Marking (1,2) Points of the Brownian Web and Applications
aa r X i v : . [ m a t h . P R ] J un Marking (1,2) Points of the Brownian Web and Applications
C. M. Newman
Courant Inst. of Mathematical Sciences, NYU, New York, NY 10012
K. Ravishankar
Dept. of Mathematics, SUNY College at New Paltz, New Paltz, NY 12561
E. Schertzer
Dept. of Mathematics, Columbia University, New York, NY 10027
October 31, 2018
Abstract
The Brownian web (BW), which developed from the work of Arratia and then T´oth and Werner,is a random collection of paths (with specified starting points) in one plus one dimensional space-timethat arises as the scaling limit of the discrete web (DW) of coalescing simple random walks. Tworecently introduced extensions of the BW, the Brownian net (BN) constructed by Sun and Swart, andthe dynamical Brownian web (DyBW) proposed by Howitt and Warren, are (or should be) scaling limitsof corresponding discrete extensions of the DW — the discrete net (DN) and the dynamical discrete web(DyDW). These discrete extensions have a natural geometric structure in which the underlying Bernoullileft or right “arrow” structure of the DW is extended by means of branching (i.e., allowing left and rightsimultaneously) to construct the DN or by means of switching (i.e., from left to right and vice-versa) toconstruct the DyDW. In this paper we show that there is a similar structure in the continuum wherearrow direction is replaced by the left or right parity of the (1,2) space-time points of the BW (pointswith one incoming path from the past and two outgoing paths to the future, only one of which is acontinuation of the incoming path). We then provide a complete construction of the DyBW and analternate construction of the BN to that of Sun and Swart by proving that the switching or branchingcan be implemented by a Poissonian marking of the (1,2) points.Le r´eseau Brownien (BW) construit `a partir des travaux de Arratia, de T`oth et de Werner est unecollection al´eatoire de chemins (avec des points de depart determin´es) dans un espace deux-dimensionnel(une dimension en temps et une autre en espace), qui est la limite d’´echelle d’un r´eseau discret (DW)de marches al´eatoires coalescentes. R´ecemment, deux extensions du BW ont ´et´e introduites: le filetBrownien (BN), construit par Sun et Swart, et le r´eseau Brownien dynamique (DyBW), propos´e parHowitt et Warren. Ces deux objets sont (ou devraient ˆetre) la limite d’´echelle de deux extensionsnaturelles du r´eseau discret—le filet discret (DN) et le r´eseau dynamique discret (DyDW). Le DN etle DyDW sont obtenus par une modification de la configuration des “fl`eches” droites ou gauches quicomposent le r´eseau discret. Pour le DN, un m´ecanisme de ramification est introduit (en permettant desfl`eches droites et gauches simultan´ement) alors que pour le DyDW, la direction des fl`eches est modifi´ee de droite `a gauche et vice-versa). Dans cet article, nous montrons qu’il existe une structure g´eom´etriqueanalogue dans le cas continu. Plus pr´ecis´ement, la direction des fl`eches dans le cas discret est remplac´eepar la direction des points (1 ,
2) du r´eseau Brownien (en un point (1 ,
2) se trouvent un chemin entrantet deux chemins sortants, l’un d’eux ´etant la continuation du chemin entrant). Nous montrons que lesramifications et changements de direction peuvent ˆetre introduits dans le cas continu par un marquagede type Poisson des points (1 , Keywords:
Brownian web, Brownian net, dynamical Brownian web, coalescing random walks, Poissonianmarking, nucleation on boundaries, sticky Brownian motion.
In [9], the present authors and L. R. Fontes obtained some results about exceptional times for a dynamicalmodel of coalescing one-dimensional random walks (the “dynamical discrete web” (DyDW)). Underlyingthose results was the idea that there should be a natural continuum limit of the DyDW, the “dynamicalBrownian web” (DyBW) for which corresponding results would be valid, provided such a continuum systemactually exists. The DyBW was also proposed in a paper of Howitt and Warren [14], where the DyDW wasfirst discussed, and some of its properties were analyzed, assuming its existence.The main purpose of the present paper is to develop a Poissonian marking of certain nongeneric points(called (1,2) points, as we will explain) in the (static) Brownian web (BW) which we then use to give thefirst complete construction of the DyBW. In a revised version [10] of [9], this construction will be used toargue that exceptional time results derived earlier for the DyDW should extend to the DyBW. As we shallsee, this marking technology is natural and has other applications besides the DyBW. One of those, whichwe explore in detail in this paper, is an alternative construction of the “Brownian net” (BN) of Sun andSwart [23]. A future application [16], which we discuss briefly in Subsection 1.2 below, is to scaling limitsof one-dimensional voter models in which there is “nucleation along boundaries.” That will extend, in anontrivial way, earlier work [8] on scaling limits in which nucleation “in the bulk” was treated by usingmarking of nongeneric (0,2) points of the BW, which are simpler to deal with than (1,2) points. Anothermodel closely related to the marking of the Brownian web is a class of stochastic flows of kernels introducedby Howitt and Warren [14]. This is the subject of ongoing work [21].In addition to direct applications of Poissonian markings of BW (1,2) points, we believe that theseconstructions are of interest as special examples of an approach that is relevant beyond the Brownian websetting. Indeed, the idea of using Poissonian marking of nongeneric double points in the context of theSchramm-Loewner Evolution
SLE (6), was proposed in [4, 5] as an approach to the continuum scaling limitsof both “near-critical” and dynamical two-dimensional percolation models. In that setting, the critical scalinglimit is analogous to the BW, dynamical percolation to the DyDW and near-critical percolation to a discrete2igure 1: Forward coalescing random walks (full lines) and their dual backward walks (dashed lines).web with small nonzero drift. Progress in applying that approach has been reported by Garban, Pete andSchramm [12, 11]; for other results on scaling limits of near-critical percolation, see [17, 18, 6].
The Discrete Web.
The discrete web is a collection of coalescing one-dimensional simple random walksstarting from every point in the discrete space-time domain Z even = { ( x, t ) ∈ Z : x + t is even } . TheBernoulli percolation-like structure is highlighted by defining ξ x,t for ( x, t ) ∈ Z even to be the increment ofthe random walk at location x at time t . These Bernoulli variables are symmetric and independent and thepaths of all the coalescing random walks can be reconstructed by assigning to each point ( x, t ) an arrow from( x, t ) to { x + ξ x,t , t + 1 } and considering all the paths starting from arbitrary points in Z even that followthe arrow configuration ℵ . We note that there is also a set of dual (or backward) paths defined by the same ξ x,t ’s but with arrows from ( x, t + 1) to ( x − ξ x,t , t ). The collection of all dual paths is a system of backward(in time) coalescing random walks that do not cross any of the forward paths (see Figure 1).There are two natural variants of the discrete web; one is the dynamical discrete web (DyDW) whichinvolves switching of arrows and the other is the discrete net (DN) which involves branching (or equivalently,adding) of arrows. Each of these is constructed by a straightforward modification of the arrow structurein the standard discrete web. The essence of this paper is a construction of analogous modifications in thecontinuum space-time setting. The Dynamical Discrete Web.
In the DyDW, there is, in addition to the random walk discretetime parameter, an additional (continuous) dynamical time parameter τ . The system starts at τ = 0 as anordinary DW and then evolves in τ by randomly switching the direction of each arrow at a fixed rate (say λ ), independently of all other arrows. This naturally defines a dynamical arrow configuration τ ℵ ( τ ) .If one follows the arrows starting from the (space-time) origin at (0 , τ = 0 as a simple3ymmetric random walk and then evolves dynamically in τ in a different way than the “dynamical randomwalks” studied in [2]. As noted in [9], the nature of exceptional dynamical times is quite different in thissituation than in that of [2]. For example, the dynamical random walk constructed from ℵ ( τ ) violates thelaw of the iterated logarithm on a set of τ ’s of Hausdorff dimension one. The Discrete Net.
In the DN, space-time points have at any point arrows of both directions withprobability p ∈ [0 , p ), corresponding to points where there is branching of paths, or only a left arrow (withprobability (1 − p ) /
2) or only a right arrow (with probability (1 − p ) / τ and otherwise declaring that there is only one arrow whose direction isthat of the DyDW at dynamical time 0. This yields the DN with p = 1 − e − λτ .Under diffusive scaling, individual random walk paths converge to Brownian motions and the entirecollection of discrete paths in the DW converges in an appropriate sense (see [7]) to the continuum Brownianweb (BW). We review in Section 2 some of the basic features of the BW, which developed from the workof Arratia [1] and of T´oth and Werner [24], but meanwhile we briefly comment on its structure. The BWis a random collection of paths (with specified starting points) in continuum space-time with one or morepaths starting from every point. Furthermore, although generic (e.g., deterministic) space-time points haveonly m out = 1 outgoing (to later times) paths from that point and m in = 0 incoming paths passing throughthat point (from earlier times), there are non-generic points with other values of ( m in , m out ). In this paper,a dominant role is played by the (1 ,
2) points as we shall explain.It is natural that there should also exist scaling limits of the DyDW (including of the random walk fromthe origin evolving in τ ) and of the DN (with appropriate scaling of τ and p along with space-time). Indeed,this has been studied by Sun and Swart [23] for the case of the net and by Howitt and Warren [14] for thecase of the dynamical web. The focus of this paper is on how to construct these continuum objects directlyfrom the BW in a way that parallels the discrete construction. A priori, this appears difficult since thediscrete construction is entirely based on modifying the discrete arrow structure of the DW, while in theBW it is unclear whether there even is any arrow structure to modify.The main themes of this paper are thus: “Where is the arrow structure of the BW?” and “How is itmodified to yield the BN and the DyBW (including a dynamically evolving Brownian motion from theorigin)?”. As we will see, the answer to the first question is that the arrow structure of the BW comesfrom the (1 ,
2) points, each of which is equipped with a left or right parity according to which of the twooutgoing paths is the continuation of the single incoming path — see Figure 2 below. The answer to thesecond question is based on a Poissonian marking of the (1 ,
2) points, which can then be used either to createbranching or to switch parity at marked points. 4 .2 Nucleation on Boundaries
The discrete-time one-dimensional voter model starts at time zero with colors assigned to each odd integersite and then evolves in time by assigning a color to the space-time point ( i, j + 1) with i + j + 1 odd as thatof ( i − , j ) or ( i + 1 , j ) with probability 1 / q possiblecolors ( q = 2 , , . . . ); then the boundaries between sites of different colors evolve forward in time (on theeven space-time points) — in the case q = 2 as annihilating random walks, as mixed annihilating-coalescingwalks for 3 ≤ q < ∞ and in the limit q → ∞ (with each site having its own unique color at time zero)as coalescing random walks. Since the finite q case can essentially be recovered from the q = ∞ model byprojection, one can restrict attention to the case of both forward and backward coalescing random walks.Naturally, the continuum scaling limit of voter models is described by the BW. Indeed, in the votermodel as just described, it suffices to consider (as did Arratia [1]) the collection of all outgoing BW pathsfrom time zero. However, if one modifies the voter model to allow for small noise, i.e., at each space-timepoint there is a probability p that rather than take on the color of a neighboring spatial point one time stepearlier, a random color (out of q possibilities, or a wholly new color for q = ∞ ) is chosen (or nucleated), thenmuch more of the BW structure comes into play in the scaling limit (in which also p is properly scaled). Asanalyzed in [8], this model in the scaling limit is one in which new colors are nucleated on (0 ,
2) points of theBW and it can be constructed by means of a Poissonian marking of those points. The reason (0 ,
2) pointsare relevant is because a newly nucleated color in the voter model inside a cluster of some other color createstwo new boundaries which need to persist for a macroscopic amount of time before coalescing in order to beseen in the scaling limit.There are natural settings, namely the so-called q -state stochastic Potts models of Statistical Physics,such that for q ≥ q = 2 corresponds to the Ising model) one needs to consider a morecomplex noise structure in which the probability of nucleation of new colors may depend on the color of thesite in question and its neighbors. For example, one may require for nucleation that a site have a differentcolor than its left (respectively, right) neighbor. For that type of noise, it turns out that the constructionof the scaling limit naturally involves the Poissonian marking of left (respectively, right) (1 ,
2) points. Thereason (1 ,
2) points are relevant here is that the newly nucleated color in the voter model is just to the right(or left) of a previously existing boundary and creates a new boundary that needs to persist in the scalinglimit. This type of application of our marking of (1 ,
2) points will be carried out in a future paper [16].5 .3 Outline of the Paper
The remainder of the paper is organized as follows. In Section 2, we give a review of the basic structure of theBrownian web and its dual (or backward) web, with special emphasis on the (1 ,
2) points. In Section 3, weexplain precisely how to mark (1 ,
2) points, which are points where backward and forward BW paths touch,by first defining for finitely many backward and forward paths a local time measure for touching to serveas a Poisson intensity measure. The overall marking process is then the limit as the number of forward andbackward paths tends to infinity. In Subsection 3.3, we give a preliminary explanation of how the markingprocess will be used to construct the BN and the DyBW.In Section 4, we consider the special marking process (and resulting modified Brownian web path) con-structed from a single forward BW path and all backward paths that touch it from the right. In particular,we show that the resulting modified forward path is related to the original BW path by sticky reflection.Brownian motions with a sticky interaction will also play an important role in later sections as they do in[23] and [14]. In Section 5, we review the construction from [23] of the BN and then prove that our alternateconstruction using marked (1 ,
2) points is equivalent. In Section 6, we construct the DyBW and prove someelementary properties of this object. Section 7 contains the proofs of many of the results stated in previoussections along with some propositions and lemmas that are needed for those proofs. We note in particularthat Section 7.3 contains a number of key results about the structure of excursions in the Brownian webfrom a single web path.
The (forward) Brownian web is the scaling limit of the discrete web under diffusive space-time scaling; itis a random collection of paths with specified starting points in space-time. The (continuous) paths takevalues in a metric space ( ¯ R , ρ ) which is a compactification of R . (Π , d ) denotes the space whose elementsare paths with specific starting points. The metric d is defined as the maximum of the sup norm of thedistance between two paths and the distance between their respective starting points. The Brownian webtakes values in a metric space ( H , d H ), whose elements are compact collection of paths in (Π , d ) with d H theinduced Hausdorff metric. Thus the Brownian web is an ( H , F H )-valued random variable, where F H is theBorel σ -field associated to the metric d H . The next theorem, taken from [7], gives some of the key propertiesof the BW. Theorem 2.1.
There is an ( H , F H ) -valued random variable W whose distribution is uniquely determinedby the following three properties.(o) from any deterministic point ( x, t ) in R , there is almost surely a unique path B ( x,t ) starting from ( x, t ) . i) for any deterministic, dense countable subset D of R , almost surely, W is the closure in ( H , d H ) of { B ( x,t ) : ( x, t ) ∈ D} . (ii) for any deterministic n and ( x , t ) , . . . , ( x n , t n ) , the joint distribution of B ( x ,t ) , . . . , B ( x n ,t n ) is thatof coalescing Brownian motions (with unit diffusion constant). Note that (i) provides a practical construction of the Brownian web. For D as defined above, constructcoalescing Brownian motion paths starting from D . This defines a skeleton for the Brownian web. W issimply defined as the closure of this precompact set of paths. We have considered in Subsection 1.1 the backward discrete web as the set of all coalescing random walksstarting from Z odd running backward in time without crossing the forward discrete web paths. The backward(dual) BW ˆ W may be defined analogously as a functional of the (forward) BW W . More precisely for acountable dense deterministic set of space-time points, the backward BW path from each of these is the(almost surely) unique continuous curve (going backwards in time) from that point that does not cross (butmay touch) any of the (forward) BW paths; ˆ W is then the closure of that collection of paths. The firstpart of the next proposition states that the “double BW”, i.e., the pair ( W , ˆ W ), is the diffusive scaling limitof the corresponding discrete pair ( W δ , ˆ W δ ) (as the scale parameter δ → H , F H ) × ( ˆ H , ˆ F H ) was proved in [7]; convergence of finitedimensional distributions and the second part of the proposition were already contained in [24]. Proposition 2.2.
1. Invariance principle : ( W δ , ˆ W δ ) → ( W , ˆ W ) as δ → .2. For any (deterministic) pair of points ( x, t ) and (ˆ x, ˆ t ) there is almost surely a unique forward path B starting from ( x, t ) and a unique backward path ˆ B starting from (ˆ x, ˆ t ) . The next proposition, from [22], which gives the joint distribution of a single forward and single backwardBW path, has an extension to the joint distribution of finitely many forward and backward paths. We remarkthat that extension can be used to give a characterization (or construction) of the double Brownian web( W , ˆ W ) analogous to the one for the (forward) BW from Theorem 2.1 — see [22, 8] for more details. Proposition 2.3.
1. Distribution of ( B, ˆ B ) : Let ( B ind , ˆ B ind ) be a pair of independent forward and back-ward Brownian motions starting at ( x, t ) and (ˆ x, ˆ t ) and let ( R ˆ B ind ( B ind ) , ˆ B ind ) be the pair obtainedafter reflecting (in the Skorohod sense) B ind on ˆ B ind , i.e., R ˆ B ind ( B ind ) is the following function of u ∈ [ t, ˆ t ] : R ˆ B ind ( B ind ) = B ind ( u ) − ∧ min t ≤ v ≤ u ( B ind ( v ) − ˆ B ind ( v )) on { B ind ( t ) ≥ ˆ B ind ( t ) } ,B ind ( u ) − ∨ max t ≤ v ≤ u ( B ind ( v ) − ˆ B ind ( v )) on { B ind ( t ) < ˆ B ind ( t ) } . (2.1)7 hen ( R ˆ B ind ( B ind ) , ˆ B ind ) = ( B, ˆ B ) in law , (2.2) where B is the path in W starting at ( x, t ) and ˆ B is the path in ˆ W starting at (ˆ x, ˆ t ) .2. Similarly, ( B ind , R B ind ( ˆ B ind )) = ( B, ˆ B ) in law . (2.3) (1 , Points of the Brownian Web.
While there is only a single path from any deterministic point in R in both the forward and backward webs,there exist random points z ∈ R with more than one path passing through or starting from z .We now describe the “types” of points ( x, t ) ∈ R , whether deterministic or not. We say that two paths B, B ′ ∈ W are equivalent paths entering z = ( x, t ), denoted by B = zin B ′ , iff B = B ′ on [ t − ǫ, t ] for some ǫ >
0. The relation = zin is a.s. an equivalence relation on the set of paths in W entering the point z andwe define m in ( z ) as the number of those equivalence classes. ( m in ( z ) = 0 if there are no paths entering z .) m out ( z ) is defined as the number of distinct paths starting from z . For ˆ W , ˆ m in ( z ) and ˆ m out ( z ) are definedsimilarly. Definition 2.1.
The type of z is the pair ( m in ( z ) , m out ( z )) . Figure 2: A schematic diagram of a left ( m in , m out ) = (1 ,
2) point with necessarily also ( ˆ m in , ˆ m out ) = (1 , Theorem 2.4.
For the Brownian web, almost surely, every ( x, t ) has one of the following types, all of whichoccur: (0 , , (0 , , (0 , , (1 , , (1 , , (2 , . Proposition 2.5.
For the Brownian web, almost surely for every z in R , ˆ m in ( z ) = m out ( z ) − and ˆ m out ( z ) = m in ( z ) + 1 . See Figure 2. It is important to realize that points of type (1 ,
2) can be characterized in two ways, both of which willplay a crucial role in our construction of the DyBW and BN. 1) z ∈ R is of type (1 ,
2) precisely if both aforward and a backward path pass through z . 2) A single incident path continues along exactly one of the twooutward paths — with the choice determined intrinsically. It is either left-handed or right-handed accordingto whether the continuing path is to the left or the right of the incoming (from later time) backward path.For a left (1 ,
2) point z , the right (resp, left) outgoing path will be referred to as the newly born path startingfrom z . See Figure 2 for a schematic diagram of the “left-handed” case. Both varieties occur and it is known[8] that each of the two varieties, as a subset of R , has Hausdorff dimension 1. As noted in Section 1,the two varieties of (1 ,
2) points play the same role in the continuum that left and right arrows play in thediscrete setting. In particular, one can change the direction of the “continuum” arrow at a given (1 ,
2) point z by simply connecting the incoming path to the newly born path starting from z . In the discrete picture,this amounts to changing the direction of an arrow whose switching induces a “macroscopic” effect in theweb. (1 , Points on the Brownian Web.
Recall that the φ -Hausdorff outer measure of an arbitrary subset E of R for φ : (0 , ∞ ) → (0 , ∞ ) is definedas m φ ( E ) = lim δ ↓ inf { X φ ( | b i − a i | ) | E ⊂ [ i [ a i , b i ] , | b i − a i | < δ } . (3.4)In the following, we set φ ( t ) = p t log( | log( t ) | ) and we denote the Lebesgue measure of E by | E | . Restrictedto Borel subsets E of R , m φ is a measure. Proposition 3.1.
1. Let ( B, ˆ B ) be defined as in Proposition 2.3. For almost every realization of W , forevery t ≤ u ≤ ˆ t lim ǫ ↓ ǫ |{ v : t ≤ v ≤ u, | B ( v ) − ˆ B ( v ) |√ ≤ ǫ }| (3.5) exists and will be denoted by L B, ˆ B ( u ) . . For a Borel set A ⊂ R Z u ∈ A dL B, ˆ B ( u ) = m φ (cid:16) { u ∈ A | B ( u ) = ˆ B ( u ) } (cid:17) (3.6)
3. Distribution of L B, ˆ B : L B, ˆ B is a stochastic process on [ t, ˆ t ] which is identical in law to ¯ L B, ˆ B defined asfollows: ¯ L B, ˆ B ( u ) = − ∧ min t ≤ v ≤ u ( B ind ( v ) − ˆ B ind ( v )) / √ on { B ind ( t ) ≥ ˆ B ind ( t ) } , ∨ max t ≤ v ≤ u ( B ind ( v ) − ˆ B ind ( v )) / √ on { B ind ( t ) < ˆ B ind ( t ) } , (3.7) where ( B ind , ˆ B ind ) are defined as in Proposition 2.3. Note that the third statement is analogous to the famous property discovered by L´evy that the localtime (at the origin) of a one-dimensional Brownian motion is identical in law with its record time process(see, e.g., [15]). Statement 2 is analogous to the fact that the measure induced by the local time at 0 of astandard Brownian motion coincides with the φ -Hausdorff measure of its zero-set (see Theorem 1 in [19]).Let us consider a family of n forward paths { B i } n − i =0 and a family of m backward paths { ˆ B j } m − j =0 . Wewill generally choose theses paths so that B i and ˆ B i have the same starting point z i with D = { z i } ∞ i =0 somedense deterministic set of points in R as defined in Subsection 2.1; also for consistency with other notation,we will generally assume that z is the origin in R . In non-ambiguous contexts, { B i } n − i =0 and { ˆ B j } m − j =0 willalso refer to the union of their respective traces in R .The expression for L B, ˆ B given in (3.6) can be easily generalized to the family { B i } n − i =0 and { ˆ B j } m − j =0 .E.g., for a Borel A ⊂ R , we simply define L n,m ( A ) by Z u ∈ A dL n,m ( u ) = m φ (cid:16) { t ∈ A | ∃ x ∈ R s.t. ( x, t ) ∈ { B i } n − i =0 ∩ { ˆ B j } m − j =0 } (cid:17) = m φ (cid:16) A ∩ P ( { B i } n − i =0 ∩ { ˆ B j } m − j =0 ) (cid:17) , where P denotes the projection onto the t-axis.Finally, we can extend L n,m to be a measure acting on R in the following way, which implicitly uses thea.s. property of W that if a forward and a backward family meet at some t , they do so only at a single valueof x . Definition 3.1. [Local time measure]
For the forward family { B i } n − i =0 and the backward family { ˆ B j } m − j =0 , we define the local time (outer)measure L n,m on R as follows. For a general space-time domain O , L n,m ( O ) = m φ (cid:16) P ( { B i } n − i =0 ∩ { ˆ B j } m − j =0 ∩ O ) (cid:17) . (3.8) In particular, L n,m is supported on the space-time points where the forward family touches the backwardfamily. Finally, we define an outer measure L ( O ) = m φ (cid:16) P ( { B i } ∞ i =0 ∩ { ˆ B j } ∞ j =0 ∩ O ) (cid:17) . (3.9)10 ( O ) will be referred to as the local time outer measure of O . Both L n,m and L are measures when restricted to Borel sets but may take the value ∞ . We note thatfor any open set O ⊂ R , L ( O ) = ∞ . However, we will later encounter (see e.g., Subsection 7.7) some verynatural subsets O ⊂ R with finite L -measure. See Section 4i of [25] for a similar discussion. Let us consider the Poisson point process on R × R + with intensity measure I n,m ( O × [0 , τ ]) = √ L n,m ( O ) · τ, where O is any open subset of R . We define the partial marking process τ → M n,m ( τ ) as M n,m ( τ ) = { z ∈ R : ( z, u ) is a Poisson point for some u ≤ τ } . (3.10)Heuristically, M n,m ( τ ) consists of the locations of the switching (in the DyBW) between dynamical times0 and τ if one restricts the dynamics to the “arrows” at the intersection of the forward family { B i } n − i =0 andthe backward family { ˆ B j } m − j =1 , while other arrows remain frozen. In order to introduce a “full dynamics” wewill couple the sequences {M n,m ( τ ) } n,m in such way that for n ′ ≥ n and m ′ ≥ m , M n,m ( τ ) ⊆ M n ′ ,m ′ ( τ ).To achieve this, we define the point process M as follows: Definition 3.2. M is the four-dimensional Poisson point process on R × R + × N × N with (locally finiteand random) intensity measure I defined by I ( O × [0 , τ ] × { , ..., n − } × { , ..., m − } ) = √ L n,m ( O ) · τ, (3.11) where O is any open subset of R . We can then define M ( τ ) as M ( τ ) = { z : ( z, s, n ′ , m ′ ) is in M for some n ′ , m ′ and some s ≤ τ } . (3.12)and M n,m ( τ ) is simply obtained by adding the restriction to (3.12) that n ′ ≤ n − m ′ ≤ m − {M ( τ ) } τ ≥ can be seen as a Poisson Point Process on R × R with intensity measure √ L ( dz ) × dτ . In particular, for a Borel O ⊂ R with L ( O ) < ∞ , M ( τ ) ∩ O is a Poisson point Process on R × R withintensity measure √ z ∈ O L ( dz ) × dτ . Let τ >
0. We define a partial Brownian net N n,m ( τ ) by having branching at the points of the partialmarking M n,m ( τ ) .(Later we will write N n ( τ ) for N n,n ( τ ).) For example, if the (1 ,
2) point in Figure 2 is11arked, then the Brownian net will include not only paths that connect to the left outgoing path (as inthe original web) but also ones that connect to the right outgoing path. More formally, the set of paths in N n,m ( τ ) starting from z ∈ R is the set of paths interpolating the set S of points M n,m ( τ ) ∪ { z } ∪ { + ∞} with paths in W —i.e., between any consecutive pair of points in π ∩ S , π coincides with a path in W .Finally, we define N mark ( τ ) as the closure of S ∞ n,m =1 N n,m ( τ ). In other words, N mark ( τ ) is defined byallowing branching at every marked (1 ,
2) points in the Brownian web W . Analogously, we can define abackward partial Brownian Net ˆ N n,m ( τ ) by allowing branching at the points M n,m ( τ ) in the dual Web ˆ W and define ˆ N mark ( τ ) as the closure of S ∞ n,m =1 ˆ N n,m ( τ ). In Section 5, we prove the equivalence of N mark ( τ )to the Brownian net construction of Sun and Swart [23], which by their results (see Theorem 1.1 in [23])then implies convergence of the properly rescaled discrete net to N mark ( τ ) in an appropriate topology. We can construct a partial dynamical Brownian web W n,m ( τ ), at dynamical time τ , to replace the original W by switching the direction of all the marked (1 ,
2) points in M n,m ( τ ). Formally, π is in W n,m ( τ ) iff π is in the the partial net N n,m ( τ ) and at each time t = ¯ t i that π hits a point (¯ x i , ¯ t i ) ∈ M n,m ( τ ), it thenfollows B inew , the newly born path of W starting from (¯ x i , ¯ t i ), on [¯ t i , ¯ t i + a ] for some a >
0. A nontrivialquestion is the existence of a limit for W n,m ( τ ) as n, m → ∞ . It will be shown in Section 6 that for almostall realizations of the web and its marking, a limit W ( τ ) exists for every τ (see Proposition 6.1). From here through Section 6, τ will denote a fixed deterministic number and the marking will refer to M ( τ ).We first recall the definition of a one-dimensional sticky (at the origin) Brownian motion. Definition 4.1. B stick,x is a (1 / ¯ τ ) -sticky Brownian motion starting at x iff there exists a one-dimensionalstandard Brownian motion B s.t. ∀ t ≥ , dB stick,x ( t ) = 1 B stick,x ( t ) =0 dB ( t ) + ¯ τ B stick,x ( t )=0 dt. (4.13) and B is constrained to stay positive as soon it first hits zero. It is known that (4.13) has a unique (weak) solution. Furthermore, for x = 0 this solution can beconstructed from a time-changed reflected Brownian motion. More precisely, consider t | ¯ B | ( C ( t )) , with C − ( t ) = t + 1¯ τ L ( t ) , where | ¯ B | is a reflected Brownian motion and L is its local time at the origin. Then there exists a Brownianmotion B such that ( | ¯ B | ( C ( · )) , B ) is a solution of (4.13) (see, e.g., [26]). In words, the sticky Brownian12otion is obtained from the reflected one by “transforming” local time into real time. In particular, itspends a positive Lebesgue measure of time at the origin and the larger the “degree of stickiness” 1 / ¯ τ is, themore the path sticks to the origin.In this section we consider the path [1] r z starting at z ∈ D and constructed by switching only the directionof the left (1 ,
2) points in M ( τ ) on B , the path of W starting from the origin. As we shall see, unlike in thecomplete DyBW, it is not difficult to construct [1] r z and the law of the pair ( [1] r z , B ) can be characterizedexplicitly. In particular, if we set [1] r ≡ [1] r z for z = (0 ,
0) then it readily follows from Proposition 4.1 belowthat ( [1] r − B ) / √ √ /τ )-sticky Brownian motion. This will be very useful in the rest of the paper(see Sections 5 and 6) where the analysis of paths that result from switching left and right (1 ,
2) pointsis a direct extension of the analysis here. Our construction of a sticky Brownian motion using the markedexcursions defined next is similar to Warren’s construction in [27] using the excursions of a single Brownianmotion.
Definition 4.2. [Excursions]
Let B new be the newly born path emerging from a (1 , point z = ( x, t ) onany path B ∈ W . The segment of B new before coalescence with B is called an excursion from B . D ( e ) is the time duration of the excursion e , | e | ≡ sup {| B − e | ( s ) : t ≤ s ≤ t + D ( e ) } is its diameter, T ( e ) ≡ t its starting time, ( T ( e ) , T ( e ) + D ( e )) its lifespan.If an excursion e starts from a marked point, e is called a marked excursion.A right marked excursion e is called nested iff there exists another right marked excursion e ′ s.t. T ( e ) belongs to the lifespan of e ′ . An analogous definition holds for left marked excursions.If a marked excursion e is not nested, e is said to be a maximal excursion. [1] r may be defined as the path obtained after joining together all the right maximal excursions from B . Stated differently, [1] r is the path whose excursions (in the standard sense) from B coincide with theright maximal excursions from B in the marked Brownian web. We note that everytime [1] r hits a left(1 ,
2) point on B it then follows the newly born path starting from it. (Among all the marked left (1 , [1] r only hits the starting points of maximal excursions since nested excursions are “straddled” bysome maximal excursions). Thus [1] r is consistent with the informal definition in terms of switching givenearlier in this section.Next, we recall that for any deterministic point z ∈ R , B z ∈ W is the path starting from z . We define [1] r z as the path starting from z obtained by switching all the left marked (1 ,
2) points on B ∩ B z . (Thisinformal definition may be made precise as was done for [1] r by using the right maximal excursions from B z ′ , where z ′ is the coalescing point between B and B z .) Note that [1] r z is a continuous path. To provethat, it is clearly enough to show that for fixed T, ǫ ∈ (0 , ∞ ) the process [1] r z only performs finitely manyexcursions of diameter ≥ ǫ away from B on the interval [0 , T ]. If that were not the case, there would exista sequence of marked excursions { e k } from B such that e k would make an excursion away from B with13iameter ǫ and duration t k , with t k →
0. But that would violate the compactness of W .We now set up some notation. For a path π in (Π , d ) starting from z , we denote by t π , the starting timeof π . For two paths π , π , T π ,π ≡ inf { t > t π ∨ t π : π ( t ) = π ( t ) } denotes the first meeting time of π and π , which may be + ∞ . In Subsection 7.4 we show the following proposition. Proposition 4.1.
For any deterministic z ∈ R , almost surely, there exists B (1) z , a standard Brownianmotion starting at z so that [1] r z satisfies the following SDE. d [1] r z ( t ) = dB (1) z ( t ) + 1 [1] r z ( t )= B ( t ) τ dt,dB ( t ) dB (1) z ( t ) = 1 [1] r z ( t )= B ( t ) dt, ∀ t ≥ T [1] r z ,B , [1] r z ( t ) ≥ B ( t ) (4.14)Here dB ( t ) dB (1) z ( t ) denotes d h B , B (1) z i ( t ), where h B , B (1) z i ( t ) is the cross-variation process of B and B (1) z at time t . The second part of Equation (4.14) amounts to saying that away from the diagonal { t : [1] r z ( t ) = B ( t ) } , B and [1] r z evolve independently while on the diagonal they are perfectly correlated. Inparticular, without the drift on the diagonal to “unstick” [1] r z from B , [1] r z and B would coalesce ratherthan stick when they meet.Adopting the usual terminology, we will say that [1] r z is distributed as a Brownian motion stickilyreflected off B with a degree of stickiness 1 /τ . In particular, for z = ( x,
0) the process { ( [1] r z − B ) / √ } isa ( √ /τ )-sticky Brownian motion (see Definition 4.1).In [23], Sun and Swart studied a similar equation but with the difference that [1] r (resp., B ) is replacedby a right (resp., left) drifting Brownian motion (see Equation (5.15)). For that equation, they establishedexistence and uniqueness of a weak solution (see Proposition 2.1 in [23]). Since (4.14) and (5.15) only differby their drift terms, existence and uniqueness for (4.14) follows from their result and the Girsanov Theorem.In particular, a (weak) solution ( [1] r , B ) of (4.14) is a strong Markov process. In Subsection 5.1, we outline the construction of the Brownian net given by Sun and Swart [23] and statesome related results. The presentation we give of that construction is taken from [20]. As will be seen, thisconstruction of the Brownian net is different in spirit to the one using marking given in Subsection 3.3.1.However, we will show in Theorem 5.5 that the two constructions lead to the same object.14 .1 The Brownian Net as Introduced by Sun and Swart
We now recall the left-right Brownian web ( W l , W r ), which is the key intermediate object in the constructionof the Brownian net in [23]. Following [23], we call ( l , . . . , l m ; r , . . . , r n ) a collection of left-right coalesc-ing Brownian motions , if ( l , . . . , l m ) is distributed as coalescing Brownian motions each with drift − τ ,( r , . . . , r n ) is distributed as coalescing Brownian motions each with drift + τ , paths in ( l , . . . , l m ; r , . . . , r n )evolve independently when they are apart, and the interaction between l i and r j when they meet is a formof sticky reflection. More precisely, for any L ∈ { l , . . . , l m } and R ∈ { r , . . . , r n } , the joint law of ( L, R ) attimes t > t L ∨ t R is characterized as the unique weak solution of dL ( t ) = dB l − τ dt,dR ( t ) = dB r + τ dt,d h B l , B r i ( t ) = 1 L ( t )= R ( t ) dt, ∀ t ≥ T R,L , R ( t ) ≥ L ( t ) , (5.15)where B l , B r are two standard Brownian motions. We then have the following characterization of theleft-right Brownian web from [23]. [Characterization of the left-right Brownian web] There exists an ( H , F H )-valued random variable ( W l , W r ), called the standard left-right Brownian web(with parameter τ > W l , resp. W r , is distributed as the standard Brownian web, except tilted with drift − τ , resp. + τ .(b) For any finite deterministic set z , . . . , z m , z ′ , . . . , z ′ n ∈ R , the subset of paths in W l starting from z , . . . , z m , and the subset of paths in W r starting from z ′ , . . . , z ′ n , are jointly distributed as a collectionof left-right coalescing Brownian motions.Similar to the Brownian web, the left-right Brownian web ( W l , W r ) admits a natural dual ( ˆ W l , ˆ W l ) whichis equidistributed with ( W l , W r ) modulo a rotation by 180 o of R . In particular, ( W l , ˆ W l ) and ( W r , ˆ W r ) arepairs of tilted double Brownian webs.Based on the left-right Brownian web, [23] gave three equivalent characterizations of the Brownian net,which are called respectively the hopping, wedge, and mesh characterizations . We first recall what is meantby hopping, wedges and meshes. Hopping:
Given two paths π , π ∈ Π, let t and t be the starting times of those paths. For t > t ∨ t (note the strict inequality), t is called an intersection time of π and π if π ( t ) = π ( t ). By hopping from15 to π , we mean the construction of a new path by concatenating together the piece of π before and thepiece of π after an intersection time. Given the left-right Brownian web ( W l , W r ), let H ( W l ∪ W r ) denotethe set of paths constructed by hopping a finite number of times between paths in W l S W r . Wedges:
Let ( ˆ W l , ˆ W r ) be the dual left-right Brownian web almost surely determined by ( W l , W r ). For apath ˆ π ∈ ˆΠ, let t ˆ π denote its (backward) starting time. Any pair ˆ l ∈ ˆ W l , ˆ r ∈ ˆ W r with ˆ r ( t ˆ l ∧ t ˆ r ) < ˆ l ( t ˆ l ∧ t ˆ r )defines an open set W (ˆ r, ˆ l ) = { ( x, u ) ∈ R : T < u < t ˆ l ∧ t ˆ r , ˆ r ( u ) < x < ˆ l ( u ) } , (5.16)where T = sup { t < t ˆ l ∧ t ˆ r : ˆ r ( t ) = ˆ l ( t ) } is the first (backward) hitting time of ˆ r and ˆ l , which might be −∞ .Such an open set is called a wedge of ( ˆ W l , ˆ W r ). Meshes:
By definition, a mesh of ( W l , W r ) is an open set of the form M = M ( r, l ) = { ( x, t ) ∈ R : t l < t < T l,r , r ( t ) < x < l ( t ) } , (5.17)where l ∈ W l , r ∈ W r are paths such that t l = t r , l ( t l ) = r ( t r ) and r ( s ) < l ( s ) on ( t l , t l + ǫ ) for some ǫ > l ( t l ) , t l ) the bottom point, t l the bottom time, ( l ( T l,r ) , T l,r ) the top point, T l,r the top time, r theleft boundary, and l the right boundary of M .Given an open set A ⊂ R and a path π ∈ Π, we say π enters A if there exist t π < s < t such that π ( s ) / ∈ A and π ( t ) ∈ A . We say π enters A from outside if there exists t π < s < t such that π ( s ) / ∈ ¯ A , theclosure of A , and π ( t ) ∈ A . We now recall the following characterization of the Brownian net from [23]. Theorem 5.1. [Characterization of the Brownian net]
There exists an ( H , F H ) -valued random variable N , the standard Brownian net (with parameter τ ), whosedistribution is uniquely determined by property (a) and any of the three equivalent properties (b1)–(b3) below: (a) There exist W l , W r ⊂ N such that ( W l , W r ) is distributed as the left-right Brownian web. (b1) Almost surely, N is the closure of H ( W l ∪ W r ) in (Π , d ) . (b2) Almost surely, N is the set of paths in Π which do not enter any wedge of ( ˆ W l , ˆ W r ) from outside. (b3) Almost surely, N is the set of paths in Π which do not enter any mesh of ( W l , W r ) . As pointed out in [20], the construction of the Brownian net from the left-right Brownian web can beregarded as an outside-in approach because W l and W r are the “outermost” paths among all paths in N .On the other hand, the marking construction of the Brownian net can be regarded as an inside-out approach.We start from a standard Brownian web, which consist of the “innermost” paths in the Brownian net, andconstruct the rest of the Brownian net paths by allowing branching at a Poisson set of marked points in theBrownian web. 16 .2 Equivalence of the Constructions The main ingredient in the construction we just described is the pair ( W r , W l ). In order to prove theequivalence between the two constructions we first prove that the sets of leftmost and rightmost paths of N mark (as defined in Subsection 3.3.1) are distributed as such a pair (see Proposition 5.4).In Section 4, [1] r z was constructed from B z by switching all the marked left (1 ,
2) points on B , the pathof W starting from the origin. Analogously, we can define [ n ] r z after switching all the marked left (1 ,
2) pointson B , B , ..., B n − , where B k is the path starting from z k . As can easily be seen, the interaction between [ n ] r z and the family { B i } n − i =0 is local. Hence, Proposition 4.1 implies that [1] r z evolves like an independentBrownian motion away from { B i } n − i =0 and the interaction between [ n ] r z and B i when they meet is a stickyreflection. More precisely, we have the following immediate generalization of Proposition 4.1. Proposition 5.2.
For any deterministic z , there exists B ( n ) z , a standard Brownian motion starting at z , sothat [ n ] r z , { B k } n − k =0 satisfy the following SDE. d [ n ] r z = dB ( n ) z ( t ) + 1 S n − k =0 { [ n ] r z ( t )= B k ( t ) } τ dt,dB k ( t ) dB ( n ) z ( t ) = 1 [ n ] r z ( t )= B k ( t ) dt, ∀ t ≥ T [ n ] r z ,B k , [ n ] r z ( t ) ≥ B k ( t ) . (5.18)We now motivate the next proposition. As n → ∞ , { B k } n − k =0 “fills” more and more space of R andbecause [ n ] r z sticks to the family { B k } n − k =0 it is intuitively clear that 1 S n − k =0 { [ n ] r z = B k } ≈ n , the first part of (5.18) becomes d [ n ] r z ( t ) = dB ( n ) z ( t ) + 1 S n − k =0 { [ n ] r z = B k } τ dt (5.19) ≈ dB ( n ) z ( t ) + τ dt. (5.20)Hence, for any k ∈ N , we expect ( [ n ] r z , B k ) to converge as n → ∞ in distribution to a pair ( r z , B k ) satisfyingthe following SDE. dr z = dB rz + τ dt,dB k dB rz ( t ) = 1 r z ( t )= B k ( t ) dt, ∀ t ≥ T r z ,B k , r z ≥ B k , (5.21)where B rz is a Brownian motion starting from z .We recall that { z i } ∞ i =0 is a dense deterministic subset of R . Let i ∈ N . In the following, we write [ n ] r i for [ n ] r z i . Since { [ n ] r i } n is clearly increasing in n , the sequence { [ n ] r i } n actually converges pathwise and the limitis a drifting Brownian motion. (Although it is not even clear a priori that the sequence of paths is bounded,this will follow from the fact, as motivated by (5.19)-(5.21), that there is convergence in distribution.) This17athwise limit will be referred to as r i ; it corresponds to the rightmost path of the net N mark starting from z i . In particular, any path of any partial net N n (= N n,n ) starting at z i is always to the left of r i (i.e., ≤ r i ).This motivates the following proposition, whose proof is given in Subsection 7.5. Proposition 5.3. [ n ] r i converges pointwise to a continuous path r i starting from z i with ( r i , B k ) satisfyingthe three-part SDE (5.21). Analogously, using the set of marked right (1 ,
2) points of W , we can define { l j } j a family of left-driftingBrownian motions reflected in a sticky way on the paths of W . In Subsection 7.5 we prove the followingextension of Proposition 5.3. Proposition 5.4. { r j } j (resp. { l j } j ) is a family of coalescing right- (resp., left-) drifting Brownian motionswith drift τ (resp., − τ ). The pair ( W l , W r ) , defined as the closures of { l j } j , { r j } j respectively, is distributedas a left-right Brownian web. Now, let N hop denote the net obtained from ( W r , W l ) by the hopping construction given in Section 5.1.In Subsection 7.5, we prove Theorem 5.5. N hop = N mark . (5.22) In order to describe the dynamical web, we will need the following notion of stickiness.
Definition 6.1. [Stickiness]
Let π , π be in the net N with x = π ( t ) = π ( t ) . We say that π sticks to π at z = ( x, t ) , or equivalently π ∼ z π , iff for any ǫ > , Z t + ǫt π ( u )= π ( u ) du > and Z tt − ǫ π ( u )= π ( u ) du > . We now set up some notation. We say that a path enters a point z = ( x, t ) if t π < t and π ( t ) = x . Let z be a (1 ,
2) point in N mark . For any B ∈ W entering z , we denote by B switch the path obtained from B afterswitching the direction of z . Since for any paths π ∈ N mark and ¯ B, B ∈ W entering z , π ∼ z B iff π ∼ z ¯ B ,we will sometimes write π ∼ z B without specifying B to mean that there exists a B ∈ W such that π ∼ z B .Analogously, we will write π ∼ z B switch , without specifying the path B from which B switch was constructed.Recall the partial dynamical web {W n,m ( τ ) } τ ≥ given in Subsection 3.3.2. In the following, N mark ( τ )is the net constructed from M ( τ ). The proof of the next proposition is given in Subsection 7.7.1. Thatproof makes clear that the three parts of Proposition 6.1 correspond to three alternative constructions of thedynamical Brownian web. 18 roposition 6.1. (1) There exists {W ( τ ) } τ ≥ in ( H , d H ) s.t. almost surely ∀ τ ≥ , lim n,m ↑∞ d H ( W n,m ( τ ) , W ( τ )) = 0 . (2) W ( τ ) = { π ∈ N mark ( τ ) : every time π enters a point z in M ( τ ) , π ∼ z B switch } . (3) Almost surely, W ( τ ) satisfies the two following conditions (of Theorem 2.1) for every τ ≥ .(o) From any deterministic point z in R , there is a unique path B τz ∈ W ( τ ) starting from z .(i) W ( τ ) is the closure in ( H , d H ) of { B τi } where B τi is the unique path in W ( τ ) starting from z i ∈ D . To motivate item (2), note that in the partial dynamical web W n,m ( τ ), any path π entering a point z ∈ M n,m ( τ ) locally coincides with any path B ∈ W entering z and then connects to the newly born pathstarting from z . Hence, π locally coincides with B switch and therefore obviously sticks to it. However, if z belongs to M ( τ ) \ M n,m ( τ ), then π ∼ z B . In the limit n, m → ∞ , π ∼ z B switch for every z in M ( τ ).We now turn to the description of some properties of the dynamical Brownian web. We start with adefinition. Definition 6.2. ( B, B ′ ) is a (1 /τ ) -sticky pair of Brownian motions iff1. B and B ′ are both Brownian motions starting at ( x B , t B ) and ( x B ′ , t B ′ ) that move independently whenthey do not coincide.2. For t ≥ , define B stick ( t ) ≡ | B − B ′ | ( t + t B ∨ t B ′ ) / √ . Conditioned on x = B stick (0) , { B stick ( t ) } t ≥ is a ( √ /τ ) -sticky Brownian motion (see Definition 4.1). We call ( B , . . . , B m ; B ′ , . . . , B ′ n ) a collection of ( /τ )-sticking-coalescing Brownian motions , if ( B , . . . , B m )and ( B ′ , . . . , B ′ n ) are each distributed as a set of coalescing Brownian motions and for any B ∈ { B , . . . , B m } and B ′ ∈ { B ′ , . . . , B ′ n } , ( B, B ′ ) is a (1 /τ )-sticky pair of Brownian motions.We will say that ( W , W ′ ) is a 1 /τ -sticky pair of Brownian webs iff ( W , W ′ ) satisfies the following properties(a) W , resp. W ′ , is distributed as the standard Brownian web.(b) For any finite deterministic set z , . . . , z m , z ′ , . . . , z ′ n ∈ R , the subset of paths in W starting from z , . . . , z m , and the subset of paths in W ′ starting from z ′ , . . . , z ′ n , are jointly distributed as a collectionof (1 /τ )-sticking-coalescing Brownian motions.Note that ( W , W ′ ) is defined in a similar way as ( W l , W r ) except that in (a) there is no drift and in (b) thecollection of left-right coalescing Brownian motions is replaced by the collection of (1 /τ )-sticking-coalescingBrownian motions. We are now ready to state the main result of this section whose proof is postponed toSubsection 7.7. 19 heorem 6.2. (a) ( W , W ( τ )) is a / (2 τ ) -sticky pair of Brownian webs.(b) (a Markov property). For τ ≤ τ and given ( W , {M ( τ ) } τ ≤ τ ) , the distribution of the pair ( W ( τ ) , W ( τ )) only depends on W ( τ ) .(c) (Stationarity). For τ ≤ τ , ( W ( τ ) , W ( τ )) and ( W , W ( τ − τ )) are equidistributed.(d) For any fixed deterministic time t > , the process τ → B τ ( t ) is piecewise constant. We remark that existence of a consistent family of finite dimensional distributions for the process W ( τ )follows from the results of [14]— see in particular Theorem 9 there. This section is organized as follows. In Subsection 7.1, we recall some useful properties of the Brownian web.In Subsection 7.2, we complete the construction of the local time measure outlined in Subsection 3.1. InSubsection 7.3, we carefully study some quantities related to the marked excursions of the web. Those results,whose proofs can be skipped at first reading, will be the key ingredients in the proofs of Proposition 4.1 (inSubsection 7.4) and Theorem 6.2 (in Subsection 7.7). In Subsection 7.5, we provide a proof of the resultsfrom Section 5 on the equivalence between the marking and the hopping constructions of the Brownian net.In Subsection 7.6, we give a proof of a basic fact relating the BN to (1 ,
2) points of the BW — that every“point of separation” in the BN is (in our coupling of the BW and BN) also a (1 ,
2) point of the BW. Westudy some elementary properties of the separation points in the Brownian net, and apply those results inSubsection 7.7 to prove Proposition 6.1 about the existence of the dynamical Brownian web. We note thatthe results about separation points of the Brownian net had already been derived by one of us (E. S.) jointlywith Sun and Swart and will also appear in a paper [20] by those three authors.
We start by defining the age of a point ( x, t ) assup { t − t B : B ∈ W and B ( t ) = x } . (7.23)The γ -age truncation of the Brownian web is the set of paths obtained after shortening every path of W byremoving (if necessary) the initial segment consisting of those points of age less than γ . In [FINR06] it wasproved that: Proposition 7.1.
The γ -age truncation of W is “locally sparse” in the sense that for every bounded set U ,the intersection between U and the γ -age truncation of W only consists of finitely many path segments. Two corollaries of that proposition can be formulated as follows:20 orollary 7.1.
Given B and { B n } in W so that B n → B (in (Π , d ) ) then the coalescence time of B n and B converges to the starting time of B .Proof. Let t be the starting time of B and take any ¯ t > t . Let us consider the points z n (resp., z ) where B n (resp., B ) intersect the line R × { ¯ t } . The toplogy of (Π , d ) (see [FIN05]) implies that the starting time of B n converges to t . Hence, for n ≥ n with n large enough, z n has an age larger than (¯ t − t ) / >
0. Moreover,since z n → z , the sequence { z n } belongs to a bounded segment of the line. By Proposition 1, we get that { z n } n ≥ n consist of only finitely many points. Therefore, z n is fixed after a certain n and B n coincides with B at ¯ t . Since this is valid for any ¯ t > t , the claim of Corollary 7.1 follows. Corollary 7.2.
Let B be a path in W starting at t . For any D as in Theorem 2.1 and t > t , on [ t, ∞ ) the path B coincides with a path of the skeleton (determined by D ).Proof. By definition, there exists B n in the skeleton converging to B . The conclusion immediately followsfrom the previous corollary. In this section, we prove Proposition 3.1 on which is based the construction of the local time measure. Forsimplicity of notation, we assume ( x, t ) = (0 ,
0) .Let ( ¯ B , ¯ B ) be two independent standard Brownian motion paths starting at (0 , B ind , ˆ B ind )as B ind ( u ) = ¯ B ( u ) , ˆ B ind ( u ) = ˆ x + ¯ B ( u ) − ¯ B (ˆ t ) for u ∈ [0 , ˆ t ] . (7.24)Clearly, ( B ind , ˆ B ind ) is a pair of independent forward and backward Brownian motions and we construct thesystem of refelected paths ( B, ˆ B ) as in Proposition 2.3, i.e ( B, ˆ B ) = ( R ˆ B ind ( B ind ) , ˆ B ind ).In the following, we will assume that ˆ B (0)(= ˆ B ind (0) = ˆ x − ¯ B (ˆ t )) <
0. The case ˆ B (0) > B (0) = 0 has zero probability. Let R ( ¯ B − ¯ B ) (resp., R ( B ind − ˆ B ind )) be the Skorohodreflection of ¯ B − ¯ B (resp., B ind − ˆ B ind ) at zero, i.e., R ( ¯ B − ¯ B )( u ) = ( ¯ B − ¯ B )( u ) − min [0 ,u ] ( ¯ B − ¯ B ) (7.25) R ( B ind − ˆ B ind )( u ) = ( B ind − ˆ B ind )( u ) − ∧ min [0 ,u ] ( B ind − ˆ B ind ) (7.26)= ( B − ˆ B )( u ) . (7.27)Let T be the first time ( B ind − ˆ B ind ) hits 0. Since ( B ind − ˆ B ind ) is a translation of ¯ B − ¯ B by − ˆ B ind (0) > R ( B ind ( u ) − ˆ B ind )( u ) = R ( ¯ B − ¯ B )( u ) ∀ u ≥ T , (7.28) R ( B ind ( u ) − ˆ B ind )( u ) = 0 ∀ u < T . (7.29)( ¯ B − ¯ B ) / √ R ( ¯ B − ¯ B ) / √ L ( u ) = lim ǫ ↓ ǫ |{ v ≤ u : R ( ¯ B − ¯ B )( v ) √ < ǫ }| (7.30)is equal to − min [0 ,u ] ( ¯ B − ¯ B ) / √
2. This implies that the quantity L B, ˆ B ( u ) = lim ǫ ↓ ǫ |{ v ≤ u : 1 √ R ( B ind − ˆ B ind )( v ) = 1 √ B − ˆ B )( v ) < ǫ }| (7.31)is well defined and moreover L B, ˆ B ( u ) = L ( u ∨ T ) − L ( T ) (7.32)= − min [0 ,u ∨ T ] ¯ B − ¯ B √ [0 ,T ] ¯ B − ¯ B √ − ∧ min [0 ,u ] ( B ind − ˆ B ind ) √ . (7.34)This completes the proof of items 1 and 3 of Proposition 3.1.Finally, item 2 follows from the fact (see Theorem 1 in [19]) that almost surely, the local time measureat zero of a Brownian motion is the φ -Hausdorff measure of its zero-set. To motivate this section, let us consider the pair ( [1] r , B ) (see Section 4). On any interval of { t : B ( t ) = [1] r ( t ) } , [1] r coincides with some path of the Brownian web other than B . Therefore, away from B , [1] r evolves as a Brownian motion independent of B (this is part of the proof in Subsection 7.4 belowof Proposition 4.1, which describes the distribution of ( [1] r , B )). Hence, to determine the distribution of( [1] r , B ), we will need to analyze how [1] r escapes from the diagonal { t : [1] r ( t ) = B ( t ) } .Let us define t rǫ = inf { s : ( [1] r − B )( s ) = √ ǫ } , the first time the pair ( [1] r , B ) escapes from the √ ǫ -neighborhood of the diagonal. By construction, t rǫ is also the first time any right marked excursion is ata spatial distance √ ǫ from B . In Subsection 7.3.1, we give an explicit expression for the distribution of t rǫ .In Subsection 7.3.2, we obtain asymptotics for E ( t rǫ ) for small ǫ . This will be used to prove Proposition 4.1.Finally, we present Proposition 7.4 in Subsection 7.3.3—a result relating left and right excursions from B .It will be used to prove Theorem 6.2(a) which describes the joint distribution of the dynamical Brownianweb at two different dynamical times. 22 .3.1 Distribution of t rǫ In this subsection, we will prove the following proposition.
Proposition 7.2.
Let | B | ǫ ( t ) be a Brownian motion on [0 , ǫ ] , starting at and reflected at and ǫ and let l ǫ ( t ) be its local time at level ǫ . Then P ( t rǫ ≤ t ) = P ( l ǫ ( t ) ≥ Exp [ 1 √ τ ]) , (7.35) where Exp [1 / ( √ τ )] is an exponential random variable with mean / ( √ τ ) , independent of | B | ǫ . By definition, t rǫ ≤ t iff a marked excursion enters the region U ǫ,t = { ( x, u ) : 0 ≤ u ≤ t, B ( u ) + √ ǫ ≤ x } . (7.36)Equivalently, this condition can be re-expressed using the dual Brownian web. Lemma 7.1. t rǫ ≤ t iff there exists a backward path ˆ B starting from U ǫ,t and hitting B at a marked point.Proof. To show the only if part of the lemma, assume there exists a right marked excursion e r from B and0 ≤ s ≤ t such that ( e r ( s ) , s ) ∈ U ǫ,t . One can then construct a sequence { ˆ B n } in ˆ W such that ˆ B n startsat (ˆ x n , ˆ t n ) with B (ˆ t n ) < ˆ x n < e r (ˆ t n ) and (ˆ x n , ˆ t n ) → ( e r ( s ) , s ). Since paths of the web and its dual do notcross, ˆ B n is squeezed between e r and B and thus enters the marked starting point z of e r . By compactnessof ˆ W , ˆ B n converges (along a subsequence) to some path ˆ B ∈ ˆ W starting at ( e r ( s ) , s ) ∈ U ǫ,t and entering thepoint z . The converse argument to prove the if part of the lemma is similar.We denote by L ǫ,t ([ t , t ]) the local time measure of all the points in R × [ t , t ] where B meets abackward path starting from U ǫ,t . This naturally defines a measure L ǫ,t on R and we set L ǫ,t ([0 , t ]) ≡ ˜ l ǫ ( t ).By definition, the set of marked points at the intersection between B and the set of backward paths startingfrom U ǫ,t is a Poisson point process with intensity √ τ ˜ l ǫ ( t ). Hence, P ( t rǫ ≤ t ) = P (˜ l ǫ ( t ) ≥ Exp[ 1 √ τ ]) , (7.37)where Exp[1 / ( √ τ )] is independent of W .To study the measure L ǫ,t , we introduce the (backward) process I tǫ (see Figure 3) defined as ∀ s ∈ [0 , t ] , I tǫ ( s ) = inf { ˆ B ( s ) : ˆ B ∈ ˆ W , z ( ˆ B ) ∈ U ǫ,t } , (7.38)where z ( ˆ B ) denotes the starting point of ˆ B .Not surprisingly, the set of times when I tǫ and B coincide is the support of L ǫ,t . This claim can beverified as follows. Because of the compactness of ˆ W , the time it takes for a path in ˆ W starting from U ǫ,t toreach the curve B is uniformly bounded away from 0. This means that the (backward) age of those paths(see (7.23)) is strictly positive and the claim follows directly from Proposition 7.1. Proposition 7.2 directlyfollows from (7.37) and the following lemma. 23igure 3: The process I tǫ is the left envelope of all the backward paths starting from the region U tǫ . Lemma 7.2.
The process | B | ǫ defined on [0 , t ] by | B | ǫ ( s ) ≡ − √ (cid:16) I tǫ ( t − s ) − B ( t − s ) − √ ǫ (cid:17) is a Brownian motion on [0 , ǫ ] starting at and reflected at and ǫ .Proof. Let { ˆ B k,n } n ∈ N ,k ∈{ ,..., n } be the family of backward paths starting from points of the form z k,n =( B ( kt/ n ) + √ ǫ, kt/ n ). We define { π k,n } = { − √ (cid:16) ˆ B k,n ( t − s ) − B ( t − s ) − √ ǫ (cid:17) : t − kt/ n ≤ s ≤ t } . (7.39)Clearly, { π k,n } starts from { (0 , t − kt/ n ) } and is identical in law with a family of forward coalescing Brownianmotions Skorohod reflected at ǫ .As can easily be seen, the process n | B | ǫ ( u ) ≡ sup { π k,n ( u ) : k ∈ { , . . . , n }} (7.40)converges pointwise to | B | ǫ as n goes to ∞ .Now, let us decompose the process | B | ǫ into its up and downcrossings (the first upcrossing is the sectionof the path on [0 , t ǫ ], where t ǫ is the first time | B | ǫ hits ǫ ; the first downcrossing is the section of the pathbetween t ǫ and its return time to 0). We aim to prove that an upcrossing (resp., downcrossing) is a copy ofan independent Brownian motion starting at 0 (resp., ǫ ), reflected at 0 (resp., ǫ ) and stopped when it hits ǫ (resp., 0). It is straightforward to show the equidistribution and independence of the up and downcrossings.The downcrossings have the required distribution because | B | ǫ coincides with π k,n for some n and k during adowncrossing. It remains to determine the law of the upcrossings. Let u (resp., u ,n ) be the first upcrossingof the process | B | ǫ (resp., n | B | ǫ ). u ,n is simply made of pieces of Brownian motions stopped if they hit ǫ .Let B (which depends on n ) be the continuous process starting at 0 and obtained by gluing those pieces at24igure 4: The continuous dashed path B is constructed from the plain path n | B | ǫ .their endpoints (see Figure 4). By a simple induction, it is easy to see that ∀ t ∈ [ k/ n , ( k + 1) / n ) , u ,n ( t ) = B ( t ) − inf { B ( j n ) : j = 0 , / n , ..., k/ n } . (7.41)and by the Markov property, B is a Brownian motion stopped when u ,n hits ǫ . As n → ∞ , the right handside of (7.41) converges in law to B ( t ) − inf [0 ,t ] B (7.42)where B is a Brownian motion stopped when B ( t ) − inf [0 ,t ] B hits ǫ . On the other hand, the left hand sideof (7.41) converges almost surely to u . Hence, the first upcrossing of | B | ǫ ( s ) is identical in distributionwith that of a Brownian motion starting at 0, Skorohod reflected at 0 and stopped when it hits 1. B In this subsection, we prove
Proposition 7.3. lim ǫ ↓ E ( t rǫ ) /ǫ = √ τ and E ([ t rǫ ] ) = o ( ǫ ) as ǫ ↓ . We only prove the first claim. The second one can be proved along the same lines.Let P W denote the probability distribution of the marked Brownian web conditioned on the web W . ByProposition 7.2, we have the following. E ( t rǫ ) /ǫ = R ∞ P ( t rǫ > ǫt ) dt = Z ∞ E ( P W ( t rǫ > ǫt )) dt = R ∞ E ( P W (cid:16) l ǫ ( ǫt ) < Exp[ √ τ ] (cid:17) ) dt = Z ∞ E (cid:16) exp[ −√ τ · l ǫ ( ǫt )] (cid:17) dt. To take the limit as ǫ →
0, we will use the following lemma.25 emma 7.3.
Let t, γ > . There exist c, C ∈ (0 , ∞ ) such that P ( | l ǫ ( ǫt ) − t | > γt ) ≤ C exp( − c tγǫ ) . (7.43) Hence, l ǫ ( ǫt ) converges in probability to t/ as ǫ → .Proof. We need to show that P ( l ǫ ( ǫt ) − t > tγ ) ≤ C exp( − c tγǫ ) , (7.44) P ( t − l ǫ ( ǫt ) > tγ ) ≤ C exp( − c tγǫ ) . (7.45)We only prove the first inequality. The second one can be obtained using analogous arguments. Using thescaling invariance of Brownian motion, the first inequality reduces to P ( ǫl ( t/ǫ ) − t > tγ ) ≤ C exp( − c tγǫ ) , (7.46)where l ( u ) is identical in distribution to the local time accumulated on the set { x = 2 j + 1 } j ∈ Z at time u by a standard Brownian motion B . Define t = inf { s : B ( s ) = ± } and for k ≥ t k = inf { t ≥ t k − : | B ( t ) − B ( t k − ) | = 2 } . ∆ t k = t k +1 − t k has mean 4. Furthermore, by excursion theory, the local times ∆ l k accumulated on { x = 2 j + 1 } j ∈ Z during the time intervals [ t k , t k +1 ], for k ≥ N ǫ ( t ) = inf { k : t k ≥ t/ǫ } . Then, if we set γ ′ = (1 + γ ) and n = t ǫ (1 + γ ) = tγ ′ ǫ , P (cid:18) ǫl ( t/ǫ ) − t > tγ (cid:19) ≤ P ( N ǫ ( t ) > n ) + P [ ǫ X k ≤ n ∆ l k ] − t > tγ ≤ P X k ≤ n ∆ t k < tǫ + P ǫ X k ≤ n [∆ l k − > tγ ≤ P n X k ≤ n [4 − ∆ t k ] ≥ γγ ′ − ǫtγ ′ + P n X k ≤ n [∆ l k − > γγ ′ . Equation (7.46) follows by classical large deviation estimates.To complete the analysis of lim ǫ → E ( t rǫ ) /ǫ , we use Lemma 7.3 with γ = 1 / E (exp( −√ τ · l ǫ ( ǫt ))) ≤ exp( − τ √ t P ( l ǫ ( ǫt ) ≤ t ≤ exp( − τ √ t C exp( − c t ǫ ) . (7.48)It follows that the family { P ( t rǫ ≥ ǫ · ) } ǫ ≤ is uniformly integrable. Therefore, by Lemma 7.3lim ǫ ↓ Z ∞ E (exp( −√ τ · l ǫ ( ǫu )) du = Z ∞ lim ǫ ↓ E (exp( −√ τ · l ǫ ( ǫu )) du = R ∞ e −√ τ · u/ du = √ τ . This completes the proof of Proposition 7.3. 26 .3.3 Marked Right and Left Excursions
Let e l be a left marked excursion from B . We say that T ( e l ) (the starting time of e l ) is straddled by theright excursion e r iff T ( e r ) < T ( e l ) < T ( e r ) + D ( e r ). In this subsection, we prove the following. Proposition 7.4.
Let e l,ǫ be the first left marked excursion from B with diameter (see Definition 4.2)greater or equal to √ ǫ . Then P ( T ( e l,ǫ ) is straddled by some right marked excursion e r ) → , as ǫ ↓ . (7.49)Define A ǫ ≡ { T ( e l,ǫ ) is straddled by some right marked excursion e r } and t lǫ as the left analog of t rǫ (so that t lǫ is the first time t that B ( t ) − e l,ǫ ( t ) = √ ǫ and therefore t lǫ ≥ T ( e l,ǫ )).For any H >
0, lim sup ǫ ↓ P ( A ǫ ) ≤ lim sup ǫ ↓ P ( A ǫ , t lǫ ≤ ǫH ) + lim sup ǫ ↓ P ( t lǫ > ǫH ) ≤ lim sup ǫ ↓ P ( A ǫ , t lǫ ≤ ǫH ) + lim sup ǫ ↓ E ( t lǫ ) ǫH = lim sup ǫ ↓ P ( A ǫ , t lǫ ≤ ǫH ) + √ τ H where the equality follows from Proposition 7.3 and the the identity E ( t rǫ ) = E ( t lǫ ). Since H can be madearbitrarily large, in order to prove Proposition 7.4 it suffices to show that for any H > ǫ ↓ P (cid:0) A ǫ , t lǫ ≤ ǫH (cid:1) = 0 . (7.50)Let ǫ n = ( ǫH ) / n for n ≥ ǫ − = + ∞ . Breaking up A ǫ accordingly to the duration of the excursion e r straddling T ( e l,ǫ ), we have P ( A ǫ , t lǫ ≤ ǫH ) = P ( ∃ a right marked excursion e r with (7.51) T ( e r ) ≤ T ( e l,ǫ ) ≤ T ( e r ) + D ( e r ) , t lǫ ≤ ǫH )= X n ≥− P ( C ′ n ) ≤ X n ≥− P ( C n ) , (7.52)where C ′ n = {∃ a right marked excursion e r with D ( e r ) ∈ [ ǫ n +1 , ǫ n ) s.t. T ( e r ) ≤ T ( e l,ǫ ) ≤ T ( e r ) + ǫ n , t lǫ ≤ ǫH } , (7.53) C n = {∃ a right marked excursion e r with D ( e r ) ≥ ǫ n +1 s.t. T ( e r ) ≤ T ( e l,ǫ ) ≤ T ( e r ) + ǫ n , T ( e l,ǫ ) ≤ ǫH } . (7.54)Let P L, W be the probability distribution of the marked Brownian web conditioned on W and the markingof the left (1 ,
2) points. Since given the Brownian web, the markings of the left and the right (1 ,
2) points27re independent, we get P L, W ( C n ) = √ τ L D ( e r ) ≥ ǫ n +1 ( R × [ T ( e l,ǫ ) − ǫ n , T ( e l,ǫ )]) (7.55)= √ τ L D ( e r ) ≥ ǫ n +1 ([ T ( e l,ǫ ) − ǫ n , T ( e l,ǫ )]) (7.56)where L D ( e r ) ≥ ǫ n +1 is the local time measure on the possible starting points in R of a right excursion from B with D ( e r ) ≥ ǫ n +1 , and L D ( e r ) ≥ ǫ n +1 is the projection of that measure along the t -axis. Let n ≥
0. Since T ( e l,ǫ ) ∈ [0 , ǫH ], there exists k ∈ {− , , ..., n − } such that[ T ( e l,ǫ ) − ǫ n , T ( e l,ǫ )] ⊂ T k,n , with T k,n = [ kǫ n , ( k + 2) ǫ n ] . (7.57)Hence, P L, W ( C n ) ≤ √ τ max − ≤ k ≤ n − L D ( e r ) ≥ ǫ n +1 ( T k,n ) , (7.58) ≤ √ τ max ≤ k ≤ n L D ( e r ) ≥ ǫ n +1 ( T k,n ) (7.59)where we used the equality L ( R × T k,n ) = L ( R × [ kǫ n ∨ , ( k + 2) ǫ n ∨ T , − = [0 , ǫH ], the formula above also remains valid for n = −
1. Averagingover the realizations of W and the marking of left (1 ,
2) points, we obtain that for any p ≥ P ( C n ) ≤ √ τ E (max ≤ k ≤ n L D ( e r ) ≥ ǫ n +1 ( T k,n )) (7.60) ≤ C p | max ≤ k ≤ n L D ( e r ) ≥ ǫ n +1 ( T k,n ) | p (7.61)where C p is a finite positive constant and | X | p denotes the L p norm of X w.r.t. P . Lemma 7.4.
For any p ≥ there exists K < ∞ s.t. for n ≥ − , | max ≤ k ≤ n L { D ( e r ) ≥ ǫ n +1 } ( T k,n ) | p ≤ K n ( p − ) √ ǫH. (7.62) Proof.
We prove the lemma for n ≥
0. The case n = − T , − = [0 , ǫH ]) can be treated analogously.By translation invariance of the marked Brownian web, P (cid:2) L { D ( e r ) ≥ ǫ n +1 } ([ kǫ n , ( k + 2) ǫ n ]) > x (cid:3) = P (cid:2) L { D ( e r ) ≥ ǫ n +1 } ([0 , ǫ n ]) > x (cid:3) . (7.63)Therefore, P ( max ≤ k ≤ n L { D ( e r ) ≥ ǫ n +1 } ( T k,n ) > x ) ≤ n +1 P ( L { D ( e r ) ≥ ǫ n +1 } ([0 , ǫ n ]) > x ) . The scaling invariance of the Brownian web under the mapping on paths, B λ − / B ( λt ), yields (for a , b ≥
0) the equidistribution of L { D ( e r ) ≥ a λ } ([0 , b λ ]) and √ λL { D ( e r ) ≥ a } ([0 , b ]). Hence L { D ( e r ) ≥ ǫ n +1 } ([0 , ǫ n ]) = d r ǫH n L { D ( e r ) ≥ } ([0 , , | X | p ) p equals R ∞ px p − P ( | X | > x ) dx , implies that | max ≤ k ≤ n L { D ( e r ) ∈ T n } ( T k,n ) | p ≤ /p n/p p Z ∞ x p − P " L { D ( e r ) > } ([0 , > x r n ǫH dx ! p = 2 /p n ( p − ) √ ǫH | L { D ( e r ) > } ([0 , | p . To complete the proof, we need to show that for any p ≥ | L { D ( e r ) > } ([0 , | p < ∞ .We use the fact (see, e.g., [7]) that for any s >
0, there are two distinct dual Brownian paths starting from( B ( s ) , s ), those two paths being separated by the path B . In order for s ∈ [0 ,
2] to be in the support of L { D ( e r ) > } , B must be hit by a (dual) path of ˆ W starting in the region { ( x, t ) : x ≥ B ( t ) , t ≥ s + 1 / } . Atany such time s , there must be an integer k in { , ..., } such that B is hit by ˆ B k/ , the dual path startingat ( B ( k/ , k/
4) and located to the right of B . This implies that L { D ( e r ) > } is bounded above by thelocal time measure induced by the finite family of backward paths { ˆ B k/ } k ≤ . From [22] (see Proposition2.3 above), the process s → ˆ B k/ ( k/ − s ) − B ( k/ − s ) (7.64)defined on [0 , k/
4] is a Brownian motion reflected at 0 and the local time measure L B , ˆ B k/ is just the usuallocal time measure at the origin of that reflecting Brownian motion. It is a standard fact that local time atthe origin has all moments and Lemma 7.4 follows.Combining (7.51), (7.61) and Lemma 7.4 for any p >
2, there exists C ′ p < ∞ s.t. P ( A ǫ , t lǫ ≤ ǫH ) ≤ C ′ p √ ǫH, (7.65)so that (7.50) and hence Proposition 7.4 follow. ( B , [1] r z ) (Proof of Proposition 4.1) First, we prove the following lemma.
Lemma 7.5.
The family { ( B , [1] r z ) } z ∈ R ×{ } of random pairs of continuous paths is a family of strongMarkov processes with stationary transition probabilities.More precisely, for any stopping time T , conditioned on the past of the paths up to T , i.e., conditionedon {F T } (where F t is the σ -field generated by { (cid:0) B ( s ) , [1] r ( s ) (cid:1) } s ≤ t and {F T } is defined accordingly), (cid:0) B ( t + T ) − B ( T ) , [1] r z ( t + T ) − B ( T ) (cid:1) t ≥ is distributed like ( B , [1] r z ( T ) ) with z ( T ) = [1] r z ( T ) − B ( T ) .Proof. We take z = (0 , z .29 eak Markov Property : Recall that L ,n is the natural local time measure on the set E n defined as E n = B \ n − [ i =0 ˆ B i ! . For the time being,
T > E − n will denote the subset of E n T { t ≤ T } consisting of all the points on B hit by a path ˆ B i starting from z i = ( x i , t i ) with i ≤ n − t i ≤ T . E + n will refer to E n T { t ≥ T } . Finally, we define (1 ,n ) ¯ r as the path constructed from B by switching thedirection of the marked left (1 ,
2) points in E + n S E − n .Let L +(1 ,n ) (resp., L − (1 ,n ) ) be the measure L (1 ,n ) restricted to E + n (resp., E − n ). First, conditioned on theBrownian web, the markings of E + n and E − n are two independent Poisson point processes with respectiveintensity measure L +(1 ,n ) and L − (1 ,n ) . Second, ( L +(1 ,n ) , { (1 ,n ) ¯ r ( t ) } t ≥ T ) (resp., L − (1 ,n ) ) is measurable w.r.t. (cid:0) W [ T, ∞ ] , B ( T ) , (1 ,n ) ¯ r ( T ) (cid:1) (resp., W [ −∞ ,T ] ) , where W [ t ,t ] is the set of paths in W starting in the window [ t , t ] and stopped at t . By independence of W [ T, ∞ ) and W [ −∞ ,T ] , the future evolution of ( B , (1 ,n ) ¯ r ) is independent of its past given ( B ( T ) , (1 ,n ) ¯ r ( T )).Assuming momentarily that (1 ,n ) ¯ r converges pointwise to [1] r , it is straightforward to show that [1] r alsocontinues afresh at T provided that the distribution of ( B , [1] r ¯ z ), with ¯ z = (¯ x, x . This we will do next. The stationarity of transition probabilities in Lemma 7.5 then simply followsfrom the translation invariance of the marked Brownian web.We now prove that ( B , [1] r (¯ x, ) is continuous with respect to ¯ x . Let ¯ z n = (¯ x n , → ¯ z . We distinguishbetween two cases:1. ¯ z = (¯ x,
0) with ¯ x = 0. Before meeting B , [1] r ¯ z (resp., [1] r ¯ z n ) follows B ¯ z (resp., B ¯ z n ), the path in W starting from ¯ z (resp., ¯ z n ). For n large enough, B ¯ z and B ¯ z n coalesce at some time µ n before either ofthose paths meets B . Hence, [1] r ¯ z and [1] r ¯ z n coalesces at time µ n with µ n → n ↑ ∞ .2. ¯ z = (0 , γ >
0, we can always find a marked left (1 ,
2) point at ( B ( t ) , t ) for some t ∈ [0 , γ ].Let ˆ B ∈ ˆ W pass through that mark and let ( x M , t M ) be the earliest of the marks along ˆ B . Sincealmost surely (0 ,
0) is not a (1 ,
2) point, t M is strictly positive and for n large enough 0 < x n < ˆ B (0).For n large enough, B ¯ z n coalesces with B before t M . By construction, [1] r z n and [1] r can only cross ˆ B at a marked point on B ∩ ˆ B . Since t M is the earliest marked point on ˆ B , [1] r ¯ z n and [1] r are squeezedbetween B and ˆ B on [0 , t M ] and thus they meet (and coalesce) by t M ≤ γ .For the weak Markov property, it remains to prove that (1 ,n ) ¯ r converges to [1] r . Recall that theexcursions of [1] r from B coincide with the maximal excursions from B (see Definition 4.2). First, let e be a maximal excursion starting at some z . For n large enough, it is clear that z belongs to E + n S E − n . Bydefinition of a maximal excursion, (1 ,n ) ¯ r hits z and then follows e . Second, let z ′ be the starting point ofa marked excursion e ′ which is not maximal and hence is nested in some maximal excursion. For n large30nough, (1 ,n ) ¯ r follows that maximal excursion and therefore misses the excursion e ′ . Hence, in the limit,the excursions of (1 ,n ) ¯ r coincide with the maximal excursions from B , and thus (1 ,n ) ¯ r converges pointwiseto [1] r . Strong Markov Property : Now let T be a stopping time with respect to the right-continuous filtration {F t } and let T n be the following discrete approximation of T :if T ∈ [ k n , k + 12 n ) , T n = k + 12 n . (7.66) T n is a discrete stopping time and the weak Markov property implies that Lemma 7.5 is also valid for T n . { ( B ( t + T n ) − B ( T n ) , [1] r ( t + T n )) − B ( T n ) } t ≥ converges pathwise to { ( B ( t + T ) − B ( T ) , [1] r ( t + T ) − B ( T )) } t ≥ as n → ∞ . The result now follows from the distributional continuity of ( B , [1] r (¯ x, ) withrespect to ¯ x that we have already established.Next, we claim that the pair ( B , [1] r z ) satisfies the three following properties.(1) B is a standard Brownian path starting at (0 , [1] r z starts at z .(2) Away from the diagonal { t : [1] r z ( t ) = B ( t ) } , the two processes evolve as two independent Brownianmotions.(3) Defining t rǫ ≡ inf { t > | [1] r − B | ( t ) = √ ǫ } satisfies(i) P (cid:0) ( [1] r − B )( t rǫ ) = + √ ǫ (cid:1) = 1,(ii) lim ǫ ↓ E ( t rǫ ) /ǫ = √ /τ and E ([ t rǫ ] ) = o ( ǫ ) as ǫ ↓ B , [1] r z ) away from the diagonal. (3) describes the splittingmechanism when ( B , [1] r z ) is on the diagonal. 3(i) says that [1] r z always escapes the diagonal to the right.(Note that the definition of t rǫ given in (3) is consistent with the one given in Section 7.4 as the first time [1] r − B hits + √ ǫ .) 3(ii) specifies the rate at which ( B , [1] r z ) escapes the diagonal. We note that thisapproach is very similar to the one in [13].We now turn to the verification of (1)-(3) for ( B , [1] r z ). Property (1) is obviously satisfied. Property(2) follows directly from Lemma 7.5 and the definition of [1] r z . Property (3)(i) is obvious. Property 3(ii) isgiven by Proposition 7.3 above.Next, we verify that if ( ¯ B , [1] ¯ r z ) is a solution of the SDE (4.14), it also satisfies conditions (1)-(3). Lemma 7.6.
Let ( ¯ B , [1] ¯ r z ) be a solution of the SDE (4.14). Then ( ¯ B , [1] ¯ r z ) is a strong Markov processwith stationary transition probabilities and it satisfies conditions (1)-(3). roof. As discussed in Section 4, the SDE (4.14) has a unique weak solution which implies that ( ¯ B , [1] ¯ r z ) isa strong Markov process. The stationarity property is obvious and ( ¯ B , [1] ¯ r z ) obviously satisfies properties(1)-(2) and (3)(i). It remains to verify 3(ii).Since B stick ≡ ( [1] ¯ r − ¯ B ) / √ √ /τ )-sticky Brownian motion, it is identical in law with t | B | ( C ( t )) , where C − ( t ) = t + √ τ L ( t ) , where | B | is a reflected Brownian motion and L is its local time at the origin. Therefore, t rǫ (for ( [1] ¯ r − ¯ B ))is distributed like √ τ L ( T ǫ ) + T ǫ , where T ǫ is the first time | B | hits ǫ . By excursion theory, L ( T ǫ ) is an exponential random variable withmean ǫ . Since the distribution of T ǫ is that of ǫ T , we indeed get E ( t rǫ ) /ǫ → √ τ and E ([ t rǫ ] ) = o ( ǫ ) . (7.67)Finally, we prove the following uniqueness result which is the last ingredient needed to prove Proposition4.1. This result is analog to Proposition 16 in [13]. Lemma 7.7.
Let { ( B , [1] r z ) } z ∈ R ×{ } and { ( ¯ B , [1] ¯ r z ) } z ∈ R ×{ } be two families of strong Markov processes,with stationary transition probabilities, satisfying properties (1)-(3). For z = ( x, , B stick,x ≡ ( B − [1] r z ) / √ and ¯ B stick,x ≡ ( ¯ B − [1] ¯ r z ) / √ are equidistributed.Proof. By stationarity of the transition probabilities and the Markov property, B stick,x or ¯ B stick,x can bedecomposed into two independent parts. The first part is a Brownian motion stopped when it hits zero whilethe second one is distributed like B stick, or ¯ B stick, . Hence, it is enough to show that B stick ≡ B stick, and¯ B stick ≡ ¯ B stick, are equidistributed. Also by the Markov property and the stationarity of the transitionprobabilities, it is enough to show that for any s ≥ B stick ( s ) and ¯ B stick ( s ) are equidistributed. We alsonote that by property (3)(i), B stick and ¯ B stick are ≥ X denotes either B stick or ¯ B stick and f is a positive bounded continuous functionvanishing on the interval [0 , ǫ ], with ǫ >
0. For any ǫ < ǫ , define t ǫ = 0 and, for any k ≥ t k +1 ǫ ≡ inf { t > t kǫ : | X | ( t ) = ǫ } , t k +2 ǫ ≡ inf { t > t k +1 ǫ : X ( t ) = 0 } . (7.68)We have E ( Z ∞ e − λs f ( X ( s )) ds ) = ∞ X k =1 E Z t kǫ t k − ǫ f ( X ( s )) e − λs ds ! , (7.69)= E Z t ǫ t ǫ f ( X ( s )) e − λs ds ! ∞ X k =0 E ( e − λt kǫ ) . (7.70)32ext, t kǫ = k − X i =0 (cid:0) [ t i +2 ǫ − t i +1 ǫ ] + [ t i +1 ǫ − t iǫ ] (cid:1) . By stationarity and the Markov property, we get that E ( e − λt kǫ ) = (cid:16) E ( e − λt ǫ ) (cid:17) k (cid:16) E ( e − λ [ t ǫ − t ǫ ] ) (cid:17) k . (7.71)This implies that E ( Z ∞ e − λs f ( X ( s )) ds ) = E Z t ǫ t ǫ f ( X ( s )) e − λs ds ! − E ( e − λt ǫ ) · E ( e − λ [ t ǫ − t ǫ ] ) . (7.72)Moreover, since lim ǫ ↓ E ( t rǫ ) ǫ = √ τ and E ([ t rǫ ] ) = o ( ǫ ) (7.73)and t ǫ = t rǫ , it follows that E ( e − λt ǫ ) = 1 − √ λτ ǫ + o ( ǫ ) . (7.74)During [ t ǫ , t ǫ ], the process coincides with a Brownian motion starting at ǫ and stopped when it hits 0.By standard computations, we get that E ( e − λ [ t ǫ − t ǫ ] ) = e −√ λǫ . (7.75)Combining Equations (7.72),(7.74) and (7.75), we obtain E ( Z ∞ e − λs f ( X ( s )) ds ) = E (cid:16)R t ǫ t ǫ f ( X ( s )) e − λs ds (cid:17) ǫ √ (cid:18) √ λ + λτ + o (1) (cid:19) − . (7.76)Since the left-hand side of the equality does not depend on ǫ , E (cid:16)R t ǫ t ǫ f ( X ( s )) e − λs ds (cid:17) ǫ (7.77)has a limit l ( X ), depending on f , as ǫ → E ( Z ∞ e − λs f ( X ( s )) ds ) = Z ∞ e − λs E ( f ( X ( s ))) ds = √ l ( X ) (cid:18) √ λ + λτ (cid:19) − . (7.78)Futhermore, using the various defining properties of B stick and ¯ B stick , l ( X ) = lim ǫ ↓ ǫ − E e − λt ǫ Z t ǫ − t ǫ f ( X ( u + t ǫ )) e − λu du ! = lim ǫ ↓ ǫ − E (cid:16) e − λt ǫ (cid:17) E Z t ǫ − t ǫ f ( X ( u + t ǫ )) e − λu du ! = lim ǫ ↓ ǫ − E Z T f ( ǫ + B ( v )) e − λv dv ! B is a standard Brownian motion and T is the first time it hits − ǫ . (The second equality follows fromthe strong Markov property while the third one follows from lim ǫ ↓ E (cid:16) e − λt ǫ (cid:17) = 1 and also from the factthat on [ t ǫ , t ǫ ], X evolves like a Brownian motion.)Thus l ( B stick ) = l ( ¯ B stick ) and therefore Z ∞ e − λs E ( f ( B stick ( s ))) ds = Z ∞ e − λs E ( f ( ¯ B stick ( s ))) ds. Inverting the Laplace transform yields that for every s and every positive bounded continuous function f vanishing on the interval [0 , ǫ ], E ( f ( B stick ( s ))) = E ( f ( ¯ B stick ( s ))). By the monotone convergence theorem,we can remove the constraint f ( x ) = 0 for x ∈ [0 , ǫ ] which implies that B stick ( s ) and ¯ B stick ( s ′ ) areequidistributed.Lemma 7.7 shows that the distribution of ( B − [1] r z ) / √ [1] r − B ) / √ √ /τ )-sticky Brownian motion. The proof ofProposition 4.1 is a consequence of the following observation. Let z = ( x, B , [1] r z ) satisfyingproperties (1)-(2) and such that ( [1] r z − B ) / √ √ /τ )-sticky Brownian motion satisfies the SDE (4.14).Proposition 4.1 being now established, we end this section with a possibly surprising theorem about theexit time of a sticky Brownian motion. Combining Proposition 7.2 and Proposition 4.1, we have Theorem 7.5.
Let B stick be a Brownian motion starting at and stickily reflected at with an amount ofstick ¯ τ . If t ǫ is the first ǫ hitting time of B stick : P ( t ǫ ≤ t ) = P ( l ǫ ( t ) ≥ Exp [ 12¯ τ ]) where l ǫ is the local time at level ǫ at time t of a Brownian motion on [0 , ǫ ] , starting at , reflected at and ǫ , and Exp (1 / (2¯ τ )) is an independent exponential random variable with mean / (2¯ τ ) . The heuristics described in Subsection 5.2 are made rigorous in this subsection. [Proof of Proposition 5.3]
We set i = 0 (with z i = 0) as the proof for general i is essentially the same. Recall that [ n ] r and { B k } k ≤ n − are coupled via the SDE (5.18). We start by proving that for such a coupling we have Lemma 7.8. ∀ t ≥ , P ( t ∈ T n − j =0 { s : r ( s ) = B j ( s ) } ) → as n → ∞ . Proof.
Let ǫ be a fixed positive number. We define x ǫn = sup { B j ( t − ǫ ) : B j ( t − ǫ ) ≤ r ( t − ǫ ) for j ≤ n − } . B ǫ be the path in { B i } n − i =0 such that B ǫ ( t − ǫ ) = x ǫn . For any s ≥ t − ǫ , we define∆ ǫ ( s ) = 1 √ r − B ǫ )( s ) . By (5.21), conditioned on the past of ( r , B , ..., B n − ) up to time t − ǫ , ∆ ǫ solves the following SDE, where B is a standard Brownian motion. d ∆ ǫ ( s ) = 1 ∆ ǫ =0 dB ( s ) + τ √ ds , ∆ ǫ ( t − ǫ ) = x ǫn . (7.79)∆ ǫ is a drifting Brownian motion stickily reflected at 0 and P ( t ∈ n − \ j =0 { s : r ( s ) = B j ( s ) } ) ≤ P ( r ( t ) = B ǫ ( t )) = P (∆ ǫ ( t ) = 0 | ∆ ǫ ( t − ǫ ) = x ǫn ) . Since x ǫn → n → ∞ ,lim sup n →∞ P ( t ∈ n − \ j =0 { s : r ( s ) = B j ( s ) } ) ≤ P (∆ ǫ ( t ) = 0 | ∆ ǫ ( t − ǫ ) = 0) = P (∆ ǫ ( ǫ ) = 0 | ∆ ǫ (0) = 0) . Note that the process ˜∆ ǫ defined by d ˜∆ ǫ = d ∆ ǫ − τ √ ∆ ǫ =0 dt is a ( √ /τ )-sticky Brownian motion. For sucha process, it is known (see e.g., [3]) that P ( ˜∆ ǫ ( ǫ ) = 0 | ˜∆ ǫ (0) = 0) → ǫ →
0. By a straightforwardapplication of the Girsanov theorem, we see that P (∆ ǫ ( ǫ ) = 0 | ∆ ǫ (0) = 0) → ǫ → t > P t [ n ] r ,B k be the probability measure induced by the pair ( [ n ] r , B k ) on the space ofcontinuous functions on [0 , t ] endowed with its usual Borel σ − algebra. P tr ,B k is defined analogously as thedistribution of the pair satisfying (5.21). We first prove that P t [ n ] r ,B k = ⇒ P tr ,B k as n → ∞ . (7.80)We define n χ ( t ) = 1 t ∈ T n − j =0 { s : r ( s ) = B j ( s ) } . Lemma 7.8 above and Fubini’s Theorem imply that E ( Z t n χ ( t ′ ) dt ′ ) → . (7.81)For n ≥ k , the SDEs (5.18) and (5.21) only differ by their drift term. By the Girsanov Theorem, P t [ n ] r ,B k isabsolutely continuous with respect to P tr ,B k and d P t [ n ] r ,B k = d P tr ,B k exp (cid:18) − τ Z t n χ ( t ′ ) dr ( t ′ ) + τ Z t n χ ( t ′ ) dt ′ (cid:19) . (7.82)Since r is a (drifting) Brownian motion, (7.81) and standard arguments imply that the term in the expo-nential tends to zero in probability. It follows that P t [ n ] r ,B k = ⇒ P tr ,B k as n → ∞ . (7.83)35he pointwise convergence of [ n ] r to r was already explained in Section 5.2 by the fact that [ n ] r ismonotonic in n . This completes the proof of Proposition 5.3. [Proof of Proposition 5.4] It is easy to see from Proposition 5.3 that W r (resp., W l ) is a right-drifting (resp., left-drifting) Brownianweb (it is enough to check that two paths in W r evolve independently when they are apart; this can beendone by simple locality arguments). It remains to prove that W r and W l interact in the sticky way of aleft-right Brownian web (see [23] and Subsection 5.1 above). This boils down to proving that ( r i , l j ) satisfiesthe four-part SDE (5.15). For simplicity, let us take i = j = 0. Other cases can be treated similarly. Wealready know that r and l satisfy dr = dB r + τ dt (7.84) dl = dB l − τ dt, (7.85)and that r ≥ l . It remains to show that d h B r , B l i ( t ) = 1 r = l ( t ) d h B , B i ( t ) = 1 r = l ( t ) dt. As caneasily be seen, r and l evolve independently away from each other. Therefore, d h B r , B l i ( t ) = 1 r = l ( t ) d h B r , B l i ( t ) + 1 r = l ( t ) d h B r , B l i ( t ) (7.86)= 1 r = l ( t ) d h B r , B l i ( t ) . (7.87) B is squeezed between r and l . Hence, r ( t ) = l ( t ) implies that B ( t ) = r ( t ) = l ( t ). Since, byProposition 5.3, d h B , B r i ( t ) = 1 r ( t )= B ( t ) dt (7.88) d h B , B l i ( t ) = 1 l ( t )= B ( t ) dt, (7.89)(7.87) implies, as desired, that d h B r , B l i ( t ) = 1 r = l ( t ) d h B , B i ( t ) = 1 r = l ( t ) dt. (7.90) [Proof of Theorem 5.5] In the proof we will also consider N wedge , the net obtained from ( W r , W l , ˆ W r , ˆ W l ) by the wedge construc-tion of Subsection 5.1. Here ˆ W r and ˆ W l are respectively the dual (backward) webs of W r , W l (constructedby marking) which can be constructed using the dual versions of Propositions 5.3 and 5.4.Since N hop = N wedge (see Theorem 5.1), it suffices to show that (i) N mark ⊃ N hop and (ii) N mark ⊂N wedge .In order to prove (i), we need to show that a path obtained by hopping from W r to W l (or W l to W r )is still in N mark . Take two paths r i and l j intersecting at time t ; we need to show that the concatenation of36 i (before t ) with l j (after t ) is in N mark and similarly for the other concatenation. First, if we consider theanalogous question in a partial net N n the result is obviously true. Indeed, if n r i and n l j are respectively theright- and left-most paths of N n starting from z i and z j , the path constructed by hopping from one path tothe other at some meeting point is in N n . Let ǫ > u ∈ [ t, t + ǫ ] suchthat r i ( u ) > l j ( u ). Taking n large enough, we get n r i ( u ) > n l j ( u ). On the other hand, n r i ≤ r i and n l j ≥ l j so that n r i ( t ) ≤ n l i ( t ). Consequently, there exists v ∈ [ t, u ] where n l j and n r i intersect. Now consider thepath obtained by hopping from n r i to n l j at time v . This path is in N n and approximates the one obtainedby hopping from r i to l j at time t except on [ t, t + ǫ ]. Since ǫ is arbitrary, the latter path is approximatedby paths in S n N n and therefore it also belongs to N mark .We now prove (ii). Consider a wedge constructed from a pair (ˆ r i , ˆ l j ) starting at (( x i , t ) , ( x j , t )) with x i < x j and let us assume there exists π ∈ N mark entering this wedge from outside and show that this leadsto a contradiction. Again, we can approximate (ˆ r i , ˆ l j ) by ( n ˆ r i , n ˆ l j ) ∈ ˆ N n × ˆ N n and π by π n ∈ N n . Since n ˆ r i ≥ ˆ r i and n ˆ l j ≤ ˆ l j , the pair ( n ˆ r i , n ˆ l j ) forms a “partial wedge” approximating the original wedge frominside. Hence, for n large enough, π n would enter this partial wedge from outside. By considering separatelythe cases where the putative entering is at a marked (1 ,
2) point of M n or not, such an entry is seen to beimpossible. In Section 3.3 we defined the dynamical Brownian web as the limit of partial dynamical webs. In thissubsection, we give a series of results which will guarantee the existence of such a limit. These are essentiallyidentical to results in [20]. However, in [20] the proofs rely on the hopping construction of the Brownian net,while in this paper we show the results by using the marking approach. As we shall see, the two points ofview are rather different. We start with a definition.
Definition 7.1. [Separation points]
Two paths π and π in N starting respectively at ( x , t ) and ( x , t ) separate at z = ( x, t ) iff t > t ∨ t with π ( t ) = π ( t ) and there exists a > such that π , π do not touchon ( t, t + a ] . A point z is called a separation point of N iff there is some π , π ∈ N that separate at z . Note that in the partial Brownian net N n paths separate at marked (1 ,
2) points. That remains valid inthe Brownian net (i.e. when n → ∞ ). Indeed, in Subsection 7.6.1, we prove the following result. Proposition 7.6.
The set of separation points in N mark and the set of marked (1 , points of the Brownianweb coincide. Furthermore, in Subsection 7.6.2 we prove the following proposition, which uses the notation π ∼ z B and π ∼ z B switch introduced in Section 6.. 37 roposition 7.7. Let z = ( x, t ) be a separation point in N mark , B be any path of W passing through z ,and N ≤ t − ǫ be the set of paths in N mark starting before or at time t − ǫ . For any ǫ ≥ , define the following(which will not depend on the choice of B ∈ W ). [ ∼ zǫ B switch ] = { π ∈ N ≤ t − ǫ : π enters z and π ∼ z B switch } , [ ∼ zǫ B ] = { π ∈ N ≤ t − ǫ : π enters z and π ∼ z B } , ∦ zǫ = { π ∈ N ≤ t − ǫ : π does not enter z } .
1. Let E z be the set of paths in N mark entering z . ∼ z is an equivalence relation on E z , and E z can bedecomposed into the two equivalence classes [ ∼ z B switch ] and [ ∼ z B ] .2. For ǫ > (note the strict inequality), [ ∼ zǫ B switch ] , [ ∼ zǫ B ] and ∦ zǫ are disjoint elements of H .3. ∃ ¯ z ∈ R and ǫ > s.t. every path of W starting in the ball B (¯ z, ǫ ) enters z . We note that in the partial net, each path entering a marked point z coincides either with B or B switch for a positive interval of time. In the full net limit, a path coincides either with B or B switch for a positiveLebesgue measure of time. By construction, marked points are separation points so we only need to prove the converse.
Definition 7.2. [ ( T , T ) Separation Points] ( x, t ) with T < t < T is said to be a ( T , T ) separationpoint iff there are two paths π and π in the net starting from R × { T } and separating at ( x, t ) which donot touch on ( t, T ] . Let T , T be two rational numbers. It suffices to prove that if ( x, t ) is a ( T , T ) separation point of N mark , then it is a marked (1 ,
2) point. Let π and π be two paths as described in Definition 7.2. Sincethe net is closed under hopping, we can assume without loss of generality that π and π have been chosento coincide up to t .By construction, there exist { π ni } i =1 , with π ni in the partial net N n (= N n,n ; see Subsection 3.3.1) sothat { π ni } converges to π i . Let us take two numbers T < q < q ≤ t where q is arbitrarily close to t .Proposition 7.8 below (for S = q and T = q ), implies that π n ( q ) = π n ( q ) for large enough n .Hence, for large enough n , π n and π n start below R × { q } and separate at a point arbitrarily closeto ( x, t ). Since the set of ( q , T ) separation points is locally finite (see Proposition 7.9 below), π n and π n separate at ( x, t ) for large enough n . By construction, π n and π n only separate at marked points andProposition 7.6 follows. Proposition 7.8. ([23]) For any
S, T with
S < T , the set of intersection points between the line R × T andthe set paths of N starting on or below R × { S } is (almost surely) locally finite. roposition 7.9. ([20]) For any S, T with
S < T , the set of ( S, T ) -separation points is (almost surely)locally finite. In the following, for any paths π , π in (Π , d ) entering a point z , we will write π ∼ zout π (resp., π ∼ zin π )iff for any ǫ > Z t + ǫt π ( u )= π ( u ) du > Z tt − ǫ π ( u )= π ( u ) du > . Note that π ∼ z π iff π ∼ zout π and π ∼ zin π . In order to prove Proposition 7.7, we will use the followingresult from [20]. Since this result is part of a much larger theorem there, we provide a direct proof. For a“pictorial” representation of the result, see Figure 5.Figure 5: Structure of meshes around a separation point. Theorem 7.10. ([20]) Let z = ( x, t ) be a separation point in N and let ǫ > . There exist three distinctmeshes M l ( r, l ) , M r ( r ′ , l ′ ) and M top ( r ′′ , l ′′ ) such that1. The bottom times of M l ( r, l ) , M r ( r ′ , l ′ ) are in ( t − ǫ, t ) and their top times are in ( t, ∞ ) . Moreover, l ≤ r ′ (at coexistence times of M r and M l ), l ( t ) = r ′ ( t ) = x and l ∼ zin r ′ .2. z is the bottom point of M top ( r ′′ , l ′′ ) . M top ( r ′′ , l ′′ ) is squeezed between M l ( r, l ) and M r ( r ′ , l ′ ) (i.e., l ≤ r ′′ and l ′′ ≤ r ′ at respective coexistence times). Moreover, r ′′ ∼ zout l , r ′ ∼ zout l ′′ .Proof. In the following, we say that two paths π and π meet at time ¯ t iff π (¯ t ) = π (¯ t ) but π < π or π > π on (¯ t − a, ¯ t ) for some a > [Construction of M r and M l ] left-drifting Brownian web W l , with drift − τ . Mark the (1 ,
2) points of W l and construct N l by branching at all theleft (1 ,
2) points (of W l ) in M l (2 τ ), the set of marks whose dynamical time coordinate is ≤ τ (the factor2 compensates for the − τ drift in W l ). On the one hand, repeating step by step what was done in Section5 (see Theorem 5.5), one can show that N l is identical in law to the usual N hop , as in Subsection 5.1. Onthe other hand, following the proof of Proposition 7.6, separation points of the net N l must be marked left(1 ,
2) points of W l . Hence, separation points of the net N hop are left (1 ,
2) points of W l and symmetricallythey are also right (1 ,
2) points of W r .One consequence is that z = ( x, t ) must be a separation point for two paths ¯ l ∈ W l and ¯ r ∈ W r startingfrom deterministic points. Lemma 6.5 in [23] analyzes meshes to the left of a path ¯ l ∈ W l . Using that lemmaand the fact that points on ¯ l where other paths from W l coalesce with ¯ l from the left are dense in ¯ l (alongwith the analogous results for ¯ r ), it follows that there exists a mesh M l ( r, l ) (resp., M r ( r ′ , l ′ )) with bottomtime in ( t − ǫ, t ) and top time > t such that l ( t ) = ¯ l ( t ) = x (resp., r ′ ( t ) = ¯ r ( t ) = x ).By Corollary 7.2, l and r ′ coalesce with some paths l i and r j (in the skeleton of W l and W r respectively)before entering the point z . The pair ( l i , r j ) satisfies the SDE (5.15) and in particular, l i ≤ r j from the firsttime they meet. It is clear that l i and r j do not meet and separate at the same point. Hence, there exists a ′ > l i ≤ r j on [ t − a ′ , ∞ ) and a sequence t n ↑ t s.t. l i ( t n ) = r j ( t n ). It immediately follows thatthere exists a ′′ with a ′ ≥ a ′′ > l ≤ r ′ on [ t − a ′′ , ∞ ) and a sequence t ′ n ↑ t s.t. l ( t ′ n ) = r ′ ( t ′ n ).Lemma 7.9 below then immediately implies that l ∼ zin r ′ . [Construction of M top ] Up to reversal of the time coordinate, the backward Brownian net is distributed as the Brownian net (seeSubsection 5.1). Hence, by what has been just proved, z is a separation point for two paths (ˆ l, ˆ r ) ∈ ( ˆ W l , ˆ W r )and there exists a > r ≤ ˆ l on ( −∞ , t + a ]. Let r ′′ (resp., l ′′ ) be the newly born path of W r (resp. W l ) starting from z . Since ( x, t ) is a right (1 ,
2) point for W r and a left (1 ,
2) point for W l , we get that on( t, t + a ] r ′′ ≤ ˆ r ≤ ˆ l ≤ l ′′ and r ′′ ≤ r ′ , l ≤ l ′′ . (7.91) M top is defined as the mesh M top ( r ′′ , l ′′ ) formed by r ′′ and l ′′ .The second part of (7.91) implies that M top is either squeezed between M r and M l or it contains either l or r ′ . Since paths of N do not enter meshes from outside, we get that on ( t, t + a ] l ≤ r ′′ ≤ ˆ r ≤ ˆ l ≤ l ′′ ≤ r ′ . (7.92)Recall the construction of the net N l (described at the beginning of this proof) based on the marking of aleft drifting Brownian web and let ( l, ˆ l ) be a pair of paths in ( W l , ˆ W l ). As can be easily seen, the set { ( x, t ) : l ( t ) = ˆ l ( t ) = x and ∃ a > ∀ s ∈ ( t, t + a ) , l ( s ) < ˆ l ( s ) } l and ˆ l , this follows fromthe fact that, for a standard Brownian motion B , the set { t : B ( t ) = 0 and ∃ a > ∀ s ∈ ( t, t + a ) , | B ( s ) | > } has zero local time measure). Since in N l , separation points are left marked (1 ,
2) points, the argument justgiven implies that for every marked point, there exists t n ↓ t such that l ( t n ) = ˆ l ( t n ). By (7.92), l ≤ r ′′ ≤ ˆ l ,implying that l ( t n ) = r ′′ ( t n ). By Lemma 7.9 below we have that l ∼ zout r ′′ and by a similar argument, weget r ′ ∼ zout l ′′ . Lemma 7.9.
Let ( l, r ) ∈ ( W l , W r ) be such that for some t > t r ∨ t l , l ( t ) = r ( t ) . For any ǫ > , R t + ǫt − ǫ l ( s )= r ( s ) ds > .Proof. Choose any t ′ with t r ∨ t l < t ′ < t − ǫ . By Corollary 7.2, on [ t ′ , ∞ ), the pair ( l, r ) coincideswith a pair ( L, R ) of ( W l , W r ), starting from deterministic points and satisfying the SDE (5.15). Lemma7.9 then follows from the fact (see Proposition 3.1 in [23]) that the support of the measure µ , defined as µ ([ t , t ]) = |{ t ∈ [ t , t ] : L ( t ) = R ( t ) }| , coincides with { t : L ( t ) = R ( t ) } .We now prove the first two claims of Proposition 7.7 for a separation point z = ( x, t ). Note that if claim2 holds for a given ǫ , it immediately holds for any ǫ ′ > ǫ . Hence, w.l.o.g., we can take ǫ > B ∈ W entering z and starting at t ′ ≤ t − ǫ . In the following, ˜ E ǫ will denote thesubset of N ≤ t − ǫ consisting of all the paths entering z .Recall that paths of N do not enter meshes (see Theorem 5.1 (b3) in Subsection 5.1). Hence, for anygiven mesh M with bottom time in ( t − ǫ, ∞ ), we can partition N ≤ t − ǫ into { R ( M ) , L ( M ) } , where R ( M )(resp., L ( M )) is the compact subset of N ≤ t − ǫ consisting of all the paths passing to the right (resp., left) of M . Let M r , M l and M top be as in Theorem 7.10 and let us define˜ E rǫ = [ L ( M r ) ∩ R ( M l )] ∩ R ( M top ) , ˜ E lǫ = [ L ( M r ) ∩ R ( M l )] ∩ L ( M top ) , (7.93)˜ E cǫ = R ( M r ) ∪ L ( M l ) . (7.94)In particular { ˜ E rǫ , ˜ E lǫ } (resp., { ˜ E rǫ , ˜ E lǫ , ˜ E cǫ } ) defines a natural partition of ˜ E ǫ (resp., N ≤ t − ǫ ) into elements of H . By definition, paths in ˜ E lǫ are squeezed between l and r ′ below z while they are squeezed between l and r ′′ above z . Hence, Theorem 7.10 immediately implies that for any two paths π , π ∈ ˜ E lǫ , π ∼ z π . Thesame property holds for ˜ E rǫ . Conversely, if π l ∈ ˜ E lǫ and π r ∈ ˜ E rǫ , the two paths separate at z . This impliesthat ∼ z is an equivalence relation on ˜ E ǫ and the corresponding equivalence classes are given by ˜ E rǫ and ˜ E lǫ .Since B and B switch separate at z they do not belong to the same equivalence class and claims 1 and 2 ofProposition 7.7 follow. 41ext, we say that the ball B (¯ z, ¯ ǫ ) with ¯ z = (¯ x, ¯ t ) is squeezed between between l and r ′ iff ¯ t − ¯ ǫ ≥ t l ∨ t r ′ and for every ( x ′ , t ′ ) ∈ B (¯ z, ¯ ǫ ), l ( t ′ ) ≤ x ′ ≤ r ′ ( t ′ ). It is clear that one can find such a ball below the point z and that any path starting from that ball is squeezed between l and r ′ and so is forced to enter the point z .Claim 3 of Proposition of 7.7 follows. In the following, we use the notation of Proposition 7.7.By compactness of N ( τ ), {W ( n,m ) ( τ ) } ( n,m ) is a precompact subset of H . Let W be any subsequentiallimit of {W ( n,m ) ( τ ) } ( n,m ) as n, m → ∞ and let W ( τ ) = { π ∈ N mark ( τ ) : every time π enters a point z in M ( τ ), π ∼ z B switch } . be as in item (2) of Proposition 6.1. We first prove (i) W ( τ ) ⊂ W ( τ ) . Let z = ( x, t ) ∈ M ( τ ), and let π ∈ W ( τ ) start at t − ǫ with ǫ >
0, and pass through z . By definition,there exists a sequence { π N } N ≥ so that π N belongs to ∪ n,m>N W n,m ( τ ) and { π N } converges to π . Taking N large enough, we can assume w.l.o.g. that π N belongs to N ≤ t − ǫ and ( x, t ) ∈ M n,m ( τ ) for n, m > N . ByProposition 7.7(2), { π N } enters z for N large enough. By construction, π N ∼ z B switch and since π N → π ,Proposition 7.7(2) implies that π ∼ z B switch . Hence, W ( τ ) ⊂ W ( τ ).Next, we prove that W ( τ ) satisfies (3)(o). We first claim that when two paths of W ( τ ) meet, theycoalesce. Let π , π ∈ W ( τ ) start at t , t respectively and meet at t ′ > t ∨ t and let us assume that π and π separate at z = ( x, t ) with t ≥ t ′ . By Proposition 7.7(1), either π ∼ z B or π ∼ z B . This contradictsthe definition of W ( τ ) and we conclude that W ( τ ) is a coalescing collection of paths. Let z i ∈ D . Anypath in W ( τ ) starting at z i is squeezed between r i and l i , the paths in W r and W l respectively startingfrom z i = ( x i , t i ). Since there exists a sequence t ′ n ↓ t i s.t. l i ( t ′ n ) = r i ( t ′ n ) and since paths in W ( τ ) coalesce,there must be a unique path in W ( τ ) starting from z i . We call this path B τi and define W ( τ ) as { B τi } .We continue to prove: (ii) W ( τ ) ⊂ W ( τ ) . Let π ∈ W ( τ ) start at ( x ′ , t ′ ) and let ǫ >
0. We claim that π hits a path in W r ∪ W l in ( t ′ , t ′ + ǫ ]. Tosee this, let a ∈ ( t ′ , t ′ + ǫ ) and let { r n } n ⊂ W r (resp., { l n } n ⊂ W r ) start at z rn (resp., z ln ) with z rn (resp., z ln ) converging to ( π ( a ) , a ) from the left (resp., from the right) of π . If there is not any path in { r n , l n } meeting π on ( a, t ′ + ǫ ), { r n } and { l n } converge (along a subsequence) to r ∈ W r and l ∈ W l respectively,42oth starting at ( π ( a ) , a ) and s.t. r < π < l on ( a, t ′ + ǫ ). In other words, π enters a mesh from outside,yielding a contradiction to Theorem 5.1. M ( τ ), or equivalently the set of separation points in N ( τ ), is dense along any path π ′ in W r ∪ W l . Sinceonce π touches some π ′ , they can only separate at a point in M ( τ ), it follows that π enters some point z ∈ M ( τ ) before t + ǫ . By virtue of Proposition 7.7 (3), there exists a ball B (¯ z, ǫ ′ ) such that any path in N ( τ ) starting in B (¯ z, ǫ ′ ) enters the point z . Hence, any path B τi such that z i belongs to D ∩ B (¯ z, ǫ ′ ) hits z .It follows that π coalesces with some B τi before time t ′ + ǫ . As a consequence, W ( τ ) ⊂ W ( τ ). Finally, weprove: (iii) W ( τ ) ⊂ W ( τ ) . It is clear that there is at least one path π i ∈ W ( τ ) starting from z i . Since W ( τ ) ⊂ W ( τ ), property3(o) for W ( τ ) (which we have already proved) implies that π i = B τi . Since W ( τ ) is compact, it followsthat W ( τ ) ⊂ W ( τ ) and from (i), (ii) above, we get that W ( τ ) = W ( τ ) = W ( τ ). This shows that allsubsequence limits of {W n,m } agree and Proposition 6.1 follows. ( W , W ( τ )) is a / (2 τ ) -Sticky Pair of Brownian Webs In the remaining subsections of the paper we prove the four parts of Theorem 6.2. In this subsection andthe next, the term marking will refer to the set M ( τ ). We already showed in the proof of Proposition 6.1that W ( τ ) is a coalescing set of paths. By a simple locality argument, it is not hard to see that for i = j , B τi and B τj move independently when they are apart. In the following, we prove that ( B i , B τj ) is a 1 / (2 τ )-stickypair of Brownian motions. This ensures that each B τj is a Brownian motion and since the paths of W ( τ )are coalescing, it follows that W ( τ ) is a Brownian web and furthermore that the interaction between W and W ( τ ) is (1 / τ )-sticky as claimed.We now prove that ( B i , B τj ) is a 1 / (2 τ )-sticky pair of Brownian motions. Since the distribution of theBrownian net is invariant under translation in the space time domain, Proposition 6.1(2) implies that W ( τ )is also translation invariant. Hence, it suffices to prove that ( B , B τj ) is a 1 / (2 τ )-sticky pair of Brownianmotions.Define ( n,m ) B τj as the path obtained from B j after switching the directions of the points in M ( n,m ) ( τ ).By parts (1) and (3)-(o) of Proposition 6.1, we havelim n ↑∞ lim m ↑∞ d ( ( n,m ) B τj , B τj ) = 0 . (7.95)In the following, we will denote by [ n ] B j ≡ [ n ] B τj the limit of ( n,m ) B τj as m → ∞ . Informally, [ n ] B j is thepath constructed from B j after switching the direction of all the (left and right) (1 ,
2) points in M ( τ ) thatlie on { B i } n − i =0 .In order to prove that ( B , B τj ) is a 1 / (2 τ )-sticky pair of Brownian motions, we claim that it is enoughto prove the following lemma (which is done in Subsection 7.7.3 below).43 emma 7.10. ( B , [1] B j ) is / (2 τ ) -sticky pair of Brownian motions. The sufficiency of Lemma 7.10 follows from the observation that the law of ( B , [ n ] B j ) is identical to theone of ( B , [1] B j ). For example, for n = 2, one may consider a revised marked Brownian web W ∗ in whichall the marked (1 ,
2) points along the finite segment of B before it coalesces with B have been switched.In W ∗ the marks along B ∗ ( ≡ B ) are the same as in the original web. The following lemma (for k = 1 and l = 0) states that this W ∗ is equidistributed with the original Brownian web. On the other hand, the pair( B ∗ , [1] B ∗ j ) for W ∗ is identical to the pair ( B , [2] B j ) for the original marked web. Since [ n ] B j almost surelyconverges to B τj , ( B , B τj ) is 1 / (2 τ )-sticky pair of Brownian motions. Lemma 7.11.
Let [1] W denote the web resulting from switching all the marked (1 , points in the originalweb W along B ; then [1] W is equidistributed as the original web. Similarly, if for some fixed k, l with k = l , W ∗ denotes the marked web resulting from switching the original web along the finite segment of B k beforeit coalesces with B l . Then W ∗ is equidistributed with W .Proof. To prove the first part of the lemma, it suffices to show that { [1] B j } are coalescing Brownian motions.Lemma 7.10 implies that each individual [1] B j is a Brownian motion and their construction shows that theyare independent before meeting. The proof that they coalesce upon meeting is basically the same as thatgiven for the paths of W ( τ ) in Subsection 7.7.1. For the second part of the lemma w.l.o.g., set k = 0. Thenthe paths B ∗ j ∈ W ∗ starting from z j coincide with [1] B j for times before the coalescence time of B and B l and afterward coincide with paths in W . It follows that { B ∗ j } are coalescing Brownian motions and thusthat W ∗ is equidistributed with W . We prove the result for j = 0. The result can then be trivially extended to any j . Our proof follows alongthe lines of the proof of Proposition 4.1 given in Subsection 7.4, except of course that here both right andleft marked (1 ,
2) points along B are switched leading to [1] B rather than [1] r . Here, it is enough to provethat ( B , [1] B z ) z ∈ R ×{ } is a family of strong Markov processes with stationary transition probabilities andthat the pair ( B , [1] B z ) satisfies the following three properties.(1) B is a standard Brownian path starting at (0 , [1] B z starts at z .(2) Away from the diagonal { t : [1] B z ( t ) = B ( t ) } , the two processes evolve as two independent Brownianmotions.(3) Defining t ǫ = inf { t > | [1] B − B | ( t ) = √ ǫ } , one has(i) P (cid:0) ( [1] B − B )( t ǫ ) = √ ǫ (cid:1) = (ii) lim ǫ ↓ E ( t ǫ ) /ǫ = √ / (2 τ ) and E ([ t ǫ ] ) = o ( ǫ ).44he strong Markov property and the stationarity of the transition probabilities can be shown as in Lemma7.5. Those two properties and the definition of [1] B z easily imply Properties (1)-(2). Property (3)-(i) isclearly true by right-left symmetry. It remains to prove (3)-(ii). Recall the definition of [1] r given in Section4. We define [1] l analogously, i.e., [1] l is obtained from B by switching all the marked right (1 ,
2) pointsin M ( τ ) ∩ B . We also define t lǫ = inf { t : [1] l ( t ) = B ( t ) − √ ǫ } , (7.96) t rǫ = inf { t : [1] r ( t ) = B ( t ) + √ ǫ } . (7.97) t rǫ , which was carefully studied in Subsection 7.3, (resp., t lǫ ) is the first time a right (resp., left) markedexcursion away from B hits B + √ ǫ (resp., B − √ ǫ ). In order to verify the first part of (3)-(ii), we willprove that lim ǫ ↓ E ( t ǫ ) /ǫ coincides with lim ǫ ↓ E ( t rǫ ∧ t lǫ ) /ǫ and that E ( t rǫ ∧ t lǫ ) /ǫ has the desired limit. Thesecond part can be proved similarly. We first use the following lemma. Lemma 7.12. [1] B is obtained by joining together marked excursions from B .Proof. Let z be a point at which [1] B separates from B . By Proposition 7.6, z is a marked point of theoriginal Brownian web W and there is a marked excursion e from B starting at z . By the structure of theseparation points given in Proposition 7.7 and since (1 ,n ) B τ → [1] B as n → ∞ , we see that (1 ,n ) B τ followsthe excursion e for sufficiently large n . As a consequence, [1] B also follows e . Since this is true for everysuch z , the lemma follows.Lemma 7.12 immediately implies that t ǫ ≥ t rǫ ∧ t lǫ . (7.98)Continuing with our proof of Property (3)-(ii), we define T ǫ = inf { t ≥ t rǫ ∧ t lǫ : | [1] B ( t ) − B ( t ) | = 0 } . Using T (0) ǫ ≡ T ǫ as a (first) stopping time increment, denoting the segments of { B , [1] r , [1] l , [1] B } up totime T (0) ǫ by { B (0)0 , [1] r , [1] l , [1] B } and then translating ( B ( T ǫ ) , T ǫ ) onto (0 , { B ( n )0 , [1] r n ) , [1] l n ) , [1] B n ) , T ( n ) ǫ } , which, as in the proof of Lemma 7.5, are i.i.d. Next, define K ǫ = inf { k : ∃ t ∈ [0 , T ( k ) ǫ ] , | [1] B k ) − B ( k )0 | ( t ) = √ ǫ } (7.99)and also, ˜ T ( n ) ǫ = T ( n ) ǫ ∧ inf { t ∈ [0 , T ( n ) ǫ ] : | [1] B n ) ( t ) − B ( n )0 ( t ) | = √ ǫ } . T ǫ ≡ ˜ T (0) ǫ ) we have E ( t ǫ ) = X n ≥ E ( ˜ T ( n ) ǫ K ǫ ≥ n ) = X n ≥ E ( ˜ T ǫ ) P ( K ǫ ≥ n ) (7.100)= E ( ˜ T ǫ ) X n ≥ P (cid:16) ∀ t ∈ [0 , T ǫ ] , | [1] B − B | ( t ) < √ ǫ (cid:17) n (7.101)= E ( ˜ T ǫ ) P (cid:0) ∃ t ∈ [0 , T ǫ ] , | [1] B − B | ( t ) = √ ǫ (cid:1) (7.102) ≤ E ( ˜ T ǫ ) P ( t lǫ ∧ t rǫ = t ǫ ) , (7.103)Next we prove the following three lemmas. Lemma 7.13. E ( ˜ T ǫ − t rǫ ∧ t lǫ ) /ǫ → as ǫ ↓ .Proof. The path [1] B evolves like a Brownian motion when it is away from B . It follows that E ( ˜ T ǫ − t rǫ ∧ t lǫ ) ≤ sup x ∈ [0 ,ǫ ] ( E ( S x )) where S x is the time a standard Brownian motion starting at x exits the interval [0 , ǫ ].This yields the claimed result. Lemma 7.14. E ( t rǫ ∧ t lǫ ) /ǫ → √ / (2 τ ) . Proof.
Conditioned on W (but not the marking M ( τ )), t rǫ and t lǫ are independent. If we denote by P W theprobability distribution of the marked Brownian web conditioned on a realization of the web W , and by E expectation with respect to the distribution P of W , we have E ( t rǫ ∧ t lǫ ) /ǫ = Z ∞ E ( P W ( t rǫ ∧ t lǫ ≥ ǫt )) dt = Z ∞ E ( P W ( t rǫ ≥ ǫt ) · P W ( t lǫ ≥ ǫt )) dt. (7.104)By Proposition 7.2, P W ( t rǫ ≥ ǫt ) = P W (cid:16) L ǫ,ǫt ([0 , ǫt ]) ≤ Exp(1 / ( √ τ )) (cid:17) = exp( −√ τ l ǫ ( ǫt )) , (7.105)By Lemma 7.3, we know that in probability l ǫ ( ǫt ) → t/
2, implying that P W ( t rǫ ≥ ǫt ) → e − τt/ √ . Bysymmetry, P W ( t lǫ ≥ ǫt ) → e − τt/ √ . In Subsection 7.3.2, we showed that { P ( t rǫ ≥ ǫ · ) = E ( P W ( t rǫ ≥ ǫ · )) } ǫ ≤ isuniformly integrable. Since E ( P W ( t rǫ ≥ ǫt ) P W ( t lǫ ≥ ǫt )) ≤ E ( P W ( t rǫ ≥ ǫt )) = P ( t rǫ ≥ ǫt ) , we have thatlim ǫ ↓ E ( t rǫ ∧ t lǫ ) /ǫ dt = Z ∞ lim ǫ ↓ E ( P W ( t rǫ ≥ ǫt ) · P W ( t rǫ ≥ ǫt )) dt = Z ∞ exp( − tτ / √
2) = √ τ . (7.106) Lemma 7.15. lim ǫ ↓ P (cid:0) t lǫ ∧ t rǫ = t ǫ (cid:1) = 0 . roof. By symmetry, it suffices to prove thatlim ǫ ↓ P (cid:0) t lǫ = t lǫ ∧ t rǫ , t lǫ = t ǫ (cid:1) = 0 . (7.107)Assume that t lǫ = t rǫ ∧ t lǫ and t ǫ = t lǫ . Then there exists a left marked excursion e l,ǫ from B starting at T ( e l,ǫ ) which is at a distance √ ǫ from B at time t lǫ . Since t ǫ = t lǫ , [1] B avoids this excursion implyingthat [1] B follows a right marked excursion e r during the time interval [ T ( e r ) , T ( e r ) + D ( e r )] such that T ( e r ) < T ( e l,ǫ ) < T ( e r ) + D ( e r ). In other words, T ( e l,ǫ ) is straddled by a marked right excursion. Thelemma follows from Proposition 7.4.By (7.103) and Lemmas 7.13, 7.14 and 7.15, we have lim sup ǫ ↓ E ( t ǫ ) /ǫ ≤ lim ǫ ↓ E ( t rǫ ∧ t lǫ ) /ǫ = √ / (2 τ ).By (7.98), Property (3)-(ii) and hence Lemma 7.10 follow. We conclude that ( W , W ( τ )) has the requireddistribution. We continue with the second and third properties of Theorem 6.2. W ( τ ) is constructed by modifying W = W ( τ = 0) according to the marking M ( τ ). In order to prove the Markov property and stationarity,it suffices to prove that this is distributionally equivalent to the following procedure: (1) construct W ( τ )from ( W , M ( τ )); then (2) construct W ( τ ) from ( W ( τ ) , M τ (∆ τ )) where M τ (∆ τ ) is a marking of W ( τ )with intensity ∆ τ ≡ τ − τ which, given the past ( W , {M ( τ ) } τ ≤ τ ), only depends on W ( τ ) .Recall that given W ,(i) for any measurable subset O ⊂ R with L ( O ) < ∞ (where L is the local time outer measure—see Definition 3.1), [ M ( τ ) \ M ( τ )] ∩ O is a Poisson Point Process on R with intensity measure( τ − τ ) L ( · ∩ O ), and(ii) {M ( τ ) } τ ≤ τ and ˜ M (∆ τ ) ≡ M ( τ ) \ M ( τ ) are independent.˜ M (∆ τ ) induces a natural marking on W ( τ ). Indeed, for every n ≥
0, we can define ˜ M τ n,n (∆ τ ) as ˜ M (∆ τ ) ∩ E n where E n = { B τ i } n − i =0 ∩ { ˆ B τ j } n − j =0 and M τ (∆ τ ) ≡ lim n ↑∞ M τ n,n (∆ τ ). We will denote by W τ n,n (∆ τ ) theweb obtained from W ( τ ) by switching the direction of all the points in M τ n,n (∆ τ ).We have already proved that W ( τ ) is a Brownian web. Hence, L ( E n ) < ∞ and, by item (i) above,conditioned on W , M τ n,n (∆ τ ) is a Poisson Point Process with intensity measure ( τ − τ ) L ( · ∩ E n ). Lemma 7.16.
Let L τ n,n be the local time measure on R induced by { B τ i } n − i =0 ∪ { ˆ B τ j } n − j =0 , i.e., L τ n,n ( O ) = m φ (cid:16) P ( { B τ i } n − i =0 \ { ˆ B τ j } n − j =0 ∩ O ) (cid:17) (where P is the projection on the t -axis). Then L ( O ∩ E n ) = L τ n,n ( O ) , where L is the usual local timemeasure of (3.9). roof. For a web W ′ , let W ′ ∩ ˆ W ′ denote the set of (1 ,
2) points of W ′ . By definition, L τ n,n ( O ) = m φ ( P ( E n ∩ O )) L ( O ∩ E n ) = m φ (cid:16) P ([ W ∩ ˆ W ] ∩ E n ∩ O ) (cid:17) Hence, in order to prove our lemma it is sufficient to prove that for every Borel Om φ (cid:16) P ([ E n ∩ O ] \ [ W ∩ ˆ W ]) (cid:17) = 0which will follow if we can prove that m φ (cid:16) P ([ W ( τ ) ∩ ˆ W ( τ )] \ [ W ∩ ˆ W ]) (cid:17) = 0 . (7.108)In order to prove (7.108) we prove m φ (cid:16) P ([ W ∩ ˆ W ] \ [ W ( τ ) ∩ ˆ W ( τ )]) (cid:17) = 0 (7.109)instead. The lemma will follow from the equidistribution of ( W , W ( τ )) and ( W ( τ ) , W ). (Recall that inSubsections 7.7.2-7.7.3 we already proved that ( W , W ( τ )) is a sticky pair of webs whose distribution isinvariant under permutation of the two webs.)We now prove (7.109). For a given realization of ( W , W ( τ )), let us assume that m φ (cid:16) P ([ W ∩ ˆ W ] \ [ W ( τ ) ∩ ˆ W ( τ )]) (cid:17) > M ( τ ), there would be strictly positive probability that M ( τ ) \ [ W ( τ ) ∩ ˆ W ( τ )] = ∅ .Let z be any point in M ( τ ). Then z is a separation point of N ( τ ). Proposition 7.7(3) directly impliesthat for some i the path B τ i ∈ W ( τ ), from z i , enters z . Since up to a reversal of the t -axis N ( τ ) and ˆ N ( τ )are equidistributed, there is a path ˆ B τ j ∈ ˆ W ( τ ) meeting B τ at z and hence that z is in W ( τ ) ∩ ˆ W ( τ ). Itwould follow that M ( τ ) ⊂ W ( τ ) ∩ ˆ W ( τ ), yielding a contradiction. This ends the proof of the lemma.Lemma 7.16 implies that M τ n,n (∆ τ ) only depends on W ( τ ). Moreover, W ( τ ) being a Brownian web,we also have the distributional identities,( W ( τ ) , M τ n,n (∆ τ )) = d ( W , ˜ M n,n (∆ τ )) , (7.110)( W ( τ ) , W τ n,n ( τ )) = d ( W , W n,n (∆ τ )) , (7.111)where W n,n (∆ τ ) and ˜ M n,n (∆ τ ) are defined as in Section 3. It remains to prove that W τ n,n (∆ τ ) converges(in ( H , d H )) to W ( τ ). Lemma 7.17. M τ (∆ τ ) and ˜ M (∆ τ ) coincide. roof. By construction, M τ (∆ τ ) ⊂ ˜ M (∆ τ ) since M τ (∆ τ ) is the marking induced by ˜ M (∆ τ ) on W ( τ ).Analogously, we define M ′ (∆ τ ) as the marking induced by M τ (∆ τ ) (= lim n ↑∞ M τ n,n (∆ τ )) on W . Wealready proved that ( W , W ( τ )) is a (1 / τ )-sticky pair of webs. Therefore, ( W , W ( τ )) is equidistributedwith ( W ( τ ) , W ) and, by (7.110), ( W , W ( τ ) , M τ (∆ τ )) is equidistributed with ( W ( τ ) , W , ˜ M (∆ τ )). Thus,( W , M ′ n,n (∆ τ )) = d ( W ( τ ) , M τ n,n (∆ τ )) . By (7.110), we conclude that M ′ n,n (∆ τ ) is distributed like ˜ M n,n (∆ τ ). Since by construction, M ′ n,n (∆ τ ) ⊂ ˜ M n,n (∆ τ ), it follows that M ′ n,n (∆ τ ) = ˜ M n,n (∆ τ ) and M ′ (∆ τ ) = ˜ M (∆ τ ). Since M ′ ( τ ) ⊂ M τ (∆ τ ), wededuce that ˜ M (∆ τ ) ⊂ M τ (∆ τ ) and hence M τ (∆ τ ) = ˜ M (∆ τ ).Let W ′ be any subsequential limit of {W τ n,n (∆ τ ) } . We next prove that W ′ = W ( τ ) via two inclusions,which completes this subsection. (i) W ′ ⊆ W ( τ ).Let z = ( x, t ) ∈ M ( τ ), and let π ∈ W ′ start at t − ǫ with ǫ >
0, and pass through z . By Proposition6.1, what we need to show is that π ∼ z B switch . By construction, there exists a sequence { π N } N ≥ so that π N belongs to ∪ n,m>M W τ n,m (∆ τ ) and { π N } converges to π . Taking N large enough, we can assume w.l.o.g.that π N belongs to N ≤ t − ǫ and ( x, t ) ∈ M n,m ( τ ) for n, m > N . Moreover, by Proposition 7.7(2) we can alsoassume that π N enters the point z . We distinguish between two cases.1. z ∈ M ( τ ). Here, z / ∈ M τ (∆ τ ) and by construction, π N ∼ z B τ . Since B τ ∼ z B switch , Proposition7.7(1) implies that π N ∼ z B switch . By Proposition 7.7(2), π ∼ z B switch .2. z ∈ ˜ M (∆ τ ). Since M τ (∆ τ ) = ˜ M (∆ τ ), we get that π N ∼ z B τ switch . We claim that B τ switch ∼ z B switch ,implying that π ∼ z B switch as desired. The claim can be verified as follows. Let us assume that B τ switch ∼ z B (and show that this leads to a contradiction). Then B τ ∼ z B switch , implying that B and B τ separate at z , or equivalently that z ∈ M ( τ ). Since M ( τ ) and ˜ M (∆ τ ) = M ( τ ) \ M ( τ )are disjoint, the claim follows. (ii) W ′ ⊇ W ( τ ).There is at least one path B ′ in W ′ starting from z i . By Proposition 6.1(3)(o), there is a unique path B τ i ∈ W ( τ ) starting from there. Since W ′ ⊆ W ( τ ) we get B ′ = B τ i . Hence, W ′ ⊇ { B τ i } = W ( τ ) (seeProposition 6.1(3)(ii)). τ → B τ ( t ) is Piecewise Constant For any τ ≤ τ , the path B τ belongs to N ( τ ). Given N ( τ ), B τ ( t ) only depends on the direction of the (1 , W ( τ ) which are located at the (0 , t )-separation points of N ( τ ). Since the set of (0 , t )-separationpoints is locally finite (see Proposition 7.9), τ → B τ ( t ) is piecewise constant.49 cknowledgements. The research reported in this paper was supported in part by N.S.F. grantsDMS-01-04278 and DMS-06-06696. The authors thank Rongfeng Sun and Jan Swart for very useful com-munications and discussions concerning their work on the Brownian net. They also thank the anonymousreferee for a number of very helpful suggestions which have led to a clearer presentation of some of ourarguments.
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