The Brownian Web

arXiv:math/0610625v5 [math.PR] 13 Jun 2008 ak,bacigcaecn on set. point branching-coalescing walks, .Telf-ih rwinwb ...... web. Brownian left-right The 2...... results main and Introduction 1. .Poete ftelf-ih rwinwb...... web Brownian left-right the of Properties 3. 08 o.3,N.3 1153–1208 3, No. DOI: 36, Vol. Probability2008, of Annals The

c aiainadtpgahcdetail. typographic 1153–1208 and 3, pagination No. 36, publis Vol. article 2008, original Statistics the Mathematical of of reprint Institute electronic an is This nttt fMteaia Statistics Mathematical of Institute e od n phrases. and words Key 1 M 00sbetclassifications. subject 2000 AMS 2007. June revised 2006; October Received ..Poete ftelf-ih D...... SDE. . . left-right . . . . the . . . . of . . . . Properties ...... 3.1...... web . . . . . Brownian ...... left-right ...... The . . . . . motions . . . . Brownian . . . . coalescing . 2.3. . . . Left-right ...... SDE . . . . left-right . . 2.2. . . The ...... 2.1...... problems . . . Outline. . . . open ...... 1.12. . and ...... applications ...... Discussion, ...... 1.11...... backbone set . . . . The point . . net . branching-coalescing . 1.10. Brownian . The . . the . . . . . of . . . characterization . . . 1.9. . . . . Dual . . . . . web . . meshes . . Brownian . . . using . dual . . 1.8. . . net The . . . Brownian . . . . the . . . . of . 1.7...... Characterization . . . . . web . . hopping . . Brownian . . using . . left-right . 1.6. . net . . The . . Brownian . . . . the . . . . of . . 1.5. . . Characterization ...... web . . Brownian 1.4. . . The . . . convergence and 1.3. random Topology branching-coalescing and configurations 1.2. Arrow 1.1. upre yteDGadb GA by and DFG the by Supported 10.1214/07-AOP357 oeo t lmnayproperties. elementary its of some admwls eso hti spsil ooti nontrivi a a with obtain the branch object to addition limiting in possible the walks call is random We probability. the it if that object show collection limiting a We of limit walks. scaling diffusive random the s as in arises point It each time. from starting motions, Brownian dimensional h (standard) The yRnfn u n a .Swart M. Jan and Sun Rongfeng By UBri and Berlin TU rwinnt rwinwb rnhn-olsigrandom branching-coalescing web, Brownian net, Brownian H RWINNET BROWNIAN THE rwinweb Brownian 2008 , hsrpitdffr rmteoiia in original the from differs reprint This . rmr 22;scnay6K5 01,60D05. 60F17, 60K35, secondary 82C21; Primary Rgat 0/612 n 201/07/0237. and 201/06/1323 grants CR ˇ in Contents h naso Probability of Annals The 1 TAPrague UTIA sacleto fcaecn one- coalescing of collection a is ´ rwinnet Brownian .1169 ...... 1154 ...... 1170 ...... e ythe by hed .1175 ...... 1166 ...... 1170 ...... 1175 ...... 1164 ...... 1173 ...... 1155 ...... 1174 ...... 1160 ...... 1167 ...... fcoalescing of 1158 ...... n study and , 1 1162 ...... 1164 ...... 1161 ...... 1159 ...... aeand pace ak 1154 ...... walks , small al 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.N Characterizations⊂ 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 Tóth 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, ℵrespectively. 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 defined 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) . ∨ t σπ σπ | ∨ − ∨ | ≥ 1 ∧ 2 The space (Π, d) is complete and separable. Note that paths converge in (Π, d) if and only if their starting times converge and the functions converge locally uniformly on R. If fact, one gets the same topology on Π (though not the same uniform structure) if one views paths as compact subsets of 2 Rc and then equips Π with the Hausdorff metric. Recall that if (E, d) is a metric space and (E) is the space of all compact subsets of E, then the Hausdorff metric d Kon (E) is defined by H K (1.6) dH(K1, K2)= sup inf d(x1,x2) sup inf d(x1,x2). x1 K1 x2 K2 ∨ x2 K2 x1 K1 ∈ ∈ ∈ ∈ If (E, d) is complete and separable, then so is ( (E), d ). For a given topol- K H ogy on E, the Hausdorff topology generated by dH depends only on the topology on E and not on the choice of the metric d. The Brownian net b and web are (Π)-valued random variables. We define scaling maps SN: R2 R2 byW K ε c → c (1.7) S (x,t) := (εx, ε2t) ((x,t) R2). ε ∈ c 2 2 2 We adopt the convention that if f : Rc Rc and A Rc , then f(A) := f(x) : x A denotes the image of A under→ f. Likewise,⊂ if K is a set of sub- {sets of R2∈(e.g.,} a set of paths), then f(K)= f(A) : A K is the image of K c { ∈ } under the map A f(A). So, for example, Sε( β) is the set of all β-paths 7→ 2 U ℵ (viewed as subsets of Rc ), diffusively rescaled with ε. This will later also ap- ply to notation such as A := x : x A and A + y := x + y : x A . We − {− ∈ } { ∈ } will sometimes also use the shorthand f(A1,...,An) := (f(A1),...,f(An)) 2 when f is a function defined on Rc and A1,...,An are elements of, or subsets 2 of, or sets of subsets of Rc . Recall from Section 1.1 the definition of an arrow configuration β and the set of all -paths. Note that is a random subset of Π. In orderℵ to Uβ ℵβ Uβ make β compact, from now on, we modify our definition of β by adding all trivialU paths π that satisfy σ Z and π(t)= Uor π(t)= π ∈ {±∞}∪ −∞ ∞ for all t [σπ, ]. The main result of this paper is the following convergence theorem.∈ ∞ 6 R. SUN AND J. M. SWART

Theorem 1.1 (Convergence to the Brownian net). There exist (Π)- K valued random variables b (b 0) such that, if εn, βn 0 and βn/εn b 0, then S ( ) are (Π)N-valued≥ random variables, and→ → ≥ εn Uβn K (1.8) (Sε ( β )) = ( b), n n n L U →∞⇒ L N where ( ) denotes law, and denotes weak convergence. The random variablesL (· ) satisfy the scaling⇒ relation Nb b>0 (1.9) (S ( )) = ( ) (ε,b> 0). L ε Nb L Nb/ε We have ( 0)= ( ), where is the Brownian web. However, the random variablesL N withL Wb> 0 areW diﬀerent from . Nb W

For βn = 0, that is, the case without branching, Theorem 1.1 follows from [8], Theorem 6.1. In the next sections we will give three equivalent characterizations of the random variables with b> 0. In view of the scaling Nb relation (1.9), it suffices to consider the case b = 1. We call b the Brownian net with branching parameter b and := the (standard)N Brownian net. N N1 1.3. The Brownian web. In order to prepare for our first characterization of the Brownian net , we start by recalling from [8], Theorem 2.1, the characterization of theN Brownian web . For any K (Π) and A R2, W ∈ K ⊂ c we let K(A) := π K : zπ A denote the collection of paths in K with { ∈ ∈ } 2 starting points zπ = (π(σπ),σπ) in A, and for z Rc , we write K(z) := K( z ). ∈ { } Theorem 1.2 (Characterization of the Brownian web). There exists a (Π)-valued random variable , the so-called (standard) Brownian web, Kwhose distribution is uniquely determinedW by the following properties: (i) For each deterministic z R2, (z) consists a.s. of a single path ∈ W (z)= πz . W { } 2 (ii) For any finite deterministic set of points z1,...,zk R , (πz1 ,...,πzk ) is distributed as a system of coalescing Brownian motions∈ starting at space- time points z1,...,zk. (iii) For any deterministic countable dense set R2, D⊂ (1.10) = ( ) a.s., W W D where denotes closure in (Π, d).

Note that by properties (i) and (iii), for any deterministic countable dense set R2, the Brownian web is almost surely determined by the countable D⊂ system of paths ( )= πz : z , whose distribution is uniquely determined by propertyW (ii).D We{ call ∈( D}) a skeleton of the Brownian web (relative to the countable dense set W). SinceD skeletons may be constructed using D THE BROWNIAN NET 7

Kolmogorov’s extension theorem, Theorem 1.2 allows a direct construction of the Brownian web. Although (z) consists of a single path for each deterministic z R2, as a result of theW closure in (1.10), there exist random points z where∈ (z) contains more than one path. These are points where the map z Wπ is 7→ z discontinuous, that is, the limit limn πzn depends on the choice of the →∞ sequence zn with zn z. These special points of the Brownian web are classiﬁed according∈ D to the→ number of disjoint incoming and distinct outgoing paths at z, and play an important role in understanding the Brownian web, and, later on, also the Brownian net. We recall the classiﬁcation of the special points of the Brownian web in Section 3.2.

1.4. Characterization of the Brownian net using hopping. Our first characterization of the Brownian net will be similar to the characterization of the Brownian web in Theorem 1.2. A difficulty is that in the Brownian net , there is a multitude of paths starting at any site z = (x,t) R2. There is,N however, a.s. a well-defined left-most path and right-most path∈ in (z), N that is, there exist lz, rz (z) such that lz(s) π(s) rz(s) for any s t and π (z). These left-most∈N and right-most paths≤ will≤ play a key role≥ in our characterization.∈N Our first task is to characterize the distribution of a finite number of left- most and right-most paths, started from deterministic starting points. Thus, 2 for given deterministic z ,...,z , z ,...,z ′ R , we need to characterize 1 k 1′ k′ ∈ ′ ′ the joint law of (lz1 ,...,lzk , rz ,...,rz ). It turns out that (lz1 ,...,lzk ) is a 1 k′ collection of coalescing Brownian motions with drift one to the left, while (rz′ ,...,rz′ ) is a collection of coalescing Brownian motions with drift one to 1 k′ the right. Moreover, paths evolve independently when they do not coincide. ′ ′ Therefore, in order to characterize the joint law of (lz1 ,...,lzk , rz ,...,rz ), it 1 k′ suffices to characterize the interaction between one left-most path lz = l(x,s) and one right-most path rz′ = r(x′,s′). The joint evolution of such a pair after time s s′ can be characterized as the unique weak solution of the two-dimensional∨ left-right SDE l s dLt = 1 Lt=Rt dBt + 1 Lt=Rt dBt dt, (1.11) { 6 } { } − r s dRt = 1 Lt=Rt dBt + 1 Lt=Rt dBt + dt, { 6 } { } l r s where Bt,Bt ,Bt are independent standard Brownian motions, and Lt and Rt are subject to the constraint that Lt Rt for all t T := inf u s s : L R . These rules uniquely determine≤ the joint≥ law of (l {,...,l≥ ∨, ′ u ≤ u} z1 zk rz′ ,...,rz′ ). We call such a system a collection of left-right coalescing Brow- 1 k′ nian motions. See Figure 5 for a picture. We refer to Sections 2.1 and 2.2 for the proof that solutions to (1.11) are weakly unique, and a more careful definition of left-right coalescing Brownian motions. 8 R. SUN AND J. M. SWART

Since we are not only interested in left-most and right-most paths, but in all paths in the Brownian net, we need a way to construct general paths from left-most and right-most paths. The method we choose in this section is based on hopping, that is, concatenating pieces of paths together at times when the two paths are at the same position. We call t an intersection time of two paths π ,π Π if σ σ

Theorem 1.3 (Characterization of the Brownian net using hopping). There exists a (Π)-valued random variable , which we call the (standard) Brownian net,K whose distribution is uniquelyN determined by the following properties: (i) For each deterministic z R2, (z) a.s. contains a unique left-most ∈ N path lz and right-most path rz. 2 (ii) For any ﬁnite deterministic set of points z ,...,z , z ,...,z ′ R , 1 k 1′ k′ ∈ ′ ′ the collection of paths (lz1 ,...,lzk , rz ,...,rz ) is distributed as a collection 1 k′ of left-right coalescing Brownian motions. (iii) For any deterministic countable dense sets l, r R2, D D ⊂ (1.13) = ( l : z l r : z r ) a.s. N Hcros { z ∈ D } ∪ { z ∈ D } Instead of hopping at crossing times, we could also have built our construction on hopping at intersection times. In fact, a much stronger statement is true.

Proposition 1.4 (The Brownian net is closed under hopping). We have ( )= . Hint N N THE BROWNIAN NET 9

We note, however, that as a result of the existence of special points in the Brownian web with one incoming and two outgoing paths, the Brownian net is not closed under hopping at times t such that π1(t)= π2(t) but t =

σπ1 σπ2 (t). Thus, it is generally not allowed to hop onto paths at their starting∨ times.

1.5. The left-right Brownian web. Given a Brownian net , if we take the closures of the sets of all left-most and right-most paths, startedN respectively from deterministic countable dense sets l, r R2, then we obtain two Brownian webs, tilted respectively with driftD D1 and⊂ +1, that are coupled in a special way. Our next theorem introduces− this object in its own right.

Theorem 1.5 (Characterization of the left-right Brownian web). There exists a (Π)2-valued random variable ( l, r), which we call the (standard) left-rightK Brownian web, whose distributionW W is uniquely determined by the following properties: (i) For each deterministic z R2, l(z) and r(z) a.s. each contain a l r∈ W W single path (z)= lz and (z)= rz . W { } W { } 2 (ii) For any ﬁnite deterministic set of points z ,...,z , z ,...,z ′ R , 1 k 1′ k′ ∈ ′ ′ the collection of paths (lz1 ,...,lzk ; rz ,...,rz ) is distributed as a collection 1 k′ of left-right coalescing Brownian motions. (iii) For any deterministic countable dense sets l, r R2, D D ⊂ (1.14) l = l : z l and r = r : z r a.s. W { z ∈ D } W { z ∈ D } + l Note that if we deﬁne titling maps by Tilt±(x,t) := (x t,t), then Tilt ( ) r ± W and Tilt−( ) are distributed as the (standard) Brownian web. The following lemma,W the proof of which can be found in Section 4, is an easy consequence of Theorem 1.3.

Lemma 1.6 (Associated left-right Brownian web). Let be the Brown- ian net. Then a.s. uniquely determines a left-right BrownianN web ( l, r) N 2 l r W W such that, for each deterministic z R , (z)= lz and (z)= rz , where l and r are respectively the∈ left-mostW and right-most{ } pathW in {(z)}. z z N If ( l, r) and are coupled as in Lemma 1.6, then we say that ( l, Wr) isW the left-rightN Brownian web associated with the Brownian net W. TheoremW 1.3 shows that, conversely, a left-right Brownian web uniquely Ndetermines its associated Brownian net a.s. In the next section we give another way to construct a Brownian net from its associated left-right Brownian web. Since the left-right Brownian web is characterized by Theorem 1.5, this yields another way to characterize the Brownian net. 10 R. SUN AND J. M. SWART

Fig. 3. A mesh M(r, l) with bottom point z and a wedge W (ˆr, ˆl) with bottom point z.

1.6. Characterization of the Brownian net using meshes. If for some z = (x,t) R2, there exist l l(z) and r r(z) such that r(s) < l(s) on (t,t +∈ε) for some ε> 0,∈W then denoting T∈W:= inf s>t : r(s)= l(s) , we call the open set (see Figure 3) { } (1.15) M = M(r, l) := (y,s) R2 : tσπ.

Theorem 1.7 (Characterization of Brownian net with meshes). Let ( l, r) be the left-right Brownian web. Then almost surely, W W = π Π : π does not enter any mesh of (1.16) N { ∈ ( l, r) with bottom time t>σ W W π} is the Brownian net associated with ( l, r). W W The next proposition implies that paths in the net do not enter meshes of ( l, r) at all (regardless of their bottom times),N and hence, formula (1.16W) staysW true if one drops the restriction that the bottom time of the mesh should be larger than σπ.

Proposition 1.8 (Containment by left- and right-most paths). Let be the Brownian net and let ( l, r) be its associated left-right BrownianN web. Then, almost surely, thereW existW no π and l l such that l(s) ∈N ∈W ≤ π(s) and π(t) < l(t) for some σπ σl

Fig. 4. Dual arrow conﬁguration with no branching.

Remark. Theorem 1.7 and Proposition 1.8 have analogues for the Brow- nian web. Indeed, generalizing our earlier deﬁnition, we can deﬁne a left-right l r Brownian web ( b, b ) with drift b 0 by replacing the drift terms +dt and dt in theW left-rightW SDE (1.11≥) with +b dt and b dt, respectively. − l r − Then 0 = 0 = 0 a.s. is distributed as the (standard) Brownian web, and TheoremW N1.7 andW Proposition 1.8 hold for any b 0. The meshes of the Brownian web are called bubbles in [9]. ≥

1.7. The dual Brownian web. Arratia [1] observed that there is a natural dual for the arrow configuration , the graphical representation of discrete ℵ0 time coalescing simple random walks. More precisely, 0 uniquely determines a dual arrow configuration ˆ defined as follows (seeℵ Figure 4): ℵ0 ˆ 2 2 0 := ((x,t + 1), (x 1,t)) Zodd Zodd : (1.17) ℵ { ± ∈ × ((x,t), (x 1,t + 1)) . ∓ ∈ℵ0} Observe that directed edges in and ˆ do not cross, and and ˆ ℵ0 ℵ0 ℵ0 ℵ0 uniquely determine each other. A dual arrow configuration ˆ0 is the graphical representation of a system of coalescing simple randomℵ walks running backward in time, and ˆ + (0, 1) is equally distributed with . In anal- −ℵ0 ℵ0 ogy with 0, let ˆ0 denote the set of backward paths along arrows in ˆ0. It follows fromU resultsU in [8, 9] that ℵ

(1.18) (Sε( 0, ˆ0)) = ( , ˆ ), ε 0 L U U →⇒ L W W where is the standard Brownian web, and ˆ is the so-called dual Brow- nian webW associated with . One has W W (1.19) ( ( , ˆ )) = ( ˆ , ). L − W W L W W In particular, ˆ is equally distributed with , the Brownian web rotated W −W 180◦ around the origin. It was shown in [9, 15] that the interaction between paths in and ˆ is that of Skorohod reﬂection. W W 12 R. SUN AND J. M. SWART

A Brownian web and its dual ˆ a.s. uniquely determine each other. There are several waysW to construct W from ˆ . We will describe one such way here, since this construction generalizesW W to the Brownian net. For any dual pathsπ ˆ1, πˆ2 ˆ that are ordered asπ ˆ1(s) < πˆ2(s) at the time s := σˆ σˆ , whereσ ˆ∈ Wdenotes the starting time ofπ ˆ (i = 1, 2), we let T := πˆ1 ∧ πˆ2 πi i sup t

1.8. Dual characterization of the Brownian net. Let ( l, r) be a left- right Brownian web. Then l and r each a.s. determineW a dualW web, which we denote respectively by Wˆ l and Wˆ r. It will be proved in Section 5.2 below that W W (1.22) ( ( l, r, ˆ l, ˆ r)) = ( ˆ l, ˆ r, l, r). L − W W W W L W W W W In particular, the dual left-right Brownian web ( ˆ l, ˆ r) is equally dis- l r W W tributed with ( , ), the left-right Brownian web rotated by 180◦ around the origin. − W W For anyr ˆ ˆ r and ˆl ˆ l that are ordered asr ˆ(s) < ˆl(s) at the time ∈ W ∈ W s :=σ ˆ σˆ , we let T := sup t

Theorem 1.10 (Dual characterization of the Brownian net). Let ( l, r, W W ˆ l, ˆ r) be a left-right Brownian web and its dual. Then, almost surely, W W (1.24) = π Π : π does not enter any wedge of ( ˆ l, ˆ r) from outside N { ∈ W W } is the Brownian net associated with ( l, r). W W THE BROWNIAN NET 13

We note that there exist paths in (even in l and r) that enter N W W wedges of ( ˆ l, ˆ r) in the sense deﬁned just before Theorem 1.7. Therefore, the conditionW inW (1.24) that π enters from outside cannot be relaxed.

1.9. The branching-coalescing point set. Just as the arrow conﬁgura- tion β is the graphical representation of a discrete system of branching- coalescingℵ random walks, the Brownian net is the graphical representation of a Markov process taking values in theN space of compact subsets of [ , ], which we call the branching-coalescing point set. In analogy with (−∞1.2),∞ for any compact A R2, we denote ⊂ c (1.25) ξA := π(t) : π (A) (t R, A (R2)). t { ∈N } ∈ ∈ K c We set R := [ , ] and let (R) denote the space of compact subsets of R, equipped with−∞ the∞ HausdorﬀK topology, under which (R) is itself a compact space. We recall that if E is a compact metrizableK space, then a Feller process in E is a time-homogeneous Markov process in E, with cadlag sample paths, whose transition probabilities Pt(x,dy) have the property that the map (x,t) Pt(x, ) from E [0, ) into the space of probability measures on E is continuous7→ · with respect× to∞ the topology of weak convergence. Feller processes are strong Markov processes [6], Theorem 4.2.7.

Theorem 1.11 (Branching-coalescing point set). Let be the Brown- ian net. Then for any s R and K (R), N ∈ ∈ K K s (1.26) ξ := ξ ×{ } (s t< ) t t ≤ ∞ defines a Feller process (ξt)t s in (R) with continuous sample paths, started ≥ K from the initial state K at time s. For each deterministic t>s, the set ξt is a.s. locally finite in R. If K := K (R) : K = K R , then ∈ K′ { ∈ K ∩ } (1.27) P[ξ ′ t s] = 1. t ∈ K ∀ ≥ Note that excludes sets in which either or is an isolated point, K′ −∞ ∞ and hence, ′ can in a natural way be identified with the space of all closed subsets of RK. Thus, property (1.27) says that we can view the branching- coalescing point set as a Markov process taking values in the space of closed subsets of R. The branching-coalescing point set ξt arises as the scaling limit of the branching-coalescing random walks ηt introduced in (1.2). The scaling regime considered in Theorem 1.1 allows us to interpret ξt heuristically as a collection of Brownian particles which coalesce instantly when they meet but branch with an infinite rate. The infinite branching rate makes it difficult, however, to develop a good intuition from this simple picture. In particular, even for the process started at time zero from just one point, there is a dense 14 R. SUN AND J. M. SWART collection of random times t> 0 such that ξt is not locally finite. The proof of this fact is not difficult, but for lack of space, we defer it to a future paper. For the branching-coalescing point set started from the whole extended real line R, we can explicitly calculate the expected density at any t> 0. 2 Below, A denotes the cardinality of a set and Φ(x)= 1 x e y /2 dy. √2π − | | R−∞ Proposition 1.12 (Density of branching-coalescing point set). We have t R 0 e− (1.28) E[ ξ ×{ } [a, b] ] = (b a) + 2Φ(√2t) | t ∩ | − · √πt for all [a, b] R, t> 0. ⊂

R 0 1/2 Note that the density of ξt ×{ } is proportional to t− as t 0. This is consistent with the behavior of the Brownian web, but the dec↓ay is faster than is known for other coalescents such as Kingman’s coalescent or the branching-coalescing particle systems in [3], Theorem 2(b). On the other hand, the density approaches the constant 2 as t , in contrast to the Brownian web. →∞ Our next proposition shows that it is possible to recover (R 0 ) from N × { } R 0 2 (ξt ×{ })t 0. Below, for any K (Rc ), we let ≥ ⊂ K (1.29) K = z R2 : A K s.t. z A ∪ { ∈ c ∃ ∈ ∈ } denote the union of all sets in K. We call K the image set of K. For ∪ t [ , ], let Πt := π Π : σπ = t denote the space of all paths with ∈ −∞ ∞ { ∈ } R 0 starting time t. Note that ( Π )= (x,t) : t 0,x ξ ×{ } ( , ) . ∪ N ∩ 0 { ≥ ∈ t } ∪ { ∗ ∞ }

Proposition 1.13 (Image set property). Let be the Brownian net. Then, almost surely for all t [ , ], N ∈ −∞ ∞ (1.30) Π = π Π : π ( Π ) . N ∩ t { ∈ t ⊂ ∪ N ∩ t } 1.10. The backbone. In this section we study ( , ), the set of paths in the Brownian net starting at time , andN its∗ −∞ discrete counterpart −∞ β( , ). These sets are relevant in the study of ergodic properties of theU ∗ branching-coalescing−∞ point set and the branching-coalescing random walks. Borrowing terminology from branching theory, we call ( , ) and ( , ) respectively the backbone of the Brownian net andN the∗ −∞backbone Uof∗ an−∞ arrow conﬁguration.

~~Proposition 1.14 (Backbone of an arrow conﬁguration). For β 0, the set of -paths, , satisﬁes the following properties: ≥ ℵβ Uβ THE BROWNIAN NET 15~~

(i) π(0) : π β( , ) is a Bernoulli random ﬁeld on Zeven with in- { 4β ∈U ∗ −∞ } tensity ρ := (1+β)2 . (ii) ( , ) and ( , ) are equal in law. Uβ ∗ −∞ −Uβ ∗ −∞ (iii) Almost surely, β(xn,tn) β( , ) in (Π) for any sequence n U −→→∞ U ∗ −∞ K 2 xn (x ,t ) Z satisfying t and lim sup | | < β. n n even n n tn ∈ →−∞ →∞ | | Note that [recall (1.2)]

( , ) (1.31) η ∗ −∞ = π(t) : π ( , ) (t Z) t { ∈Uβ ∗ −∞ } ∈ defines, modulo parity, a stationary system of branching-coalescing ran- ( , ) dom walks (ηt∗ −∞ )t Z. Thus, property (i) implies that, modulo parity, ∈ 4β Bernoulli product measure with intensity (1+β)2 is an invariant measure for the branching-coalescing random walks with branching probability β. This is perhaps surprising, unless one is familiar with other branching-coalescing particle systems such as Schlögl models (see, e.g., [3, 5, 13]). Property (ii) says that this invariant law is, moreover, reversible in a rather strong sense. Note that an arrow configuration β is not symmetric with respect to time reversal, so this statement is not asℵ obvious as it may seem. Property (iii) implies that the branching-coalescing random walks (ηt)t 0 exhibit complete convergence, that is, for any nonempty initial state η ≥ Z , as t , 0 ⊂ even →∞ η2t (resp. η2t+1) converges in law to a Bernoulli product measure on Zeven Z 4β (resp. odd) with intensity ρ = (1+β)2 .

~~Fig. 5. Left-right coalescing Brownian motions and the backbone of the Brownian net. 16 R. SUN AND J. M. SWART~~

~~For the Brownian net, we have the following analogue of Proposition 1.14.~~

Proposition 1.15 (Backbone of the Brownian net). The Brownian net satisfies the following properties: N (i) π(0) : π ( , ) is a Poisson point process on R with intensity{ 2. ∈N ∗ −∞ }\{±∞} (ii) ( , ) and ( , ) are equal in law. N ∗ −∞ −N ∗ −∞ (iii) Almost surely, (xn,tn) ( , ) in (Π) for any sequence n N −→→∞ N ∗ −∞ K 2 xn (x ,t ) R satisfying t and lim sup | | < 1. n n n n tn ∈ →−∞ →∞ | | In analogy with the branching-coalescing random walks, it follows that the law of a Poisson point set on R with intensity 2 is an invariant law for the branching-coalescing point set, that the latter exhibits complete convergence, and hence, this is its unique nontrivial invariant law. See Figure 5 for a picture of the backbone, or rather its image set ( , ). Note that by Proposition 1.13, any path starting at time ∪Nthat∗ stays−∞ in ( , ) is a path in ( , ). −∞ ∪N ∗ −∞ N ∗ −∞ 1.11. Discussion, applications and open problems. This article began with the question of whether it is possible to add a small branching probability to the arrow configuration 0, which scales to the Brownian web, in such a way that one still obtains aℵ nontrivial limit. At first sight, this may not seem possible because of the instantaneous coalescing of paths in the Brow- nian web. At second thought, for arrow configurations β with branching probability β, if we rescale space and time by ε and ε2 andℵ let ε 0, then → for the left-most and right-most β-path starting from the origin to have a nontrivial limit, we need β/ε bℵfor some b> 0. It seems a coincidence that exactly the same scaling of β→and ε is needed for the invariant measures of the branching-coalescing random walks from Proposition 1.14(i) to have a nontrivial limit. It was the observation of this coincidence that started off the present article. Arratia’s [1, 2] original motivation for studying the Brownian web came from one-dimensional voter models. In fact, coalescing simple random walks are dual to the one-dimensional nearest-neighbor voter model in two ways. They represent the genealogy lines of the voter model, and they also characterize the evolution of boundaries between domains of different types in an infinite type voter model. Voter models are used in population genet- ics to study the spread of genes in the absence of selection and mutation. They can also be viewed as the stochastic dynamics of zero-temperature one- dimensional Potts models. These points of view suggest several extensions of the Brownian web. THE BROWNIAN NET 17

In [9] the marked Brownian web was introduced for the study of one- dimensional Potts models at small positive temperature. There, with small probability, a site may change its type, giving rise to a “nucleation event.” In the biological context, such an event may be interpreted as a mutation. For the dual system of coalescing random walks, this results in a small death rate. The diffusive scaling limit of such a system is characterized by a Poisson marking of paths in the dual Brownian web, according to their length measure, where marks indicate deaths of particles. There are at least two motivations for studying the Brownian net. First, in the biological interpretation, if instead of mutation, one adds a small selection rate, then one ends up with a biased voter model, which is dual to branching-coalescing random walks (compare [3]). Near the completion of this article, we learned that Newman, Ravishankar and Schertzer have been studying a differently motivated model that also leads to the Brown- ian net. Their model is a one-dimensional infinite-type Potts model, where, in contrast to the model in [9], nucleation events can only occur at the boundaries between different types. These boundaries then evolve as a system of continuous-time branching-coalescing random walks, which leads to the Brownian net. Rather than starting from the left-right Brownian web, their construction of the Brownian net is based on allowing hopping in the (standard) Brownian web at points that are chosen according to a Poisson marking of the set of intersection points between paths in the Brownian web and its dual ˆ . This construction will be published in [12]. WThere are a numberW of questions about the Brownian net that are worth investigating. First, we would like to give a complete classification of all special points in the Brownian net, in analogy with the classification of special points in the Brownian web. We have some results in that direction and will present them in a forthcoming paper [14]. An important ingredient in that work is to characterize the interaction between forward left-most and dual right-most paths, which can be used to give an alternative characterization of the left-right Brownian web not discussed in the present paper. The second question regards the universality of the Brownian net and the branching-coalescing point set. Our convergence result is for the simplest system of branching-coalescing random walks. It is plausible that the same result holds for more general branching-coalescing systems, such as Schlögl models or the biased annihilating branching process from [16]. Related to this is the question of how to characterize the branching-coalescing point set by means of a generator or well-posed martingale problem. The third question is to study the marked Brownian net, which can be defined by a Poisson marking of paths in the Brownian net in the same spirit as the marked Brownian web introduced in [9]. In the biological context, such a model describes genealogies in the presence of small selection and rare mutations. It can be shown that the resulting branching-coalescing point 18 R. SUN AND J. M. SWART set with deaths undergoes a phase transition of contact-process type as the death rate is increased. This model might therefore be of some relevance in the study of the one-dimensional contact process. Finally, it needs to be investigated how the Brownian net relates to certain other objects that have been introduced in the literature. In particular, it seems that a subclass of the stochastic flows of kernels introduced by Howitt and Warren in [10] is supported on the Brownian net. Also, it would be interesting to know if the branching-coalescing point set is related to some field theory used in theoretical physics. The physicist’s way of viewing this process would probably be to say that these are coalescing Brownian motions with an infinite branching rate, but, due to the coalescence, most of this branching is not effective, so at macroscopic space scales one only observes the ‘renormalized’ branching rate, which is finite.

1.12. Outline. The rest of the paper is organized as follows. In Section 2 we construct and characterize the left-right Brownian web (Theorem 1.5) by first characterizing the left-right SDE and left-right coalescing Brownian motions described in Section 1.4. In Section 3 we establish some basic properties for the left-right SDE, recall some properties of the Brownian web and its dual, and prove some basic properties for the left-right Brownian web and its dual. In Section 4 we prove the equivalence of the hopping construction (The- orem 1.3) and the dual construction (Theorem 1.10) of the Brownian net. In Section 5 we prove Theorem 1.1, our main convergence result. In fact, we prove something more: denoting the collections of all left-most and right- most paths in an arrow configuration by l and r , respectively, we show ℵβ Uβ Uβ that S ( l , r , ) converges to ( l, r, ), where ( l, r) is a left-right ε Uβ Uβ Uβ W W N W W Brownian web and is the associated Brownian net. Here the hopping and N dual characterizations of the Brownian net serve respectively as a stochastic lower and upper bound on limit points of S ( ). ε Uβ In Section 6 we carry out two density calculations. The first of these yields Proposition 1.12, while the second estimates the density of the set of times when the left-most path starting at the origin first meets some path in the Brownian net starting at time 0 to the left of the origin. This second calculation is used in Section 7 to establish the characterization of the Brownian net using meshes (Theorem 1.7) and Proposition 1.8. These two results then in turn imply Propositions 1.4 and 1.13. Finally, in Section 8 we prove Theorem 1.11 on the branching-coalescing point set, and in Section 9 we prove Propositions 1.14 and 1.15 on the backbones of arrow configurations and the Brownian net. THE BROWNIAN NET 19

2. The left-right Brownian web. In Section 2.1 we characterize the left- right SDE described in Section 1.4 as the unique weak solution of the SDE (1.11). In Section 2.2 we give a rigorous deﬁnition of a collection of left-right coalescing Brownian motions described in Section 1.4. Finally, in Section 2.3 we construct the left-right Brownian web and prove Theorem 1.5.

2.1. The left-right SDE. Recall that a Markov transition probability kernel Pt(x,dy) on a compact metrizable space has the Feller property if the map (x,t) Pt(x, ) from E [0, ) into the space of probability measures on E is continuous7→ · with respect× to∞ the topology of weak convergence. Each Feller transition probability kernel gives rise to a strong Markov process with cadlag sample paths [6], Theorem 4.2.7. If E is not compact, but locally compact, then let E = E denote the one-point compactiﬁcation of E. ∞ ∪ {∞} In this case, one says that a Markov transition probability kernel Pt(x,dy) on E has the Feller property if the extension of Pt(x,dy) to E deﬁned by ∞ putting Pt( , ) := δ (t 0) has the Feller property. The corresponding Markov process∞ · is called∞ a≥ Feller process.

Proposition 2.1 (Well-posedness and stickiness of the left-right SDE). 2 For each initial state (L0, R0) R , there exists a unique weak solution to the SDE (1.11) subject to the constraint∈ that L R for all t T := inf s t ≤ t ≥ { ≥ 2 0 : Ls = Rs . The family of solutions (Lt, Rt)t 0 (L0,R0) R deﬁnes a Feller } { ≥ } ∈ process. The law of the total time that Lt and Rt are equal is given by

~~∞ Bt (L0 R0) 0 (2.1) 1 Lt=Rt dt = sup t + − ∧ , LZ0 { } L t 0 √2 − 2 ≥ where Bt is a standard Brownian motion (started at zero).~~

Denote R2 := (x,y) R2 : x y . A weak R2 -valued solution to (1.11) ≤ { ∈ ≤ } ≤ is a quintuple (L,R,Bl,Br,Bs), where Bl,Br,Bs are independent Brownian motions and (L, R) is a continuous, adapted R2 -valued process such that (1.11) holds in integral form (where the stochastic≤ integrals are of Itˆo-type). We rewrite the SDE (1.11) into a diﬀerent equation, which has a pathwise unique solution. (In contrast, we believe that solutions to (1.11) are not pathwise unique; see [18] and the references therein for a similar equation where this has been proved.) Consider the following equation: (i) dL = dB˜l + dB˜s dt, t Tt St − ˜r ˜s (ii) dRt = dBTt + dBSt + dt, (2.2) (iii) Tt + St = t, t

~~(iv) 1 Ls~~

Note that (2.2)(iv) says that St increases only when Lt = Rt. By deﬁ- nition, by a weak R2 -valued solution to (2.2), we will mean a 7-tuple ≤ (L,R,S,T, B˜l, B˜r, B˜s), where B˜l, B˜r, B˜s are independent Brownian motions, S, T are nonnegative, nondecreasing, continuous, adapted processes such that (2.2)(iii) and (iv) hold, and (L, R) is a continuous, adapted R2 -valued process such that (2.2)(i) and (ii) hold in integral form. ≤ Proposition 2.1 follows from the following lemma.

Lemma 2.2 (Space-time SDE). (a) There is a one-to-one correspon- dence in law between weak R2 -valued solutions of (1.11) and weak R2 -valued solutions of (2.2). ≤ ≤ 2 (b) For each initial state (L0, R0) R , equation (2.2) has a pathwise unique solution. ∈ ≤ t (c) Solutions to (2.2) satisfy St := 0 1 Ls=Rs ds, R { } 1 ˜l ˜r (2.3) St = sup ( 2 (L0 + Bs R0 Bs) s) a.s., 0 s Tt − − − ≤ ≤ and limt Tt = . →∞ ∞

Proof of Proposition 2.1. Since Lt and Rt evolve independently until they meet, it suffices to consider the case L0 R0. The existence and uniqueness of weak solutions to (1.11) under the given≤ constraint follow from Lemma 2.2(a) and (b), while (2.1) follows from Lemma 2.2(c). To prove the Feller property, by the continuity of sample paths, it suffices to show that the law on path space of solutions to (1.11) depends continuously on the initial state. Since the first meeting time and position of two independent Brownian motions depend continuously on their initial states, it suffices to show continuity of R2 -valued solutions to (2.2) in the initial state. Fix ≤ Brownian motions B˜l, B˜r and B˜s, and let (Ln, Rn,Sn, T n) be a sequence n n 2 of solutions to (2.2) with initial states (L0 , R0 ) = (ln, rn) R , such that 2 n n ∈ ≤ n n (ln, rn) (l, r) R . Since L and R are Brownian motions and S , T increase→ with slope∈ ≤ at most 1, the sequence (Ln, Rn,Sn, T n) is tight. It is not hard to see that any subsequential limit solves (2.2) (compare the proof of Proposition 5.1 in Section 5.1), and therefore, (Ln, Rn) converges to the pathwise unique solution of (2.2) with initial state (l, r).

Proof of Lemma 2.2. We start with the proofs of parts (b) and (c). Our approach is to transform an equation with a sticky boundary into a SDE with immediate reﬂection, which is a standard technique to deal with sticky reﬂection. Given a solution to (2.2), put

~~Dt := Rt Lt, (2.4) − W := R + B˜r L B˜l. t 0 t − 0 − t THE BROWNIAN NET 21~~

~~Then~~

~~(2.5) dDt = dWTt + 2 dt.~~

~~It is easy to see from (2.2) that Dt leaves 0 immediately, that is, there exist no s~~

(2.6) dD −1 = dW + 2 dτ + 2 dS −1 , Tτ τ Tτ where D −1 is constrained to be nonnegative for all τ > 0, and 2S −1 is a Tτ Tτ nondecreasing process that increases only when D −1 = 0. Equation (2.6) Tτ is an SDE with instant reﬂection, known as the Skorohod equation (see, e.g., Section 3.6.C of [11]). It can be solved (pathwise) uniquely for 2S −1 , Tτ yielding

(2.7) 2S −1 = inf (Ws + 2s). Tτ − 0 s τ ≤ ≤ Time changing back and remembering the deﬁnition of W , we arrive at (2.3). By the fact that St + Tt = t, we ﬁnd that 1 ˜l ˜r (2.8) t = Tt + sup ( 2 (L0 + Bs R0 Bs) s). 0 s Tt − − − ≤ ≤ Since the function 1 ˜l ˜r (2.9) τ τ + sup ( 2 (L0 + Bs R0 Bs) s) 7→ 0 s τ − − − ≤ ≤ is strictly increasing and continuous, it has a unique inverse, which is t Tt. This proves that S and T are pathwise unique, and therefore, by (2.2)(i)7→ and (ii), also L and R are pathwise unique. Since the solution D −1 of (2.6) spends zero Lebesgue time at 0, time- Tτ changing τ = Ts, we see that

~~Tt t~~

~~(2.10) 0= 1 D −1 =0 dτ = 1 Ds=0 dTs. Z0 { Tτ } Z0 { } t By (2.2)(iii) and (iv), we conclude that St = 0 1 Ls=Rs ds and Tt = t R { } 0 1 Ls~~

~~˜r ˜s (ii) dRt = dBTt + dBSt + dt, (2.11) t~~

~~(iii) Tt = 1 Ls~~

(iv) St = 1 Ls=Rs ds. Z0 { } Conversely, solutions to (2.11) obviously solve (2.2). Given a R2 -valued solution to (2.2), setting ≤ t l ˜l ˆl Bt := BTt + 1 Ls=Rs dBs, Z0 { } t r ˜r ˆr (2.12) Bt := BTt + 1 Ls=Rs dBs, Z0 { } t s ˜s ˆs Bt := BSt + 1 Ls

2.2. Left-right coalescing Brownian motions. In this section we give a rig- ′ ′ orous definition of a collection lz1 ,...,lzk , rz ,...,rz of paths of left-right 1 k′ 2 coalescing Brownian motions, started at points z ,...,z , z ,...,z ′ R . 1 k 1′ k′ ∈ Write zi = (xi,ti) and zi′ = (xi′ ,ti′ ). The times t1,...,tk,t1′ ,...,tk′ ′ divide R into a finite number of intervals. It suffices to define a Markov process that specifies the time evolution of the left-right coalescing Brownian motions during each such interval. Thus, we need to construct a Markov process (L1,t,...,Lk,t; R1,t,...,Rk′,t)t 0 k+k′ ≥ in R such that (L1,t,...,Lk,t) and (R1,t,...,Rk′,t) are each distributed THE BROWNIAN NET 23 as coalescing Brownian motions with drift 1 and +1 respectively, and − the interaction between paths in (L1,t,...,Lk,t) and (R1,t,...,Rk′,t) is that of the left-right SDE (1.11). Instead of characterizing the joint process (L1,t,...,Lk,t; R1,t,...,Rk′,t) as the unique weak solution of a system of SDEs, which is rather laborious, we give an inductive construction using the distribution of (Lt, Rt). We first construct the system up to the first time two left Brownian motions coalesce, or two right Brownian motions coalesce, or a right Brownian motion hits a left Brownian motion from the left. In the last case, the right Brownian motion has to continue on the right of the left Brownian motion, so we call this a crossing. If our left and right coalescing Brownian motions are initially ordered as LRRLRLRLLLRRLR, say, then we partition them as LR R LR LR L L LR R LR , letting all pairs of a left Brownian{ motion}{ }{ followed}{ }{ by}{ a right}{ Brownian}{ }{ } motion constitute a partition element with two members, and putting all remaining Brownian motions into partition elements with one member. We let the partition elements evolve independently until the first coalescing or crossing time. Here partition elements containing two members evolve according to the left-right SDE (1.11), while partition elements containing one member are just Brow- nian motions with drift +1 or 1. At the first coalescing or crossing time, we respectively coalesce or cross− the motions that have hit each other, repar- tition the remaining Brownian motions and continue the process. Note that there can be at most k + k′ coalescence events and at most kk′ crossings, so this procedure is iterated at most finitely often and eventually leads to a single pair (L, R). The above construction uniquely defines the system of left-right coalescing ′ ′ Brownian motions lz1 ,...,lzk , rz ,...,rz . By the Feller property of coalesc- 1 k′ ing Brownian motions and solutions to the left-right SDE, it is clear that the ′ ′ law of (lz1 ,...,lzk , rz ,...,rz ) depends continuously on the starting points 1 k′ z1,...,zk, z1′ ,...,zk′ ′ , and the marginal distribution of a subset of paths in ′ ′ lz1 ,...,lzk , rz ,...,rz is also a system of left-right coalescing Brownian { 1 k′ } motions. This consistency property allows the definition of a countable system of left-right coalescing Brownian motions.

~~2.3. The left-right Brownian web. We now construct the left-right Brow- nian web and prove Theorem 1.5.~~

Proof of Theorem 1.5. We ﬁrst show uniqueness. Fix countable dense sets l, r R2. Suppose that there exists a (Π) (Π)-valued random variableD D (⊂ l, r) satisfying properties (i)–(iii)K in Theorem× K 1.5. By W W l l property (i), let lz, z , denote the almost sure unique element in starting from z, and let∈r D, z r, denote the almost sure unique elementW in z ∈ D 24 R. SUN AND J. M. SWART

r l r starting from z. Then by property (ii), ((lz)z l , (rz)z r ) is a ΠD ΠD - Wvalued random variable whose finite-dimensional∈D distribut∈D ions are that× of left-right coalescing Brownian motions. Hence, the law of ((lz)z l , (rz)z r ) is uniquely determined, and by property (iii), so is the law of∈D ( l, r).∈D We now construct a (Π) (Π)-valued random variable ( Wl, Wr) satisfying properties (i)–(iii)K in Theorem× K 1.5. By our constructionW of left-rightW coalescing Brownian motions in Section 2.2 and their consistency property when more paths are added, we can invoke Kolmogorov’s extension l r theorem to assert that there exists a Π Π -valued random variable D × D ((lz)z l , (rz)z r ) whose finite-dimensional distributions are that of left- right∈D coalescing∈D Brownian motions. Define

l l r r (2.14) := l : z , := r ′ : z . W { z ∈ D } W { z ′ ∈ D } Note that l and r are each distributed as a standard Brownian web with drift W1 and +1W respectively. Properties (i) and (iii) then follow from the analogous− properties for the standard Brownian web. It only remains to show (ii). Let u1,...,uk and u1′ ,...,uk′ ′ be deterministic ﬁnite subsets 2 { } { } l of R . By (i), almost surely, a unique path lui starts from each ui, r ∈W 1 i k, and a unique path ru′ starts from each u′ , 1 j k′. Choose ≤ ≤ j ∈W j ≤ ≤ l r zn,i , zn,j′ such that zn,i ui and zn,j′ uj′ as n . Since the Brownian∈ D web∈ is D a.s. continuous at→ deterministic points→ (see Proposition→∞ 3.2), we have lz lu and rz′ ru′ in Π, and hence, n,i → i n,j → j

′ ′ ′ ′ (2.15) (lzn,1 ,...,lzn,k , rz ,...,rz ) = (lu1 ,...,luk , ru ,...,ru ). n,1 n,k′ n 1 k′ L →∞⇒ L By the continuity of left-right coalescing Brownian motions in its starting ′ ′ points, it follows that (lu1 ,...,luk , ru ,...,ru ) is distributed as a system of 1 k′ left-right coalescing Brownian motions starting from u1,...,uk, u1′ ,...,uk′ ′ , verifying property (ii).

3. Properties of the left-right Brownian web. In Sections 3.1–3.3 below we collect some properties of solutions to the left-right SDE, the Brownian web and its dual, and the left-right Brownian web and its dual, respectively.

~~3.1. Properties of the left-right SDE. Recall that a set X is perfect if X is closed and x X x for all x X, that is, X has no isolated points. ∈ \{ } ∈~~

Proposition 3.1 (Properties of the left-right SDE). Let (Lt, Rt)t 0 be ≥ the unique weak solution of the SDE (1.11) with initial condition (L0, R0) R2, subject to the constraint that L R for all t T := inf s 0 : L =∈ t ≤ t ≥ { ≥ s Rs . Let I := t 0 : Lt = Rt and let µI be the measure on R deﬁned by µ (}A) := ℓ(I {A)≥, where ℓ denotes} Lebesgue measure. Then: I ∩ THE BROWNIAN NET 25

~~(a) Almost surely, I is a nowhere dense perfect set. (b) Almost surely, I is the support of µI .~~

Proof. If T = , the lemma is vacuous. Since (Lt, Rt)t 0 is a strong Markov process and∞T is a stopping time, we may assume without≥ loss of generality that T = 0, that is, L0 = R0. Define W as in (2.4), put W˜ τ := W + 2τ (τ 0), and τ ≥ (3.1) Xτ := W˜ τ + Rτ where Rτ := inf W˜ s (τ 0). − 0 s τ ≥ ≤ ≤ Then X is a Brownian motion with diffusion constant 2 and drift 2, instan- taneously reflected at zero. It is well known (and not hard to prove) that τ 0 : Xτ = 0 is the support of dR. { Setting≥ D :=} R L (t 0), we see by (2.6), (2.7) and (2.11)(iii) that t t − t ≥ t

~~(3.2) Dt = XTt where Tt := 1 Ds>0 ds (t 0). Z0 { } ≥~~

It follows that I = t 0 : Dt = 0 is the image of τ 0 : Xτ = 0 under the map τ T 1. Since{ by≥ (2.7) and} (2.11)(iv), { ≥ } 7→ τ− t 1 (3.3) St = 1 Ds=0 ds = 2 RTt (t 0), Z0 { } ≥ 1 1 the measure µI is the image of the measure 2 dR under the map T − . Since 1 1 T − is a continuous open map, it follows that supp(µI )= T − (supp(dR)) = 1 T − ( τ 0 : Xτ = 0 )= I. This proves part (b). It follows that I has no isolated{ points,≥ that is,} is perfect. To see that I is nowhere dense, by the Markov property, it suﬃces to show that Dt leaves the origin immediately. Indeed, setting σ := inf t 0 : Dt > 0 and using (2.2), we see that σ { ≥ } 0= Dσ = 0 2 dt = 2σ a.s. This proves part (a). R 3.2. Properties of the Brownian web. In this section we recall some properties of the standard Brownian web and its dual ˆ , which can all be W W found in [8, 9, 15, 17]. Recall thatσ ˆπˆ denotes the starting time of a dual pathπ ˆ. Thus, a dual path is a mapπ ˆ : [ , σˆ ] [ , ] such −∞ πˆ → −∞ ∞ ∪ {∗} thatπ ˆ : [ , σˆπˆ] R [ , ] is continuous, andπ ˆ( ) := whenever [ −∞, σˆ ]. ∩ → −∞ ∞ ±∞ ∗ ±∞∈ −∞ πˆ Proposition 3.2 (Properties of the Brownian web). Let be the Brow- W nian web and ˆ its dual. Then: W (a) ( , ˆ ) is equally distributed with ( ˆ , ). (b) AlmostW W surely, paths in coalesce− whenW W they meet, that is, for each W π,π′ and t>σπ σπ′ such that π(t)= π′(t), one has π(s)= π′(s) for all s ∈Wt. ∨ ≥ 26 R. SUN AND J. M. SWART

(c) Almost surely, paths and dual paths do not cross, that is, there exist no π , πˆ ˆ , and s,t [σπ, σˆπˆ] such that (π(s) πˆ(s)) (π(t) πˆ(t)) < 0. ∈W(d) Almost∈ W surely, paths∈ and dual paths spend zero− Lebesgue· time− together, σˆπˆ ˆ that is, we have σπ 1 π(t)=ˆπ(t) dt = 0 for all π and πˆ . { } ∈W2 ∈ W + (e) Almost surely,R for each point z = (x,t) R , xn− x, xn x, πn− + + ∈ ↑ ↓ ∈ (xn−,t), and πn (xn ,t), the limits πz := limn πn− and πz+ := W + ∈ W − →∞ limn πn exist and do not depend on the choice of πn− (xn−,t) and π+ →∞(x+,t). ∈W n ∈W n Points z R2 in the Brownian web are classiﬁed according to the number of disjoint∈ incoming and distinct outgoing paths at z. By deﬁnition, an incoming path at z = (x,t) is a path π such that σπ

Lemma 3.3 (Classiﬁcation of points in the Brownian web). (a) Almost surely, all z R2 are in ( , ˆ ) of one of the types (0, 1)/(0, 1), ∈ W W (0, 2)/(1, 1), (0, 3)/(2, 1), (1, 1)/(0, 2), (1, 2)l/(1, 2)l, (1, 2)r/(1, 2)r and (2, 1)/ (0, 3). See Figure 6. (b) For each deterministic t R, almost surely each point on R t is of either type (0, 1), (0, 2) or (1,∈1) in . × { } W

~~Fig. 6. Types of points in the Brownian web and its dual (W, Wˆ ). THE BROWNIAN NET 27~~

~~(c) Each deterministic point z R2 is almost surely of type (0, 1) in . ∈ W The next lemma shows that convergent sequences of paths in converge W in a rather strong sense.~~

~~Lemma 3.4 (Convergence of paths). Let be the standard Brownian web. Then: W~~

(a) Almost surely, for any πn n N,π such that πn π, one has { } ∈ ∈W → σπ σπ and sup t σπ σπ : πn(t) = π(t) σπ. n → { ≥ n ∨ 6 }n −→ →∞ 2 (b) Let be a deterministic countable dense subset of R . Let πz z be the skeletonD of relative to the starting set . Then almost surely,{ } for∈D W D all π and ε> 0, there exists z = (x,t) such that t (σπ ε, σπ + ε) and π∈W(s)= π(s) for all s σ + ε. ∈ D ∈ − z ≥ π Proof. By [8], Proposition 4.1, t,δ := γ(t) : γ ,σ t δ is a.s. W { ∈W γ ≤ − } locally ﬁnite for each t, δ Q with δ> 0. Therefore, π π implies that, for ∈ n → each σπ

In applications of Lemma 3.4, one mostly needs part (b). Typically, a property is proved ﬁrst for skeletal paths, and then extended to all paths in the web by Lemma 3.4(b). We say that a path π1 crosses a path π2 from left to right if there exist σ σ s

~~σπ1 s~~

Lemma 3.5 (Equivalence of crossing). Let ( , ˆ ) be the Brownian web W W and its dual. A path γ Π crosses some π from left to right if and only if it also crosses some∈ πˆ ˆ from left to∈W right. The same is true if we interchange left and right. ∈ W

~~Proof. Assume γ Π crosses π from left to right, that is, γ(s) < π(s) and γ(t) >π(t) for∈ some σ ∈Wσ s~~

3.3. Properties of the left-right Brownian web. In this section we collect some basic properties of the left-right Brownian web ( l, r) and its dual W W ( ˆ l, ˆ r). Recall the deﬁnitions of intersection times and crossing times from SectionW W 1.4. For any π ,π Π, we let 1 2 ∈ (3.4) I(π ,π ) := t (σ σ , ) : π (t)= π (t) 1 2 { ∈ π1 ∨ π2 ∞ 1 2 } denote the set of intersection times of π1 and π2.

Proposition 3.6 (Properties of the left-right Brownian web). Let ( l, W r, ˆ l, ˆ r) be the standard left-right Brownian web and its dual. Then, almostW W surely,W the following statements hold: l r (a) For each l and r such that σl σr < , one has Tcros := inf t>σ σ : l(t∈W) < r(t) = inf∈Wt>σ σ : l(t) ∨r(t) <∞ , and l(t) r(t) { l ∨ r } { l ∨ r ≤ } ∞ ≤ for all t Tcros. (b) For≥ each l l and r r, I(l, r) is a (possibly empty) nowhere dense perfect set. ∈W ∈W (c) For each l l and r l such that σ σ < , the set I(l, r) ∈W ∈W l ∨ r ∞ is the support of the measure µI on (σl σr, ) deﬁned by µI(l,r)(A) := ℓ(I(l, r) A), where ℓ denotes Lebesgue measure.∨ ∞ ∩ (d) Paths in l cannot cross paths in ˆ r from left to right, that is, W W there exist no l l, rˆ ˆ r, and σ s

Proof. Let l and r be deterministic countable dense subsets of R2, D D l r and let lz z l and rz z r be the corresponding skeletons of and . By Theorem{ } ∈D1.5 and Lemma{ } ∈D3.4(b), it suﬃces to prove parts (a)–(c)W for skele-W l r tal paths, and hence for deterministic pairs (lz, rz′ ) where z and z′ . Since such deterministic pairs satisfy the SDE (1.11) by Theorem∈ D 1.5,∈ parts D (a)–(c) follow readily from Proposition 3.1(a) and (b). Property (d) is a consequence of (a) and Lemma 3.5.

4. The Brownian net. Let ( l, r, ˆ l, ˆ r) be a left-right Brownian web and its dual, and set W W W W (4.1) := ( l r). Nhop Hcros W ∪W Note that if l, r R2 are deterministic countable dense sets, then by D D ⊂ l l r r Lemma 3.4(b), we also have hop = cros( ( ) ( )). Deﬁne mesh and by formulas (1.16N) and (H1.24),W respectively.D ∪W InD Sections 4.1N and Nwedge 4.2 we prove the inclusions hop wedge and wedge hop, respectively. As an application, in SectionN 4.3⊂Nwe establishN Theorems⊂N1.3 and 1.10, as THE BROWNIAN NET 29 well as Lemma 1.6. In addition, we prove Theorem 1.9 in Section 4.2, and, as a preparation for the characterization of the Brownian net using meshes, we prove the inclusion hop mesh in Section 4.1. The proof of the other inclusion is more diﬃcult,N and⊂N will be postponed to Section 7.

~~4.1. . Set Nhop ⊂Nwedge l noncros := π Π : π does not cross paths in from right (4.2) P { ∈ W to left or paths in r from left to right . W }~~

Lemma 4.1 (Closedness of constructions). The sets wedge, mesh and are closed. N N Pnoncros Proof. Note that if a path π Π enters a mesh with bottom time t> ∈ l σπ, then it must enter from outside. Likewise, if π crosses a dual path ˆl ˆ R2 ∈ˆ W from right to left, then it enters the open set (x,t) : t< σˆˆl, x< l(t) from outside. Thus, taking into account Lemma{ 3.5,∈ all statements follow} from the fact that if π ,π Π satisfy π π, and π enters an open set A n ∈ n → from outside, then for n suﬃciently large, πn also enters A from outside.

Lemma 4.2 (Noncrossing property). We have a.s. Nhop ⊂ Pnoncros l Proof. It suﬃces to show that no path π hop crosses paths in from right to left. By Lemma 4.1, it suﬃces to verify∈N the statement for pathsW l r l r in cros( ). By Propositions 3.2(b) and 3.6(a), paths π H W ∪W ∈W l∪W have the stronger property that there exist no σπ

Let A be either a mesh or wedge with (ﬁnite) bottom point z = (x,t). We say that a path π Π enters A through z if σπ t such that (π(s),s) ∈A and (π(u), u) A for all u [t,s]. Note that if a path enters a mesh (wedge)∈ from outside,∈ then it must∈ either cross a left-most or right-most (dual) path in the wrong direction, or enter the mesh (wedge) through its bottom point.

Lemma 4.3 (Finite wedges contained in meshes). For every wedge W with bottom point z, there exists a mesh M with bottom point z such that W M. ⊂ Proof. Write z = (x,t) and letr, ˆ ˆl be the left and right boundary of W . By Lemma 3.3, there exist r r(z) and l l(z) such that r(s) rˆ(s) ˆ ∈W ∈W ≤ for all s (t, σˆrˆ) and l(s) l(s) for all s (t, σˆˆl). It follows that r and l are the left and∈ right boundary≤ of a mesh containing∈ W (see Figure 3). 30 R. SUN AND J. M. SWART

Lemma 4.4 (Hopping contained in mesh construction). We have hop a.s. N ⊂ Nmesh l r Proof. By Lemma 4.1, it suffices to show that cros( ) mesh. H l W r∪W ⊂N We will show that, even stronger, paths in cros( ) do not enter meshes regardless of their bottom times. It isH easy toW see∪W that this stronger property is preserved under hopping, so it suffices to show that paths in l r do not enter meshes. By symmetry, it suffices to show this for paths inW ∪Wl. By Propositions 3.2(b) and 3.6(a), it suffices to show that paths in lWcannot enter meshes through their bottom point. Let M = M(r, l) bea Wmesh with left and right boundary r and l and bottom point z = (x,t). Let l l′ := lz and r′ := rz+ be the left-most path in (z) and the right-most path in− r(z), respectively, in the sense of PropositionW 3.2(e). Then, by W Proposition 3.6(a), l′(s) r(s) and l(s) r′(s) for all s t (see Figure 3). l ≤ ≤ ≥ If some l′′ enters M through z, then by Lemma 3.3, z must be of the type (1, 2)∈Wor (1, 2) in l, and therefore, l continues along either l or l . l r W ′′ ′ In either case, l′′ does not enter M.

Lemma 4.5 (Hopping contained in wedge construction). We have hop a.s. N ⊂ Nwedge l r Proof. By Lemma 4.1, it suﬃces to show that cros( ) wedge. Thus, we must show that paths in ( l Hr) doW not∪W cross⊂N paths in Hcros W ∪W ˆ l, ˆ r in the wrong direction or enter wedges through their bottom points. TheW W ﬁrst assertion follows from Lemmas 3.5 and 4.2, while the second assertion is a result of Lemmas 4.3 and 4.4.

~~4.2. wedge hop. In this section we prove that wedge hop. We start withN a preparatory⊂N lemma. N ⊂N~~

~~Lemma 4.6 (Compactness of ). (Π) a.s. Nhop Nhop ∈ K~~

Proof. Recall (Θ1, Θ2) from (1.4). From the deﬁnition of the topology on Π introduced in Section 1.2, by Arzela–Ascoli, we note that a set K Π is precompact if and only if the set of functions deﬁned by the images⊂ of the graphs of π K under the map (Θ1, Θ2) is equicontinuous, that is, the modulus of continuity∈ of K,

~~mK (δ) := sup Θ1(π(t),t) Θ1(π(s),s) : (4.3) {| − | π K, s,t σ , Θ (s) Θ (t) δ ∈ ≥ π | 2 − 2 |≤ } satisﬁes m (δ) 0 as δ 0. K ↓ ↓ THE BROWNIAN NET 31~~

Lemma 4.2 implies that, for each π and s σ , we have l π r ∈Nhop ≥ π ≤ ≤ on [s, ), where l := l(π(s),s) and r := r(π(s),s)+ denote respectively the left- ∞ − most and the right-most paths in l(π(s),s) and r(π(s),s), in the sense of Proposition 3.2(e). It follows that,W for any t>s,W

~~Θ1(π(t),t) Θ1(π(s),s) (4.4) | − | Θ (l(t),t) Θ (l(s),s) Θ (r(t),t) Θ (r(s),s) . ≤ | 1 − 1 | ∨ | 1 − 1 | Taking the supremum over all π and σ s~~

~~Lemma 4.7 (Hopping contains wedge construction). We have wedge a.s. N ⊂ Nhop~~

Proof. We must show that any path π wedge can be approximated by a sequence of paths π ( l r).∈N By the compactness of n ∈Hcros W ∪W Nhop (Lemma 4.6), it suffices to show that, for any π wedge, ε> 0, and σπ < ε l ∈Nr t1 < >n > 1 andr ˆ ,..., rˆ such that 1 · · · m 1 m ˆ r rˆk (π(tnk ) ε, tnk ) and (4.5) ∈ W − n := sup i : n >i> 1, rˆ (t ) <π(t ) ε . k+1 { k k i i − } This process terminates ifr ˆk(ti) π(ti) ε for all nk >i> 1. In this case we set m := k. We definer ˆ :=r ˆ on (≥t ,t− ] (j = 1,...,m 1) andr ˆ :=r ˆ on j nj+1 nj − m [t ,t ]. By left-right symmetry, we define n = n > >n ′ > 1, ˆl ,..., ˆl ′ , 1 nm 1′ · · · m′ 1 m and ˆl analoguously. We make the following claims:

~~(1)r ˆ π ˆl on [t1,tn]. ≤ ≤ ˆ (2) ε′ := infs [t1,tn](l(s) rˆ(s)) > 0. ∈ − (3) rˆ(t ) π(t ) ˆl(t ) π(t ) ε for i = 2,...,n. | i − i | ∨ | i − i |≤ 32 R. SUN AND J. M. SWART~~

~~Fig. 7. Steering a hopping path in the “ﬁsh-trap” (ˆr, ˆl).~~

(4) limt t rˆ(t) rˆ(ti) and limt t lˆ(t) lˆ(ti) for i = 2,...,n 1, which are ↓ i ≤ ↓ i ≥ − the only possible discontinuities ofr ˆ and ˆl. Properties (1) and (2) follow from our assumption that π does not enter wedges whose left and right boundaries are any of the dual pathsr ˆ1,..., rˆm and ˆl1,..., ˆlm′ . Properties (3) and (4) are now obvious from our construction. The pair (ˆr, ˆl) resembles a ﬁsh-trap (see Figure 7). ε l r We now construct a path π ( ) such that σ ε (σ ,t ), ∈Hcros W ∪W π ∈ π 1 πε(t ) π(t ) ε, andr ˆ(s) πε(s) lˆ(s) for all s [t ,t ]. To this aim, | 1 − 1 |≤ ≤ ≤ ∈ 1 n we inductively choose l , l , l ,... l, r , r , r ,... r, and τ ,τ ,... such 1 3 5 ∈W 2 4 6 ∈W 1 2 that τi is a crossing time of li and ri+1 if i is odd and a crossing time of ri and li+1 if i is even, in the following way. First, we choose l1 such that σ (σ ,t ) and l (t ) (ˆr(t ), ˆl(t )) [π(t ) ε, π(t )+ ε]. Assuming that l1 ∈ π 1 1 1 ∈ 1 1 ∩ 1 − 1 we have already chosen l1,...,li and r2,...,ri 1, we proceed as follows. If − rˆ(s) < li(s) lˆ(s) for all s [τi 1,tn] (where τ0 := t1), the process termi- ≤ ∈ − nates. Otherwise, since paths cannot cross dual paths [Proposition 3.2(c)], li must hitr ˆ before time tn. In this case, we set σi := inf s [τi 1,tn] : li(s)= { ∈ − rˆ(s) . Using Propositions 3.2(c) and 3.6(a), we can choose δ> 0 suﬃciently } small and r r started in (x,s) : σ δ

~~Deﬁning πε ( l r) by hopping between the paths l , l ,... and ∈Hcros W ∪W 1 3 r2, r4,... at the times τ1,τ2,..., we have found the desired approximation of π by hopping paths.~~

~~Since it is very similar to the proof of Lemma 4.7, we include here the proof of Theorem 1.9.~~

Proof of Theorem 1.9. Let be defined by the right-hand Wwedge side of (1.21). Since paths in cannot cross paths in ˆ , to show that W W W⊂ wedge, it suffices that paths in cannot enter wedges of ˆ through their bottomW points. This can be provedW by mimicking the proofsW of Lemmas 4.3 and 4.4. The inclusion wedge can be proved in the same way as the proof of Lemma 4.7. SinceW is⊂W compact, it suffices to show that path that does not enter wedges fromW outside can be approximated by paths in . We can define a “fish-trap” whose left and right boundary are constructedW by piecing dual paths together. In this case, any path in entering the fish- trap from below must stay between its left and right boundaryW , so no hopping is necessary.

~~4.3. Characterizations with hopping and wedges.~~

Proofs of Theorem 1.3, Lemma 1.6 and Theorem 1.10. Consider a left-right Brownian web and its dual ( l, r, ˆ l, ˆ r), and let be W W W W Nwedge defined as in (1.24) and hop be defined as in (4.1). By Lemmas 4.5 and 4.7, = . It followsN from Lemma 4.2 that, for every z = (x,t) R2, we Nhop Nwedge ∈ have lz (s) π(s) rz+(s) for all π hop(z) and s t, where lz , rz+ are defined− for ≤ l, r≤as in Proposition∈N3.2(e). In particular,≥ for deterministic− z, the a.s. uniqueW W paths l l(z) and r r(z) are respectively the left- z ∈W z ∈W most and right-most paths in hop(z). Setting := hop = wedge, we have found a (Π)-valued (by LemmaN 4.6) randomN variableN thatN satisfies conditions (i)–(ii)K of Theorem 1.3. To see that condition (iii) is also satisfied, note l l r r that by Lemma 3.4(b), hop = cros( ( ) ( )) for any deterministic countable dense setsN l, r HR2. SinceW D a∪W randomD variable satisfying the conditions of Theorem 1.3D isD obviously⊂ unique in distribution, the proof of Theorem 1.3 is complete. l Since for each deterministic z, the a.s. unique paths lz (z) and rz r(z) are the left-most and right-most paths in , this∈W also shows that∈ toW each Brownian net, there exists an associated left-rightN Brownian web, which is obviously unique by properties (i) and (ii) of Theorem 1.3. This proves Lemma 1.6. Finally, since = , we have also proved Theorem 1.10. N Nwedge 34 R. SUN AND J. M. SWART

5. Convergence. In this section we prove Theorem 1.1. In fact, we prove something more: we prove the joint convergence under diffusive scaling of the collections of all left-most and right-most paths (and their dual) in the arrow configuration to the left-right Brownian web (and its dual), and of ℵβ the collection of all β-paths to the associated Brownian net. Throughout this section, denotesℵ the (standard) Brownian net, defined by the hopping or dual characterizationN (Theorem 1.3 or 1.10), which have been shown to be equivalent. We will not use the mesh characterization of the Brownian net (Theorem 1.7, yet to be proved) in this section. In Section 5.1 we prove the convergence of a single pair of left-most and right-most paths in the arrow configuration β to a solution of the left-right SDE (1.11). In Section 5.2 we prove the convergenceℵ of all left-most and right-most paths and their dual to the left-right Brownian web and its dual. Finally, in Section 5.3, we prove Theorem 1.1.

5.1. Convergence to the left-right SDE. Recall the deﬁnition of β and from Section 1.1. Let l (resp. r ) denote the set of left-mostℵ (resp. Uβ Uβ Uβ right-most) paths in β, that is, β-paths which follow arrows to the left (resp. right) at branchingU points. Weℵ have the following convergence result l r n for a single pair of paths in ( βn , βn ). Below, R [0, ) denotes the space of continuous functions fromU [0, U) to Rn, equippedC ∞ with the topology of uniform convergence on compacta.∞

Proposition 5.1 (Convergence of a pair of left and right paths). Let (n) (n) (n) βn, εn 0 with βn/εn 1. Let x ,y Zeven be points such that (εnx , (n) → → 2 ∈ (n) l εny ) (x,y) for some (x,y) R . Let (Lt )t 0 denote the path in β → ∈ ≥ U n (n) (n) r (n) starting at (x , 0), and (Rt )t 0 the path in β starting at (y , 0). Then ≥ U n (n) (n) (5.1) ((εnL 2 , εnR 2 )t 0) = ((Lt, Rt)t 0), t/εn t/εn n ≥ L ≥ →∞⇒ L where denotes weak convergence of probability laws on 2 [0, ), and ⇒ CR ∞ (Lt, Rt)t 0 is the unique weak solution of (1.11) with initial state (L0, R0)= ≥ (x,y), subject to the constraint that Lt Rt for all t T := inf s 0 : Ls = R . ≤ ≥ { ≥ s}

(n) (n) (n) Proof. Set Tn := inf s 0 : Ls = Rs . Since up to time Tn, L and (n) { ≥ } R are independent random walks with drift βn and +βn respectively, it follows from Donsker’s invariance principle and− the almost sure continuity of the ﬁrst intersection time between two independent Brownian motions with drift 1 that ± (n) (n) (5.2) ((εnL 2 , εnR 2 )t 0) = ((Lt T , Rt T )t 0). t/εn Tn t/εn Tn n ∧ ∧ ≥ L ∧ ∧ ≥ →∞⇒ L THE BROWNIAN NET 35

Therefore, it suffices to prove Proposition 5.1 for the case x(n) = y(n). By translation invariance, we may take x(n) = y(n) = 0. (n) (n) Note that (εnL 2 )t 0 and (εnR 2 )t 0 individually converge weakly to a t/εn ≥ t/εn ≥ Brownian motion with drift 1, respectively, +1. This implies tightness for − (n) (n) the family of joint processes (L , R ) n N. Our strategy is to represent (n) (n) { } ∈ (Lt , Rt )t 0 as the solution of a difference equation, which in the limit yields an SDE≥ with a unique solution. Since the discontinuous coefficients of the SDE (1.11) are problematic, we prefer to work with (2.2), which behaves better under limits. l r s Let (Vt )t N0 , (Vt )t N0 and (Vt )t N0 be independent discrete-time simple symmetric random∈ walks∈ starting at∈ the origin at time zero. For α = l,r,s, (n),α, α let (Dt −)t N0 be a process such that whenever Vt jumps one step to the (n),α, ∈ right, Dt − with probability βn jumps two steps to the left. Likewise, let (n),α,+ (Dt )t N0 be the process that with probability βn jumps two steps to ∈ α α (n),α, the right whenever Vt jumps one step to the left. As a result, Vt + Dt − is a random walk with drift β , and V α + D(n),α,+ is a random walk with − n t t drift +βn. (n) (n) The unscaled process (Lt , Rt ) at integer times can be constructed as the solution of (n) l (n),l, s (n),s, Lt = V (n) + D (n) − + V (n) + D (n) −, Tt Tt St St (n) r (n),r,+ s (n),s,+ Rt = V (n) + D (n) + V (n) + D (n) , Tt Tt St St (5.3) t 1 (n) − Tt = 1 (n) (n) , Ls

~~(n) (n) α (n),α, (n) (n) [compare with (2.11)]. We deﬁne Lt , Rt , Vt , Dt ±, Tt and St at (n) noninteger times by linear interpolation. Note that dT = 1 (n) (n) dt. t L~~

t 2 (n) (iv) 1 (n) (n) d(εnSs/ε2 ) = 0, Z εnR 2 εnL 2 >εn n 0 { s/εn − s/εn } where in the indicator event in (iv), we impose the lower bound of εn instead (n) (n) of 0 for εnR 2 εnL 2 to compensate the eﬀect of linearly interpolating s/εn − s/εn S(n) between integer times. Clearly,

l r s (n),l, (εnV 2 , εnV 2 , εnV 2 , εnD 2 −, t/εn t/εn t/εn t/εn (5.5) − (n),r,+ (n),s, (n),s,+ − εnDt/ε2 , εnDt/ε2 , εnDt/ε2 )t 0 n − n n ≥ converge weakly in law on 7 [0, ) to CR ∞ ˜l ˜r ˜s (5.6) (Bt, Bt , Bt ,t,t,t,t)t 0. ≥ (n) (n) We have noted that the laws of (εnL 2 , εnR 2 )t 0 n N are tight. Since { t/εn t/εn ≥ } ∈ 2 (n) 2 (n) t εnT 2 increases with slope at most 1, the laws of (εnT 2 )t 0 n N 7→ t/εn { t/εn ≥ } ∈ 2 (n) are also tight. The same is true for (εnS 2 )t 0 n N. Therefore, for n N, { t/εn ≥ } ∈ ∈ the laws of the 11-tuple, which consists of the 7-tuple in (5.5) joint with (n) (n) 2 (n) 2 (n) (εnL 2 , εnR 2 , εnT 2 , εnS 2 )t 0, are also tight. By going to a subse- t/εn t/εn t/εn t/εn ≥ quence, we may assume that the 11-tuple converges weakly to some limiting process

˜l ˜r ˜s (5.7) (Bt, Bt , Bt ,t,t,t,t,Lt, Rt, Tt,St)t 0. ≥ By Skorohod’s representation theorem (see, e.g., Theorem 6.7 in [4]), we can couple the 11-tuples for n N and the limiting process in (5.7), such that ∈ the convergence is almost sure in 11 [0, ). CR ∞ Assuming this coupling, we claim that (Lt, Rt, Tt,St)t 0 solves the equa- ≥ tion (2.2), and is therefore determined uniquely in law by Lemma 2.2. Indeed, (2.2)(i)–(iii) follow immediately by taking the limit n in (5.4)(i)–(iii). →∞ We claim that (2.2)(iv) follows from (5.4)(iv). For each δ> 0, choose a continuous nondecreasing function ρ : [0, ) R, such that ρ (u)=0 for u δ δ ∞ → δ ≤ and ρ (u)=1 for u 2δ. Then, using (5.4)(iv) and taking the limit n , δ ≥ →∞ we ﬁnd that

~~t (5.8) ρδ(Rs Ls) dSs = 0 Z0 − for each δ> 0. Letting δ 0, we arrive at (2.2)(iv). ↓ THE BROWNIAN NET 37~~

5.2. Convergence to the left-right Brownian web. In this section we prove the convergence, under diffusive scaling, of the collections of all left-most and right-most paths in the arrow configuration β (and their dual) to the left- right Brownian web (and its dual). As a corollary,ℵ we also prove formula (1.22). Recall the scaling map Sε defined in (1.7).

Proposition 5.2 (Convergence of multiple left-right paths). Let βn, εn (n) (n) (n) (n) 2 → 0 with βn/εn 1. Let z1 ,...,zk , z1′ ,...,zk′ ′ Zeven be such that (n) → (n) ∈ (n) S (z ) z and S (z′ ) z for i = 1,...,k and j = 1,...,k . Let l εn i → i εn j → j′ ′ i l (n) r denote the path in βn starting from zi, and let rj denote the path in βn U k+k′ U starting from zj′ . Then on the space Π , (n) (n) (n) (n) (5.9) (Sε (l ,...,l , r ,...,r ′ )) = (l1,...,lk, r1,...,rk′ ), n 1 k 1 k n L →∞⇒ L where (l1,...,lk, r1,...,rk′ ) is a collection of left-right coalescing Brownian motions as deﬁned in Section 2.2, starting from (z1,...,zk, z1′ ,...,zk′ ′ ).

Proof. Recall the inductive construction of (l1,...,lk, r1,...,rk′ ) from (n) (n) (n) (n) Section 2.2. Note that (l1 ,...,lk , r1 ,...,rk′ ) can be constructed using the same inductive approach. Since the inductive construction pieces together independent evolutions of sets of paths, where each set consists of either a single left-most or right-most path or a pair of left-right paths, the proposition follows easily from Proposition 5.1 and the observation that the stopping times used in the inductive construction are almost surely contin- ′ uous functionals on Πk+k with respect to the law of independent evolutions of paths in diﬀerent partition elements.

Let ˆ denote the arrow conﬁguration dual to , deﬁned exactly as in ℵβ ℵβ (1.17), and let ˆ denote the set of all ˆ -paths. Let ˆl (resp. ˆr ) denote Uβ ℵβ Uβ Uβ the set of ˆ -paths dual to l (resp. r ), that is, the set of all left-most ℵβ Uβ Uβ (resp. right-most) paths in ˆ after rotating the graph of ˆ by 180 . Let Uβ Uβ ◦ Πˆ := π : π Π , the image space of Π under the rotation map , while preserving{− the∈ metric.} We have the following result: −

Theorem 5.3 (Convergence to the left-right Brownian web and its dual). l r l r 2 Let βn, εn 0 with βn/εn 1. Then Sε ( , , ˆ , ˆ ) are (Π) → → n Uβn Uβn Uβn Uβn K × (Π)ˆ 2-valued random variables, and K l r l r l r l r (5.10) (Sε ( , , ˆ , ˆ )) = ( , , ˆ , ˆ ), n βn βn βn βn n L U U U U →∞⇒ W W W W where ( l, r, ˆ l, ˆ r) is the left-right Brownian web and its dual. W W W W 38 R. SUN AND J. M. SWART

Proof. It follows from Theorem 6.1 of [8], Theorem 1.2 and Proposi- tion 3.2 that l l l l (Sε ( , ˆ )) = ( , ˆ ) and L n Uβn Uβn n ⇒ L W W (5.11) →∞ r r r r (Sε ( , ˆ )) = ( , ˆ ). n βn βn n L U U →∞⇒ L W W l r ˆl ˆr l r ˆ l ˆ r Therefore, Sεn ( β , β , β , β ) n N is a tight family. Let (X , X , X , X ) { U n U n U n U n } ∈ be any weak limit point. Then (Xl, Xˆ l) and (Xr, Xˆ r) are distributed as ( l, ˆ l) and ( r, ˆ r) respectively. Therefore, (Xl, Xr) satisﬁes conditions (i)W andW (iii) of TheoremW W 1.5. By Proposition 5.2, (Xl, Xr) also satisﬁes condition (ii) of Theorem 1.5, and therefore, (Xl, Xr) has the same distribution as the standard left-right Brownian web ( l, r). Since l and r deter- W W W W mine their duals ˆ l and ˆ r almost surely, (Xl, Xr, Xˆ l, Xˆ r) has the same W W distribution as ( l, r, ˆ l, ˆ r). W W W W Proof of formula (1.22). Since the analogue of (1.22) obviously holds in the discrete setting, (1.22) is a consequence of the convergence in (5.10).

5.3. Convergence to the Brownian net. In this section we prove Theo- rem 1.1. It suﬃces to prove (1.8) for b =1 and b = 0. The general case b> 0 follows the same proof as for b = 1 if we set ( b) := (S1/b( )), which automatically gives the scaling relation (1.9). Thus,L N TheoremL 1.1Nis implied by the following stronger result.

Theorem 5.4 (Convergence to the associated Brownian net). Let β , ε n n → 0 with β /ε b 0, 1 . Then S ( , l , r , ˆl , ˆr ) are (Π)3 n n → ∈ { } εn Uβn Uβn Uβn Uβn Uβn K × (Π)ˆ 2-valued random variables. If b = 1, then K l r l r l r l r (5.12) (Sε ( β , , , ˆ , ˆ )) = ( , , , ˆ , ˆ ), n n βn βn βn βn n L U U U U U →∞⇒ L N W W W W where is the (standard) Brownian net and ( l, r, ˆ l, ˆ r) is its asso- ciatedN left-right Brownian web and its dual. If bW= 0W, thenW W l r l r (5.13) (Sε ( β , , , ˆ , ˆ )) = ( , , , ˆ , ˆ ), n n βn βn βn βn n L U U U U U →∞⇒ W W W W W where ( , ˆ ) is the Brownian web and its dual. W W Proof. We start with the case b = 1 and then say how our arguments can be adapted to cover also the case b = 0. Recall the modulus of continuity mK( ) of K (Π) from (4.3). Just as in the proof of Lemma 4.6, we see that · ∈ K

~~l r (5.14) mSεn ( β )(δ) mSε ( )(δ), U n ≤ n Uβn ∪Uβn THE BROWNIAN NET 39~~

l hence, the tightness of Sεn ( βn ) n N follows from the tightness of Sεn ( βn ) and S ( r ) (n N). Thus,{ U by going} ∈ to a subsequence, we may assumeU that εn Uβn ∈ the laws in (5.12) converge to a limit ( , l, r, ˆ l, ˆ r), where by The- L N ∗ W W W W orem 5.3, ( l, r, ˆ l, ˆ r) is the left-right Brownian web and its dual. We W W W W l r l r need to show that ∗ is the Brownian net associated with ( , , ˆ , ˆ ). Our strategy willN be to show that ,W whereW W Wand Nhop ⊂N ∗ ⊂Nwedge Nhop wedge are defined as in Section 4. It then follows from the equivalence of Nthe hopping and dual constructions of the Brownian net (Theorems 1.3 and 1.10) that ∗ = . Let l, Nr RN2 be deterministic countable dense sets. For each z l D D r⊂ 2 2 ∈ D (resp. z′ ), we fix a sequence zn Zeven (resp. zn′ Zeven) such that ∈ D ∈ ˆ(n) (n∈) Sεn (zn) z (resp. Sεn (zn′ ) z′), and we let lz (resp.r ˆz′ ) denote the path →l r → in Sε ( ˆ ) (resp. Sε ( ˆ )) starting in Sε (zn) (resp. Sε (z′ )). Let n Uβn n Uβn n n n (5.15) τ(ˆπ , πˆ ):= sup t< σˆ σˆ :π ˆ (t) =π ˆ (t) 1 2 { πˆ1 ∧ πˆ2 1 2 } denote the first meeting time of the two dual pathsπ ˆ1, πˆ2. Since, up to ˆ(n) (n) their first meeting time, lz andr ˆz′ are independent random walks, and since random walk paths joint with their first meeting time converge under diffusive scaling to Brownian motions joint with their first meeting time, we have l r ˆl ˆr ˆ(n) (n) (Sεn ( βn , βn , βn , βn , βn ), (τ(lz , rˆz′ ))z l, z′ r ) (5.16) L U U U U U ∈D ∈D l r l r ˆ = ( ∗, , , ˆ , ˆ , (τ(lz, rˆz′ )) l ′ r ). n z , z →∞⇒ L N W W W W ∈D ∈D By Skorohod’s representation theorem, we can construct a coupling such that the convergence in (5.16) is almost sure. Assuming such a coupling, we will show that hop ∗ wedge. N ⊂N ⊂N l l r r To show that hop ∗, it suffices to show that cros( ( ) ( )) N ⊂N l l r H r W D ∪W D is contained in ∗. Any π cros( ( ) ( )) is constructed by hopping at crossingN times between∈H left-mostW D ∪W and right-mostD skeletal paths l r l r π1,...,πm as in (1.12). By the a.s. convergence of Sε ( , ) to ( , ), n Uβn Uβn W W (n) l r (n) there exist πi Sεn ( βn βn ) such that πi πi (i = 1,...,m). By the structure of crossing∈ timesU ∪U [Proposition 3.6(a)],→ the crossing time between (n) (n) πi and πi+1 converges to the crossing time between πi and πi+1 for all i = 1,...,m 1. Therefore, the path π(n) that is constructed by hopping at − (n) (n) (n) crossing times between π1 ,...,πm converges to π. Since π Sεn ( βn ) ∈ l Ul by the nearest-neighbor nature of βn -paths, this proves that cros( ( ) r r ℵ H W D ∪ ( )) ∗. WToD show⊂N that , we need to show that a.s. no path π N ∗ ⊂Nwedge ∈N ∗ enters a wedge W (ˆr, ˆl) from outside. If π enters a wedge W (ˆr, ˆl) from ∈N ∗ 40 R. SUN AND J. M. SWART outside, then by Lemma 3.4(b), π must enter some skeletal wedge W (ˆrz′ , ˆlz), l r with z and z′ , from outside. By the a.s. convergence of Sεn ( βn ) to ∈ D (n∈) D (n) U ∗, there exist π Sεn ( βn ) such that π π. By the a.s. convergence N (n) ˆ(n) ∈ Û → ofr ˆz′ and lz tor ˆz′ and lz and the convergence of their first meeting time, for n large enough, π(n) must enter a discrete wedge from outside, which is impossible. This concludes the proof for b =1. The proof for b = 0 is similar. Note that if in the left-right SDE (1.11), one removes the drift terms dt, then solutions (L, R) are just coalescing Brownian motions. Using this± fact, it is not hard to generalize Propositions 5.1 and 5.2 in the sense that if βn/εn 0, then left-most and right-most paths converge to coalescing Brownian→ motions (with zero drift). Modifying Theorem 5.3 appropriately, we find that

~~l r l r (5.17) (Sε ( , , ˆ , ˆ )) = ( , , ˆ , ˆ ). n βn βn βn βn n L U U U U →∞⇒ W W W W~~

By going to a subsequence if necessary, we may assume that Sεn ( βn ) converges to some limit . The inclusion is now trivial,U while the W∗ W⊂W∗ other inclusion can be obtained by showing that no path in ∗ enters a wedge of ˆ from outside, applying Theorem 1.9. W W 6. Density calculations. In this section we carry out two density calculations for the Brownian net , based on the hopping and dual characterizations (Theorem 1.3 and TheoremN 1.10), which have been shown in Section 4 to be equivalent. In Section 6.1 we calculate the density of the set of points on R t that are on the graph of some path in starting at time 0, that is, we× prove { } Proposition 1.12. In Section 6.2 we estimateN the density of the set of times that are the first meeting times between l l(0, 0) and some ∈W path in hop starting to the left of 0 at time 0. Our calculations show that both setsN are a.s. locally finite. The second density calculation gives infor- mation on the configuration of meshes on the left of a general left-most path l, which will be used in Section 7 to prove that paths in mesh cannot enter the area to the left of l. From this, we then readily obtainN Theorem 1.7, as well as Propositions 1.4, 1.8 and 1.13.

6.1. The density of the branching-coalescing point set. In this section we prove Proposition 1.12. Let be the Brownian net, deﬁned by the hopping or dual characterization (TheoremN 1.3 and Theorem 1.10). Set (6.1) ξ := π(t) : π ,σ = 0 (t> 0). t { ∈N π } R 0 Note that ξt = ξt ×{ }, the branching-coalescing point set (deﬁned in Sec- tion 1.9) started at time zero from R. The exact computation of the density of ξt is based on the following two lemmas. THE BROWNIAN NET 41

Lemma 6.1 (Avoidance of intervals). Almost surely, for each s,t,a,b R with ss. } Proof. Ifr, ˆ ˆl with the described properties exist, then by the dual characterization of the Brownian net (Theorem 1.10), no path in starting at time s can pass through (a, b) t . Conversely, if there existsN no π (R s ) such that π(t) (a, b×), { then} for each ε> 0 and for each ∈N × { } ∈ rˆ ˆ r(a + ε, t) and ˆl ˆ l(b ε, t), we must have τ := sup us. For if τε s, then by the steering argument used in the proof of Lemma} 4.7 (see Figure≤ 7), for each δ > 0 we can construct a path in ( l r) starting at time s + δ in (ˆr (s + δ), ˆl (s + δ)) and passing Hcros W ∪W ε ε ˆ ˆ ′ through [a + ε, b ε] t. Lettingr, ˆ l denote any limits of pathsr ˆεn , lεn along − × ˆ sequences εn, εn′ 0, we see that τ := sup us. In fact, by Lemma 3.4(a), we↓ must have τ>s. { }

Lemma 6.2 (Hitting probability of a pair of left-right SDE). Let Ls and Rs be the solution of (1.11) with initial condition L0 = 0 and R0 = ε for some ε> 0. Let T = inf s 0 : L = R . Then ε { ≥ s s} ε 2ε ε (6.2) 1 Ψε(t) := P[Tε

~~−y2/2 where Φ(x)= x e dy. √2π R−∞~~

~~Proof. Let Yt = Bt + √2t with Y0 = 0, and let Mt = inf0 s t Ys. − ≤ ≤ Clearly, Rt Lt ε is equally distributed with √2Yt before it reaches level ε. Therefore,− P−[T~~

~~P[Mt′ dx,Bt′ dy] (6.4) ∈ ∈ 1 2(2x + y) (2x+y)2/(2t) = e− dx dy x 0,y x. √2πt · t · ≥ ≥− 42 R. SUN AND J. M. SWART~~

By Girsanov’s formula, the measure for (Ys)0 s t is absolute continuous ≤ ≤ √2B′ t with respect to the measure for (Bs′ )0 s t with density e t− . Therefore, ≤ ≤ ε P Mt ≥ √2 (6.5) ∞ ∞ √2y t 1 2(2x + y) (2x+y)2/(2t) = e − e− dy dx. Zε/√2 Z x √2πt · t · − ε/√2 Split the integral into two regions: I = ∞ε/√2 dy ε/∞√2 dx;andII= − dy R− R R−∞ × ∞y dx. Then we have R− √2y t ∞ e ∞ 2(2x + y) (2x+y)2/(2t) I= e− dy e− dx Z ε/√2 √2πt Zε/√2 t · − t ∞ 1 √2y (y+√2ε)2/(2t) (6.6) = e− e − dy Z ε/√2 √2πt − 2ε ∞ 1 (y+√2ε √2t)2/(2t) 2ε ε = e− e− − dy = e− Φ √2t . Z ε/√2 √2πt − √2t − Similarly, ε/√2 t − 1 √2y y2/(2t) II= e− e − dy Z √2πt (6.7) −∞ ε/√2 − 1 (y √2t)2/(2t) ε = e− − dy =Φ √2t . Z √2πt − − √2t −∞ This concludes the proof.

Proof of Proposition 1.12. It follows from Lemmas 6.1 and 6.2, and the continuity of ε Ψ (t) that 7→ ε (6.8) P[ξt (a, b) = ∅]= P[ξt [a, b] = ∅]=Ψb a(t) (t> 0) ∩ 6 ∩ 6 − for deterministic a < b. Since the law of ξt is clearly translation invariant in space, to prove (1.28), without loss of generality, we may assume [a, b]= [0, 1]. Let = i : n N, 0 i 2n denote the dyadic rationals. By (6.8), R { 2n ∈ ≤ ≤ } P[x ξt] = 0 for each deterministic x R. Since is countable, almost surely ξ ∈ = ∅. Therefore, ∈ R t ∩ R n i 1 i (6.9) ξt [0, 1] = lim 1 i 2 : ξt − , = ∅ a.s. n n n | ∩ | →∞ ≤ ≤ ∩ 2 2 6

~~By monotone convergence and translation invariance,~~

~~n 1 ∂ (6.10) E[ ξt [0, 1] ] = lim 2 P ξt 0, = ∅ = Ψε(t) , n n ε=0 | ∩ | →∞ ∩ 2 6 ∂ε | which yields equation (1.28). THE BROWNIAN NET 43~~

6.2. The density on the left of a left-most path. Let be the Brownian net, defined by the hopping or dual characterization (TheoreN m 1.3 and The- orem 1.10), and let ( l, r, ˆ l, ˆ r) be its associated left-right Brownian web and its dual. ForW eachWl W lW, let ∈W C(l) := t>σl : π s.t. σπ = σl, (6.11) { ∃ ∈N π(t)= l(t),π(s) < l(s) s [σ ,t) ∀ ∈ l } be the set of times when some path in , started at the same time as l and to the left of l, first meets l. We willN prove that, almost surely, C(l) l is a locally finite subset of (σl, ) for each l . By Lemma 3.4(b), it suffices to verify this property for∞l l with deterministic∈W starting points, in particular, l started at (0, 0), which∈W is implied by the following lemma.

~~Proposition 6.3 (Density on the left of a left-most path). Let l be the a.s. unique path in l starting at the origin. Then, for each 0~~

Proof. By a similar argument as in the proof of Proposition 1.12, it suﬃces to show that 1 (6.13) lim sup P[C(l) [t,t + ε] = ∅] 2ψ(t)2. ε 0 ε ∩ 6 ≤ → For t> 0, letr ˆ[t] be the left-most [viewed with respect to the graph of r ˆ r ˆ r ˆ ˆ l ( , )] path in (l(t),t) and let l[t] be the right-most path in (l(t),t) thatW liesW on the leftW of l. Note that, by Lemma 3.3(b), for each deterministicW t> 0, ˆ l(l(t),t) almost surely contains two paths, one lying on each side of l. SimilarW arguments as in the proof of Lemma 6.1 show that P[C(l) [t,t + ε] = ∅] (6.14) ∩ 6 = P[ˆr (s) < ˆl (s) s (0,t)]. [t+ε] [t] ∀ ∈ Set L := l(t + ε) l(t s), s [ ε, t], s − − ∈ − (6.15) Lˆ := l(t + ε) ˆl (t s), s [0,t], s − [t] − ∈ Rˆ := l(t + ε) rˆ (t s), s [ ε, t]. s − [t+ε] − ∈ − 44 R. SUN AND J. M. SWART

It has been shown in [15] (see also [9]) that paths in and ˆ interact by W W Skorohod reflection. Similar arguments show that if a pathr ˆ ˆ r is started on the left of a path l l, thenr ˆ is Skorohod reflected off l∈. Therefore,W on ∈W the time interval [ ε, 0], the process (Ls, Rˆs) satisfies L Rˆ and solves the SDE − ≤ l dLs = dBs ds, (6.16) − ˆ ˆr dRs = dBs + ds + d∆s′ , l ˆr where B and B are independent Brownian motions, and ∆s′ is a reflection term that increases only when L = Rˆ . Set σ := inf s> 0 : Lˆ = Rˆ t. s s { s s}∧ Then on the time interval [0,σ], the process (Ls, Lˆs, Rˆs) satisfies L Lˆ Rˆ and solves the SDE ≤ ≤ dL = dBl ds, s s − ˆ (6.17) dLˆ = dBl ds + d∆ , s s − s ˆ ˆr dRs = dBs + ds, l ˆl ˆr where B ,B ,B are independent Brownian motions and ∆s increases only when Ls = Lˆs. By Lemma 6.4 below,

~~2 (6.18) P[Lˆs < Rˆs s (0,t)] P[Rˆ0 L0 dη]Ψη(t) . ∀ ∈ ≤ Z − ∈~~

Set Xs := Rˆs ε Ls ε (s [0, ε]). Then X is a Brownian motion with diﬀu- sion constant− 2− and− drift 2,∈ Skorohod reﬂected at 0, which has the generator 2 ∂ + 2 ∂ with boundary condition ∂ f(η) = 0. Therefore, ∂η2 ∂η ∂η |η=0 1 2 lim ε− P[Rˆ0 L0 dη]Ψη(t) ε 0 Z → − ∈ 1 2 (6.19) = lim ε− E[ΨX (t) ] ε 0 ε → 2 ∂ ∂ 2 2 = 2 + 2 (Ψη(t) ) = 2ψ(t) , ∂η ∂η η=0

~~2 where we have used that, for ﬁxed t> 0, η Ψη(t) is a bounded twice continuously diﬀerentiable function satisfying7→ our boundary condition.~~

Lemma 6.4 (Hitting estimate). Let (L, L,ˆ Rˆ) be a solution to the SDE (6.17) started at (L0, Lˆ0, Rˆ0) = (0, 0, η). Then (6.20) P[Lˆ < Rˆ s (0,t)] Ψ (t)2, s s ∀ ∈ ≤ η where Ψη(t) is deﬁned in (6.2). THE BROWNIAN NET 45

~~Proof. We introduce new coordinates:~~

Vt := Lˆt Lt, (6.21) − W := Rˆ L . t t − t The process (V,W ) lives in the space (v, w) R2 : 0 v w up to the time τ := inf t> 0 : V = W and solves the{ SDE ∈ ≤ ≤ } { t t} ˆl l dVt := dBs dBs + d∆s, (6.22) − dW := dBˆr dBl + 2 ds, t s − s where ∆s is a reﬂection term, increasing only when Vs = 0. Changing coordinates once more, we set

Xt := Wt Vt, (6.23) − Yt := Wt + Vt. Then (X,Y ) takes values in (x,y) R2 : 0 x y up to the time τ := inf t> 0 : X = 0 and solves the{ SDE∈ ≤ ≤ } { t } ˆr ˆl dXs := dBs dBs + 2 ds d∆s, (6.24) − − ˆ dY := dBˆr + dBl 2 dBl + 2 ds + d∆ , s s s − s s where ∆s increases only when Xs = Ys. Our strategy will be to compare 1 2 1 2 (X,Y ) with a process (X′,Y ′) of the form X′ = U U and Y ′ = U U , ∧ 2 ∨ where U 1, U 2 are independent processes with generator ∂ + 2 ∂ . We will ∂u2 ∂u show that X hits zero before X . Note that if U i = u, then P[U i > 0 s ′ 0 s ∀ ∈ [0,t]]=Ψu(t), which is deﬁned in (6.2). Therefore, ∂ ∂2 ∂ (6.25) Ψu(t)= + 2 Ψu(t). ∂t ∂u2 ∂u 1 Moreover, if (X′,Y ′) is started in (x,y), then P[Xs′ > 0 s [0,t]] = P[Us > 0 s [0,t]]P[U 2 > 0 s [0,t]]=Ψ (t)Ψ (t). With this∀ in∈ mind, we set ∀ ∈ s ∀ ∈ x y (6.26) F (t,x,y):=Ψx(t)Ψy(t). Let G be the operator ∂2 ∂ ∂2 ∂ (6.27) G := + 2 + 3 + 2 . ∂x2 ∂x ∂y2 ∂y By Itˆo’s formula,

dF (t s, Xs τ ,Ys τ ) − ∧ ∧ ∂ (6.28) = + 1 s<τ G F (t s, Xs τ ,Ys τ ) ds −∂t { } − ∧ ∧ ∂ ∂ + 1 s<τ F (t s, Xs τ ,Ys τ ) d∆s { } ∂y − ∂x − ∧ ∧ 46 R. SUN AND J. M. SWART

~~Fig. 8. Meshes stack up on the left of a left-most path l ∈ Wl.~~

∂ plus martingale terms. It follows from the deﬁnition of Ψu(t) that ∂t Ψu(t) 2 ≤ 0 and ∂ Ψ (t) 0, and therefore, by (6.25), ∂ Ψ (t) 0. As a result, using ∂u u ≥ ∂u2 u ≤ (6.25) and (6.26), we see that ( ∂ ∂ )F (t,x,y) =0 and ∂y − ∂x |x=y ∂ ∂2 (6.29) + G F (t,x,y) = 2 (Ψx(t)Ψy(t)) 0. −∂t ∂y2 ≤

~~Inserting this into (6.28), we ﬁnd that (F (t s, Xs τ ,Ys τ ))s [0,t τ] is a local − ∧ ∧ ∈ ∧ supermartingale, which implies that~~

2 (6.30) P[τ>t]= E[F (t t τ, Xt τ ,Yt τ )] F (t, X0,Y0)=Ψη(t) . − ∧ ∧ ∧ ≤ As a corollary to Proposition 6.3, we obtain the following lemma, which describes the conﬁguration of meshes on the left of a left-most path. (See Figure 8.) THE BROWNIAN NET 47

Lemma 6.5 (Meshes on the left of a left-most path). Almost surely, l the set C(l) in (6.11) is a locally ﬁnite subset of (σl, ) for each l . For each consecutive pair of times t, u C(l) [i.e., t < u∞and C(l) (t,∈W u)= ∅], there exists a mesh M(r , l ) with∈ bottom time s (σ ,t) and∩ top point ′ ′ ∈ l (l(u), u), such that l′ < l on [s,t) and l′ = l on [t, u]. If C(l) has a minimal element t, then there exists a mesh M(r′, l) with right boundary l, bottom point (l(σl),σl) and top point (l(t),t).

ε Proof. For any path π and ε> 0, define a trunctated path by πh i := (π(t),t) : t [σπ + ε, ] . Let l(0,0) be the a.s. unique left-most path start- { ∈ ∞ } ε ing in the origin. The proof of Proposition 6.3 applies to l(0h ,i0) as well; ε in particular, C(l(0h ,i0)) has the same density as C(l(0,0)) for each ε> 0. ε By Lemma 3.4(b), if follows that a.s., C(lh i) is a locally finite subset of (σ + ε, ) for each l l and ε> 0. Since C(l ε ) (σ + δ, ) decreases to l ∞ ∈W h i ∩ l ∞ C(l) (σl + δ, ) as ε 0, for each fixed δ> 0, it follows that a.s., C(l) is a ∩ ∞ ↓ l locally finite subset of (σl, ) for each l . l ∞ ∈W For any l (see Figure 8), consider t, u C(l) σl such that t < u and C(l) (t,∈W u)= ∅, that is, either t, u is a consecutive∈ ∪ pair{ } of times in C(l), ∩ or t = σl and u is the minimal element of C(l). By an argument similar to the proof of Lemma 6.1, there existr ˆ ˆ r(l(u), u) and ˆl ˆ l(l(t),t) such [u] ∈ W [t] ∈ W thatr ˆ l on [σ , u], lˆ l on [σ ,t], and τ := sup s t :r ˆ (s)= lˆ (s) [u] ≤ l [t] ≤ l t,u { ≤ [u] [t] } satisfies τt,u >σl if τt,u W t,u t,u ′ { τt,u : r[u](s)= l(s) and t′ := inf s>τt,u : l[t](s)= l(s) . By Proposition 3.2(c) and (e), r rˆ} on [τ , u],{ and therefore, by Propositions} 3.6(a), (b), [u] ≤ [u] t,u u u. Likewise, since l ˆl on [τ ,t], we have t t. Now r and l ′ ≥ [t] ≥ [t] t,u ′ ≤ [u] [t] are the left and right boundary of a mesh M(r[u], l[t]) with bottom time τt,u and top point (l(u′), u′), such that l[t] < l on (τt,u,t′) and l[t] = l on [t′, u′]. Since hop mesh (Lemma 4.4) and both t (if tσl) and u are times when a pathN in ⊂Nstarting at time σ first meets l from the left, it follows that Nhop l t′ = t and u′ = u. (If t = σl, then obviously τt,u = σl = t = t′.) To complete the proof, we must show that τt,u σl. This follows from Lemma 6.6 below.

~~Lemma 6.6 (Top and bottom points of meshes). Almost surely, no bottom point of one mesh is the top point of another mesh.~~

Proof. Assume that z R2 is the bottom point of a mesh M(r, l) and ∈ the top point of another mesh M(r′, l′). By Propositions 3.2(c) and 3.6(d), 48 R. SUN AND J. M. SWART anyr ˆ ˆ r starting in M(r, l) must pass through z (and likewise for lˆ l). ∈ W ∈W Therefore, l′, r′ andr ˆ are three paths entering z disjointly. This can be ruled out just as in the proof of Theorem 3.11 in [9], where it is argued that a.s. there is no point z R2 where two forward and one backward path in ( , ˆ ) enter z disjointly. ∈ W W

7. Characterization with meshes. In this section, we prove Theorem 1.7, as well as Propositions 1.4, 1.8 and 1.13. We ﬁx a left-right Brownian web l r l r and its dual ( , , ˆ , ˆ ) and deﬁne hop, wedge and mesh as in Sec- tion 4. The keyW technicalW W W result is the followingN N lemma, whichN states that Proposition 1.8 holds for . Nmesh Lemma 7.1 (Containment by left-most and right-most paths). Almost surely, there exist no π and l l such that l(s) π(s) and π(t) < ∈Nmesh ∈W ≤ l(t) for some σπ σl

Proof. Without loss of generality, we may assume that σl >σπ; otherwise, consider a left-most path starting at any time in (σπ,s) that is the continuation of l. By Lemma 6.5, there exists a locally ﬁnite collection of meshes on the left of l, with bottom times in [σ , ), that block the way of l ∞ any path in mesh trying to enter the area to the left of l. (See Figure 8.) N

Proof of Theorem 1.7 and Proposition 1.8. We start by proving that . Since, by Lemma 4.3, paths in do not enter Nmesh ⊂Nwedge Nmesh wedges through their bottom points, it suﬃces to show that paths in mesh do not cross dual left-most and right-most paths in the wrong direction.N By Lemma 3.5, it suﬃces to show that paths in mesh do not cross forward left-most and right-most paths in the wrong direction.N This follows from Lemma 7.1. Since it has already been proved in Lemmas 4.4, 4.5 and 4.7 that Nmesh ⊃ hop = wedge, it follows that all these sets are a.s. equal. This proves The- oremN 1.7N. Lemma 7.1 now translates into Proposition 1.8.

Proof of Proposition 1.4. By Theorem 1.7 and Proposition 1.8, the Brownian net associated with a left-right Brownian web ( l, r) consists exactly of thoseN paths in Π that do not enter meshes. It isW easyWto see that this set is closed under hopping.

~~Proof of Proposition 1.13. Let ( l, r) be the left-right Brownian web associated with . We have to showW that,W for each t [ , ] and π N ∈ −∞ ∞ ∈ THE BROWNIAN NET 49~~

Πt such that π ( Πt), we have π . By Theorem 1.7, each mesh of l r ⊂ ∪ N∩ ∈N ( , ) with bottom time in (t, ) has empty intersection with ( Πt), andW therefore,W π does not enter∞ any such mesh. Again by Theorem∪ N∩1.7, it follows that π . ∈N 8. The branching-coalescing point set. In this section we prove Theo- rem 1.11. We start with two preparatory lemmas.

Lemma 8.1 (Hopping paths starting from a closed set). Let be the Brownian net. Let K R2 be closed. Let l, r R2 be deterministicN countable dense sets such that,⊂ moreover, l DK Dis dense⊂ in K. Then D ∩ (8.1) (K) Π(K) ( l( l) r( r)). N ⊂ ∩Hcros W D ∪W D Proof. By the dual characterization of the Brownian net (Theorem 1.10), it suffices to show that any path π starting at some point z = (x,t) K that does not enter wedges from outside can be approximated by paths∈ in ( l( l) r( r)), also starting in K. By the compactness of , it suf- Hcros W D ∪W D N fices to show that, for each t b, such that V K = ∅ but∈ Wz D∂V ∈,W whereD we define rˆ ˆl (a,b) ∩ ∈ (a,b) (8.3) ∂V := (x,s) R2 : a

Z :=r ˆ(b) ˆl(b), (8.4) − R := (ˆr(s) rˆ(b))s [a,b], − ∈ ˆ ˆ L := (l(s) l(b))s [a,b]. − ∈ We claim that, for any y R and continuous functions ωr,ωl : [a, b] R, the conditional probability ∈ → (8.5) P[V K = ∅ and ∂V K = ∅ Y = y, R = ω , L = ω ] (a,b) ∩ (a,b) ∩ 6 | r l is zero. Indeed, for given Y, R and L, there can be at most one value of Z for which the event V K = ∅ and ∂V K = ∅ occurs. Since conditioned (a,b) ∩ (a,b) ∩ 6 on S < b, which is necessary for ∂V(a,b) = ∅, the distribution of the random variable Z is absolute continuous with6 respect to Lebesgue measure, the conditional probability in (8.5) is zero. Integrating over the distributions of Y, R and L, we arrive at our result.

Lemma 8.2 (Almost sure continuity). Let be the Brownian net, and 2 N let Kn, K (Rc ) be deterministic sets satisfying Kn K. Assume that ( , ) / ∈K K. Then (K ) (K) a.s. → ∗ −∞ ∈ N n →N Proof. Using the compactness of , by going to a subsequence if nec- N essary, we may assume that (Kn) for some compact subset . Obviously, all paths in haveN starting→A points in K, so (K).A⊂N Write K := K R2 and K :=AK K . Since (z) is trivial for z A⊂NK , it is easy to ′ ∩ ′′ \ ′ N ∈ ′′ see that (K′′). We are left with the task to show (K′). Choose a deterministicA⊃N countable dense set R2 such that, moreover,A⊃N K is D⊂ D∩ ′ dense in K′. For each z K′, choose zn Kn such that zn z. Then ∈D ∩ r ∈ → lzn lz. If lz crosses a path r , then for n large enough, lzn also crosses r. Therefore,→ it is not hard to∈W see that l r (8.6) Π(K′) ( ( ) ( )). A⊃ ∩Hcros W D ∪W D By Lemma 8.1, it follows that (K ). A⊃N ′ Remark. If ( , ) K, then by Proposition 1.15(iii), the conclusion of Lemma 8.2 still∗ holds,−∞ ∈ provided there exist (x ,t ) K such that t n n ∈ n n → and lim supn xn / tn < 1. Here some of the (xn,tn) maybe( , ), with−∞ the convention→∞ that| | | |/ := 0. As the proof of Proposition 1.15∗ −∞(iii) shows, this condition cannot| ∗ | be|∞| relaxed very much.

Proof of Theorem 1.11. The continuity of sample paths of (ξt)t 0 is a direct consequence of the deﬁnition of ξ and the fact that is≥ a t N (Π)-valued random variable. The fact that ξt is a.s. locally ﬁnite in R for Kdeterministic t>s follows from Proposition 1.12. THE BROWNIAN NET 51

For t 0, the transition probability kernel Pt on (R) associated with ξ is given≥ by K K s (8.7) P (K, ) := P[ξ ×{ } ], K (R). t · s+t ∈ · ∈ K Note that the right-hand side of (8.7) does not depend on s R by the ∈ translation invariance of the Brownian net. By Lemma 8.2, if Kn K and t t, then → n → Kn tn K t (8.8) Pt (Kn, )= P[ξ ×{− } ] = P[ξ ×{− } ]= Pt(K, ), n 0 n 0 · ∈ · →∞⇒ ∈ · · proving the Feller property of (Pt)t 0. We still have to show that (Pt)t 0 is a Markov transition probability kernel.≥ This is not completely obvious,≥ but it follows, provided we show that, for any s

t0 K s (8.10) P[ (K s ) ∞ (K s ) ]= P[ ′(ξ ×{ } t ) ], N × { } |t0 ∈·|N × { } |s N t0 × { 0} ∈ · l r where ′ is an independent copy of . Let ( , ) be the left-right Brow- nian webN associated with . By theN propertiesW ofW left-right coalescing Brow- l r t0 N l r nian motions, ( , ) and ( , ) t∞0 are independent, and therefore, W W |−∞ W W | t0 by the hopping construction, it follows that and t∞0 are independent. K s N |−∞ N | ×{ } t0 R In particular, ξt0 and (K s ) s are independent of ( t0 ). To show (8.10), it thereforeN suﬃces× { to} show| that N × { }

K s (8.11) (K s ) ∞ = (ξ ×{ } t ) a.s. N × { } |t0 N t0 × { 0} The inclusion is trivial. To prove the converse, we need to show that any K⊂ s ×{ } path π (ξt0 t0 ) is the continuation of a path in (K s ); this follows∈N from Lemma×8.3 { }below. N × { } To prove (1.27), note that K ′(R) if and only if supK< , or sup(K R)= and K, and likewise∈ K at . Thus, by symmetry,∞ it suﬃces to∩ show∞ that, almost∞∈ surely: −∞ (i) sup(ξ ) < implies sup(ξ ) < t s, s ∞ t ∞ ∀ ≥ (8.12) (ii) sup(ξ R)= implies sup(ξ R)= t s, s ∩ ∞ t ∩ ∞ ∀ ≥ (iii) ξ implies ξ t s. ∞∈ s ∞∈ t ∀ ≥ Formula (i) follows from the fact that (sup(ξt))t s is the right-most path in (ξ s ), which is a Brownian motion with≥ drift +1. Formula (ii) is N s × { } 52 R. SUN AND J. M. SWART easily proved by considering the right-most paths starting at a sequence of points in ξs R tending to ( ,s). Last, formula (iii) follows from the fact that ( ,s∩) contains the trivial∞ path π(t) := (t s). N ∞ ∞ ≥ In the proof of Theorem 1.11 we have used the following lemma, which is of some interest on its own.

Lemma 8.3 (Hopping at deterministic times). Let be the Brownian N net and t R. Then almost surely, for each π,π such that σ σ ′ t ∈ ′ ∈N π ∨ π ≤ and π(t)= π′(t), the path π′′ deﬁned by

~~(8.13) π′′ := (π(s),s) : s [σ ,t] (π′(s),s) : s [t, ] { ∈ π } ∪ { ∈ ∞ } satisﬁes π . ′′ ∈N~~

~~Proof. If σπ = t, there is nothing to prove, so without loss of generality we may assume that σ s for some deterministic s~~

We end this section with a proposition that will be used in the proof of Lemma 9.2, and that is of interest in its own right. Note that the statement below implies that, provided that the initial states converge, systems of branching-coalescing random walks, diﬀusively rescaled, converge in an appropriate sense to the branching-coalescing point set.

Proposition 8.4 (Convergence of paths started from subsets). Let βn, 2 2 εn 0 with βn/εn 1. Let Kn Zeven, K (Rc ) satisfy Sεn (Kn) K, where→ denotes convergence→ in⊂ (R2). Assume∈ K ( , ) / K. Then → → K c ∗ −∞ ∈ (8.14) (Sε ( β (Kn))) = ( (K)). n n n L U →∞⇒ L N Proof. By going to a subsequence if necessary, we may assume that

(Sεn ( βn , βn (Kn))) ( , ) for some compact subset (K). Write KL := KU RU2 and K ⇒L:= KNKA. Since (z) is trivial for zA⊂NK , it is easy ′ ∩ ′′ \ ′ N ∈ ′′ to see that (K′′). We are left with the task to show (K′). Choose a deterministicA⊃N countable dense set R2 such that,A⊃N moreover, D⊂ c K′ is dense in K′. By the same arguments as those used in the proof of TheoremD∩ 5.4 to show that , we have Nhop ⊂N ∗ l r (8.15) Π(K′) ( ( ) ( )) . ∩Hcros W D ∪W D ⊂A By Lemma 8.1, it follows that (K ) . N ′ ⊂A THE BROWNIAN NET 53

~~9. The backbone. In Sections 9.1 and 9.2 we prove Propositions 1.14 and 1.15, respectively.~~

9.1. The backbone of branching-coalescing random walks. Let β be an A ℵ arrow configuration. Recall the definition of ηt from (1.2). Let Zeven := 2Z and Zodd := 2Z + 1. For any s Z and A Zeven or A Zodd depending on whether s is even or odd, setting∈ ⊂ ⊂ A s (9.1) η := η ×{ } (t Z, t s) t t ∈ ≥ defines a Markov chain (ηt)t s taking values, in turn, in the spaces of subsets ≥ of Zeven and Zodd, started at time s in A. We call η = (ηt)t s a system of branching-coalescing random walks. We call a probability law≥µ on the space of subsets of Z an invariant law for η if (η )= µ implies (η )= µ, even L 0 L 2 and a homogeneous invariant law if µ is translation invariant and (η0)= µ implies (η +1)= µ. Note that we shift η by one unit in space toL stay on L 1 1 Zeven. ( , ) It is easy to see that (η0∗ −∞ ) defines a homogeneous invariant law for η. Our strategy for provingL Proposition 1.14 will be as follows. First we prove 4β that the Bernoulli measure µρ with intensity ρ = (1+β)2 is a homogeneous invariant law for η, and that µρ is reversible in a sense that includes informa- tion about the arrow configuration β. Next, we prove Proposition 1.14(iii). From this, we derive that there existsℵ only one nontrivial invariant law for ( , ) η, hence, (η ∗ −∞ )= µ , which proves part (i). Last, part (ii) follows from L 0 ρ the reversibility of µρ. We first need to add additional structure to the branching-coalescing random walks that also keeps track of the arrows in that are used by the ℵβ walks. To this aim, if (ηt)t=s,s+1,... is defined as in (9.1) with respect to an arrow configuration , then we define ℵβ ηt+1/2 := x,x′ : x ηt, ((x,t), (x′,t + 1)) β (9.2) {{ } ∈ ∈ℵ } (t [s, ) Z). ∈ ∞ ∩ Note that ηt+1/2 keeps track of which arrows in β are used by the branching- coalescing random walks between the times t andℵ t +1. It is not hard to see that (ηs+k/2)k N0 is a Markov chain. ∈

Lemma 9.1 (Product invariant law). The Bernoulli product measure µρ Z 4β on even with intensity ρ = (1+β)2 is a reversible homogeneous invariant law for the Markov chain (ηs+k/2)k N0 deﬁned above, in the sense that, if (η )= µ , then for all even t 0, ∈ L 0 ρ ≥ (9.3) (η0, η1/2,...,ηt 1/2, ηt)= (ηt, ηt 1/2,...,η1/2, η0). L − L − The same holds for all odd t 1, provided that the conﬁgurations on the right-hand side of (9.3) are shifted≥ in space by one unit. 54 R. SUN AND J. M. SWART

~~Proof. It suﬃces to prove the statement for t = 1, that is, we need to prove that if (η )= µ , then L 0 ρ (9.4) (η , η , η )= (η + 1, η + 1, η + 1). L 0 1/2 1 L 1 1/2 0~~

Indeed, since (ηt/2)t N0 is Markov, (η0,...,ηs 1/2) and (ηs+1/2,...,ηt) are ∈ − conditionally independent given ηs for all s [1,t] Z. The identity (9.3) for general even t 0, and its analogue for odd∈ t 0,∩ then follow easily from (9.4) by induction.≥ ≥ Note that η1/2 determines η0 and η1 a.s. Indeed,

η0 = x Zeven : x′ Zodd s.t. x,x′ η1/2 , (9.5) { ∈ ∃ ∈ { }∈ } η = x′ Z : x Z s.t. x,x′ η . 1 { ∈ odd ∃ ∈ even { }∈ 1/2} Therefore, (9.4) follows, provided we show that (9.6) (η )= (η + 1). L 1/2 L 1/2 We will prove (9.6) by showing that if (η )= µ with ρ = 4β , then L 0 ρ (1+β)2 (η ) is a Bernoulli product measure on the set of all nearest neighbor L 1/2 pairs of integers. Note that, for x Zeven, the event x,x 1 η1/2 means that the arrow from (x, 0) to (x ∈1, 1) is used by a{ random± }∈ walker. Since ± (η0) is a product measure, arrows going out of diﬀerent x,x′ Zeven are obviouslyL independent. Thus, it suﬃces to show that, for x ∈Z , the ∈ even events x,x 1 η1/2 and x,x + 1 η1/2 are independent. Now, for x Z { , − }∈ { }∈ ∈ even (9.7) P[ x,x 1 η and x,x + 1 η ]= ρβ, { − }∈ 1/2 { }∈ 1/2 while 1 β (9.8) P[ x,x 1 η ]= P[ x,x + 1 η ]= ρ − + β . { − }∈ 1/2 { }∈ 1/2 2 1+β 2 Thus, we obtain the desired independence, provided that ρβ = (ρ 2 ) , 4β which has ρ = (1+β)2 as its unique nonzero solution.

Proof of Proposition 1.14(iii). By going to a subsequence if necessary, we may assume that β(xn,tn) for some β. Since all paths in start at ( , ), U( , ).→A To prove the otherA⊂U inclusion, it suffices A ∗ −∞ A⊂Uβ ∗ −∞ to show that, for each π β( , ) and t Zeven, for n sufficiently large, we can find π (x ,t∈U) such∗ −∞ that π = π∈on [t, ) Z. By hopping, it ′ ∈Uβ n n ′ ∞ ∩ suffices to show that, for each even N> 0 and t Zeven, there exists n0 such that, for all n n , ∈ ≥ 0 [ N, N] π(t) : π β(xn,tn) (9.9) − ∩ { ∈U } [ N, N] π(t) : π ( , ) . ⊃ − ∩ { ∈Uβ ∗ −∞ } THE BROWNIAN NET 55

ˆ ˆ Let l := l( N 1,t) andr ˆ :=r ˆ(N+1,t) be the dual left-most and right-most paths − − in ˆβ started from ( N 1,t) and (N + 1,t), respectively. By the strong lawℵ of large numbers,− almost− surely,

~~lˆ(s) rˆ(s) (9.10) lim = β and lim = β. s s s s − →−∞ − →−∞ − Therefore, by our assumptions on (xn,tn), we haver ˆ(tn)~~

~~Proof of Proposition 1.14(i) and (ii). It is not hard to see that (0, ) (η0 −∞ ) is the maximal invariant law of η with respect to the usual stochasticL order. Proposition 1.14(iii) implies that~~

(0,0) (0, 2n) (0, ) (9.11) P[η ]= P[η − ] = P[η −∞ ]. 2n 0 n 0 ∈ · ∈ · →∞⇒ ∈ · (0, ) Using monotonicity, it is easy to see from (9.11) that (η −∞ ) is the limit L 0 law of η2n as n for any nonempty initial state η0. In particular, this →∞(0, ) implies that (η0 −∞ ) is the unique invariant law of η that is concentrated L (0, ) on nonempty states, and therefore, by Lemma 9.1, (η −∞ )= µ . L 0 ρ Part (ii) is now a consequence of the reversibility of µρ as formulated in Lemma 9.1.

~~9.2. The backbone of the branching-coalescing point set. In this section we prove Proposition 1.15.~~

~~Proof of Proposition 1.15(iii). This can be proved by the same arguments as in the proof of Proposition 1.14(iii), except we now need Propo- sition 1.4 to hop between paths in the net.~~

~~We will derive parts (i) and (ii) of Proposition 1.15 from their discrete counterparts, by means of the following lemma.~~

~~Lemma 9.2 (Convergence of the backbone). If βn, εn 0 with βn/εn 1, then → →~~

~~(9.12) (Sε ( β ( , ))) = ( ( , )). n n n L U ∗ −∞ →∞⇒ L N ∗ −∞ 56 R. SUN AND J. M. SWART~~

~~Proof. By going to a subsequence if necessary, using Theorem 1.1, we may assume that~~

(9.13) (Sε ( β , β ( , ))) = ( , ), n n n n L U U ∗ −∞ →∞⇒ L N A where is the Brownian net and . Since all paths in start in ( , N), obviously ( , ). ToA⊂N prove the other inclusion,A it suﬃces to∗ show−∞ that (usingA⊂N notation∗ −∞ introduced in the proof of Theorem 1.11)

(9.14) ( , ) ∞ = ∞, N ∗ −∞ |t A|t for all t R. As a ﬁrst step, we will show that ∈ (9.15) π(t) : π ( , ) = π(t) : π . { ∈N ∗ −∞ } { ∈ A} The inclusion is clear. Taking the limit in Proposition 1.14(i), we see that, for all t ⊃R, π(t) : π is a Poisson point set with intensity 2. On the other hand,∈ taking{ the∈ limit A} in Proposition 1.12, we see that π(t) : π ( , ) is a translation invariant point set, also with intensity{ 2. Hence,∈ N(9.15∗ )−∞ follows.} Since the inclusion in (9.14) is clear, it suﬃces to show that ( , ) ⊃ N ∗ −∞ |t∞ and t∞ are equal in law. Let P be the random set in (9.15). By Lemma 8.3, A| t ( , ) t∞ = (P t ). By the independence of and t∞ (see Nthe∗ proof−∞ | of TheoremN ×1.11 { } ) and what we have just proved,N |−∞ it followsN | that (P t ) is equally distributed with (P ′ t ), where P ′ is a Poisson Npoint× set { with} intensity 2, independent ofN . By× { Proposition} 8.4, the law of is the same as that of (P t ), andN we are done. A|t∞ N ′ × { } Proof of Proposition 1.15(i) and (ii). The statements follow by a passage to the limit in Propositions 1.14(i) and (ii), using Lemma 9.2.

APPENDIX: DEFINITIONS OF PATH SPACE In this appendix we compare the definition of the path space Π and its topology used in the present paper with the definitions used in [7, 8]. Let be the space of all functions π : [σπ, ] [ , ], with σπ [ , ], suchP that t Θ (π(t),t) is continuous∞ on→ ( −∞, ∞). For π ,π ∈ −∞, define∞ 7→ 1 −∞ ∞ 1 2 ∈ P d(π1,π2) by (1.5) and define d′ in the same way, but with the supremum over all t σπ1 σπ2 replaced by an unrestricted supremum over all t R. Call two≥ elements∧ π ,π d-equivalent (resp. d -equivalent) if d(π∈,π ) = 0 1 2 ∈ P ′ 1 2 [resp. d′(π1,π2) = 0], and let Π (resp. Π′) denote the spaces of d-equivalence classes (resp. d -equivalence classes) in . Then (Π, d) is in a natural way ′ P isomorphic to the set of paths defined in Section 1.2, while (Π′, d′) is the space of paths used in [7, 8]. The difference between these two spaces is small. Indeed, two paths π1,π2 are d-equivalent if and only if (A.1) σ = σ and π (t)= π (t) σ t< , π1 π2 1 2 ∀ π ≤ ∞ THE BROWNIAN NET 57 while they are d′-equivalent if and only if (A.2) σ = σ < and π (t)= π (t) σ t< . π1 π2 ∞ 1 2 ∀ π ≤ ∞ Thus, the only difference between Π and Π′ is that, while the former has only one path with starting time , the latter has a one-parameter family (r) ∞ (π )r [ , ] of such paths, given by ∈ −∞ ∞ (r) (A.3) σ (r) := , π ( ) := r (r [ , ]). π ∞ ∞ ∈ −∞ ∞ A sequence of paths π converges in d to the limit π(r) if and only if σ n ′ πn → and πn(σπn ) r. Both the spaces (Π, d) and (Π′, d′) are complete and ∞separable, and the→ former is the continuous image of the latter under a map (r) that identifies the family of paths (π )r [ , ] with a single path. Of course, it is more natural to identify∈ −∞ all∞ paths starting at infinity. In fact, it seems that the authors of [8] used the metric in (1.5) in earlier versions of their manuscript, but then by accident dropped the restriction that t σ σ in the supremum [C. M. Newman, personal communication]. ≥ π1 ∧ π2 Acknowledgments. We thank Charles Newman, Krishnamurthi Ravis- hankar and Emmanuel Schertzer for answering questions about [8] and [12]. We thank Peter Mörters for answering questions about special points of Brownian motion and the referee for helpful suggestions leading to a better exposition. R. Sun thanks UTIA´ and J. M. Swart thanks EURANDOM for their hospitality during short visits, which were supported by RDSES short visit grants from the European Science Foundation. R. Sun was a postdoc at EURANDOM from Oct 2004 to Oct 2006, when this work was completed.

REFERENCES [1] Arratia, R. (1979). Coalescing Brownian motions on the line. Ph.D. thesis, Univ. Wisconsin, Madison. [2] Arratia, R. Coalescing Brownian motions and the voter model on Z. Unpublished partial manuscript. Available from [email protected]. [3] Athreya, S. R. and Swart, J. M. (2005). Branching-coalescing particle systems. Probab. Theory Related Fields 131 376–414. MR2123250 [4] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. J. Wiley, New York. MR1700749 [5] Ding, W., Durrett, R. and Liggett, T. M. (1990). Ergodicity of reversible reaction diﬀusion processes. Probab. Theory Related Fields 85 13–26. MR1044295 [6] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Convergence. Wiley, New York. MR0838085 [7] Fontes, L. R. G., Isopi, M., Newman, C. M. and Ravishankar, K. (2002). The Brownian web. Proc. Natl. Acad. Sci. 99 15888–15893. MR1944976 58 R. SUN AND J. M. SWART

[8] Fontes, L. R. G., Isopi, M., Newman, C. M. and Ravishankar, K. (2004). The Brownian web: Characterization and convergence. Ann. Probab. 32 2857–2883. MR2094432 [9] Fontes, L. R. G., Isopi, M., Newman, C. M. and Ravishankar, K. (2006). Coarsening, nucleation, and the marked Brownian web. Ann. Inst. H. Poincaré Probab. Statist. 42 37–60. MR2196970 [10] Howitt, C. and Warren, J. Consistent families of Brownian motions and stochastic flows of kernels. ArXiv: math.PR/0611292. [11] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York. MR1121940 [12] Newman, C. M., Ravishankar, K. and Schertzer, E. Marking (1, 2) points of the Brownian web and applications. In preparation. [13] Schlogl,¨ F. (1972). Chemical reaction models and non-equilibrium phase transi- tions. Z. Phys. 253 147–161. [14] Schertzer, E., Sun, R. and Swart, J. M. Special points of the Brownian net. In preparation. [15] Soucaliuc, F., Toth,´ B. and Werner, W. (2000). Reflection and coalescence between one-dimensional Brownian paths. Ann. Inst. H. PoincaréProbab. Statist. 36 509–536. MR1785393 [16] Sudbury, A. (1999). Hunting submartingales in the jumping voter model and the biased annihilating branching process. Adv. in Appl. Probab. 31 839–854. MR1742697 [17] Toth,´ B. and Werner, W. (1998). The true self-repelling motion. Probab. Theory Related Fields 111 375–452. MR1640799 [18] Warren, J. (2002). The noise made by a Poisson snake. Electron. J. Probab. 7 1–21. MR1943894

TU Berlin UTIA´ AV CRˇ MA 7-5, Fakultat¨ II—Institut fur¨ Mathematik Pod vodarenskou´ veˇˇz´ı 4 Straße des 17. Juni 136 18208 Praha 8 10623 Berlin Czech Republic Germany E-mail: [email protected] E-mail: [email protected] URL: http://staﬀ.utia.cas.cz/swart/