 Fig. 2. The compactification Rc of R . , which we call the Brownian net with branching parameter b. Here Nb N0 is equal to in distribution, while b with b> 0 differ from , but are related to eachW other through scaling.N W 1.2. Topology and convergence. To formulate our main results, we first need to define the space in which our random variables take values and the topology with respect to which we will prove convergence. Our topology is essentially the same as the one used in [7, 8], except for a slight (and in most applications irrelevant) detail, as explained in the Appendix. 2 2 Let Rc be the compactification of R obtained by equipping the set 2 2 Rc := R ( ,t) : t R ( , ) with a topology such that (xn,tn) ( ,t) if∪{x ±∞ and∈ t}∪{t∗ ±∞R, and} (x ,t ) ( , ) if t (re-→ ±∞ n →±∞ n → ∈ n n → ∗ ±∞ n →±∞ gardless of the behavior of xn). In [7, 8], such a compactification is achieved by taking the completion of R2 with respect to the metric (1.3) ρ((x ,t ), (x ,t )) = Θ (x ,t ) Θ (x ,t ) Θ (t ) Θ (t ) , 1 1 2 2 | 1 1 1 − 1 2 2 | ∨ | 2 1 − 2 2 | where the map Θ=(Θ1, Θ2) is defined by tanh(x) (1.4) Θ(x,t)=(Θ (x,t), Θ (t)) := , tanh(t) . 1 2 1+ t | | 2 2 We can think of Rc as the image of [ , ] under the map Θ. Of course, ρ and Θ are by no means the only choices−∞ ∞ that achieve the desired compact- 2 ification. See Figure 2 for a picture of Rc (for a somewhat different choice of Θ). By definition, a (continuous) path in R2 is a function π : [σ , ] [ , ] c π ∞ → −∞ ∞ ∪ , with σπ [ , ], such that π : [σπ, ] R [ , ] is continuous, {∗} ∈ −∞ ∞ ∞ ∩ → −∞ ∞ 2 and π( )= whenever [σπ, ]. Equivalently, if we identify Rc with the image±∞ of [ ∗ , ]2 under±∞∈ the map∞ Θ, then π is a continuous map from −∞ ∞ THE BROWNIAN NET 5 2 [Θ2(σπ), Θ2( )] to R whose graph is contained in Θ([ , ] ). Throughout ∞ −∞ ∞ 2 the paper we identify a path π with its graph (π(t),t) : t [σπ, ] Rc . { 2 ∈ ∞ }⊂ Thus, we often view paths as compact subsets of Rc . We stress that the starting time is part of the definition of a path, that is, paths that are de- fined by the same function but have different starting times are considered to be different. Note that both the function defining a path and its starting time can be read off from its graph. 2 We let Π denote the space of all paths in Rc , equipped with the metric d(π1,π2) := Θ2(σπ1 ) Θ2(σπ2 ) (1.5) | − | sup Θ1(π1(t σπ1 ),t) Θ1(π2(t σπ2 ),t) .
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