Special points of the Brownian net

The Brownian net, which has recently been introduced by Sun and Swart [SS08], and independently by Newman, Ravishankar and Schertzer [NRS08], generalizes the Brownian web by allowing branching. In this paper, we study the structure of the Brownian net in more detail. In particular, we give an almost sure classification of each point in $R^2$ according to the configuration of the Brownian net paths entering and leaving the point. Along the way, we establish various other structural properties of the Brownian net.


Introduction
The Brownian web, , is essentially a collection of one-dimensional coalescing Brownian motions starting from every point in space and time 2 . It originated from the work of Arratia [Arr79; Arr81] on the scaling limit of the voter model, and arises naturally as the diffusive scaling limit of the system of one-dimensional coalescing random walks dual to the voter model; see also [FINR04] and [NRS05]. In the language of stochastic flows, the coalescing flow associated with the Brownian web is known as the Arratia flow. A detailed analysis of the Brownian web was carried out by Tóth and Werner in [TW98]. More recently, Fontes, Isopi, Newman and Ravishankar [FINR04] introduced a by now standard framework in which the Brownian web is regarded as a random compact set of paths, which (in a suitable topology) becomes a random variable taking values in a Polish space. It is in this framework that the object initially proposed by Arratia in [Arr81] takes on the name the Brownian web.
Recently, Sun and Swart [SS08] introduced a generalization of the Brownian web, called the Brownian net, , in which paths not only coalesce, but also branch. From a somewhat different starting point, Newman, Ravishankar and Schertzer independently arrived at the same object. Their alternative construction of the the Brownian net will be published in [NRS08]. The motivation in [SS08] comes from the study of the diffusive scaling limit of one-dimensional branching-coalescing random walks with weak branching, while the motivation in [NRS08] comes from the study of onedimensional stochastic Ising and Potts models with boundary nucleation. The different constructions of the Brownian net given in [SS08] and [NRS08] complement each other and give different insights into the structure of the Brownian net.
In the Brownian web, at a typical, deterministic point in 2 , there is just a single path leaving the point and no path entering the point. There are, however, random, special points, where more than one path leaves, or where paths enter. A full classification of these special points is given in [TW98], see also [FINR06]. The special points of the Brownian web play an important role in the construction of the so-called marked Brownian web [FINR06], and also in the construction of the Brownian net in [NRS08]. A proper understanding of the Brownian net thus calls for a similar classification of special points of the Brownian net, which is the main goal of this paper. Along the way, we will establish various properties for the Brownian net , and the left-right Brownian web ( l , r ), which is the key intermediate object in the construction of the Brownian net in [SS08].
Several models have been studied recently which have close connections to the Brownian net. One such model is the so-called dynamical Browian web, which is a Browian web evolving in time in such a way that at random times, paths switch among outgoing trajectories at points with one incoming, and two outgoing paths. Such a model is similar in spirit to dynamical percolation, see e.g. [Hag98]. In [HW07], Howitt and Warren characterized the two-dimensional distributions of the dynamical Brownian web. This leads to two coupled Brownian webs which are similar in spirit to the left-right Brownian web ( l , r ) in [SS08]. Indeed, there is a close connection between these objects. In [NRS08], the dynamical Brownian web and the Brownian net are constructed in the same framework, and questions of exceptional times (of the former) are investigated. A discrete space-time version of the dynamical Brownian web was studied in [FNRS07].
A second model closely related to the Brownian net is a class of stochastic flows of kernels introduced by Howitt and Warren [HW06]. These stochastic flows are families of random transition kernels, describing a Brownian motion evolving in a random space-time environment. It turns out that these stochastic flows can be constructed through a random switching between outgoing paths in a 'reference' Brownian web, which plays the role of the random environment, similar to the construction of the dynamical Brownian web and the Brownian net in [NRS08]. A subclass of the stochastic flows of kernels of Howitt and Warren turns out to be supported on the Brownian net. This is the subject of the ongoing work [SSS08]. Results established in the present paper, as well as [NRS08], will provide important tools to analyze the Howitt-Warren flows.
Finally, there are close connections between the Brownian net and low temperature scaling limits of one-dimensional stochastic Potts models. In [NRS09], these scaling limits will be constructed with the help of a graphical representation based on a marking of paths in the Brownian net. Their construction uses in an essential way one of the results in the present paper (the local finiteness of relevant separation points proved in Proposition 2.9 below).
In the rest of the introduction, we recall the characterization of the Brownian web and its dual from [FINR04;FINR06], the characterization of the left-right Brownian web and the Brownian net from [SS08], the classification of special points of the Brownian web from [TW98;FINR06], and lastly we formulate our main results on the classification of special points for the left-right Brownian web and the Brownian net according to the configuration of paths entering and leaving a point.
In the rest of the paper, for K ∈ and A ⊂ R 2 c , let K(A) denote the set of paths in K with starting points in A. When A = {z} for z ∈ R 2 c , we also write K(z) instead of K({z}). We recall from [FINR04] the following characterization of the Brownian web.

Theorem 1.1. [Characterization of the Brownian web]
There exists a ( , )-valued random variable , called the standard Brownian web, whose distribution is uniquely determined by the following properties: (a) For each deterministic z ∈ 2 , almost surely there is a unique path π z ∈ (z).
To each Brownian web , there is associated a dual Brownian webˆ , which is a random set of paths running backward in time [Arr81; TW98; FINR06]. The pair ( ,ˆ ) is called the double Brownian web. By definition, a backward pathπ, with starting time denoted byσπ, is a function π : [−∞,σπ] → [−∞, ∞] ∪ { * }, such that t → (t,π(t)) is a continuous map from [−∞,σπ] to R 2 c . We let (Π,d) denote the space of backward paths, which is in a natural way isomorphic to (Π, d) through time reversal, and we let (ˆ , dˆ ) denote the space of compact subsets of (Π,d), equipped with the Hausdorff metric. We say that a dual pathπ crosses a (forward) path π if there exist σ π ≤ s < t ≤σπ such that (π(s) −π(s))(π(t) −π(t)) < 0. The next theorem follows from [FINR06, Theorem 3.7]; a slightly different construction of the dual Brownian web can be found in [SS08, Theorem 1.9].

Theorem 1.2. [Characterization of the dual Brownian web]
Let be the standard Brownian web. Then there exists aˆ -valued random variableˆ , defined on the same probability space as , called the dual Brownian web, which is almost surely uniquely determined by the following properties: (a) For any deterministic z ∈ 2 , almost surelyˆ (z) consists of a single pathπ z , which is the unique path inΠ(z) that does not cross any path in .
It is known that, modulo a time reversal,ˆ is equally distributed with . Moreover, paths in andˆ interact via Skorohod reflection. (This follows from the results in [STW00], together with the standard discrete aproximation of the double Brownian web.) We now recall the left-right Brownian web ( l , r ), which is the key intermediate object in the construction of the Brownian net in [SS08]. Following [SS08], we call (l 1 , . . . , l m ; r 1 , . . . , r n ) a collection of left-right coalescing Brownian motions, if (l 1 , . . . , l m ) is distributed as coalescing Brownian motions each with drift −1, (r 1 , . . . , r n ) is distributed as coalescing Brownian motions each with drift +1, paths in (l 1 , . . . , l m ; r 1 , . . . , r n ) evolve independently when they are apart, and the interaction between l i and r j when they meet is described by the two-dimensional stochastic differential equation where B l t , B r t , B s t are independent standard Brownian motions, and (L, R) are subject to the constraint that L t ≤ R t for all t ≥ τ L,R , (1.5) where, for any two paths π, π ∈ Π, we let τ π,π := inf{t > σ π ∨ σ π : π(t) = π (t)} (1.6) denote the first meeting time of π and π , which may be ∞. It can be shown that subject to the condition (1. We cite the following characterization of the left-right Brownian web from [SS08, Theorem 1.5].

Theorem 1.3. [Characterization of the left-right Brownian web]
There exists a ( 2 , 2 )-valued random variable ( l , r ), called the standard left-right Brownian web, whose distribution is uniquely determined by the following properties: (a) For each deterministic z ∈ 2 , almost surely there are unique paths l z ∈ l and r z ∈ r .
(c) For any deterministic countable dense subset ⊂ 2 , almost surely l is the closure of {l z : z ∈ } and r is the closure of {r z : z ∈ } in the space (Π, d).
Comparing Theorems 1.1 and 1.3, we see that l and r are distributed as Brownian webs tilted with drift −1 and +1, respectively. Therefore, by Theorem 1.2, the Brownian webs l and r a.s. uniquely determine dual websˆ l andˆ r , respectively. It turns out that (ˆ l ,ˆ r ) is equally distributed with ( l , r ) modulo a rotation by 180 o .
Based on the left-right Brownian web, [SS08] gave three equivalent characterizations of the Brownian net, which are called respectively the hopping, wedge, and mesh characterizations. We first recall what is meant by hopping, and what are wedges and meshes.
Hopping: Given two paths π 1 , π 2 ∈ Π, any t > σ π 1 ∨ σ π 2 (note the strict inequality) is called an intersection time of π 1 and π 2 if π 1 (t) = π 2 (t). By hopping from π 1 to π 2 , we mean the construction of a new path by concatenating together the piece of π 1 before and the piece of π 2 after an intersection time. Given the left-right Brownian web ( l , r ), let H( l ∪ r ) denote the set of paths constructed by hopping a finite number of times among paths in l ∪ r . Wedges: Let (ˆ l ,ˆ r ) be the dual left-right Brownian web almost surely determined by ( l , r ). Recall thatσπ denotes the starting time of a backward pathπ. Any pairl ∈ˆ l ,r ∈ˆ r witĥ r(σl ∧σr ) <l(σl ∧σr ) defines an open set (see Figure 2) where, in analogy with (1.6),τr ,l := sup{t <σl ∧σr :r(t) =l(t)} denotes the first (backward) hitting time ofr andl. Such an open set is called a wedge of (ˆ l ,ˆ r ). Ifτr ,l > −∞, we call τr ,l the bottom time, and (l(τr ,l ), τr ,l ) the bottom point of the wedge W (r,l).
Meshes: By definition, a mesh of ( l , r ) is an open set of the form (see Figure 2) where l ∈ l , r ∈ r are paths such that σ l = σ r , l(σ l ) = r(σ r ) and r(s) < l(s) on (σ l , σ l + ε) for some ε > 0. We call (l(σ l ), σ l ) the bottom point, σ l the bottom time, (l(τ l,r ), τ l,r ) the top point, τ l,r the top time, r the left (!) boundary, and l the right boundary of M .
Given an open set A ⊂ 2 and a path π ∈ Π, we say π enters A if there exist σ π < s < t such that π(s) / ∈ A and π(t) ∈ A. We say π enters A from outside if there exists σ π < s < t such that π(s) / ∈Ā and π(t) ∈ A. We now recall the following characterization of the Brownian net from [SS08, Theorems 1.3, 1.7, 1.10].

Theorem 1.4. [Characterization of the Brownian net]
There exists a ( , )-valued random variable , the standard Brownian net, whose distribution is uniquely determined by property (a) and any of the three equivalent properties (b1)-(b3) below: (a) There exist l , r ⊂ such that ( l , r ) is distributed as the left-right Brownian web.
(b1) Almost surely, is the closure of H( l ∪ r ).
(b2) Almost surely, is the set of paths in Π which do not enter any wedge of (ˆ l ,ˆ r ) from outside.
(b3) Almost surely, is the set of paths in Π which do not enter any mesh of ( l , r ).
Remark. Properties (b1)-(b3) in fact imply that the left-right Brownian web ( l , r ) contained in a Brownian net is almost surely uniquely determined by the latter, and for each deterministic z ∈ 2 , the path in l , resp. r , starting from z is just the left-most, resp. right-most, path among all paths in starting from z. Since ( l , r ) uniquely determines a dual left-right Brownian web (ˆ l ,ˆ r ), there exists a dual Brownian netˆ uniquely determined by and equally distributed with (modulo time reversal).
The construction of the Brownian net from the left-right Brownian web can be regarded as an outside-in approach because l and r are the "outermost" paths among all paths in . On the other hand, the construction of the Brownian net in [NRS08] can be regarded as an inside-out approach, since they start from a standard Brownian web, which may be viewed as a collection of "innermost" paths, to which new paths are added by allowing branching to the left and right. More precisely, they allow hopping at a set of marked points, which is a Poisson subset of the set of all special points with one incoming and two outgoing paths. One may call this construction the marking construction of the Brownian net. In this paper, we will only use the characterizations provided by Theorem 1.4.

Classification of special points
To classify each point of 2 according to the local configuration of paths in the Brownian web or net, we first formulate a notion of equivalence among paths entering, resp. leaving, a point, which provides a unified framework. We say that a path π ∈ Π enters a point z = (x, t) ∈ 2 if σ π < t and π(t) = x. We say that π leaves z if σ π ≤ t and π(t) = x.
Note that, on Π, ∼ z in and ∼ z out are not equivalence relations. However, almost surely, they define equivalence relations on the set of all paths in the Brownian web entering or leaving z. Due to coalescence, almost surely for all π 1 , π 2 ∈ and z = (x, t) ∈ 2 , (1.9) Let m in (z), resp. m out (z), denote the number of equivalence classes of paths in entering, resp. leaving, z, and letm in (z) andm out (z) be defined similarly for the dual Brownian webˆ . For the Brownian web, points z ∈ 2 are classified according to the value of (m in (z), m out (z)). Points of type (1,2) are further divided into types (1, 2) l and (1, 2) r , where the subscript l (resp. r) indicates that the left (resp. right) of the two outgoing paths is the continuation of the (up to equivalence) unique incoming path. Points in the dual Brownian webˆ are labelled according to their type in the Brownian web obtained by rotating the graph ofˆ in 2 by 180 o . We cite the following result from [TW98, Proposition 2.4] or [FINR06, Theorems 3.11-3.14].
We do not give a picture to demonstrate Theorem 1.6. However, if in the first row in Figure 3, one replaces each pair consisting of one left-most (green) and right-most (red) path by a single path, and likewise for pairs of dual (dashed) paths, then one obtains a schematic depiction of the 7 types of points in /ˆ .
We now turn to the problem of classifying the special points of the Brownian net. We start by observing that also in the left-right Brownian web, a.s. for each z ∈ 2 , the relations ∼ z in and ∼ z out define equivalence relations on the set of paths in l ∪ r entering, resp. leaving z. This follows from (1.9) for the Brownian web and the fact that a.s. for each l ∈ l , r ∈ r and σ l ∨ σ r < s < t such that l(s) = r(s), one has l(t) ≤ r(t) (see Prop. 3.6 (a) of [SS08]). Moreover, by the same facts, the equivalence classes of paths in l ∪ r entering, resp. leaving, z are naturally ordered from left to right.
Our classification of points in the Brownian net is mainly based on the equivalence classes of incoming and outgoing paths in l ∪ r . To denote the type of a point, we first list the incoming equivalence classes of paths from left to right, and then, separated by a comma, the outgoing equivalence classes of paths from left to right. If an equivalence class contains only paths in l resp. r we will label it by l, resp. r, while if it contains both paths in l and in r we will label it by p, standing for pair. For points with (up to equivalence) one incoming and two outgoing paths, a subscript l resp. r means that all incoming paths belong to the left one, resp. right one, of the two outgoing equivalence classes; a subscript s indicates that incoming paths in l belong to the left outgoing equivalence class, while incoming paths in r belong to the right outgoing equivalence class. If at a point there are no incoming paths in l ∪ r , then we denote this by o or n, where o indicates that there are no incoming paths in the net , while n indicates that there are incoming paths in (but none in l ∪ r ).
Thus, for example, a point is of type (p, lp) r if at this point there is one equivalence class of incoming paths in l ∪ r and there are two outgoing equivalence classes. The incoming equivalence class is of type p while the outgoing equivalence classes are of type l and p, from left to right. All incoming paths in l ∪ r continue as paths in the outgoing equivalence class of type p.
Points in the dual Brownian netˆ , which is defined in terms of the dual left-right Brownian web (ˆ l ,ˆ r ), are labelled according to their type in the Brownian net and left-right web obtained by rotating the graphs ofˆ and (ˆ l ,ˆ r ) in 2 by 180 o . With the notation introduced above, we can now state our first main result on the classification of points in 2 for the Brownian net.

Theorem 1.7. [Classification of special points of the Brownian net]
Let ( l , r ) be the standard left-right Brownian web, let (ˆ l ,ˆ r ) be its dual, and let andˆ be the associated Brownian net and its dual. Then almost surely, each point in 2 is of one of the following 20 types in /ˆ (see Figure 3):  (2) (p, pp) s /(p, pp) s ; (3) (l,p)/(o,lp), (o,lp)/(l,p), (r,p)/(o,pr), (o,pr)/(r,p); (4) (l, pp) r /(p, lp) r , (p, lp) r /(l, pp) r , (r, pp) l /(p, pr) l , (p, pr) l /(r, pp) l ; (5) (l, lp) r /(l, lp) r , (r, pr) l /(r, pr) l ; Remark 1 Our classification is mainly based on the configuration of equivalence classes of paths in the left-right Brownian web l ∪ r entering and leaving a point. For points of types (o, p) and (n, p), however, we also use information about paths which belong to the Brownian net but not to l ∪ r . By distinguishing notationally between these types, we achieve that the type of a point inˆ is uniquely determined by its type in . Moreover, by counting n as an incoming path (and counting equivalence classes of types l, r, p as one incoming resp. outgoing path each), we achieve that m out (z) =m in (z) + 1 andm out (z) = m in (z) + 1, in analogy with the Brownian web.
Remark 2 Modulo symmetry between left and right, and between forward and dual paths, there are only 9 types of points in Theorem 1.7: four types from group (1), and one type each from groups (2)-(6). Group (1) corresponds to the 7 types of points of the Brownian web, except that each equivalence class of paths is now of type p. We call points of type (pp, p) meeting points, and points of type (p, pp) s (from group (2)) separation points. Points in groups (3)-(6) are 'cluster points' that arise as the limit of a nested sequence of excursions between paths in the left-right Brownian web, or its dual (see Proposition 3.11 and Figure 6 below).

Structure of special points
When contemplating the special points of the Brownian net as depicted in Figure 3, one is struck by the fact that points from groups (3)-(6) (the 'cluster points') look just like certain points from group (1), except that some paths are missing. For example, points of type (l, p)/(o, lp) look like points of type (p, p)/(o, pp), points of type (l, pp) r /(p, lp) r look like points of (p, pp) r /(p, pp) r , and points of type (n, p)/(o, lp) look like points of type (p, p)/(o, pp). In most cases, when one member of a (dual) left-right pair seems to be missing, the other member of the pair is still present, but for points of group (6), a whole (dual) incoming pair seems to have disappeared. In the present section, we will show how to make these 'missing' paths visible in the form of reflected left-most (or right-most) paths. These reflected left-most (resp. right-most) paths are not elements of l (resp. r ), but they turn out to be countable concatenations of paths in l (resp. r ).
Except for fulfilling the aesthetic role of bringing to light what we feel is missing, these reflected paths also serve the more practical aim of putting limitations on how general paths in the Brownian net (and not just the left-right Brownian web l ∪ r ) can enter and leave points of various types. Recall that our classicifation theorem (Theorem 1.7) primarily describes how left-most and right-most paths enter and leave points in the plane. In Theorems 1.11 and 1.12 below, we first describe the structure of a special set of reflected left-most and right-most paths near the special points, and then we use this to describe the local structure of the Brownian net at these points. Our results will show that, with the exception of outgoing paths at points of type (o, lr), all Brownian net paths must enter or leave a point squeezed between a pair consisting of one (reflected) left-most and (reflected) right-most path.
We start with some definitions. By definition, we say that a set of paths, all starting at the same time t, has a maximum (resp. minimum) if there exists a path π ∈ such that π ≤ π (resp. π ≤ π ) on [t, ∞) for all π ∈ . We denote the (necessarily unique) maximum (resp. minimum) of by max( ) (resp. min( )).
We call the path l z,π (resp. r z,π ) in (1.10) the reflected left-most (resp. right-most) path relative tô π. See Figure 5 below for a picture of a reflected right-most path relative to a dual left-most path. Lemma 1.8 says that such reflected left-most (resp. right-most) paths are well-defined, and are concatenations of countably many left-most (resp. right-most) paths. Indeed, it can be shown that in a certain well-defined way, a reflected left-most path l z,π always 'turns left' at separation points, except those in the countable set (l z,π ), where turning left would make it crossπ. We call the set (l z,π ) in (1.11) the set of reflection times of l z,π , and define (r z,π ) analogously. In analogy with (1.10), we also define reflected dual pathsl z,π andr z,π relative to a forward path π ∈ .
For a given point z in the plane, we now define special classes of reflected left-most and right-most paths, which extend the classes of paths in l and r entering and leaving z. (1.12) Similar definitions apply to r ,ˆ l ,ˆ r , , andˆ . We define and define r in (z),ˆ l in (z),ˆ r in (z) analogously, by symmetry. Finally, we define l out (z) := l z,r ∈ out (z) :r ∈ˆ r in (z ), z = (x , t), x ≤ x , (1.14) and we define r out (z),ˆ l out (z),ˆ r out (z) analogously. We call the elements of l in (z) and l out (z) extended left-most paths entering, resp. leaving z. See Figure 4 for an illustration.
As we will see in Theorem 1.11 below, the extended left-and right-most paths we have just defined are exactly the 'missing' paths in groups (3)-(6) from Theorem 1.7. Finding the right definition of extended paths that serves this aim turned out to be rather subtle. Note that paths in l in (z) and r in (z) reflect off paths inˆ r out (z) andˆ l out (z), respectively, but paths in l out (z) and r out (z) could be reflected off reflected paths. If, instead of (1.14), we would have defined l out (z) as the set of all reflected left-most paths that reflect off paths inˆ r in (z) orˆ in (z), then one can check that for points of type (o, lr), in the first case we would not have found all 'missing' paths, while in the second case we would obtain too many paths. Likewise, just calling any countable concatenation of left-most paths an extended left-most path would, in view of our aims, yield too many paths.
To formulate our results rigorously, we need one more definition.
Note that by (1.9), a.s. for each l 1 , l 2 ∈ l and z ∈ 2 , l 1 ∼ z in l 2 implies l 1 = z in l 2 and l 1 ∼ z out l 2 implies l 1 = z out l 2 . Part (a) of the next theorem shows that our extended left-most and right-most paths have the same property. Parts (b) and (c) say that extended left-and right-most always pair up, and are in fact the 'missing' paths from groups (3)-(6) of Theorem 1.7.
We finally turn our attention to the way general paths in (and not just our extended paths) enter and leave the special points from Theorem 1.7. The next theorem shows that with the exception of outgoing paths at points of type (o, lr), all Brownian net paths must enter and leave points squeezed between a pair consisting of one extended left-most and one extended right-most path. l l l r r ′r r ′l l ′ r r l ′ Figure 4: Local structure of outgoing extended left-most and right-most paths and incoming extended dual left-most and right-most paths in points of types ( · , lp) and (o, lr). In the picture on the left, r andr are reflected paths that are not elements of r andˆ r , respectively. In the picture on the right, only l and r are 'true' left-most and right-most paths, while all other paths are 'missing' paths that are not visible in the schematic picture for points of type (o, lr) in Figure 3.
(a) The relation ∼ z in is an equivalence relation on in (z). Each equivalence class of paths in in (z) contains an l ∈ l in (z) and r ∈ r in (z), which are unique up to strong equivalence, and each π ∈ satisfies l ≤ π ≤ r on [t − , t] for some > 0.
(b) If z is not of type (o, lr), then the relation ∼ z out is an equivalence relation on out (z). Each equivalence class of paths in out (z) contains an l ∈ l out (z) and r ∈ r out (z), which are unique up to strong equivalence, and each π ∈ satisfies l ≤ π ≤ r on [t, t + ] for some > 0.
(c) If z is of type (o, lr), then there exist π ∈ such that l ∼ z out π ∼ z out r while l ∼ z out r, where l and r are the unique outgoing paths at z in l and r , respectively.
(d) At points of types with the subscript l (resp. r), all incoming paths in continue in the left (resp. right) outgoing equivalence class. Except for this restriction, any concatenation of a path in in (z) up to time t with a path in out (z) after time t is again a path in .

Outline and open problems
In this section, we outline the main structure of our proofs and mention some open problems.
In Section 2 we study separation points, i.e., points of type (p, pp) s from Theorem 1.7. In a sense, these are the most important points in the Brownian net, since at these points paths in the Brownian net have a choice whether to turn left or right. Also, these are exactly the marked points in [NRS08], and their marking construction shows that the Brownian net is a.s. determined by its set of separation points and an embedded Brownian web.
In order to prepare for our study of separation points, in Section 2.1, we investigate the interaction between forward right-most and dual left-most paths. It turns out that the former are Skorohod reflected off the latter, albeit they may cross the latter from left to right at some random time. In Section 2.2, the results from Section 2.1 are used to prove that crossing points between forward right-most and dual left-most paths are separation points between left-most and right-most paths, and that these points are of type (p, pp) s .
In Section 2.3, we study 'relevant' separation points. By definition, a point z = (x, t) ∈ 2 is called an (s, u)-relevant separation point for some −∞ ≤ s < u ≤ ∞, if s < t < u, there exists a π ∈ such that σ π = s and π(t) = x, and there exist incoming l ∈ l and r ∈ r at z such that l < r on (t, u). At (s, u)-relevant separation points, a Brownian net path going from time s to time u has a choice whether to turn left or right that may be relevant for where it ends up at time u. The main result of Section 2.3 says that for deterministic −∞ < s < t < ∞, the set of (s, u)-relevant separation points is a.s. a locally finite subset of × (s, u). This fact has several useful consequences. As a first application, in Section 2.4, we prove Lemma 1.8.
In Section 3, we study the image set of the Brownian net started at a fixed time T . Let T := {π ∈ : σ π = T } denote the set of paths in the Brownian net starting at a given time T ∈ [−∞, ∞], and let N T be the image set of T in R 2 c , i.e., In view of (1.16), much can be learned about the Brownian net by studying the closed set N T .
In Section 3.1, it is shown that the connected components of the complement of N T relative to {(x, t) ∈ R 2 c : t ≥ T } are meshes of a special sort, called maximal T -meshes. In Section 3.2, it is shown that N T has a local reversibility property that allows one, for example, to deduce properties of meeting points from properties of separation points. Using these facts, in Section 3.3, we give a preliminary classification of points in 2 based only on the structure of incoming paths in . In Section 3.4, we use the fractal structure of N T to prove the existence, announced in [SS08], of random times t > T when {x ∈ : (x, t) ∈ N T } is not locally finite.
It turns out that to determine the type of a point z ∈ 2 in , according to the classification of Theorem 1.7, except for one trivial ambiguity, it suffices to know the structure of the incoming paths at z in both andˆ according to the preliminary classification from Section 3.3. Therefore, in order to prove Theorem 1.7, we need to know which combinations of types in andˆ are possible according to the latter classification. In particular, proving the existence of points in groups (4) and (5) from Theorem 1.7 depends on showing that there are points where the incoming paths in both andˆ form a nested sequence of excursions. Section 4 contains some excursion theoretic arguments that prepare for this. In particular, in Section 4.1, we prove that there are many excursions between a given left-most path l and dual left-most pathl on the left of l that are entered by some dual right-most path. In Section 4.2, we prove that there are many points wherel hits l while at the same time some right-most path makes an excursion away from l.
In Section 5, we finally prove our main results. Section 5.1 contains the proof of Theorem 1.7, while Section 5.2 contains the proofs of Theorems 1.11 and 1.12.
We conclude the paper with two appendices (Appendices A.1 and A.2) containing some facts and proofs that are not used in the main argument but may be of independent interest.
Our investigations leave open a few questions that we believe are important for understanding the full structure of the Brownian net, and which we hope to settle in future work. The first question we would like to mention concerns the image set in (1.15). Fix −∞ ≤ s < u ≤ ∞, and say that a point z ∈ 2 is n-connected to s and u if z ∈ {(x, t) ∈ N s : s < t < u} and one needs to remove at least n points from N s to disconnect z from {(x, t) ∈ N s : t = s or t = u}. Note that the notion of connectedness in N s is graph theoretic and does not respect the time direction inherent in the Brownian net. Here is a conjecture:

Conjecture 1.13. [3-connected points]
Almost surely for each −∞ ≤ s < u ≤ ∞, the set of 3-connected points to s and u is a locally finite subset of × (s, u), and all 3-connected points are either meeting or separation points.
It is easy to see that all points in {(x, t) ∈ N s : s < t < u} are 2-connected to s and u. The results in our present paper imply that meeting and separation points are not 4-connected. All (s, u)-relevant separation points are 3-connected, but not all 3-connected separation points are (s, u)-relevant. If Conjecture 1.13 is correct, then the structure of all Brownian net paths going from time s to u can be described by a purely 3-connected, locally finite graph, whose vertices are the 3-connected points.
Other open problems concern the reflected left-most and right-most paths described in Theorem 1.11. Here is another conjecture:
Conjecture 1.14 says that near t, all reflection points ofr off l lie onl (and hence, by symmetry, a similar statement holds for the reflection points ofl off r). If this is true, then the picture in Figure 4 simplifies a lot. In particular, there exists an > 0 such that on [t, t + ], the paths are eventually ordered as l ≤ r ≤r ≤l ≤ l ≤ r. Note from Figure 4 that at present, for points of type (o, lr), we cannot even rule out that l (u n ) < r (u n ) for a sequence of times u n ↓ t.
It seems that Conjectures 1.13 and 1.14 cannot be proved with the methods developed in the present paper and [SS08]. Instead, we hope to tackle these problems by calling in the marking construction of the Brownian net developed in [NRS08].
Another open problem is to determine the Hausdorff dimension of the sets of points of each of the types from Theorem 1.7. For the Brownian web, the Hausdorff dimensions of all types of points are known, see [FINR06, Theorem 3.12]. We believe that points from group (1) of Theorem 1.7 have the same Hausdorff dimension as the corresponding points in the Brownian web. Separation points (group (2)) are countable. About the Hausdorff dimensions of points from groups (3)-(6), we know nothing.

List of global notations
For ease of reference, we collect here some notations that will be used throughout the rest of the paper. In the proofs, some new notation might be derived from the global notations listed here, such as by adding superscripts or subscripts, in which case the objects they encode will be closely related to what the corresponding global notation stands for.

General notation:
( · ) : law of a random variable.
: a deterministic countable dense subset of 2 . S : the diffusive scaling map, applied to subsets of 2 , paths, and sets of paths, defined as S (x, t) := ( x, 2 t) for (x, t) ∈ 2 . τ : a stopping time; in Section 4: a time of increase of the reflection process (see (4.84)). τ π,π : the first meeting time of π and π , defined in (1.6). (L t , R t ) : a pair of diffusions solving the left-right SDE (1.4). ∼ z in , ∼ z out : equivalence of paths entering, resp. leaving z ∈ 2 , see Definition 1.5. = z in , = z out : strong equivalence of paths entering, resp. leaving z ∈ 2 , see Definition 1.10.

Paths, Space of Paths:
(R 2 c , ρ) : the compactification of 2 with the metric ρ, see (1.1). z = (x, t) : point in R 2 c , with position x and time t. s, t, u, S, T, U : times.
(Π, d) : the space of continuous paths in (R 2 c , ρ) with metric d, see (1.2). Π(A), Π(z) : the set of paths in Π starting from a set A ⊂ R 2 c resp. a point z ∈ R 2 c . The same notatation applies to any subset of Π such as , l , . π : a path in Π. σ π : the starting time of the path π π(t) : the position of π at time t ≥ σ π .

Brownian webs:
( ,ˆ ) : a double Brownian web consisting of a Brownian web and its dual. π z ,π z : the a.s. unique paths in resp.ˆ starting from a deterministic point z ∈ 2 . ( l , r ) : the left-right Brownian web, see Theorem 1.3. in (z), out (z) : the set of paths in entering, resp. leaving z. l in (z), l out (z) : the sets of extended left-most paths entering, resp. leaving z, see Definition 1.9. l, r,l,r : a path in l , resp. r ,ˆ l ,ˆ r , often called a left-most, resp. right-most, dual left-most, dual right-most path. l z , r z ,l z ,r z : the a.s. unique paths l (z), resp. r (z),ˆ l (z),ˆ r (z) starting from a deterministic point z ∈ 2 . l z,π , r z,π : reflected left-most and right-most paths starting from z, and reflected offπ ∈ˆ , see (1.10). W (r,l) : a wedge defined by the pathsr ∈ˆ r andl ∈ˆ l , see (1.7). M (r, l) : a mesh defined by the paths r ∈ r and l ∈ l , see (1.8).

Interaction between forward and dual paths
We know from [STW00] that paths in l andˆ l , resp. r andˆ r , interact by Skorohod reflection. More precisely, conditioned onl ∈ˆ l with deterministic starting pointẑ = (x,t), the path l ∈ l with deterministic starting point z = (x, t), where t <t, is distributed as a Brownian motion with drift −1 Skorohod reflected offl. It turns out that conditioned onl ∈ˆ l , the interaction of r ∈ r starting at z withl is also Skorohod reflection, albeit r may crossl at a random time.

the path r is given (in distribution) by the unique solution to the Skorohod equation
where B is a standard Brownian motion, ∆ is a nondecreasing process increasing only when , and r is subject to the constraint thatl(s) ≤ r(s) for all t ≤ s ≤t.
(b) Conditioned on (l(s)) t≤s≤t with x <l(t), the path r is given (in distribution) by the unique solution to the Skorohod equation where B is a standard Brownian motion, ∆ is a nondecreasing process increasing only when r(s) =l(s), T is an independent mean 1/2 exponential random variable, and r is subject to the constraints that r(s) The interaction between paths in l andˆ r is similar by symmetry. If z = (l(t), t), where t is a deterministic time, then exactly two paths r 1 , r 2 ∈ r start from z, where one solves (2.17) and the other solves (2.18). Conditional onl, the paths r 1 and r 2 evolve independently up to the first time they meet, at which they coalesce.
Remark Lemma 2.1 gives an almost sure construction of a pair of paths (l, r) starting from deterministic pointsẑ and z. By the same argument as in [STW00], we can extend Lemma 2.1 to give an almost sure construction of a finite collection of paths in (ˆ l , r ) with deterministic starting points, and the order in which the paths are constructed is irrelevant. (The technical issues involved in consistently defining multiple coalescing-reflecting paths in our case are the same as those in [STW00].) Therefore, by Kolmogorov's extension theorem, Lemma 2.1 may be used to construct (ˆ r , l ) restricted to a countable dense set of starting points in 2 . Taking closures and duals, this gives an alternative construction of the double left-right Brownian web ( l , r ,ˆ l ,ˆ r ), and hence of the Brownian net.
Proof of Lemma 2.1 We will prove the following claim. Letẑ = (x,t) and z = (x, t) be deterministic points in 2 with t <t and letl ∈ˆ l and r ∈ r be the a.s. unique dual left-most and forward right-most paths starting fromẑ and z, respectively. We will show that it is possible to construct a standard Brownian motion B and a mean 1/2 exponential random variable T such thatl, B, and T are independent, and such that r is the a.s. unique solution to the equation ∆ is a nondecreasing process increasing only at times s ∈ [t,t] when r(s) =l(s), and r is subject to the constraints that r(s) ≤l(s) resp.l(s) ≤ r(s) for all s ∈ [t,t] such that s < τ resp. τ ≤ s. Note that (2.19) is a statement about the joint law of (l, r), which implies the statements about the conditional law of r givenl in Lemma 2.1.
which may be infinite. Note that for a.e. pathl, the time τ is a stopping time for W . If τ < ∞, then for s ≥ τ our equation is again a Skorohod equation, with reflection in the other direction, so To prove that there exist W and T such that r solves (2.19), we follow the approach in [STW00] and use discrete approximation. First, we recall from [SS08] the discrete system of branchingcoalescing random walks on 2 even = {(x, t) : x, t ∈ , x + t is even} starting from every site of 2 even . Here, for (x, t) ∈ 2 even , the walker that is at time t in x jumps at time t + 1 with probability 1−ε 2 to x − 1, with the same probability to x + 1, and with the remaining probability ε branches in two walkers situated at x − 1 and x + 1. Random walks that land on the same position coalesce immediately. Following [SS08], let ε denote the set of branching-coalescing random walk paths on 2 even (linearly interpolated between integer times), and let l ε , resp. r ε , denote the set of left-most, resp. right-most, paths in ε starting from each z ∈ 2 even . There exists a natural dual system of branching-coalescing random walks on 2 odd = 2 \ 2 even running backward in time, where (x, t) ∈ 2 even is a branching point in the forward system if and only if (x, t + 1) is a branching point in the backward system, and otherwise the random walk jumping from (x, t) in the forward system and the random walk jumping from (x, t + 1) in the backward system are coupled so that they do not cross. Denote the dual collection of branching-coalescing random walk paths byˆ , and letˆ l , resp.ˆ r , denote the dual set of left-most, resp. right-most, paths inˆ . Let S : 2 → 2 be the diffusive scaling map S (x, t) = ( x, 2 t), and define S applied to a subset of 2 , a path, or a set of paths analogously.
Let n be a sequence satisfying n ↓ 0. Choose z n = (x n , t n ) ∈ 2 even andẑ n = (x n ,t n ) ∈ 2 odd such that S ε n (z n ,ẑ n ) → (z,ẑ) as n → ∞. If we denote byl n the unique path inˆ l n (ẑ n ) and by r n the unique path in r n (z n ), then by Theorem 5.2 of [SS08], we have where denotes law and ⇒ denotes weak convergence.
The conditional law of r n givenl n has the following description. Let be the set of points where a forward path can meetl. Let I l n := {( y, s) ∈ I n :l(s + 1) <l(s)} and I r n := {( y, s) ∈ I n :l(s) <l(s + 1)} be the sets of points where a forward path can meetl from the left and right, respectively. Conditional onl n , the process (r n (s)) s≥t n is a Markov process such that if (r n (s), s) ∈ I n , then r n (s +1) = r n (s)+1 with probability (1+ n )/2, and r n (s +1) = r n (s)−1 with the remaining probability. If (r n (s), s) ∈ I l n , then r n (s + 1) = r n (s) + 1 with probability 2 n /(1 + n ) and r n (s + 1) = r n (s) − 1 with the remaining probability. Here 2 n /(1 + n ) is the conditional probability that (r n (s), s) is a branching point of the forward random walks given thatl n (s + 1) <l n (s). Finally, if (r n (s), s) ∈ I r n , then r n (s + 1) = r n (s) + 1 with probability 1. In view of this, we can constructr n as follows. Independently ofl n , we choose a random walk (W n (s)) s≥t n starting at (x n , t n ) that at integer times jumps from y to y −1 with probability (1− n )/2 and to y + 1 with probability (1 + n )/2. Moreover, we introduce i.i.d. Bernoulli random variables (X n (i)) i≥0 with [X n (i) = 1] = 4 n /(1 + n ) 2 . We then inductively construct processes ∆ n and r n , starting at time s in ∆ n (s) = 0 and r n (s) = x n , by putting (2.27) and r n (s + 1) := r n (s) if ∆ n (s + 1) > ∆ n (s). (2.28) This says that r n evolves as W n , but is reflected offl n with probability 1 − 4 n /(1 + n ) 2 if it attempts to cross from left to right, and with probability 1 if it attempts to cross from right to left. Note that if (r n (s), s) ∈ I l n , then r n attempts to cross with probability (1 + n )/2, hence the probability that it crosses is 4 n /(1 + n ) 2 · (1 + n )/2 = 2 n /(1 + n ), as required.
Extend W n (s), ∆ n (s), and r n (s) to all real s ≥ t by linear interpolation, and set T n := inf{i ≥ 0 : X i = 1}. Then The process r n satisfies r n (s) ≤l n (s) − 1 resp.l n (s) + 1 ≤ r n (s) for all s ∈ [t n ,t n ] such that s ≤ τ n resp. τ n + 1 ≤ s, and ∆ n increases only for s ∈ [t n ,t n ] such that |r n (s) −l n (s)| = 1.
By a slight abuse of notation, set T n := n T n , τ n := 2 n τ n , and letl n , r n , W n , and ∆ n be the counterparts of l n , r n , W n , and ∆ n , diffusively rescaled with S n . Then wherel, r are the dual and forward path we are interested in, W is a Brownian motion with drift 1, started at (x n , t n ), T is an exponentially distributed random variable with mean 1/2, andl, W, T are independent.
It follows from the convergence in (2.31) that the laws of the processes∆ n (s) := 1 2 (r n (s) − W n (s)) are tight. By (2.29)-(2.30), for n large enough, one has∆ n = ∆ n on the event thatl(t) < x, while on the complementary event,∆ n is −∆ n reflected at the level −T n . Using this, it is not hard to see that the processes ∆ n are tight. Therefore, going to a subsequence if necessary, by Skorohod's representation theorem (see e.g. Theorem 6.7 in [Bi99]), we can find a coupling such that where the paths converge locally uniformly on compacta. Since the ∆ n are nondecreasing and independent of T n , and T has a continuous distribution, one has τ n → τ a.s. Taking the limit in (2.29)-(2.30), it is not hard to see that r solves the equations (2.19)-(2.20).
If z is of the form z = (l(t), t) for some deterministic t, then the proof is similar, except that now we consider two approximating sequences of right-most paths, one started at (l n (t n ) − 1, t n ) and the other at (l n (t n ) + 1, t n ).
The next lemma is very similar to Lemma 2.1 (see Figure 5).

Lemma 2.2. [Sequence of paths crossing a dual path]
Letẑ = (x,t) ∈ 2 and t <t be a deterministic point and time. LetB, B i (i ≥ 0) be independent, standard Brownian motions, and let (T i ) i≥0 be independent mean 1/2 exponential random variables.
. Set τ 0 := t and define inductively paths (r i (s)) s≥τ i (i = 0, . . . , M ) starting at r i (τ i ) =l(τ i ) by the unique solutions to the Skorohod equation where ∆ i is a nondecreasing process increasing only when r i (s) =l(s), the process r i is subject to the constraints that Then M < ∞ a.s. and we can couple {l, r 0 , . . . , r M } to a left-right Brownian web and its dual in such a way thatl ∈ˆ l and r i ∈ r for i = 0, . . . , M .
Proof This follows by discrete approximation in the same way as in the proof of Lemma 2.1. Note that (2.34) ensures that r 0 , . . . , r M coalesce when they meet. To see that M < ∞ a.s., define (r (s)) s≥t by r : subject to the constraint that r (s) ≤l(s) for all t ≤ s ≤t. In particular, we have ∆(t) = sup s∈[t,t] (B(s) −l(s) +l(t)) < ∞ and the times τ 1 , . . . , τ M are created by a Poisson point process on [t,t] with intensity measure 2 d∆.

Structure of separation points
In this section we apply the results from the previous section to study the structure of separation points. We start with some definitions. First, we recall the definition of intersection points from [SS08], and formally define meeting and separation points.
Crossing points of two forward paths π 1 , π 2 ∈ Π have been defined in [SS08]. Below, we define crossing points of a forward path π and a dual pathπ.

Definition 2.4. [Crossing points]
We say that a path π ∈ Π crosses a pathπ ∈Π from left to right and π crossesπ either from left to right or from right to left at time t.
A disadvantage of the way we have defined crossing is that it is possible to find paths π ∈ Π and π ∈Π with σ π <σπ, π(σ π ) <π(σ π ), andπ(σπ) < π(σ π ), such that π crossesπ from left to right at no time in (σ π ,σπ). The next lemma shows, however, that such pathologies do not happen for left-most and dual right-most paths.
Proof By [SS08, Lemma 3.4 (b)] it suffices to prove the statements for paths with deterministic starting points. Set which implies our claim. By symmetry between forward and dual, and between left and right paths, we also have r(τ + n ) =l(τ + n ) for some n ↓ 0.
The next proposition says that the sets of crossing points of (ˆ l , r ) and ( l ,ˆ r ), of separation points of ( l , r ) and (ˆ l ,ˆ r ), and of points of type (p, pp) s /(p, pp) s in /ˆ as defined in Section 1.3, all coincide.

Proposition 2.6. [Separation points]
Almost surely for each z ∈ 2 , the following statements are equivalent: (i) z is a crossing point of some l ∈ l andr ∈ˆ r , (ii) z is a crossing point of somel ∈ˆ l and r ∈ r , (iii) z is a separation point of some l ∈ l and r ∈ r , (iv) z is a separation point of somel ∈ˆ l andr ∈ˆ r , Moreover, the set {z ∈ 2 : z is of type (p, pp) s in } is a.s. countable.
To prove (i)⇒(iii), let z = (x, t) be a crossing point ofr ∈ˆ r and l ∈ l . By [SS08, Lemma 3.4 (b)], without loss of generality, we can assume thatr and l start from deterministic points with σ l < t < σr . By Lemma 2.2 and the fact that paths inˆ r cannot cross, there exists anr ∈ˆ r (z) and > 0 such that l ≤r on [t − , t]. By Lemma 2.5, we can find s ∈ (t − , t) such that l(s) <r (s). Now any path r ∈ r started at a point ( y, s) with l(s) < x <r (s) is confined between l andr , hence passes through z. Since r cannot crossr we haver ≤ r on [t,σr ]. Since l and r spend positive Lebesgue time together whenever they meet by [SS08, Prop. 3.6 (c)], while r andr spend zero Lebesgue time together by [SS08, Prop. 3.2 (d)], z must be a separation point of l and r.
To prove (iii)⇒(v), let z = (x, t) be a separation point of l ∈ l and r ∈ r so that l(s) < r(s) on (t, t + ] for some > 0. Without loss of generality, we can assume that l and r start from deterministic points with σ l , σ r < t. Chooset ∈ (t, t + ] ∩ where ⊂ is some fixed, deterministic countable dense set. By Lemma 2.2, there exist uniquet > τ 1 > · · · > τ M > σ l , For each point ( y, s) with l(s) < y <r(s) we can find a path r ∈ r ( y, s) that is confined between l andr, so using the fact that r is closed we see that there exists a path r ∈ r (z) that is confined between l andr. By Lemma 2.5, there exist n ↓ 0 such that l(t + n ) =r(t + n ), so l ∼ z out r . Similarly, there exists l ∈ l starting from z which is confined betweenl and r and satisfies l ∼ z out r. The point z must therefore be of type (1, 2) l in l and of type (1, 2) r in r , and hence of type (p, pp) s in .
To prove (v)⇒(ii), let l ∈ l and r ∈ r be the left-most and right-most paths separating at z and let r ∈ r (z) and l ∈ l (z) be the right-most and left-most paths such that l ∼ z out r and l ∼ z out r. Then, by [SS08, Prop. 3.2 (c) and Prop. 3.6 (d)], anyl ∈ˆ l started in a point z = (x , t ) with t < t and r (t ) < x < l (t ) is contained between r and l , hence must pass through z. Sincê The fact that the set of separation points is countable, finally, follows from the fact that by [SS08, Lemma 3.4 (b)], each separation point between some l ∈ l and r ∈ r is the separation point of some l ∈ l ( ) and r ∈ r ( ), where is a fixed, deterministic, countable dense subset of 2 .

Relevant separation points
In a sense, separation points are the most important points in the Brownian net, since these are the points where paths have a choice whether to turn left or right. In the present section, we prove that for deterministic times s < u, there are a.s. only locally finitely many separation points at which paths in starting at time s have to make a choice that is relevant for determining where they end up at time u.
We start with a useful lemma.
Proof Part (b) follows from the steering argument used in [SS08, Lemma 4.7]. To prove part (a), choose x Since paths in the Brownian net do not enter wedges from outside (see Theorem 1.4 (b2)), one hasr n ≤ π ≤l n andr n <l n on (s, u). By monotonicity,r n ↑r andl n ↓l for somer ∈ˆ r (x − , u) andl ∈ˆ l (x + , u). The claim now follows from the nature of convergence of paths in the Brownian web (see [SS08, Lemma 3.4 (a)]).
The next lemma introduces our objects of interest.

Lemma 2.8. [Relevant separation points]
Almost surely, for each −∞ ≤ s < u ≤ ∞ and z = (x, t) ∈ 2 with s < t < u, the following statements are equivalent: (i) There exists a π ∈ starting at time s such that π(t) = x and z is the separation point of some l ∈ l and r ∈ r with l < r on (t, u).
(ii) There exists aπ ∈ starting at time u such thatπ(t) = x and z is the separation point of somê l ∈ l andr ∈ˆ r withr <l on (s, t).
Proof By symmetry, it suffices to prove (i)⇒(ii). If z satisfies (i), then by Lemma 2.7, there exists â π ∈ starting at time u such thatπ(t) = x, and there existl ∈ l (z),r ∈ r (z) such thatr <l on (s, t). Since z is the separation point of some l ∈ l and r ∈ r , by Proposition 2.6, z is also the separation point of somel ∈ l andr ∈ r . Again by Proposition 2.6, z is of type (p, pp) s , hence we must havel =l andr =r on [−∞, t].
If z satisfies the equivalent conditions from Lemma 2.8, then, in line with the definition given in Section 1.5, we say that z is an (s, u)-relevant separation point. We will prove that for deterministic S < U, the set of (S, U)-relevant separation points is a.s. locally finite. Let Φ(x) := 1 2π x −∞ e − y 2 /2 d y denote the distribution function of the standard normal distribution and set Proof It suffices to prove the statement for deterministic S < s < u < U; the general statement then follows by approximation. Set For n ≥ 1, set D n := {S + k(U − S)/n : 0 ≤ k ≤ n − 1}, and for t ∈ (S, U) write t n := sup{t ∈ D n : t ≤ t}. By our previous remarks and the equicontinuity of the Brownian net, for each z = (x, t) ∈ R S,U there exist (x n , y n ) ∈ Q t n , t n +1/n such that x n → x and y n → x. It follows that for any S < s < u < U and a < b, and therefore, by Fatou, where in the last step we have used (2.43) and Riemann sum approximation. This proves the inequality ≤ in (2.40). In particular, our argument shows that the set R S,T ∩ ((a, b) × (S, T )) is a.s. finite.
By our previous remarks, each point (x, y) ∈ Q t,t+1/n gives rise to an (S, U)-relevant separation point z ∈ × (t, t + 1/n). To get the complementary inequality in (2.40), we have to deal with the difficulty that a given z may correspond to more than one (x, y) ∈ Q t,t+1/n . For δ > 0, set and define F δ t similarly. By Lemma 2.10 below, for each > 0, we can find δ > 0 such that for all t ∈ [s, u]. Arguing as before, we find that for all s ≤ t ≤ t ≤ u. Similarly, by symmetry between forward and dual paths, (2.49) where l (x,t) and r (x,t) are the a.s. unique left-most and right-most paths starting from (x, t). Then, by (2.48) and (2.49), for each > 0 we can choose δ > 0 and K > 0 such that (2.51) By the equicontinuity of the net, (2.52) Since the random variables on the right-hand side of (2.52) are bounded from above by the integrable random variable we can take expectations on both sides of (2.52) and let → 0, to get the lower bound in (2.40).
The final statement of the proposition follows by observing that the integral on the right-hand side To complete the proof of Proposition 2.9, we need the following lemma. (2.54) the f δ are continuous functions on (0, ∞] decreasing to zero, hence lim δ↓0 sup t∈K f δ (t) = 0 for each compact K ⊂ (0, ∞].
The following simple consequence of Proposition 2.9 will often be useful.

Lemma 2.11. [Local finiteness of relevant separation points]
Almost surely, for each −∞ ≤ s < u ≤ ∞, the set R s,u of all (s, u)-relevant separation points is a locally finite subset of × (s, u).
Proof Let be a deterministic countable dense subset of . Then, by Proposition 2.9, R s,u is a locally finite subset of × [s, u] for each s < u, s, u ∈ . For general s < u, we can choose s n , u n ∈ such that s n ↓ s and u n ↑ u. Then R s n ,u n ↑ R s,u , hence R s,u is locally finite.
Proof It suffices to prove the statement for deterministic times. In that case, the expectation of the quantity in (2.55) is given by Part (b) of the next lemma is similar to Lemma 2.5.
By [SS08, Prop. 1.8],π ≤l 0 on (−∞,t], so M ≥ 1 and τ 1 ≥ t. If τ 1 = t we are done. Otherwise, we claim thatπ ≤l 1 on (−∞, τ 1 ]. To see this, assume thatl 1 (s) <π(s) for some s < τ 1 . Then we can start a left-most path l betweenl 1 andπ. By what we have proved in part (a) and [SS08, Prop. 3.2 (c)], l is contained byl 1 andπ, hence l and r form a wedge of ( l , r ) which by the characterization ofˆ using wedges (Theorem 1.4 (b2)) cannot be entered byπ, which yields a contradiction. This shows thatπ ≤l 1 on (−∞, τ 1 ]. It follows that M ≥ 2 and τ 2 ≥ t. Continuing this process, using the finiteness of M , we see that τ i = t for some i = 1, . . . , M . Conversely, if r(s) =π(s) for some s ∈ (σ r ,σπ) such that z := (r(s), s) is of type (p, pp) s , then by Lemma 2.7 (b), the pathπ is contained between the left-most and right-most paths starting at z that are not continuations of incoming left-most and right-most paths. It follows that any incoming right-most path r at z must satisfyπ ≤ r on [s,σπ]. This shows that t is the first separation point that r meets onπ.
We claim that ∀s ∈ and π ∈ (z) s.t. π(s) =π(s) and π ≤π on [t, s] ∃π ∈ (z) s.t. π = π on [t, s] and π ≤π on [t,σπ]. (2.60) To prove this, set π 0 := π and s 0 := s and observe that (π 0 (s 0 ), s 0 ) is a separation point where some right-most path r crossesπ. Let r be the outgoing right-most path at (π 0 (s 0 ), s 0 ) that is not equivalent to r. By Lemma 2.7 (b), r ≤π on [s 0 , s 0 + ] for some > 0. Let π 1 be the concatenation of π 0 on [t, s 0 ] with r on [s 0 , ∞]. Since the Brownian net is closed under hopping [SS08, Prop. 1.4], using the structure of separation points, it is not hard to see that π 1 ∈ . (Indeed, we may hop from π 0 onto the left-most path entering (π 0 (s 0 ), s 0 ) at time s and then hop onto r at some time s + and let ↓ 0, using the closedness of .) The definition of and Lemma 2.13 (b) imply that either π 1 ≤π on [t,σπ] or there exists some s 1 ∈ , s 1 > s 0 such that r crossesπ at s 1 . In that case, we can continue our construction, leading to a sequence of paths π n and times s n ∈ such that π n ≤π on [t, s n ]. Since is locally finite, either this process terminates after a finite number of steps, or s n ↑σπ. In the latter case, using the compactness of , any subsequential limit of the π n gives the desired path π . We next claim ⊂ (r z,π ), where the latter is defined as in (1.11). Indeed, if s ∈ , then (π(s), s) is of type (p, pp) s by Lemma 2.13 (b). Moreover, we can find some π ∈ (z) such that π(s) =π(s) and π ≤π on [t, s]. By (2.60), we can modify π on [s,σπ] so that it stays on the left ofπ, hence π ≤ r z,π by the maximality of the latter, which implies that r z,π (s) =π(s) hence s ∈ (r z,π ).

Set
:= ∪ {t,σπ, ∞} ifσπ is a cluster point of and := ∪ {t, ∞} otherwise. Let s, u ∈ − satisfy s < u and (s, u) ∩ = . Let be some fixed, deterministic countable dense subset of . By Lemma 2.12 we can choose t n ∈ such that t n ↓ s and r z,π (t n ) <π(t n ). Choose r n ∈ r (r z,π (t n ), t n ) such that r z,π ≤ r n on [t n , ∞]. By [SS08, Lemma 8.3], the concatenation of r z,π on [t, t n ] with r n on [t n , ∞] is a path in , hence r n can crossπ only at times in , and therefore r n ≤π on [t n , u]. Using the compactness of r , let r ∈ r be any subsequential limit of the r n . Then r z,π ≤ r on [s, ∞], r ≤π on [s, u], and, since is closed, the concatenation of r z,π on [t, s] and r on [s, ∞] is a path in .
To complete our proof we must show that ⊃ (r z,π ). To see this, observe that if s ∈ (r z,π ), then, since r z,π is a concatenation of right-most paths, by Lemma 2.13 (b), some right-most path crossesπ at s, hence s ∈ .

Maximal T -meshes
Let T := {π ∈ : σ π = T } denote the set of paths in the Brownian net starting at a given time T ∈ [−∞, ∞] and let N T be its image set, defined in (1.15). We call N T the image set of the Brownian net started at time T . In the present section, we will identify the connected components of the complement of N T relative to {(x, t) ∈ R 2 c : t ≥ T }. The next lemma is just a simple observation.

Lemma 3.1. [Dual paths exit meshes through the bottom point] If M (r, l) is a mesh with bottom point z = (x, t) andπ ∈ˆ starts in M (r, l), then r(s) ≤π(s) ≤ l(s) for all s ∈ [t,σπ].
Proof Immediate from Lemma 2.13 (a).
We will need a concept that is slightly stronger than that of a mesh.

Definition 3.2. [ * -meshes]
A mesh M (r, l) with bottom point z = (x, t) is called a * -mesh if there existr ∈ˆ r andl ∈ˆ l witĥ r ∼ z inl , such that r is the right-most element of r (z) that passes on the left ofr and l is the left-most element of l (z) that passes on the right ofl.

Lemma 3.3. [Characterization of * -meshes]
Almost surely for all z = (x, t) ∈ 2 andr ∈ˆ r ,l ∈ˆ l such thatr ∼ z inl , the set is a * -mesh with bottom point z. Conversely, each * -mesh is of this form.
Remark A simpler characterization of * -meshes (but one that is harder to prove) is given in Lemma 3.13 below. Once Theorem 1.7 is proved, it will turn out that if a mesh is not a * -mesh, then its bottom point must be of type (o, ppp). (See Figure 3.)

Proof of Lemma 3.3
Let z = (x, t) ∈ 2 andr ∈ˆ r ,l ∈ˆ l satisfyr ∼ z inl . Let r be the right-most element of r (z) that passes on the left ofr and let l be the left-most element of l (z) that passes on the right ofl. Then, obviously, M (r, l) is a * -mesh, and each * -mesh is of this form. We claim that M (r, l) = M z (r,l).
To see that M (r, l) ⊃ M z (r,l), note that if z = (x , t ) ∈ M (r, l) and t > t, then either there exists anr ∈ˆ r (z ) that stays on the left of r, or there exists anl ∈ˆ l (z ) that stays on the right of l; in either case, z ∈ M z (r,l).
To see that M (r, l) ⊂ M z (r,l), assume that z = (x , t ) ∈ M (r, l). Since each path inˆ (z ) is contained by the left-most and right-most dual paths starting in z , it suffices to show that eacĥ r ∈ˆ r (z ) satisfiesr ≤r on [t, t + ] for some > 0 and eachl ∈ˆ l (z ) satisfiesl ≤l on [t, t + ] for some > 0. By symmetry, it suffices to prove the statement forr . So imagine that r <r on (t, t ). Then there exists an r ∈ r (z) that stays betweenr andr, contradicting the fact that r is the right-most element of r (z) that passes on the left ofr.

(a) A set of the form (3.61), with z = (x, t), is a maximal T -mesh if and only if t = T or if t > T
andr <l on (T, t).

(b) The maximal T -meshes are mutually disjoint, and their union is the set
Proof Let M (r, l) and M (r , l ) be * -meshes with bottom points z = (x, t) and z = (x , t ), respectively, with t ≥ t ≥ T . Then either M (r, l) and M (r , l ) are disjoint, or there exists a z ∈ M (r, l) ∩ M (r , l ). In the latter case, by Lemma 3.3, anyr ∈ˆ r (z ) andl ∈ˆ l (z ) must pass through z and z (in this order) and we have M We have just proved that T -meshes are either disjoint or one is contained in the other, so maximal T -meshes must be mutually disjoint. Let O T := {(x, t) ∈ R 2 c : t ≥ T }\N T . It is easy to see that O T ⊂ × (T, ∞). Consider a point z = (x, t) ∈ 2 with t > T . If z ∈ O T , then by Lemma 2.7 there existr ∈ˆ r (z) andl ∈ˆ l (z) with the property that there does not exist a z = (x , t ) with t ≥ T such thatr ∼ z inl , hence by Lemma 3.3, z is not contained in any T -mesh. On the other hand, if z ∈ O T , then by Lemma 2.7 (b) and the nature of convergence of paths in the Brownian web (see [SS08, Lemma 3.4 (a)]), there existr ∈ˆ r andl ∈ˆ l starting from points (x − , t) and (x + , t), respectively, with x − < x < x + , such thatr(s) =l(s) for some s ∈ (T, t). Now, setting u := inf{s ∈ (T, t) :r(s) =l(s)} and z := (r(u), u), by Lemma 3.3, M z (r,l) is a maximal T -mesh that contains z.

Reversibility
Recall from (1.15) the definition of the image set N T of the Brownian net started at time T . It follows from [SS08, Prop. 1.15] that the law of N −∞ is symmetric with respect to time reversal. In the present section, we extend this property to T > −∞ by showing that locally on × (T, ∞), the law of N T is absolutely continuous with respect to its time-reversed counterpart. This is a useful property, since it allows us to conclude that certain properties that hold a.s. in the forward picture also hold a.s. in the time-reversed picture. For example, meeting and separation points have a similar structure, related by time reversal. (Note that this form of time-reversal is different from, and should not be confused with, the dual Brownian net.) We write µ ν when a measure µ is absolutely continuous with respect to another measure ν, and µ ∼ ν if µ and ν are equivalent, i.e., µ ν and ν µ. (3.62) Proof By the reversibility of the backbone, it suffices to prove that and (3.69)

Classification according to incoming paths
In this section, we give a preliminary classification of points in the Brownianb net that is entirely based on incoming paths. Note that if there is an incoming path π ∈ at a point z = (x, t), then z ∈ N T for some T < t, where N T is the image set of the Brownian net started at time T , defined in (1.15). Therefore in this section, our main task is to classify the special points of N T .
For a given T ∈ [−∞, ∞), let us say that a point z = ( Points that are isolated from the right are defined similarly, with the supremum replaced by an infimum and both inequality signs reversed. We now give a classification of points in 2 based on incoming paths in the Brownian net. Recall Definition 2.3 of intersection, meeting, and separation points.

Definition 3.8. [Classification by incoming paths]
We say that a point z = (x, t) ∈ 2 is of type if there is an incoming π ∈ at z, but there is no incoming π ∈ l ∪ r at z, if there is an incoming l ∈ l at z, but there is no incoming r ∈ r at z, if there is an incoming r ∈ r at z, but there is no incoming l ∈ l at z, (C s ) if z is a separation point of some l ∈ l and r ∈ r , (C m ) if z is a meeting point of some l ∈ l and r ∈ r , (C p ) if there is an incoming l ∈ l at z and an incoming r ∈ r at z, and z is not of type (C s ) or (C m ).
Note that by Lemma 2.7, for any T < t such that z ∈ N T , points of types (C l ), (C r ), and (C n ) are either not isolated from the left, or not isolated from the right, or both. In view of this, we call these points cluster points. Points of the types (C l ) and (C r ) are called one-sided cluster points and points of type (C n ) two-sided cluster points. Proposition 3.11 below shows that, among other things, cluster points are the limits of nested sequences of excursions between left-most and right-most paths.
The main result of this section is the following. (c) Each deterministic z ∈ 2 is a.s. of type (C o ).

Proof of Lemma 3.9 (b) and (c)
If t ∈ is deterministic, and T < t, then by [SS08, Prop 1.12], the set N T ∩ ( × {t}) is locally finite. In particular, if there is an incoming path π ∈ at a point z ∈ ×{t}, then z is isolated from the left and right. Since each meeting or separation point of some l ∈ l and r ∈ r is the meeting or separation point of some left-most and right-most path chosen from a fixed, deterministic countable dense set, and since paths started at deterministic starting points a.s. do not meet or separate at deterministic times, z must be of type (C p ).
Before proving Lemma 3.9 (a), we first establish some basic properties for each type of points in Definition 3.8. We start with a definition and a lemma.
By definition, we say that two paths π, π ∈ Π make an excursion from each other on a time interval (s, u) if σ π , σ π < s (note the strict inequality), π(s) = π (s), π = π on (s, u), and π(u) = π (u). The next lemma says that excursions between left-most and right-most paths are rather numerous.
Proof Choose s < u such that {l(t) : t ∈ [s, u]} ⊂ O and choose some t from a fixed, deterministic countable dense subset of such that t ∈ (s, u). By Lemma 3.9 (b), there exists a unique incoming path r ∈ r at (l(t), t), and r does not separate from l at time t. Since by [SS08, Prop. 3.6 (b)], the set {v : r(v) = l(v)} is nowhere dense, we can find u 1 > s 1 > u 2 > s 2 > · · · such that u n ↓ t and r makes an excursion from l during the time interval (s n , u n ) for each n ≥ 1. By the local equicontinuity of the Brownian net, we can choose n large enough such that {(x, t) : t ∈ [s n , u n ], x ∈ [l(t), r(t)]} ⊂ O.

Proposition 3.11. [Structure of points with incoming paths]
(a) If z = (x, t) is of type (C l ), then there exist l ∈ l and r n ∈ r (n ≥ 1), such that l(t) = x < r n (t), each r n makes an excursion away from l on a time interval (s n , u n ) t, [s n , u n ] ⊂ (s n−1 , u n−1 ), s n ↑ t, u n ↓ t, and r n (t) ↓ x. (See Figure 6). By symmetry, an analogous statement holds for points of type (C r ).
(b) If z = (x, t) is of type (C n ), then there exist l 1 ∈ l , r 2 ∈ r , l 3 ∈ l , . . ., such that l 2n+1 (t) < x < r 2n (t), each path (l n for n odd, r n for n even) in the sequence makes an excursion away from the previous path on a time interval (s n , u n ) t, [s n , u n ] ⊂ (s n−1 , u n−1 ], s n ↑ t, u n ↓ t, l 2n+1 (t) ↑ x, and r 2n (t) ↓ x (the monotonicity here need not be strict).
(c) If z = (x, t) is of type (C s ), then for each T < t with z ∈ N T , there exist maximal T -meshes M (r, l) and M (r , l ) with bottom times strictly smaller than t and top times strictly larger than t, and a maximal T -mesh M (r , l ) with bottom point z, such that l ∼ z in r , l ∼ z out r , and l ∼ z out r . , l ) with top point z, such that l ∼ z in r , l ∼ z in r , and l ∼ z out r . (e) If z = (x, t) is of type (C p ), then for each T < t with z ∈ N T , there exist maximal T -meshes M (r, l) and M (r , l ) with bottom times strictly smaller than t and top times strictly larger than t, such that l ∼ z in r and l ∼ z out r . Proof (a): Let l be an incoming left-most path at z and choose T < t such that l ⊂ N T . Choose t n from some fixed, deterministic countable dense subset of such that t n ↑ t. By Lemma 3.9 (b), for each n there is a unique incoming path r n ∈ r at (l(t n ), t n ). By assumption, r n does not pass through z, hence r n makes an excursion away from l on a time interval (s n , u n ) with t n ≤ s n < t < u n ≤ ∞. Clearly s n ↑ t and u n ↓ u ∞ for some u ∞ ≥ t. We claim that u ∞ = t. Indeed, if u ∞ > t, then (l(s n ), s n ) (n ≥ 1) are (s 1 , u ∞ )-relevant separation points, hence the latter are not locally finite on × (s 1 , u ∞ ), contradicting Lemma 2.11. By going to a subsequence if necessary, we can asssure that s n−1 < s n and u n < u n−1 . The fact that r n (t) ↓ x follows from the local equicontinuity of the Brownian net.

(d) If z = (x, t) is of type (C m ), then for each T < t with z ∈ N T , there exist maximal T -meshes M (r, l) and M (r , l ) with bottom times strictly smaller than t and top times strictly larger than t, and a maximal T -mesh M (r
(b): Let π ∈ be an incoming path at z and choose T < t such that π ⊂ N T . By Lemma 2.7 (a), there existr ∈ˆ r (z) andl ∈ˆ l (z) such thatr <l on (T, t) andr ≤ π ≤l on [T, t]. Choose an arbitrary l 1 ∈ l starting from z 1 = (x 1 , s 1 ) with s 1 ∈ (T, t) and x 1 ∈ (r(s 1 ),l(s 1 )). Since there is no incoming left-most path at z, and since l 1 cannot crossl, the path l 1 must crossr at some time s 2 < t. By Proposition 2.6, there exists r 2 ∈ r such that (l 1 (s 2 ), s 2 ) is a separation point of l 1 and r 2 , and r 2 lies on the right ofr. Since there is no incoming right-most path at z, and since r 2 cannot crossr, the path r 2 must crossl at some time s 3 < t, at which a path l 3 ∈ l separates from r 2 , and so on. Repeating this procedure gives a sequence of paths l 1 ∈ l , r 2 ∈ r , l 3 ∈ l , . . . such that the n-th path in the sequence separates from the (n − 1)-th path at a time s n ∈ (s n−1 , t), and we have l 1 (t) ≤ l 3 (t) ≤ · · · < x < · · · ≤ r 4 (t) ≤ r 2 (t). By the a.s. local equicontinuity of l ∪ r and l ∪ˆ r , it is clear that s n ↑ t, l 2n+1 (t) ↑ x and r 2n (t) ↓ x. Let u n := inf{s ∈ (s n , ∞) : l n−1 = r n } if n is even and u n := inf{s ∈ (s n , ∞) : r n−1 = l n } if n is odd. Then u n ↓ u ∞ ≥ t. The same argument as in the proof of part (a) shows that u ∞ = t.
To prepare for parts (c)-(e), we note that if there exist incoming paths l ∈ l and r ∈ r at a point z ∈ 2 and T < t is such that z ∈ N T , then by Lemma 3.7, z is isolated from the left and from the right, hence by Proposition 3.5 (b), there exist maximal T -meshes M (r, l) and M (r , l ) with bottom times strictly smaller than t and top times strictly larger than t, such that l(t) = x = r (t). We now prove (c)-(e).
(c): If z is a separation point of some left-most and right-most path, then by Proposition 2.6, z must be a separation point of the paths l and r mentioned above, and there exist paths r ∈ r (z) and l ∈ l (z) such that l ∼ z out r , and l ∼ z out r , and z is the bottom point of the mesh M (r , l ). Note that M (r , l ) is a maximal T -mesh by Definition 3.4. (e): If z is not a separation or meeting point of any left-most and right-most path, then the paths l and r mentioned above must satisfy l ∼ z in r and l ∼ z out r .
Proof of Lemma 3.9 (a) The statement that each point in 2 belongs to exactly one of the types (C o ), (C n ), (C l ), (C r ), (C s ), (C m ), and (C p ) is entirely self-evident, except that we have to show that a point cannot at the same time be a separation point of some l ∈ l and r ∈ r , and a meeting point of some (possibly different) l ∈ l and r ∈ r . This however follows from Proposition 2.6.
It follows from parts (b) and (c) of the lemma (which have already been proved) that points of the types (C o ) and (C p ) occur. Obviously, points of types (C s ) and (C m ) occur as well. To prove the existence of cluster points, it suffices to establish the existence of nested sequences of excursions, which follows from Lemma 3.10.
Proposition 3.11 yields a useful consequence.

Proof
We start by showing that if z ∈ 2 is a separation (resp. meeting) point of two paths π, π ∈ , then z is of type (C s ) (resp. (C m )). Thus, we need to show that a.s. for any z ∈ 2 , if π, π ∈ T are incoming paths at z, one has π ∼ z in π if z is not a meeting point, and π ∼ z in π if z is not a separation point.
If z is not a cluster point, these statements follow from the configuration of maximal T -meshes around z as described in Proposition 3.11 (c)-(e).
If z is a two-sided cluster point, then any path π ∈ must pass through the top points of the nested excursions around z, hence all paths in (z) are equivalent as outgoing paths at z. If z is a onesided cluster point, then by [SS08, Prop. 1.8], l ≤ π on [t, ∞) for all incoming net paths at z, hence all incoming net paths must pass through the top points of the nested excursions and therefore be equivalent as outgoing paths.
Our previous argument shows that at a cluster point z = (x, t), for any T < t such that z ∈ N T , all paths π ∈ Π such that π(t) = x and π ⊂ N T are equivalent as outgoing paths at z. By local reversibility (Proposition 3.6), it follows that all paths π ∈ Π such that π(t) = x and π ⊂ N T are equivalent as ingoing paths at z. (Note that reversing time in N T does not change the fact that z is a cluster point.) This completes the proof that if z ∈ 2 is a separation (resp. meeting) point of two paths π, π ∈ , then z is of type (C s ) (resp. (C m )). If z is of type (C s ), then by Proposition 2.6, z is of type (p, pp) s . If z is of type (C m ), then by Proposition 3.11 (d), there are exactly two incoming left-right pairs at z, and there is at least one outgoing left-right pair at z. Therefore, z must be of type (2, 1) in both l and r , hence there are no other outgoing paths in l ∪ r at z, so z is of type (pp, p).
Remark Lemma 3.12 shows in particular that any meeting point of two paths π, π ∈ l ∪ r is of type (pp, p). This fact can be proved by more elementary methods as well. Consider the Markov process (L, R, L ) given by the unique weak solutions to the SDE where B l , B r , B s , and B l are independent Brownian motions, and we require that L t ≤ R t for all t ≥ 0. Set τ = inf{t ≥ 0 : R t = L t }. (3.75) Then the claim follows from the fact that the process started in (L 0 , R 0 , L 0 ) = (0, 0, ) satisfies lim →0 (0,0, ) L τ = R τ = 1, (3.76) which can be shown by a submartingale argument. Since this proof is of interest on its own, we give it in Appendix A.2.
The next lemma is a simple consequence of Lemma 3.12.

Lemma 3.13. [Characterization of * -meshes]
A mesh M (r, l) with bottom point z = (x, t) is a * -mesh if and only if there exists no π ∈ l (z)∪ r (z), π = l, r, such that r ≤ π ≤ l on [t, t + ] for some > 0.
Proof If M (r, l) is a mesh with bottom point z = (x, t) and there exists some l = l ∈ l (z) such that r ≤ l ≤ l on [t, t + ] for some > 0, then r < l on [t, t + ] by [SS08, Prop. 3.6 (a)], hence by Lemma 3.1 we can findr ∈ˆ r andl ∈ˆ l such that r ≤r ≤l ≤ l on [t, t + ], which implies that M (r, l) is not a * -mesh. By symmetry, the same is true if there exists some r = r ∈ r (z) such that r ≤ r ≤ l on [t, t + ].
For any mesh M (r, l), by Lemma 3.1, we can findr ∈ˆ r andl ∈ˆ l such that r ≤r ≤l ≤ l on [t, t + ] for some > 0. We claim thatr ∼ z inl if M (r, l) satisfies the assumptions of Lemma 3.13; from this, it then follows that M (r, l) is a * -mesh. To prove our claim, assume thatr ∼ z inl . Then, by Lemma 3.12, there existl ∈ˆ l andr ∈ˆ r such thatr ≤l <r ≤l on [t, t + ] for some > 0. By [SS08, Prop 3.6 (d)], this implies that there exist l ∈ l (z) such thatl ≤ l ≤r on [t, t + ], hence M (r, l) does not satisfy the assumptions of Lemma 3.13.

Special times
For any closed set K ⊂ 2 , set It has been proved in [SS08, Theorem 1.11] that for any closed A ⊂ , the process (ξ A×{0} t ) t≥0 is a Markov process taking values in the space of closed subsets of . It was shown in [SS08, Prop 1.12] that ξ A×{0} t is a.s. a locally finite point set for each deterministic t > 0. It was claimed without proof there that there exists a dense set of times t > 0 such that ξ A×{0} t is not locally finite. Indeed, with the help of Lemma 3.10, we can prove the following result.

Proposition 3.14. [No isolated points]
Almost surely, there exists a dense set ⊂ (0, ∞) such that for each t ∈ and for each closed A ⊂ , the set ξ A×{0} t contains no isolated points.
Proof We claim that it suffices to prove the statement for A = . To see this, suppose that x ∈ ξ A×{0} t is not isolated from the left in ξ ×{0} t for some t > 0. It follows from the characterization of the Brownian net using meshes (see Theorem 1.4 (b3)) that the pointwise infimum π := inf{π ∈ (A×{0}) : π (t) = x} defines a path π ∈ (A×{0}). Let l be the left-most element of l (π(0), 0). By Lemma 3.7, there is no incoming left-most path at (x, t), hence we must have l(t) < x. Since x is not isolated from the left in ξ ×{0} t , there are π n ∈ starting at time 0 such that π n (t) ∈ (l(t), x) for each n, and π n (t) ↑ x. Now each π n must cross either l or π, so by the fact that the Brownian net is closed under hopping ([SS08, Prop. 1.4]), x is not isolated from the left in ξ A×{0} t .
To prove the proposition for A = , we claim that for each 0 < s < u and for each n ≥ 1, we can find s ≤ s < u ≤ u such that (3.78) To show this, we proceed as follows. If (3.78) holds for s = s and u = u we are done. Otherwise, we can find some t 1 ∈ (s, u) and In particular, x 1 is an isolated point so there is an incoming l 1 ∈ l at (x 1 , t 1 ), hence by Lemma 3.10 we can find an r 1 ∈ r such that r 1 makes an excursion from l 1 during a time interval (s 1 , u 1 ) with s < s 1 < u 1 < u, with the additional property that x 1 − 1 2n ≤ l 1 < r 1 ≤ x 1 + 1 2n on (s 1 , u 1 ). Now either we are done, or there exists some t 2 ∈ (s 1 , u 1 ) and In this case, we can find l 2 ∈ l and r 2 ∈ r making an excursion during a time interval (s 2 , u 2 ), with the property that l 2 and r 2 stay in [x 2 − 1 2n , x 2 + 1 2n ]. We iterate this process if necessary. Since x m+1 must be at least a distance 1 2n from each of the points x 1 , . . . , x m , this process terminates after a finite number of steps, proving our claim.
By what we have just proved, for any 0 < s < u, we can find s ≤ s 1 ≤ s 2 ≤ · · · ≤ u 2 ≤ u 1 ≤ u such that (3.79) Necessarily n (s n , u n ) = {t} for some t ∈ , and we conclude that ξ ×{0} t contains no isolated points.

Excursions between forward and dual paths
Excursions between left-most and right-most paths have already been studied briefly in Section 3.3. In this section, we study them in more detail. In particular, in order to prove the existence of points from groups (4) and (5) of Theorem 1.7, we will need to prove that, for a given left-most path l and a dual left-most pathl that hits l from the left, there exist nested sequences of excursions of right-most paths away from l, such that each excursion interval contains an intersection point of l andl. As a first step towards proving this, we will study excursions between l andl.
The proof of Proposition 4.1 is somewhat long and depends on excursion theory. We start by studying the a.s. unique left-most path started at the origin. Let l ∈ l (0, 0) be that path and fix some deterministic u > 0. By the structure of special points of the Brownian web (see, e.g., [SS08, Lemma 3.3 (b)]),ˆ l (l(u), u) contains a.s. two paths, one on each side of l. Letl be the one on the left of l. Set (4.82) Our first lemma, the proof of which can be found below, says that X is standard Brownian motion reflected at the origin.

Lemma 4.2. [Reflected Brownian motion]
There exists a standard Brownian motion B = (B t ) t≥0 such that Extend X t to all t ≥ 0 by (4.83), and put Then the intervals of the form (S τ− , S τ ) with τ ∈ are precisely the intervals during which X makes an excursion away from 0. Define a random point set N on (0, ∞) 2 by The following fact is well-known. Proof It follows from Brownian scaling that (S τ ) τ≥0 is a stable subordinator with exponent 1/2, and this implies that ν(dh) = ch −3/2 dh for some c > 0. The precise formula (4.86) can be found in [KS91, Sect. 6.2.D].
To explain the main idea of the proof of Proposition 4.1, we formulate one more lemma, which will be proved later. and observe that where U τ is as in (4.88). Note that modulo translation and time reversal, the triple (R τ , L τ ,L τ ) is just (r [U τ ] , l,l) during the time interval [U τ − h τ , U τ ] whenl and l make an excursion away from each other, andr [U τ ] coalesces withl upon first hittingl. Let N 3 u be the random subset of 3 ×(0, Ψ u ) defined by (4.90) We will show that N 3 u can be extended to a Poisson point process N 3 on 3 × (0, ∞). Proposition 4.1 will then be established by showing that N 3 u contains infinitely many points (R τ , L τ ,L τ , h τ , τ) with the property thatR τ crosses L τ before h τ . (Note that since we are only interested in the time when r [U τ ] crosses l, there is no need to followr [U τ ] after it meetsl. This is why we have definedR τ t in such a way that it coalesces withL τ t .)

Proof of Lemma 4.2 Set
(4.91) We know from [STW00] (see also Lemma 2.1 and [SS08, formula (6.17)]) that conditioned on l, the dual pathl is distributed as a Brownian motion with drift −1, Skorohod reflected off l. Therefore, on [0, u], the paths L andL are distributed as solutions to the SDE where B l t and Bˆl t are independent, standard Brownian motions, Φ t is a nondecreasing process, increasing only when L t =L t , and one has L t ≤L t for all t ∈ [0, u]. Extending our probability space if necessary, we may extend solutions of (4.92) so that they are defined for all t ≥ 0. Set Then B − t and B + t are independent standard Brownian motions, (4.94) where X t ≥ 0 and Ψ t := 1 2 Φ t increases only when X t = 0. In particular, setting B := B − and noting that X in (4.94) solves a Skorohod equation, the claims in Lemma 4.2 then follow.
and we define a point process N 1 on 1 × [0, ∞) by (4.96) The following facts are well-known.

Lemma 4.6. [Excursions between a left and dual left path]
The set N 2 is a Poisson point process on 2 ×[0, ∞) with intensity µ 2 h (d f )ν(dh), where ν is the measure in (4.86) and the µ 2 h are probability measures on 2 h (h > 0) given by where F h is a random variable as in (4.98) and B a Brownian motion independent of F h .
Proof This follows from the fact that, by (4.93) and (4.94), (4.102) where B + is a standard Brownian motion independent of X and Ψ. Note that restrictions of B + to disjoint excursion intervals are independent and that, since Ψ increases only at times t when L t =L t , it drops out of the formulas for L τ andL τ .
Remark General excursion theory tells us how a strong Markov process can be constructed by piecing together its excursions from a singleton. In our situation, however, we are interested in excursions of the process (L t ,L t ) from the set {(x, x) : x ∈ }, which is not a singleton. Formula (4.102) shows that apart from motion during the excursions, the process (L t ,L t ) also moves along the diagonal at times when L t =L t , even though such times have zero Lebesgue measure. (Indeed, it is possible to reconstruct (L t ,L t ) from its excursions and local time in {(x, x) : x ∈ }, but we do not pursue this here.) Together with Lemma 4.6, the next lemma implies that the point process N 3 u on 3 × (0, Ψ u ) defined in (4.90) is a Poisson point process, as claimed.

Lemma 4.7. [Distribution of crossing times]
The paths (R τ ) τ∈ u are conditionally independent given l andl, and their conditional law up to coalescence withL τ is given by the solution to the Skorohod equation where B is a standard Brownian motion, ∆ is a nondecreasing process increasing only whenR τ t = L τ t , T is an independent mean 1/2 exponential random variable, andR τ is subject to the constraints that We will prove Lemmas 4.4 and 4.7 in one stroke. To prove the other statements we use discrete approximation (compare the proof of Lemma 2.1). As in [SS08], we consider systems of branching-coalescing random walks on 2 even with branching probabilities n → 0. Diffusively rescaling space and time as (x, t) → ( n x, 2 n t) then yields the Brownian net in the limit. In the discrete system, we consider the left-most path l n starting at the origin, we choose u n ∈ such that n u n → u and consider the dual left-most pathl n started at time u n at distance one to the left of l n . For each 0 ≤ i ≤ u n , we let i + := inf{ j ≥ i : l n ( j) −l n ( j) = 1} and we letr n (i) denote the position at time i of the dual right-most path started at time i + at distance one on the right of l n . Thenr n is the concatenation of dual right-most paths, started anew immediately on the right of l n each timel n is at distance one from l n . In analogy with the definition ofR τ in (4.89), we setr n (i) :=l n (i) ∨r n (i). Now (l n ,l n ), diffusively rescaled, converges in distribution to (l,l) where l is the left-most path in the Brownian net starting at the origin andl is the a.s. unique dual left-most path starting at (l(u), u) that lies on the left of l. Moreover, the set of times whenl n is at distance one from l n , diffusively rescaled, converges to the set {s ∈ (0, u] :l(s) = l(s)}. (This follows from the fact that the reflection local time ofl n off l n converges to its continuum analogue, and the latter increases wheneverl(s) = l(s).) In the diffusive scaling limit, the pathr n converges to a pathr such that t →r(u − t) is rightcontinuous and is set back to l(u − t) each timel(u − t) meets l(u − t). Between these times, in the same way as in the proof of Lemma 2.1, we see that the conditional law ofr given l andl is as described in Lemma 4.7.
Sincer n is the concatenation of dual right-most paths, we see that at each time s + ∈ + (l,l) there starts at least one dual right-most path that lies on the right of l on [s + − , s + ] for some > 0, completing the proof of Lemma 4.4.
Proof of Proposition 4.1 We first prove the claims for the a.s. unique paths l ∈ l (0, 0) and l ∈ˆ l (l(u), u) such thatl lies on the left of l. For each τ ∈ u , set Let N be the Poisson point process in (4.85), let N u denote the restriction of N to (0, ∞) × (0, Ψ u ), and set Then N u is a thinning of N u , obtained by independently keeping a point (h τ , τ) ∈ N u with probability ρ(h τ ), where ρ : (0, ∞) → (0, ∞) is some function. Indeed, by Lemmas 4.6 and 4.7, ρ(h) has the following description. Pick a random variable F h as in Lemma 4.5, two standard Brownian motions B, B , and a mean 1/2 exponential random variable T , independent of each other. Set (4.106) Let (R , ∆) be the solution to the Skorohod equation reflected to the left off L, and set (4.108) Then (4.110) It follows from Brownian scaling (see (4.98)) that (4.112) By (4.86), it follows that, for some c > 0, i.e., the intensity measure of the thinned Poisson point process is not integrable, hence the set u := {τ : (h τ , τ) ∈ N u } is a dense subset of (0, Ψ u ), hence + (l,l) is dense in (l,l). This completes the proof for the special paths l ∈ l (0, 0) andl ∈ˆ l (l(u), u) such thatl lies on the left of l.
To prove the same statement for l ∈ l (0, 0) andr ∈ˆ r (l(u), u) wherer lies on the left of l, first note that because u is deterministic, Lemma 2.1 implies the existence of such anr. Set and let L be as in (4.91). Then, by Lemma 2.1, L andR are distributed as solutions to the SDE (compare (4.92)) dL t = dB l t − dt, dR t = dBr t + dt + dΦ t , (4.115) where B l t and Br t are independent, standard Brownian motions, Φ t is a nondecreasing process, increasing only when L t =R t , and L t ≤R t for all t ∈ [0, u]. By Girsanov, solutions of (4.115) are equivalent in law to solutions of (4.92), hence we can reduce this case to the case of a dual left-most path. In particular, by what we have already proved, almost surely for each s + ∈ + (l,r) there exists a uniquer [s + ] ∈ˆ r (l(s + ), s + ) that lies on the right of l, and the set + (l,r) := s + ∈ + (l,r) :r [s + ] crosses l at some time in (s − , s + ) (4.116) is a dense subset of (l,r).
By translation invariance, + (l,l) is dense in (l,l) for each l ∈ l started from a point z = (x, t) ∈ 2 andl ∈ˆ l (l(u), u) that lies on the left of l. By [SS08, Lemma 3.4 (b)], we can generalize this to arbitrary l ∈ l . Since any dual left-most path that hits l from the left at a time s must have coalesced with some left-most path started in (l(u), u) for some u ∈ with u > s and lying on the left of l, we can generalize our statement to arbitrary l ∈ l andl ∈ˆ l . The argument for dual right-most paths is the same.

Theorem 5.2. [Classification of points in the Brownian net]
Let be the standard Brownian net and letˆ be its dual. Then, using the classification of Definition 3.8, almost surely, each point in 2 is of one of the following 19 types in /ˆ : (2) (C s )/(C s ); (4) (C l )/(C p ), (C p )/(C l ), (C r )/(C p ), (C p )/(C r ); (5) (C l )/(C l ), (C r )/(C r ); For clarity, we split the proof into three lemmas.

Lemma 5.3. [Forward and dual types]
Almost surely, for all z ∈ 2 : (a) If z is of type (C s ) in , then z is of type (C s ) inˆ .
(d) If z is of type (C l ) in , then z is not of type (C r ) inˆ .
In particular, each point in 2 is of one of the 19 types listed in Theorem 5.2. (d) If z is of type (C p )/(C p ), then z is of type (p, pp) l /(p, pp) l or (p, pp) r /(p, pp) r .
(e) If z is of type (C s )/(C s ), then z is of type (p, pp) s /(p, pp) s .
(f) If z is of type (C l )/(C o ), then z is of type (l, p)/(o, lp).
(g) If z is of type (C l )/(C p ), then z is of type (l, pp) r /(p, lp) r .
(h) If z is of type (C l )/(C l ), then z is of type (l, lp) r /(l, lp) r .
(i) If z is of type (C n )/(C o ), then z is of type (n, p)/(o, lr).

Analogous statements hold for the remaining types in Theorem 5.2 by left-right and forward-backward symmetry.
Proof of Lemma 5.3 Part (a) follows from Proposition 2.6. If z is of type (C m ) in , then by Lemma 3.12, there is a single outgoing left-right pair at z, hence by Lemma 5.1, z is of type (C o ) inˆ , proving part (b). If z is of type (C n ) in , then each outgoing Brownian net path at z must pass through the top points of the nested excursions around z as described in Proposition 3.11 (a). Hence also in this case, there is a single outgoing left-right pair at z, so by Lemma 5.1, z is of type (C o ) inˆ , proving part (c). To prove part (d), suppose that z is of type (C l ) and thatr ∈ˆ r enters z. By Proposition 3.11 (a), there exist right-most paths r n making a sequence of nested excursions away from l. Since each r n forms with l a wedge of ( l , r ) that cannot be entered byr, the latter must satisfyr ≤ l on (t, t + ) for some > 0. Sincer reflects off l, we can find some u > t such thatr(u) < l(u). Now any path inˆ l started in (r(u), l(u)) × {u} must enter z, hence z is not of type (C r ) inˆ . It is easy to check that (a)-(d) rules out all but the 19 types listed in Theorem 5.2.
Proof of Lemma 5.4 It follows from Lemma 3.9 that each deterministic z ∈ 2 is of type ( By the structure of the Brownian web, there exist l ∈ l andl ∈ˆ l such that l ≤l on (σ l ,σl ) and the set (l,l) defined in (4.80) is not empty. Since (l,l) is uncountable and since the set of all crossing points in 2 is countable, there exist lots of points z = (x, t) ∈ (l,l) such that no path in r crosses l and no path in r crossesl at z. For such points, we can find r ∈ r andr ∈ˆ r such that l ≤ r ≤r ≤l on [t − , t + ] for some > 0. Of all types of points listed in Theorem 5.2, only points of type (C p )/(C p ) have incoming paths in l , r ,ˆ r , andˆ l , hence, by Lemma 5.3, z must be of this type.
We are left with the task of establishing the existence of points of types (C l )/(C o ), (C l )/(C p ), and (C l )/(C l ). By Lemma 3.7 and Proposition 3.11 (b), a point z is of type (C l ) in if and only if there is an incoming path l ∈ l at z and there are right-most paths r n making a nested sequence of excursions from l, as described in Proposition 3.11 (b). We need to show that we can choose these nested excursions in such a way that z is of type (C o ), (C p ), or (C l ) inˆ .
Fix a path l ∈ l and let {l n } n∈ be paths inˆ l starting from a deterministic countable dense subset of 2 . Sincel 1 is reflected off l, using Lemma 3.10, we can find a path r 1 ∈ r such that r 1 makes an excursion from l during an interval (s 1 , u 1 ) with l 1 = l on [s 1 , u 1 ]. By the same arguments, we can inductively find paths r n ∈ r such that r n makes an excursion from l during an interval (s n , u n ) with l n = l on [s n , u n ] ⊂ (s n−1 , u n−1 ). We can choose the r n such that n [s n , u n ] = {t} for some t ∈ . Then setting z := (l(t), t) yields a point such that z is of type (C l ) in and no path inˆ l enters z, hence (by Lemma 5.3 (d)), z must be of type (C l )/(C o ).
To construct points of type (C l )/(C p ), we fix paths l ∈ l andr ∈ˆ r withr ≤ l on [σ l ,σr ] such that the set (l,r) defined in (4.80) is not empty. By Proposition 4.8 and Lemma 3.10, we can inductively find paths r n ∈ r such that r n makes an excursion from l during an interval (s n , u n ) with [s n , u n ] ⊂ (s n−1 , u n−1 ) and (s n , u n ) ∩ (l,r) = . Choosing {t} = n [s n , u n ] and setting z := (l(t), t) then yields a point of type (C l ) in such thatr enters z, hence (by Lemma 5.3 (d)), z must be of type (C l )/(C p ).
Proof of Lemma 5.5 Part (e) about separation points has already been proved in Proposition 2.6. It follows from Lemma 3.12 that at points of all types except (C m ), all incoming paths in are equivalent. Therefore, points of type (C o ) are of type (o, ·), points of type (C n ) are of type (n, ·), points of type (C l ) are of type (l, ·), and points of types (C p ) are of type (p, ·), while by Lemma 3.12, points of type (C m ) are of type (pp, p). We next show that, the configuration of incoming paths in at z determines the configuration of outgoing paths inˆ at z: (p) If z is of type (C p ) in , then z is of type (·, pp) inˆ .
(m) If z is of type (C m ) in , then z is of type (·, ppp) inˆ .
(l) If z is of type (C l ) in , then z is of type (·, lp) inˆ .
(n) If z is of type (C n ) in , then z is of type (·, lr) inˆ .
To prove statement (p), we observe that if z = (x, t) is of type (C p ), then by Proposition 3.11 (e), for each T < t with z ∈ N T , there exist maximal T -meshes M − T = M (r , l) and M + T = M (r, l ) with bottom times strictly smaller than t and top times strictly larger than t, such that l ∼ z in r and l ∼ z out r.
Therefore, by the structure of the Brownian web and ordering of paths, there existl − ,l + ∈ˆ l (z) andr − ,r + ∈ˆ r (z) such that r ≤r − ≤l − ≤ l ≤ r ≤r + ≤l + ≤ l on [t − , t], for some > 0. By Lemma 3.1, the pathsl − andr − both pass through the bottom point of M − T . Since T can be chosen arbitrarily close to t, we must havel − ∼ z outr − , and similarlyl + ∼ z outr + .
The proof of statement (m) is similar, where in this case we use the three maximal T -meshes from Proposition 3.11 (d).
If z = (x, t) is of type (C l ), then by the structure of the Brownian web and ordering of paths, there existl − ,l + ∈ˆ l (z) such thatl − ≤ l ≤l + , and there exists a uniquer ∈ˆ r (z) withr ≤l − . By Lemma 3.7, z is isolated from the left in N T for any T < t with z ∈ N T , hence by Proposition 3.5 (b), there exists a maximal T -mesh M (r, l) with bottom time strictly smaller than t and top time strictly larger than t, such that l(t) = x. Now any path inˆ on the left of l must exit M (r, l) through its bottom point, hence the same argument as before shows thatl + ∼ z outr , proving statement (l). Finally, if z = (x, t) is of type (C n ), then by the structure of the Brownian web,ˆ r (z) andˆ l (z) each contain a unique path, sayr andl. By Lemma 5.1,r ∼ z outl , so by ordering of paths,r <l on (t − , t) for some > 0. This proves statement (n).
To complete our proof, we must determine how incoming paths continue for points of the types (C p )/(C p ), (C l )/(C p ), and (C l )/(C l ).
Points of type (C p )/(C p ) must be either of type (p, pp) l /(p, pp) l or (p, pp) r /(p, pp) r . Hence, points of at least one of these types must occur, so by symmetry between left and right, both types must occur.
If l ∈ l andl ∈ˆ l are incoming paths at a point z and l ≤l, then it has been shown in the proof of Lemma 5.4 that z must either be of type (C p )/(C p ) or of type (C s )/(C s ). It follows that at points of type (C l )/(C p ) and (C l )/(C l ), the incoming path l ∈ l continues on the right of the incoming pathl ∈ˆ l .

Structure of special points
In this section, we prove Theorems 1.11 and 1.12. We start with a preparatory lemma.
Proof Without loss of generality we may assume that r 1 < r 2 on (s, u). Since r 1 separates from r 2 at time s, by the structure of reflected paths (Lemma 1.8), we must have s ∈ (r 1 ) but s ∈ (r 2 ), and henceπ 1 (s) = r 1 (s) = r 2 (s) <π 2 (s). By the structure of separation points and Lemma 2.7 (a), we have r 1 ≤π 1 ≤ r 2 on [s, s + ] for some > 0. Set τ := inf{t > u : r 2 (t) <π 1 (t)}. Then τ ≤ u since r 1 does not crossπ 1 and spends zero Lebesgue time withπ 1 (see Lemma 2.12). By Lemma 2.13 (a) and the structure of reflected paths, we have τ ∈ (r 2 ), henceπ 1 (τ) =π 2 (τ). It follows thatπ 1 andπ 2 separate at some time t ∈ (s, τ]. Proof of Theorem 1.11 (a) To prove part (a), assume that l i = l z i ,r i ∈ l in (z) (i = 1, 2) satisfy l 1 ∼ z in l 2 but l 1 = z in l 2 . Then l 1 and l 2 make excursions away from each other on a sequence of intervals (s k , u k ) with u k ↑ t. By Lemma 5.6, it follows thatr i ∈ˆ r out (z) (i = 1, 2) separate at a sequence of times t k ↑ t, which contradicts (1.9). If l i = l z,r i ∈ l out (z) (i = 1, 2) satisfy l 1 ∼ z out l 2 but l 1 = z out l 2 , then the same argument shows that the pathsr i ∈ˆ r in (z i ) with z i = (x i , t) and x i ≤ x separate at a sequence of times t k ↓ t. It follows that z 1 = z 2 =: z , r 1 ∼ z in r 2 , but r 1 = z in r 2 , contradicting what we have just proved.
Before we continue we prove one more lemma.

Lemma 5.7. [Containment between extremal paths]
Almost surely, for each z = (x, t) ∈ 2 , if l is the left-most element of l (z) and r is the right-most element of r , then every π ∈ (z) satisfies l ≤ π ≤ r on [t, ∞) and everyπ ∈ˆ in (z) satisfies l ≤π ≤ r on [t,σπ].
Proof By symmetry, it suffices to prove the statements for l. The first statement then follows by approximation of l with left-most paths starting on the left of x, using the fact that paths in cannot paths in l from right to left [SS08, Prop. 1.8]. The second statement follows from Lemma 2.7 (a).
Case I In this case, by Lemma 5.7, all paths in out (z) are contained between the equivalent paths l and r. Note that this applies in particular to paths in l out (z) and r out (z), so by Theorem 1.11 (a), l out (z) = l out (z) and l out (z) = l out (z) up to strong equivalence. By Lemma 2.7 (a),ˆ in (z) = . Case II We start with points of type (·, pp)/(p, ·). Write l (z) = {l, l } and r = {r, r } where l ∼ z out r and l ∼ z out r. By Lemma 3.13, r and l form a maximal t-mesh, hence, by Proposition 3.5 (b) and Lemma 5.7, all paths in out (z) are either contained between l and r or between l and r. By Theorem 1.11 (a), this implies that l out (z) = {l, l } and r out (z) = {r, r } up to strong equivalence. To prove the statements about dual incoming paths, we note that by Lemma 5.5, points of type (·, pp)/(p, ·) are of type (C p ) or (C s ) inˆ . Therefore, by Proposition 3.11 (c) and (e), there are unique pathsr ∈ˆ r in (z) andl ∈ˆ l in (z) and all paths inˆ in (z) are eventually contained between the equivalent pathsr andl. By Theorem 1.11 (a),ˆ l in (z) = {l} andˆ r in (z) = {r} up to strong equivalence.
For points of type (·, ppp)/(pp, ·) the argument is similar, except that there are now two maximal t-meshes with bottom point z, and for the dual incoming paths we must use Proposition 3.11 (d).
Case III Our proof in this case actually also works for points of type (·, pp)/(p, ·), although for these points, the argument given in Case II is simpler.
Part (a) follows from the fact that a Brownian motion (with constant drift) almost surely does not hit a deterministic space-time point.
Proof By the strong Markov property of the unique weak solution of (1.4) (see Proposition 2.1 of [SS08]), it suffices to verify (a) and (b) for the solution of (1.4) with initial condition L 0 = R 0 = 0. Let W τ = 2B τ + 2τ, where B τ is a standard Brownian motion. Let denote W τ Skorohod reflected at 0. Then recall from the proof of Proposition 2.1 and Lemma 2.2 in [SS08] that, the process D t = R t − L t is a time change of X τ , converting the local time of X at 0 into real time. More precisely, D t is equally distributed with X T t , where the inverse of T is defined by In particular,

A.2 Meeting points
In this appendix, we give an alternative proof of the fact that any meeting point of two paths π, π ∈ l ∪ r is of type (pp, p). Recall that any pair L ∈ l and R ∈ r with deterministic starting points solve the SDE (1.4). Consider the SDE (3.74). Let us change variables and denote X t = L t − L t , Y t = R t − L t , so dX t = 1 {Y t > 0} (dB l