# Convergence of the dynamical discrete web to the dynamical Brownian web

Krishnamurthi Ravishankar  SUNY, College at New Paltz  Kumarjit Saha  Ashoka
University

###### Abstract

In this paper we study the convergence of dynamical discrete web (DyDW) to the
dynamical Brownian web (DyBW) in the path space topology. We show that almost
surely the DyBW has RCLL paths taking values in an appropriate metric space
and as a sequence of RCLL paths, the scaled dynamical discrete web converges
to the DyBW. This proves weak convergence of the DyDW process to the DyBW
process.

##  1 Introduction and main result

In this paper, we present a number of results concerning a dynamical version
of the Brownian web (BW), known as the dynamical Brownian web (DyBW) and
convergence to it for a dynamical version of coalescing random walks model
known as the dynamical discrete web (DyDW). The DyDW was introduced by Howitt
and Warren [HW09], is a system of coalescing simple symmetric one-dimensional
random walks evolving over a dynamic time interval $[0,1]$. We first describe
the DyDW model briefly.

The discrete web (DW) is a collection of one-dimensional simple symmetric
random walks starting from every point in the discrete space-time domain
${\mathbb{Z}}^{2}_{\text{even}}:=\\{(x,t)\in{\mathbb{Z}}^{2}:x+t\text{
even}\\}$. We consider an i.i.d. collection of random variables
$\\{I_{(x,t)}:(x,t)\in{\mathbb{Z}}^{2}_{\text{even}}\\}$ such that

$\displaystyle I_{(0,0)}:=\begin{cases}+1&\text{ w.p. }1/2\\\ -1&\text{ w.p.
}1/2,\end{cases}$

where $I_{(x,t)}$ gives the random walk increment at location $x$ at time $t$.
Following the walk, the walker at $(x,t)$ goes to
$(x+I_{(x,t)},t+1)\in{\mathbb{Z}}^{2}_{\text{even}}$ along the edge
$\langle(x,t),(x+I_{(x,t)},t+1)\rangle$. Thus, following the edges in DW we
get a (random) collection of paths. The motivation for calling this system as
the DW comes from the fact that under diffusive scaling as a collection of
paths, the DW converges to the BW (Theorem 6.1 of [FINR04]). For the dynamical
version, we start with an ordinary DW at dynamic time $0$ and at each lattice
point, direction of the outgoing edge switches at a fixed rate independently
of all other lattice points. This gives DyDW as a process of collection of
paths evolving dynamically over the time interval $[0,1]$.

Howitt and Warren [HW09] rightly guessed that if we slow down the rate of
switching (of outgoings edges) by $1/\sqrt{n}$, then under diffusive scaling,
there should be a non-trivial scaling limit the dynamical Brownian web process
$\\{{\mathcal{W}}(s):s\in[0,1]\\}$. Assuming existence of such a process, it’s
finite dimensional distributions were analysed in [HW09] and it was shown that
for $s_{1},s_{2}\in[0,1]$, the distribution of
$({\mathcal{W}}(s_{1}),{\mathcal{W}}(s_{2}))$ is given by sticky pair of
Brownian webs with degree of stickiness is given by $1/|2(s_{2}-s_{1})|$ (for
a definition of sticky pair of Brownian webs see Section 3.2.1). Existence of
a consistent family of finite dimensional distribution for
${\mathcal{W}}(\tau)$ follows from this and stationarity and Markov property
proved in Theorem 6.2 of [NRS10] . In [NRS10], Newman et al. proved that such
a process uniquely exists and provided a rigorous construction of the DyBW as
well. In another work [NRS09A], a sketch of the proof of convergence of finite
dimensional distributions of DyDW to that of DyBW was given. But to the best
of our knowledge, weak convergence of the DyDW process to the DyBW process has
not been shown so far. Our goal in this paper is to provide a stronger
topological setting for studying this convergence and give a proof for
convergence in that setting, namely as a process with RCLL paths. We state our
result in detail in Theorem 3.2. Towards this, we established that the DyBW
process has RCLL paths a.s. (taking values in an appropriate metric space). We
prove this in Theorem 2.2.

The paper is organised as follows. In Section 2 we prove that the DyBW has
RCLL paths taking values in an appropriate metric space a.s. Details of the
relevant metric space have also been described. In Section 3 we describe the
DyDW model and prove that it’s finite dimensional distributions converges to
that of DyBW. The main argument for the same was already developed in
[NRS09A]. Finally, in Section 4 we prove that as a sequence of RCLL paths, the
sequence of scaled dynamic discrete webs is tight and hence, we have process
level convergence.

##  2 DyBW is a.s. RCLL path process

In this section we show that the DyBW process has RCLL paths a.s. (Theorem
2.2). The standard BW originated in the work of Arratia ( see [A79] and [A81])
as the scaling limit of the voter model on ${\mathbb{Z}}$. Later Tóth and
Werner [TW98] gave a construction of a system of coalescing Brownian motions
starting from every point in space-time plane ${\mathbb{R}}^{2}$ and used it
to construct the true self-repelling motion. Intuitively, the BW can be
thought of as a collection of one-dimensional coalescing Brownian motions
starting from every point in the space time plane ${\mathbb{R}}^{2}$. Later
Fontes et. al. [FINR04] provided a framework in which the Brownian web is
realized as a random variable taking values in a Polish space, which enabled
them to prove the weak convergence to the Brownian web of coalescing random
walks starting from every point of the space-time lattice in the diffusive
scaling limit. In the following section we recall relevant details from
[FINR04] to describe the Polish space of our interest.

###  2.1 Preliminary details about Polish space of collection of paths

Let ${\mathbb{R}}^{2}_{c}$ denote the completion of the space time plane
${\mathbb{R}}^{2}$ with respect to the metric

$\rho((x_{1},t_{1}),(x_{2},t_{2}))=|\tanh(t_{1})-\tanh(t_{2})|\vee\Bigl{|}\frac{\tanh(x_{1})}{1+|t_{1}|}-\frac{\tanh(x_{2})}{1+|t_{2}|}\Bigr{|}.$

As a topological space ${\mathbb{R}}^{2}_{c}$ can be identified with the
continuous image of $[-\infty,\infty]^{2}$ under a map that identifies the
line $[-\infty,\infty]\times\\{\infty\\}$ with the point $(\ast,\infty)$, and
the line $[-\infty,\infty]\times\\{-\infty\\}$ with the point
$(\ast,-\infty)$. A path $\pi$ in ${\mathbb{R}}^{2}_{c}$ with starting time
$\sigma_{\pi}\in[-\infty,\infty]$ is a mapping
$\pi:[\sigma_{\pi},\infty]\rightarrow[-\infty,\infty]$ such that
$\pi(\infty)=\pi(-\infty)=\ast$ and $t\rightarrow(\pi(t),t)$ is a continuous
map from $[\sigma_{\pi},\infty]$ to $({\mathbb{R}}^{2}_{c},\rho)$. We then
define $\Pi$ to be the space of all paths in ${\mathbb{R}}^{2}_{c}$ with all
possible starting times in $[-\infty,\infty]$. The following metric, for
$\pi_{1},\pi_{2}\in\Pi$

$d_{\Pi}(\pi_{1},\pi_{2})=|\tanh(\sigma_{\pi_{1}})-\tanh(\sigma_{\pi_{2}})|\vee\sup_{t\geq\sigma_{\pi_{1}}\wedge\sigma_{\pi_{2}}}\Bigl{|}\frac{\tanh(\pi_{1}(t\vee\sigma_{\pi_{1}}))}{1+|t|}-\frac{\tanh(\pi_{2}(t\vee\sigma_{\pi_{2}}))}{1+|t|}\Bigr{|}$

makes $\Pi$ a complete, separable metric space. Convergence in this metric can
be described as locally uniform convergence of paths as well as convergence of
starting times. Let ${\cal H}$ be the space of compact subsets of
$(\Pi,d_{\Pi})$ equipped with the Hausdorff metric $d_{{\cal H}}$ given by,

$d_{{\cal H}}(K_{1},K_{2})=\sup_{\pi_{1}\in K_{1}}\inf_{\pi_{2}\in
K_{2}}d_{\Pi}(\pi_{1},\pi_{2})\vee\sup_{\pi_{2}\in K_{2}}\inf_{\pi_{1}\in
K_{1}}d_{\Pi}(\pi_{1},\pi_{2}).$

The space $({\cal H},d_{{\cal H}})$ is a complete separable metric space. Let
$B_{{\cal H}}$ be the Borel $\sigma-$algebra on the metric space $({\cal
H},d_{{\cal H}})$. The Brownian web ${\cal W}$ is an $({\cal H},B_{{\cal H}})$
valued random variable. Below we state Theorem 2.1 of [FINR04] characterizing
the Brownian web as a ${\cal H}$-valued random variable.

######  Theorem 2.1 (Theorem 2.1 of [FINR04]).

There exists an $({\mathcal{H}},{\mathcal{B}}_{{\mathcal{H}}})$-valued random
variable ${\mathcal{W}}$ whose distribution is uniquely determined by the
following properties:

  * $(a)$

from any deterministic point ${\mathbf{x}}\in{\mathbb{R}}^{2}$, there is
almost surely a unique path $\pi^{{\mathbf{x}}}\in{\mathcal{W}}$ starting from
${\mathbf{x}}$;

  * $(b)$

for a finite set of deterministic points
${\mathbf{x}}^{1},\dotsc,{\mathbf{x}}^{k}\in{\mathbb{R}}^{2}$, the collection
$(\pi^{{\mathbf{x}}^{1}},\dotsc,\pi^{{\mathbf{x}}^{k}})$ is distributed as
coalescing Brownian motions starting from
${\mathbf{x}}^{1},\dotsc,{\mathbf{x}}^{k}$;

  * $(c)$

for any countable deterministic dense set ${\mathcal{D}}$ of
${\mathbb{R}}^{2}$, ${\mathcal{W}}$ is the closure of
$\\{\pi^{{\mathbf{x}}}:{\mathbf{x}}\in{\mathcal{D}}\\}$ in $(\Pi,d_{\Pi})$
almost surely.

The DyBW $\\{{\mathcal{W}}(\tau):\tau\in[0,1]\\}$ is defined as a ${\cal
H}$-valued process evolving over dynamic time domain $[0,1]$ with finite
dimensional distributions as mentioned in Section 3.2.1. Newman et. al.
[NRS10] provided a rigorous construction and showed that such a process indeed
exists. Our next result shows that the DyBW process has ${\cal H}$-valued RCLL
paths a.s.

######  Theorem 2.2.

The DyBW process $\\{{\mathcal{W}}(\tau):\tau\in[0,1]\\}$ has RCLL paths a.s.

In what follows, we denote the Polish space of ${\cal H}$-valued RCLL paths
defined over $[0,1]$ with Skorohod metric as ${\cal D}^{{\cal H}}([0,1])$. In
other words, Theorem 2.2 gives us that the DyBW is a ${\cal D}^{{\cal
H}}([0,1])$ valued random variable.

While providing a a rigorous construction of the DyBW
$\\{{\mathcal{W}}(\tau):\tau\in[0,1]\\}$, Newman et. al. developed an
alternate construction of the Brownian net ${\mathcal{N}}$ as well. Their work
presents a construction of the DyBW and the Brownian net both constructed on a
common probability space. In this work we will extensively use this
correspondence and refer this as ‘the corresponding net ${\mathcal{N}}$’ of
the DyBW $\\{{\mathcal{W}}(\tau):\tau\in[0,1]\\}$ and vice versa. In the next
section we describe their common construction and the corresponding net
briefly. For details we refer the reader to Theorem 5.5 and Proposition 6.1 of
[NRS10].

###  2.2 The ‘corresponding’ Brownian net

The approach of Newman et. al. [NRS10] is based on the construction of a
certain Poissonian marking which is governed by local time of the forward web
along the backward web, i.e., construction of a three-dimensional Poisson
point process with intensity measure $L\times\ell$, where $\ell$ is Lebesgue
measure and $L$ is the local time measure of the forward web along the
backward web. For a deterministic point $(x,t)\in{\mathbb{R}}^{2}$ almost
surely there is exactly one outgoing path starting at $(x,t)$ and no incoming
path passing through $(x,t)$ in the Brownian web.

But there are random $(1,2)$ points in the BW where a single Brownian web path
enters the point from an earlier time and two paths leave from that. Among the
two outgoing paths, exactly one path is the continuation of the (incoming)
path coming from earlier time and the other one is “newly-born” (see Figure
1).

Figure 1:  A schematic diagram of a $(1,2)_{l}$ point as the incoming forward
path connects to the leftmost outgoing path. The other outgoing path
represents the “newly born path”. The dotted lines represent similar behaviour
for the backward paths. It shows that at a $(1,2)$ point, a backward path
meets a forward path.

Further, $(1,2)$ points of BW are precisely those at which a forward and a
backward path meet and the set of ‘marked points’ will be supported on the set
of $(1,2)$ points. Each $(1,2)$ point has a preferred left or right
“direction” and accordingly it is denoted as $(1,2)_{l}$ or $(1,2)_{r}$. For a
$(1,2)_{r}$ ($(1,2)_{l}$) point, the continuing path (coming in from earlier
time) is to the right (left) of the “newly-born” path. One of the major
observation of [NRS10] is that at the continuum level, the analogue of DyDW
switching will simply be change of directions at all marked (1, 2) points. In
other words, the DyBW at dynamic time $s_{0}$, denoted by
${\mathcal{W}}(s_{0})$, will be deduced from the initial BW ${\mathcal{W}}(0)$
by switching directions of all marked $(1,2)$ points, marked during dynamic
time interval $[0,s_{0}]$. We should mention that the situation is much more
complicated here since, $L$ is not locally finite as the set of $(1,2)$ points
is dense in ${\mathbb{R}}^{2}$. However, as showed in [NRS10], one can
approximate $L$ by a sequence of locally finite measures $L_{n}$ and do the
markings using $L_{n}$, and then let $n\to\infty$.

Further, it was shown that the same marked point process gives an alternate
construction of the Brownian net, viz., for a marked $(1,2)$ point, which was
originally an $(1,2)_{l}$ point say, the Brownian net includes not only paths
that connect to the left outgoing path (as in the original web) but also ones
that connect to the right outgoing path. Formally, for deterministic
$(x,t)\in{\mathbb{R}}^{2}$ consider the a.s. unique $\pi^{(x,t)}$ starting
from $(x,t)$ in the initial BW ${\mathcal{W}}(0)$. For each
$(x,t)\in{\mathbb{R}}^{2}$ consider the set of all the paths obtained by
continuing along both the right and left outgoing paths at each ‘marked’
$(1,2)$ point on the trajectory of $\pi^{(x,t)}$ and this collection gives the
standard Brownian net (Theorem 5.5 of [NRS10]). The above construction using
Poissonian marking of (1,2) points allows us to have the standard DyBW and the
standard Brownian net defined on the same probability space. In this paper, we
refer this as the “corresponding” Brownian net ${\mathcal{N}}$ corresponding
to the DyBW $\\{{\mathcal{W}}(s):s\in[0,1]\\}$ and we will use this
correspondence extensively. We note that the Brownian net uniquely defines the
dual net. This allows us to define the “the corresponding” double Brownian net
vector $({\mathcal{N}},\widehat{{\mathcal{N}}})$ where
$\widehat{{\mathcal{N}}}$ is the dual for ${\mathcal{N}}$.

We note here that for any interval (open or closed) $A\subseteq[0,1]$ we use
the same construction to construct the ‘corresponding’ net ${\mathcal{N}}_{A}$
corresponding to the DyBW process $\\{{\mathcal{W}}(\tau):\tau\in A\\}$
considering ‘markings’ only in the dynamic time interval $A$. For any
deterministic $0\leq\tau_{1}<\tau_{2}\leq 1$ we have

${\mathcal{N}}_{[\tau_{1},\tau_{2}]}={\mathcal{N}}_{[\tau_{1},\tau_{2})}={\mathcal{N}}_{(\tau_{1},\tau_{2})}={\mathcal{N}}_{(\tau_{1},\tau_{2}]}\text{
a.s. }$

Before ending this section we need to define some important quantities
associated with the corresponding Brownian net ${\mathcal{N}}$.

For any path $\pi\in\Pi$ and for $t\geq\sigma_{\pi}$, its restriction over the
time domain $[t,\infty)$ is denoted as $\pi_{[t,\infty)}$. For
$\pi_{1},\pi_{2}\in\Pi$ and for any interval
$A\subset[\sigma_{\pi_{1}}\vee\sigma_{\pi_{2}},\infty)$, the notation
$d_{\Pi}(\pi_{1},\pi_{2})_{A}$ is defined as

$d_{\Pi}(\pi_{1},\pi_{2})_{A}=\sup_{s\in
A}\Bigl{|}\frac{\tanh(\pi_{1}(s))}{1+|s|}-\frac{\tanh(\pi_{2}(s))}{1+|s|}\Bigr{|}.$

In other words, the quantity $d_{\Pi}(\pi_{1},\pi_{2})_{A}$ does not consider
distance between starting points and represents distance between restricted
trajectories of these two paths w.r.t. spatial part of $d_{\Pi}$ metric
restricted over the time interval
$A\subset[\sigma_{\pi_{1}}\vee\sigma_{\pi_{2}},\infty)$. For a general
collection of paths we describe the notion of separation points as follows.

######  Definition 2.3.

For a path family ${\cal K}\subset\Pi$, a point $(x,t)\in{\mathbb{R}}^{2}$ is
said to be a ‘separation point’ if there exist $\pi_{1},\pi_{2}\in{\cal K}$
such that the following conditions are satisfied :

  * (i)

$\sigma_{\pi_{1}}\vee\sigma_{\pi_{2}}<t$ and $\pi_{1}(t)=\pi_{2}(t)=x$, i.e.,
both the paths start strictly before time $t$ and pass through the point
$(x,t)$;

  * (ii)

The first meeting time of $\pi_{1},\pi_{2}$ after $t$ defined as

$t^{c}_{\pi_{1},\pi_{2}}:=\inf\\{s>t:\pi_{1}(s)=\pi_{2}(s)\\},$

is strictly bigger than $t$.

Let ${\cal S}={\cal S}({\cal K})$ denote the set of separation points in
${\cal K}$. Definition of a separating point suggests that for each separating
point $(x,t)\in{\cal S}({\cal K})$ there exist paths $\pi_{1},\pi_{2}\in{\cal
K}$ which get separated at $(x,t)$ and create an excursion set, which is a
subset of ${\mathbb{R}}^{2}$, in between them over the time interval
$(t,t^{c}_{\pi_{1},\pi_{2}})$. The (random) set of separation points in the
corresponding Brownian net is denoted as ${\cal S}({\mathcal{N}})$. We mention
here that our notion of separation points for ${\mathcal{N}}$ is slightly
different from the notion of ${\mathcal{N}}$ separation points as introduced
in [NRS10] which we mention below:

######  Definition 2.4 (Definition 7.2 [NRS10]).

For $-\infty<T_{1}<T_{2}<+\infty$ a (random) point $(x,t)$ with
$T_{1}<t<T_{2}$ is said to be a $(T_{1},T_{2})$ separation point iff there are
two paths $\pi_{1}$ and $\pi_{2}$ in the Brownian net ${\mathcal{N}}$ starting
from ${\mathbb{R}}\times\\{T_{1}\\}$ and separating at $(x,t)$ which do not
touch (intersect) on $(t,T_{2}]$.

Let $S^{T_{2}}_{T_{1}}$ denote the set of all $(T_{1},T_{2})$ separating
points in the corresponding Brownian net ${\mathcal{N}}$. We observe that
$S^{T_{2}}_{T_{1}}\subseteq{\cal S}({\mathcal{N}})$ a.s. for all
$T_{1}<T_{2}$.

Clearly, for Brownian net a separation point must necessarily be a marked
$(1,2)$ point. There can be several excursion sets created at a separation
point $(x,t)$. We are going to define the maximal excursion set and several
other quantities related to maximal excursion set generated at a separation
point, i.e., at a marked $(1,2)$ type point here:

  * (1)

Maximal excursion set generated at a separating point $(x,t)\in{\cal S}({\cal
K})$ is denoted by $E^{(x,t)}$ and defined as

$\displaystyle E^{(x,t)}:=\\{$
$\displaystyle(y,s):s\in[t,t^{c}_{\pi_{1},\pi_{2}}]\text{ and
}y\in[\pi_{1}(s),\pi_{2}(s)]\text{ for some }\pi_{1},\pi_{2}\in{\cal K}$
$\displaystyle\text{ with }\pi_{1}(t)=\pi_{2}(t)=x\text{ and
}t^{c}_{\pi_{1},\pi_{2}}>t\\}.$

  * (2)

Diameter/width at a separating point $(x,t)\in{\cal S}({\cal K})$ (of the
maximal excursion set at $(x,t)$) is denoted by $D^{(x,t)}$ and defined as

$\displaystyle D^{(x,t)}:=\sup\\{$ $\displaystyle
d_{\Pi}(\pi_{1},\pi_{2})_{[t,t^{c}_{\pi_{1},\pi_{2}}]}:\pi_{1},\pi_{2}\in{\cal
K},\sigma_{\pi_{1}}\vee\sigma_{\pi_{2}}<t,$ $\displaystyle\text{ and
}\pi_{1}(t)=\pi_{2}(t)=x\text{ with }t^{c}_{\pi_{1},\pi_{2}}>t\\}.$

  * (3)

Survival time of a separating point $(x,t)\in{\cal S}({\cal K})$ (of the
maximal excursion set at $(x,t)$) is denoted by $T^{(x,t)}$ and defined as

$\displaystyle T^{(x,t)}:=\sup\\{$
$\displaystyle\tanh(t^{c}_{\pi_{1},\pi_{2}})-\tanh(t):\pi_{1},\pi_{2}\in{\cal
K},\sigma_{\pi_{1}}\vee\sigma_{\pi_{2}}<t,$ $\displaystyle\text{ and
}\pi_{1}(t)=\pi_{2}(t)=x\text{ with }t^{c}_{\pi_{1},\pi_{2}}>t\\}.$

  * (4)

Age of a separating point $(x,t)\in{\cal S}({\cal K})$ (of the maximal
excursion set at $(x,t)$) is denoted by $A^{(x,t)}$ and defined as

$\displaystyle A^{(x,t)}:=\sup\\{$
$\displaystyle\tanh(t)-\tanh(\sigma_{\pi}):\pi\in{\cal K},\sigma_{\pi}<t\text{
and }\pi(t)=x\\}.$

For $\epsilon>0$, the set of separating points in ${\cal K}$ with diameter
more than $\epsilon$ is denoted as $D^{\epsilon}$ and defined as

$D^{\epsilon}=D^{\epsilon}({\cal K}):=\\{(x,t)\in{\cal S}({\cal
K}):D^{(x,t)}>\epsilon\\}.$

Similarly, the sets $A^{\epsilon}=A^{\epsilon}({\cal K}):=\\{(x,t)\in{\cal
S}({\cal K}):A^{(x,t)}>\epsilon\\}$ and $T^{\epsilon}=T^{\epsilon}({\cal
K}):=\\{(x,t)\in{\cal S}({\cal K}):T^{(x,t)}>\epsilon\\}$ are defined.

For ease of notation, for an interval $A\subseteq[0,1]$ for ease of notation,
let ${\cal S}_{A}$ denote the set of separation points ${\cal
S}({\mathcal{N}}_{A})$. Next, for a separating point $(x,t)\in{\cal S}_{A}$
based on the path family ${\mathcal{N}}_{A}$, we define the diameter, age and
survival time quantities denoted by
$D^{(x,t)}_{A},A^{(x,t)}_{A},T^{(x,t)}_{A}$ respectively. Based on the above
quantities, for $\epsilon>0$ we define the following random set of separation
points:

$\displaystyle D^{\epsilon}_{A}$
$\displaystyle:=D^{\epsilon}({\mathcal{N}}_{A})$ $\displaystyle
T^{\epsilon}_{A}$ $\displaystyle:=T^{\epsilon}({\mathcal{N}}_{A})\text{ and }$
$\displaystyle A^{\epsilon}_{A}$
$\displaystyle:=A^{\epsilon}({\mathcal{N}}_{A}).$

For deterministic $0\leq\tau_{1}<\tau_{2}\leq 1$ we have

$\displaystyle(D^{\epsilon}_{[\tau_{1},\tau_{2}]},T^{\epsilon}_{[\tau_{1},\tau_{2}]},A^{\epsilon}_{[\tau_{1},\tau_{2}]})=(D^{\epsilon}_{[\tau_{1},\tau_{2})},T^{\epsilon}_{[\tau_{1},\tau_{2})},A^{\epsilon}_{[\tau_{1},\tau_{2})})\text{
a.s.}$

Fix $m\in{\mathbb{N}}$. For $0\leq j\leq m-1$ for ease of notation we set

$\displaystyle D^{\epsilon}$ $\displaystyle:=D^{\epsilon}_{[0,1]}\text{ and
}D^{\epsilon}_{j}:=D^{\epsilon}_{[j/m,(j+1)/m]}=D^{\epsilon}_{[j/m,(j+1)/m)}$
$\displaystyle T^{\epsilon}$ $\displaystyle:=T^{\epsilon}_{[0,1]}\text{ and
}T^{\epsilon}_{j}:=T^{\epsilon}_{[j/m,(j+1)/m]}=T^{\epsilon}_{[j/m,(j+1)/m)}$
$\displaystyle A^{\epsilon}$ $\displaystyle:=A^{\epsilon}_{[0,1]}\text{ and
}A^{\epsilon}_{j}:=A^{\epsilon}_{[j/m,(j+1)/m]}=A^{\epsilon}_{[j/m,(j+1)/m]}.$

In the next section we use the correspondence between the DyBW and the
Brownian net and prove Theorem 2.2.

###  2.3 Proof of Theorem 2.2

We first start with some basics on RCLL paths. Let $(X,d_{X})$ denote a
general metric space. For a $(X,d_{X})$ valued function $f$ defined over
$[0,1]$ and for any subset $A\subset[0,1]$ we define

$\displaystyle\omega^{\prime}_{f}(A):=\sup\\{d_{X}(f(s_{1}),f(s_{2})):s_{1},s_{2}\in
A\\}.$

For $\delta>0$, the notation $\omega_{f}(\delta)$ denotes the natural
extension of modulus of continuity for a general function $f$ defined as

$\displaystyle\omega_{f}(\delta):=\inf\sup\\{\omega^{\prime}_{f}([t_{i},t_{i+1})):0\leq
i\leq n-1\\},$

where the infimum is taken over all partitions $0=t_{0}<t_{1}<\cdots<t_{n}=1$
of $[0,1]$, with $t_{i+1}-t_{i}>\delta$ for all $0\leq i\leq n-1$. We have the
following characterisation of RCLL functions (see page 123 of [B99]):

######  Lemma 2.5.

$f$ is RCLL if and only if $\lim_{\delta\downarrow 0}\omega_{f}(\delta)=0$.

Before we proceed further, we want to make an useful observation regarding the
function $\omega(\cdot)$ that just follows from the triangle inequality.

######  Remark 2.6.

For an RCLL function $f$ and for any $0\leq\tau_{1}<\tau^{\prime}<\tau_{2}\leq
1$ which is not a jump point, we have

$\omega_{f}([\tau_{1},\tau_{2}))\leq\omega_{f}([\tau_{1},\tau^{\prime}))+\omega_{f}([\tau^{\prime},\tau_{2})).$

Considering the dynamical Brownian web (DyBW)
$\\{{\mathcal{W}}(\tau):\tau\in[0,1]\\}$ as a ${\cal H}$-valued stochastic
process over the dynamic time domain $[0,1]$, with a slight of abuse of
notation, we denote corresponding quantities for the DyBW as
$\omega_{{\mathcal{W}}}(\delta)$. Because of Lemma 2.5, in order to prove
Theorem 2.2, it suffices to prove the following proposition.

######  Proposition 2.7.

For any $\epsilon>0$ a.s. there exists (random)
$\delta_{0}=\delta_{0}(\epsilon,\omega)>0$ such that for all
$\delta\leq\delta_{0}$, we have $\omega_{{\mathcal{W}}}(\delta)<\epsilon$.

In other words Proposition 2.7 tells us that $\lim_{\delta\downarrow
0}\omega_{{\mathcal{W}}}(\delta)=0$ a.s. Proposition 2.7 will be proved
through a sequence of lemmas.

We recall the notations introduced in the previous section and it is
straightforward to observe that for any $0\leq\tau_{1}<\tau_{2}\leq 1$ we have

$D^{\epsilon}_{[\tau_{1},\tau_{2}]}\subset D^{\epsilon}_{[0,1]}\text{ and
}T^{\epsilon}_{[\tau_{1},\tau_{2}]}\subset T^{\epsilon}_{[0,1]}\text{ a.s.}$

Before we proceed further, we make some important remarks.

######  Remark 2.8.

  * (i)

We observe that a $(1,2)$ type point $(x,t)$ of ${\mathcal{W}}(\tau)$ belongs
to the set of separating points ${\cal S}_{[\tau_{1},\tau_{2})}$ of the
corresponding Brownian net only if it undergoes switching during the interval
$(\tau_{1},\tau_{2})$ which means that the Poisson clock triggering switching
event associated to $(x,t)$ must ring during this interval. That is it is a
“marked” point and marking occurs during the interval $(\tau_{1},\tau_{2})$.

  * (ii)

For $0\leq\tau_{1}<\tau_{2}\leq 1$, let $\pi_{\tau_{1}}$ and $\pi_{\tau_{2}}$
denote the trajectory of a ‘skeletal’ Brownian path $\pi$ at dynamic times
$\tau_{1},\tau_{2}$ respectively. As $\pi$ is a skeletal Brownian path
(starting from a point in ${\mathbb{Q}}^{2}$), we have
$\sigma_{\pi_{\tau_{1}}}=\sigma_{\pi_{\tau_{2}}}\in{\mathbb{Q}}$. If none of
the $(1,2)$ type points on the trajectory of $\pi_{\tau_{1}}$ belongs to the
set $D^{\epsilon}_{[\tau_{1},\tau_{2}]}$, then we must have
$d_{\Pi}(\pi_{\tau_{1}},\pi_{\tau_{2}})<\epsilon$.

The next lemma shows that for any $(x,t)\in
D^{\epsilon}_{[\tau_{1},\tau_{2}]}$, the quantity
$T^{(x,t)}_{[\tau_{1},\tau_{2}]}$ must be sufficiently large as well.

######  Lemma 2.9.

Fix $\epsilon>0$. For any interval $A\subseteq[0,1]$ there exists
$0<\tilde{\epsilon}=\tilde{\epsilon}(\epsilon,\omega)<\epsilon$ (which does
not depend on $A$) such that a.s. we have $D^{\epsilon}_{A}\subseteq
T^{\tilde{\epsilon}}_{A}$.

###### Proof.

We observe that

$\pi\in{\mathcal{W}}(\tau)\text{ for some }\tau\in[0,1]\text{ only if
}\pi\in{\mathcal{N}},$

where ${\mathcal{N}}$ denotes the corresponding Brownian net. We also note
that a.s. paths in the Brownian net ${\mathcal{N}}$ form an equicontinuous
path family. This allows us to define
$\tilde{\epsilon}=\tilde{\epsilon}(\epsilon,\omega)>0$ as

$\displaystyle\tilde{\epsilon}:=\sup\\{$
$\displaystyle\delta\in(0,\epsilon/3):\text{ for any
}\pi\in{\mathcal{N}}\text{ and }u_{1},u_{2}\geq\sigma_{\pi}\text{ with
}|\tanh(u_{1})-\tanh(u_{2})|\leq\delta$ $\displaystyle\text{ we have
}|\frac{\pi(u_{1})}{1+|u_{1}|}-\frac{\pi(u_{2})}{1+|u_{2}|}|<\epsilon/3\\}.$
(1)

By definition, we have $\tilde{\epsilon}<\epsilon/3$ a.s. Equation (2.3)
ensures that for any
$\pi_{1},\pi_{2}\in{\mathcal{N}},t\geq\sigma_{\pi_{1}}\vee\sigma_{\pi_{2}}$
with $\pi_{1}(t)=\pi_{2}(t)$ and for any interval $I\subset[t,\infty)$
satisfying $|\tanh(u)-\tanh(t)|\leq\tilde{\epsilon}$ for all $u\in I$, we have
$d_{\Pi}(\pi_{1},\pi_{2})_{I}<\epsilon$. This follows from the observation
that

$\displaystyle|\frac{\pi_{1}(u_{1})}{1+|u_{1}|}-\frac{\pi_{2}(u_{2})}{1+|u_{2}|}|\leq|\frac{\pi_{1}(u_{1})}{1+|u_{1}|}-\frac{\pi_{1}(t)}{1+|t|}|+|\frac{\pi_{2}(t)}{1+|t|}-\frac{\pi_{2}(u_{2})}{1+|u_{2}|}|<\epsilon\text{
for all }u_{1},u_{2}\in I.$

For any $(x,t)\in D^{\epsilon}_{A}$ there exist paths $\pi_{1},\pi_{2}$ in the
corresponding Brownian net ${\mathcal{N}}_{A}$ passing through $(x,t)$ with
$d_{\Pi}(\pi_{1},\pi_{2})_{[t,\tanh(t^{c}_{\pi_{1},\pi_{2}})]}>\epsilon$. In
order to have
$d_{\Pi}(\pi_{1},\pi_{2})_{[t,\tanh(t^{c}_{\pi_{1},\pi_{2}})]}>\epsilon$, we
must have

$\tanh(t^{c}_{\pi_{1},\pi_{2}})>\tanh(t)+\tilde{\epsilon}.$

In other words, the paths $\pi_{1}$ and $\pi_{2}$ are allowed to intersect
only after time $t_{0}$ where $\tanh(t_{0})-\tanh(t)>\tilde{\epsilon}$. This
implies that $T^{(x,t)}_{[\tau_{1},\tau_{2}]}>\tilde{\epsilon}$ and completes
the proof. ∎

Next, for $\epsilon>0$ and for any interval $A\subset[0,1]$ we define

$\displaystyle\Xi^{\epsilon}_{A}=\Xi^{\epsilon}({\mathcal{N}}_{A}):=D^{\epsilon}_{A}\cap\text{A}^{\tilde{\epsilon}}_{A},$
(2)

where $\tilde{\epsilon}>0$ is as in (2.3). For simplicity of notations, we set
$\Xi^{\epsilon}:=\Xi^{\epsilon}_{[0,1]}$. It is not difficult to observe that
for any $0\leq\tau_{1}<\tau_{2}\leq 1$ we have
$\Xi^{\epsilon}_{[\tau_{1},\tau_{2}]}\subset\Xi^{\epsilon}_{[0,1]}$.

Lemma 2.9 gives us that

$\Xi^{\epsilon}_{[\tau_{1},\tau_{2}]}\subseteq\text{A}^{\tilde{\epsilon}}_{[\tau_{1},\tau_{2}]}\cap
T^{{\tilde{\epsilon}}}_{[\tau_{1},\tau_{2}]}.$

######  Lemma 2.10.

Fix $\epsilon>0$. There exists $M=M(\epsilon,\omega)\in{\mathbb{N}}$ such that
for any interval $A\subset[0,1]$ we have

$\displaystyle\Xi^{\epsilon}_{A}\subseteq\Xi^{\epsilon}_{[0,1]}\subset[-M,M]^{2}.$
(3)

###### Proof.

By definition, for any $(x,t)\in\Xi^{\epsilon}$ we have

$A^{(x,t)}_{[0,1]}\wedge T^{(x,t)}_{[0,1]}>\tilde{\epsilon}>0.$

Hence, we can choose $M_{1}>0$ such that for any $(x,t)\in\Xi^{\epsilon}$ both
the following conditions hold:

  * (i)

$t\in[-M_{1},M_{1}]$ as well as

  * (ii)

at least one of the two outgoing paths starting from $(x,t)$ intersects the
region $[-M_{1},M_{1}]^{2}$.

We will show that there exists $M=M(\epsilon,\omega)\geq M_{1}$ such that the
region $[-M,M]^{2}$ contains the set $\Xi^{\epsilon}$. We will present a
sketch of the proof here.

First we Consider outgoing paths starting from $(1,2)$-type points from left
of the box $[-M_{1},M_{1}]^{2}$. For $j\geq 1$ we define the rectangle $R_{j}$
as $R_{j}:=[-M_{1}-j-1,-M_{1}-j]\times[-M_{1},M_{1}]$. A non-negative integer
valued r.v. $\tilde{N}_{1}$ is defined as

$\tilde{N}_{1}=\sup_{j}\\{j:R_{j}\text{ contains at least }j^{2}\text{ many
}\Xi^{\epsilon}\text{ points}\\}.$

We need to show that $\tilde{N}_{1}$ is a.s. finite. Towards that for $j\geq
1$ we define the event:

$\displaystyle B^{1}_{j}:=\\{\text{The set }R_{j}\text{ contains at least
}j^{2}\text{ many }\Xi^{\epsilon}\text{ points}\\}.$

For $j\geq 1$, the total number of $\Xi^{\epsilon}$ points in $R_{j}$ is
dominated by the total number of $(1,2)$ type points in $R_{j}$. The total
number of $(1,2)$ type points in $R_{j}$ is of finite expectation and hence,
Markov’s inequality gives us $\sum_{j\geq 1}{\mathbb{P}}(B^{1}_{j})<\infty$.
By applying Borel Cantelli lemma we conclude that $\tilde{N}_{1}$ is finite
a.s.

Next, we define the r.v. $\tilde{N}_{2}$ as

$\displaystyle\tilde{N}_{2}=\sup\\{$ $\displaystyle j\geq 1:\text{ the box
}R_{j}\text{ contains at least }\text{ one }\Xi^{\epsilon}\text{ point with at
least one }$ $\displaystyle\text{ of the outgoing paths intersects
}[-M_{1},M_{1}]^{2}\\}.$

In order to prove Lemma 2.10 it suffices to show that the r.v. $\tilde{N}_{2}$
is also a.s. finite. Towards that, for $j\geq 1$ we define the event

$\displaystyle B_{j}^{2}=$ $\displaystyle\\{j\geq 1:\text{ at least one of the
outgoing paths starting from an element of }$
$\displaystyle\Xi^{\epsilon}\text{ in }R_{j}\text{ intersects
}[-M_{1},M_{1}]^{2}\\}.$

For $k\in\\{0,1,2,\cdots\\}$ let $\Omega_{k}$ denote the event that
$\Omega_{k}:=\\{\tilde{N}_{1}=k\\}$. Note that the collection
$\\{\Omega_{k}:k\geq 0\\}$ gives a partition of $\Omega$. For any $k\geq 0$ we
show that on the event $\Omega_{k}$ we have

$\displaystyle\sum_{j=1}^{\infty}{\mathbb{P}}(B^{2}_{j}\cap\Omega_{k})<\infty.$
(4)

By Borel Cantelli lemma, Equation (4) implies that on the event $\Omega_{k}$
the r.v. $\tilde{N}_{2}$ is a.s. finite. As $\\{\Omega_{k}:k\geq 0\\}$ forms a
partition of the whole space, this completes the proof.

The argument for showing (4) is standard and we only give a sketch here. With
a slight abuse of notation let $B(t)_{t\geq 0}$ denote a Brownian motion with
drift $1$. Application of union bound together with translation invariance
nature of our model allows us to bound the probability
${\mathbb{P}}(B^{2}_{j}\cap\Omega_{k})$ for any $j>k$ as

${\mathbb{P}}(B^{2}_{j}\cap\Omega_{k})\leq
j^{2}{\mathbb{P}}(\sup_{t\in[0,2M_{1}]}|B(t)|>j).$

As $j\to\infty$ the probability ${\mathbb{P}}(\sup_{t\in[0,2M_{1}]}|B(t)|>j)$
decays exponentially in $j$ and hence, we have
$\sum_{j=1}^{\infty}{\mathbb{P}}(B^{2}_{j}\cap\Omega_{k})<\infty$ for all
$k\geq 0$. This completes the proof. ∎

Next, we show that the set $\Xi^{\epsilon}$ must be finite a.s. Let
$\epsilon^{\prime}=\epsilon^{\prime}(\epsilon,\omega)$ be an a.s. strictly
positive random variable defined as

$\epsilon^{\prime}:=\sup\\{0<\delta<\tilde{\epsilon}:|s_{1}-s_{2}|<\delta\text{
implies that }|\tanh(s_{1})-\tanh(s_{2})|<\tilde{\epsilon}\\},$ (5)

where $\tilde{\epsilon}$ is as in (2.3). We are now ready to state the next
lemma.

######  Lemma 2.11.

Fix $\epsilon>0$. Set $\epsilon^{\prime}=\epsilon^{\prime}(\epsilon,\omega)>0$
as in (5) and $M=M(\omega,\epsilon)$ as in Lemma 2.10. For all
$0\leq\tau_{1}<\tau_{2}\leq 1$ a.s. we have

$\displaystyle\Xi^{\epsilon}_{[\tau_{1},\tau_{2}]}\subseteq\Bigl{(}\bigl{(}\cup_{j=0}^{\lfloor
2M/\epsilon^{\prime}\rfloor}S^{-M+(j+1)\epsilon^{\prime}}_{-M+j\epsilon^{\prime}}\bigr{)}\cap[-M,M]^{2}\Bigr{)},$

where for $T_{1}<T_{2}$ the set $S^{T_{2}}_{T_{1}}$ is defined as in
Definition 2.4.

###### Proof.

As argued earlier, it is enough to prove Lemma 2.11 for
$\Xi^{\epsilon}=\Xi^{\epsilon}_{[0,1]}$. Because of Lemma 2.10, it suffices to
show that any $(x,t)$ in $\Xi^{\epsilon}$ must belong to
$S^{-M+(j+1)\epsilon}_{-M+j\epsilon}$ for some $0\leq j\leq\lfloor
2M/\epsilon\rfloor$. By definition, for any $(x,t)\in\Xi^{\epsilon}$, we must
have $\text{A}^{(x,t)}_{s_{1}}>\tilde{\epsilon}$ for some $s_{1}\in[0,1]$. The
choice of $\epsilon^{\prime}$ ensures that in the corresponding Brownian net
${\mathcal{N}}$, there must be an incoming path which starts before
$t-\epsilon^{\prime}$ and passes through $(x,t)$. On the other hand, Lemma 2.9
ensures that we also have $T^{(x,t)}_{[\tau_{1},\tau_{2}]}>\tilde{\epsilon}$.

This ensures that in the corresponding Brownian net ${\mathcal{N}}$, there are
two outgoing paths which start before time $t-\epsilon^{\prime}$, get
separated at $(x,t)$ and thereafter, they are not allowed to meet before time
$t+\epsilon^{\prime}$. Finally, for any deterministic $s$, the set
${\mathbb{R}}\times\\{s\\}$ can not contain any $(1,2)$ type points. So for
any $(x,t)\in\Xi^{\epsilon}$, there must exist $0\leq j^{\prime}\leq\lfloor
2M/\epsilon^{\prime}\rfloor$ such that
$t\in(-M+j^{\prime}\epsilon^{\prime},-M+(j^{\prime}+1)\epsilon^{\prime})$ and
the outgoing paths remain separated till time
$-M+(j^{\prime}+1)\epsilon^{\prime}$. Together with Lemma 2.10, this completes
the proof. ∎

For any interval $A\subset[0,1]$ we consider the set $\Xi^{\epsilon}_{A}$ and
the set of corresponding Poisson clock rings for switching events is denoted
as $\Lambda^{\epsilon}_{A}$. Clearly, $\Lambda^{\epsilon}_{A}$ is a random
subset of $A\subseteq[0,1]$. For ease of notation the set
$\Lambda^{\epsilon}_{[0,1]}$ is simply denoted as $\Lambda^{\epsilon}$.
Clearly, for any $A\subseteq[0,1]$ we have

$\Xi^{\epsilon}_{A}\subseteq\Xi^{\epsilon}\text{ as well as
}\Lambda^{\epsilon}_{A}\subseteq\Lambda^{\epsilon}.$

Since $S^{-M+(j+1)\epsilon^{\prime}}_{-M+j\epsilon^{\prime}}$ are locally
finite (see Proposition 7.9 of [NRS10]) Lemma 2.11 gives us that the set
$\Xi^{\epsilon}$ is finite almost surely . This implies that that the set
$\Lambda^{\epsilon}$ is a.s. finite as well. The next lemma shows that for any
$0\leq\tau_{1}<\tau_{2}\leq 1$ in order to have
$\omega_{{\mathcal{W}}}([\tau_{1},\tau_{2}))>\epsilon$ the set
$\Lambda_{[\tau_{1},\tau_{2})}\cap(\tau_{1},\tau_{2})$ must be non-empty.

######  Lemma 2.12.

For $0\leq\tau_{1}<\tau_{2}\leq 1$ and for $\epsilon>0$, on the event
$\Lambda^{\epsilon}_{[\tau_{1},\tau_{2})}\cap(\tau_{1},\tau_{2})=\emptyset$ we
must have $\omega_{{\mathcal{W}}}([\tau_{1},\tau_{2}))\leq\epsilon$.

###### Proof.

We first observe that in order to have
$\omega_{{\mathcal{W}}}([\tau_{1},\tau_{2}))>\epsilon$, there must be dynamic
times $s_{1},s_{2}\in[\tau_{1},\tau_{2})$ with $s_{1}<s_{2}$ and a skeletal
Brownian path $\pi$ (i.e., starting from a point in ${\mathbb{Q}}^{2}$) such
that

$d_{{\cal H}}(\\{\pi_{s_{1}}\\},{\mathcal{W}}(s_{2}))>\epsilon,$ (6)

where $\pi_{s_{1}}$ denotes the trajectory of the skeletal path $\pi$ at
dynamic time $s_{1}$. A skeletal path starts from a point in
${\mathbb{Q}}^{2}$ (which is not a $(1,2)$ type point a.s.) and hence starting
time of a skeletal path does not change over dynamic time interval. Therefore
the starting time of a skeletal path $\pi$ does not depend on dynamic time
$s_{1}$ and denoted by $\sigma_{\pi}$.

Equation (6) necessarily implies that
$d_{\Pi}(\pi_{s_{1}},\pi_{s_{2}})>\epsilon$ where $\pi_{s_{2}}$ denotes the
trajectory of the same skeletal path $\pi$ at dynamic time $s_{2}$. Hence, in
order to have (6), there must be a separating point $(x,t)$ on the trajectory
of $\pi_{s_{1}}$, as a point of separation between $\pi_{s_{1}}$ and
$\pi_{s_{2}}$, such that

$D^{(x,t)}_{[\tau_{1},\tau_{2})}\geq D^{(x,t)}_{[s_{1},s_{2}]}>\epsilon.$

In other words, in order to have (6), the set
$D^{\epsilon}_{[\tau_{1},\tau_{2})}$ must be non-empty. We observe that for
any $(1,2)$ type point $(y,s)$ to be in the set
$D^{\epsilon}_{[\tau_{1},\tau_{2})}$, the associated Poisson clock must ring
during the interval $(\tau_{1},\tau_{2})$. Recall that the set
$\Lambda^{\epsilon}_{[\tau_{1},\tau_{2})}$ represents Poisson clock rings
associated to points in the set
$\Xi^{\epsilon}_{[\tau_{1},\tau_{2})}=D^{\epsilon}_{[\tau_{1},\tau_{2})}\cap
A^{\tilde{\epsilon}}_{[\tau_{1},\tau_{2})}$, where $\tilde{\epsilon}$ is as in
(2.3).

Let us assume that
$\Lambda^{\epsilon}_{[\tau_{1},\tau_{2})}\cap(\tau_{1},\tau_{2})=\emptyset$.
By Remark 2.8 this event equivalently implies that the set
$\Xi^{\epsilon}_{[\tau_{1},\tau_{2})}$ is empty as well. Further, the
condition $\Xi^{\epsilon}_{[\tau_{1},\tau_{2})}=\emptyset$ implies that for
any $(y,s)\in D^{\epsilon}_{[\tau_{1},\tau_{2})}$, we must have
$A^{(y,s)}_{[\tau_{1},\tau_{2})}\leq\tilde{\epsilon}$. We will show that under
this assumption, we can’t have (6) and henceforth we obtain a contradiction.

Choose $(y,s)$ to be a (random) point on the trajectory of $\pi_{s_{1}}$ such
that $\tanh(s)-\tanh(\sigma_{\pi})\in(\tilde{\epsilon},2\tilde{\epsilon})$ and
$(y,s)$ is not a $(1,2)$ type point. Under our assumption
$\Xi^{\epsilon}_{[\tau_{1},\tau_{2}]}=\emptyset$, the trajectory of
$\pi_{s_{1}}$ restricted over time domain $[s,\infty)$ can not have a point in
the set $D^{\epsilon}_{[\tau_{1},\tau_{2})}$. As $(y,s)$ is not a $(1,2)$ type
point, there exists a unique path $\pi^{\prime}$ in ${\mathcal{W}}(s_{2})$
starting from the point $(y,s)$. Our assumption also ensures that there is no
point in $D^{\epsilon}_{[\tau_{1},\tau_{2})}$ which belongs to the trajectory
of $\pi^{\prime}$. Finally, as we have
$\tanh(\sigma_{\pi^{\prime}})-\tanh(\sigma_{\pi})<2\tilde{\epsilon}$, we
obtain

$d_{\Pi}(\pi_{s_{1}},\pi^{\prime})\leq\epsilon.$

This completes the proof. ∎

Now we are ready to complete the proof of Proposition 2.7 and thereby proving
that the DyBW is RCLL a.s.

Proof of Proposition 2.7: Fix $\epsilon>0$. We choose
$\delta=\delta(\epsilon,\omega)>0$ such that for any
$s_{1},s_{2}\in\Lambda^{\epsilon}(\omega)$ we have

$\min\\{s_{1},(1-s_{2}),|s_{1}-s_{2}|\\}>2\delta.$

With the above choice of $\delta$, there exists a partition of the interval
$[0,1]$ as $0=s^{\prime}_{0}<s^{\prime}_{1}<\cdots<s^{\prime}_{m}=1$ such that
$s^{\prime}_{i+1}-s^{\prime}_{i}>\delta$ and
$\Lambda^{\epsilon}\cap(s^{\prime}_{i},s^{\prime}_{i+1})=\emptyset$ for all
$0\leq i\leq m-1$. This readily implies that
$\Lambda^{\epsilon}_{[s^{\prime}_{i},s^{\prime}_{i+1})}\cap(s^{\prime}_{i},s^{\prime}_{i+1})=\emptyset$
for all $\ 0\leq i\leq m-1$. Hence, Lemma 2.12 ensures that we have

$\displaystyle\max\\{\omega_{{\mathcal{W}}}([s^{\prime}_{i},s^{\prime}_{i+1})):0\leq
i\leq m-1\\}\leq\epsilon.$

This completes the proof. ∎

##  3 Finite dimensional distribution convergence of the DyDW to the DyBW

In this section we describe finite dimensional convergence of the DyDW to the
DyBW. As commented earlier, the main argument for the same was already done in
[NRS09A] and we have followed the same strategy. We first start with
describing the dynamic discrete web model. After that we describe finite
dimensional distributions of the DyBW and in Section 3.2 we present the
required finite dimensional convergence.

###  3.1 Dynamic discrete web (DyDW) model

The discrete web (DW) is a system of coalescing simple symmetric one-
dimensional random walks starting from everywhere on the space time even
lattice ${\mathbb{Z}}^{2}_{\text{even}}:=\\{(x,t)\in{\mathbb{Z}}^{2}:x+t\text{
even}\\}$. We described in the beginning that we have an i.i.d. collection of
Rademacher random variables
$\\{I_{(x,t)}:(x,t)\in{\mathbb{Z}}^{2}_{\text{even}}\\}$ where $I_{(x,t)}$
gives increment of the walker at location $x$ at time $t$. Following the walk,
the walker at $(x,t)$ reaches
$(x+I_{(x,t)},t+1)\in{\mathbb{Z}}^{2}_{\text{even}}$ and this next step is
denoted as $h(x,t)$. Set $h^{0}(x,t)=(x,t)$ and for $k\geq 1$, define
$h^{k}(x,t)=h(h^{k-1}(x,t))$. Joining the successive steps $\langle
h^{k-1}(x,t),h^{k}(x,t)\rangle$ we get the path $\pi^{(x,t)}\in\Pi$ starting
from $(x,t)$. The collection of all paths obtained from the DW is denoted as
${\cal X}:=\\{\pi^{(x,t)}:(x,t)\in{\mathbb{Z}}^{2}_{\text{even}}\\}$. For
$n\in{\mathbb{N}}$ and for any $(x,t)\in{\mathbb{R}}^{2}$, let $S_{n}$ denote
the $n$-th order diffusively scaled map given by

$S_{n}(x,t):=(x/\sqrt{n},t/n).$

With a slight abuse of notation for $A\subset{\mathbb{R}}^{2}$ we denote
$S_{n}(A)$ as $S_{n}(A):=\\{(x/\sqrt{n},t/n):(x,t)\in A\\}$. For $\pi\in\Pi$
and for ${\cal K}\subset\Pi$, we define $S_{n}(\pi)$ and $S_{n}({\cal K})$ by
identifying each path with it’s graph as a subset of ${\mathbb{R}}^{2}$. For
any $n\geq 1$ closure of the set $S_{n}({\cal X})$ taken in $(\Pi,d_{\Pi})$
metric is denoted as $\overline{S_{n}({\cal X})}$. This gives a ${\cal H}$
valued random variable. Fontes et. al. [FINR04] proved that as a sequence of
${\cal H}$ valued random variables, $\\{\overline{S_{n}({\cal
X})}:n\in{\mathbb{N}}\\}$ converges in distribution to the Brownian web.

######  Theorem 3.1 (Theorem 5.1 of [FINR04]).

As $n\to\infty$, $\overline{S_{n}({\cal X})}$ converges in distribution to the
Brownian web ${\mathcal{W}}$.

The DyDW is a dynamic evolution of the discrete web where the arrow
configuration evolves continuously over dynamic time such that at each space-
time point $(x,t)\in{\mathbb{Z}}^{2}_{\text{even}}$, the outgoing arrow
$\langle(x,t),h(x,t)\rangle$ switches at unit rate independent of everything
else. In order to define the DyDW we consider independent sequences of
collections of i.i.d. random variables
$\\{I^{m}_{(x,t)}:(x,t)\in{\mathbb{Z}}^{2}\\}_{m\geq 0}$. For each
$(x,t)\in{\mathbb{Z}}^{2}$ we consider a Poisson point process
$\\{N_{(x,t)}(\tau):\tau\in[0,1]\\}$ which is a Markov process with state
space $\\{0\\}\cup{\mathbb{N}}$ evolving over dynamic time $[0,1]$ with unit
intensity. In other words, for any $\tau\in[0,1]$ we have

${\mathbb{P}}(N_{(x,t)}(\tau)=\ell)=\frac{e^{-\tau}\tau^{\ell}}{\ell\times\cdots\times
1}\text{ for }\ell\geq 0.$

We further assume that these Markov processes, as $(x,t)$ varies over
${\mathbb{Z}}^{2}_{\text{even}}$, are mutually independent of each other. At
dynamic time $\tau\in[0,1]$, for each $(x,t)$ we use the increment
$I^{N_{(x,t)}(\tau)}_{(x,t)}$ and this gives us a distributionally equivalent
copy of the DW. Altogether, this represents the DyDW as an ${\cal H}$-valued
stationary Markov process evolving over dynamic time domain $[0,1]$.

Mathematically, for $(x,t)\in{\mathbb{Z}}^{2}_{\text{even}}$ and dynamic time
$\tau$, next step at that instant is denoted as $h_{\tau}(x,t)$ and given as

$h_{\tau}(x,t):=(x+I^{N_{(x,t)}(\tau)}_{(x,t)},t+1)\in{\mathbb{Z}}^{2}_{\text{even}}.$

$h^{k}_{\tau}(x,t)$, i.e., the $k$-th step at dynamic time instant $\tau$ is
similarly defined. The path $\pi^{(x,t)}_{\tau}\in\Pi$ starting from $(x,t)$
at dynamic time $\tau$ is obtained by linearly joining successive steps
$\langle h^{k-1}_{\tau}(x,t),h^{k}_{\tau}(x,t)\rangle$. The collection of all
paths obtained from DyDW at dynamic time $\tau$ is denoted as ${\cal
X}(\tau):=\\{\pi_{\tau}^{(x,t)}:(x,t)\in{\mathbb{Z}}^{2}_{\text{even}}\\}$ and
$\\{\overline{{\cal X}}(\tau):\tau\in[0,1]\\}$ represents the DyDW process
defined over the dynamic time domain $[0,1]$.

Fix $n\in{\mathbb{N}}$ and now we are going to define the $n$-th diffusively
scaled DyDW. Let
$\\{N^{n}_{(x,t)}(\tau):\tau\in[0,1]\\}_{(x,t)\in{\mathbb{Z}}^{2}}$ denote a
collection of independent Poisson processes with intensity $1/\sqrt{n}$, i.e.,

${\mathbb{P}}(N^{n}_{(x,t)}(\tau)=\ell)=\frac{e^{-\tau/\sqrt{n}}(\tau/\sqrt{n})^{\ell}}{\ell\times\cdots\times
1}\text{ for }\ell\geq 0.$

Corresponding to these Markov processes, we define

$h_{\tau,n}(x,t):=(x+I^{N^{n}_{(x,t)}(\tau)}_{(x,t)},t+1)$

as the next step at dynamic time $\tau$. The path $\pi^{(x,t),n}_{\tau}$ at
dynamic time $\tau$ is obtained by linearly joining successive steps $\langle
h^{k-1}_{\tau,n}(x,t),h^{k}_{\tau,n}(x,t)\rangle$. Let ${\cal
X}^{n}(\tau):=\\{\pi^{(x,t),n}_{\tau}:(x,t)\in{\mathbb{Z}}^{2}_{\text{even}}\\}$.
Let ${\cal X}_{n}(\tau)$ denote the $n$-th scaled collection of paths
$S_{n}({\cal X}^{n}(\tau))$ and the $n$-th scaled DyDW process is denoted by
$\\{\overline{{\cal X}_{n}}(\tau):\tau\in[0,1]\\}$.

In this paper we show that as the scaled DyDW as a process with RCLL paths
converges to the DyBW.

######  Theorem 3.2.

As $n\to\infty$, the DyDW process $\\{\overline{{\cal X}_{n}}(s):s\in[0,1]\\}$
converges to the DyBW $\\{{\mathcal{W}}(s):s\in[0,1]\\}$ where the convergence
happens as ${\cal D}^{{\cal H}}([0,1])$ valued random variables.

In the next section we show that the finite dimensional distributions of the
DyDW process converge to that of the DyBW. A detailed sketch of the argument
for the same was earlier presented in [NRS09A]. But to the best of our
knowledge, weak convergence of the DyDW process $\\{\overline{{\cal
X}_{n}}(s):s\in[0,1]\\}$ to the DyBW process has not been shown so far. In
Section 4 we prove tightness of the scaled DyDW and thereby complete the proof
of Theorem 3.2.

###  3.2 Finite dimensional distribution convergence

In this section we prove the following proposition which proves finite
dimensional distributional convergence for the scaled DyDW to the DyBW.

######  Proposition 3.3.

Fix deterministic $0\leq\tau_{1}<\tau_{2}<\cdots<\tau_{k}\leq 1$. Then as
$n\to\infty$, we have

$(\overline{{\cal X}}_{n}(\tau_{1}),\cdots,\overline{{\cal
X}}_{n}(\tau_{k}))\Rightarrow({\mathcal{W}}(\tau_{1}),\cdots,{\mathcal{W}}(\tau_{k})),$

as ${\cal H}^{k}$ valued random variables.

In the next section, we describe finite dimensional distributions of the DyBW
process. We prove Proposition 3.3 after that.

####  3.2.1 Finite dimensional distributions of the DyBW

We first recall the definition of a one-dimensional sticky (at the origin)
Brownian motion.

######  Definition 3.4.

$B_{\text{stick},x}$ is a $(1/\tau)$-sticky Brownian motion starting at $x$
iff there exists a one-dimensional standard Brownian motion $B$ s.t.

$dB_{\text{stick},x}(t)=\mathbf{1}_{B_{\text{stick},x}(t)\neq
0}dB(t)+\tau\mathbf{1}_{B_{\text{stick},x}(t)=0}dt\text{ for all }t\geq 0,$
(7)

and $B$ is constrained to stay positive as soon as it hits zero.

It is known that (7) has a unique (weak) solution. For $x=0$ this solution can
be constructed from a time-changed reflected Brownian motion as follows.
Consider

$t\mapsto|\overline{B}|(C(t))\text{ with }C^{-1}(t)=t+\frac{1}{\tau}L_{0}(t),$
(8)

where $|\overline{B}|$ is the reflected Brownian motion and $L_{0}(t)$ is its
local time at the origin. Then there exists a Brownian motion $\overline{B}$
such that $(|\overline{B}|(C(\cdot)),B)$ is a solution of (7). The sticky
Brownian motion is obtained from the reflected Brownian motion by
“transforming” the local time into real time and as a result, it spends
positive Lebesgue measure time at the origin. The larger the “degree of
stickiness” $1/\tau$ is, the more the path sticks to the origin.

We now describe finite dimensional distributions of the DyBW. We first present
the definition of sticky pair of Brownian motions starting from any two points
in ${\mathbb{R}}^{2}$ taken from [NRS10].

######  Definition 3.5.

$(B,B^{\prime})$ is a $(1/\tau)$-sticky pair of Brownian motions iff:

  * (i)

$B$ and $B^{\prime}$ are both Brownian motions starting at $(x_{B},t_{B})$ and
$(x_{B^{\prime}},t_{B^{\prime}})$ that move independently when they do not
coincide.

  * (ii)

For $t\geq 0$, define $B_{\text{stick}}(t):=|B-B^{\prime}|(t+t_{B}\vee
t_{B}^{\prime})/\sqrt{2}$. Conditioned on $x=B_{\text{stick}}(0)$, the process
$\\{B_{\text{stick}}(t):t\geq 0\\}$ is a $(\sqrt{2}/\tau)$\- sticky Brownian
motion starting at $x$ (see Definition 7).

We call $(B_{1},\cdots,B_{m};B^{\prime}_{1},\cdots,B^{\prime}_{n})$ a
collection of $(1/\tau)$-sticking-coalescing Brownian motions, if
$(B_{1},\cdots,B_{m})$ and $(B^{\prime}_{1},\cdots,B^{\prime}_{n})$ are each
distributed as a set of coalescing Brownian motions and for any
$B\in\\{B_{1},\cdots,B_{m}\\}$ and
$B^{\prime}\in\\{B^{\prime}_{1},\cdots,B^{\prime}_{n}\\}$, the pair
$(B,B^{\prime})$ is a $(1/\tau)$-sticky pair of Brownian motions.

We will say that $({\mathcal{W}},{\mathcal{W}}^{\prime})$ is a $1/\tau$-sticky
pair of Brownian webs if $({\mathcal{W}},{\mathcal{W}}^{\prime})$ satisfies
the following properties:

  * (a)

${\mathcal{W}}$, resp. ${\mathcal{W}}^{\prime}$, is distributed as the
standard Brownian web.

  * (b)

For any finite deterministic set
${\mathbf{x}}_{1},\cdots,{\mathbf{x}}_{m},{\mathbf{x}}^{\prime}_{1},\cdots,{\mathbf{x}}^{\prime}_{n}\in{\mathbb{R}}^{2}$,
the subset of paths in ${\mathcal{W}}$ starting from these points are jointly
distributed as a collection of $(1/\tau)$-sticking- coalescing Brownian
motions starting from the given sets of points.

Newman et. al. [NRS10] presented a rigorous construction of the process
$\\{{\mathcal{W}}(\tau):\tau\in[0,1]\\}$ such that for any
$0\leq\tau_{1}<\tau_{2}\leq 1$ the pair
$({\mathcal{W}}(\tau_{1}),{\mathcal{W}}(\tau_{2}))$ equidistributed as
$({\mathcal{W}}(0),{\mathcal{W}}(\tau_{2}-\tau_{1}))$ which has the same
distribution as $1/(2(\tau_{2}-\tau_{1}))$-sticky pair of Brownian webs. Our
next remark explains that the distribution of
$({\mathcal{W}}(\tau_{1}),{\mathcal{W}}(\tau_{2}))$ uniquely specifies finite
dimensional distributions of the DyBW process. Remark 3.6 essentially follows
from stationarity and Markov property of theorem 6.2 of [NRS10].

######  Remark 3.6.

Set $k\geq 1$ and fix $0\leq\tau_{1}<\cdots<\tau_{k}\leq 1$. Let $({\cal
Z}_{1},\cdots,{\cal Z}_{k})$ be such that for all $1\leq i\leq k$, we have

${\cal Z}_{i}\stackrel{{\scriptstyle d}}{{=}}{\mathcal{W}},$ (9)

where the notation $\stackrel{{\scriptstyle d}}{{=}}$ stands for same
distribution. Further, for any $k$ many deterministic points
$(x_{1},t_{1}),\cdots,(x_{k},t_{k})\in{\mathbb{R}}^{2}$, let $\pi^{i}$ denote
the a.s. unique path in ${\cal Z}_{i}$ starting from $(x_{i},t_{i})$. Equation
(9) guarantees existence and uniqueness of such $\pi^{i}$. Suppose
distribution $(\pi^{1},\cdots,\pi^{k})\in\Pi^{k}$ satisfies the following:

  * (a)

For each $1\leq i\leq k$, marginally the random path $\pi_{i}$ is distributed
as the standard Brownian motion starting from $(x_{i},t_{i})$.

  * (b)

As long as the paths are disjoint, they evolve like independent Brownian
motions.

  * (c)

As soon as any two paths intersect, due to sticky interaction they spend non-
trivial time together. Precisely, if paths $\pi^{i}$ and $\pi^{j}$ intersect,
together they evolve like a $1/(2|\tau_{i}-\tau_{j}|)$-sticky Brownian motion.

Then we must have

$({\cal Z}_{1},\cdots,{\cal Z}_{k})\stackrel{{\scriptstyle
d}}{{=}}({\mathcal{W}}(\tau_{1}),\cdots,{\mathcal{W}}(\tau_{k})),$

where ${\mathcal{W}}(\tau)$ denotes the DyBW at dynamic time $\tau$.

In the next section we use Remark 3.6 to prove Proposition 3.3.

####  3.2.2 Proof or Proposition 3.3

Proof of Proposition 3.3: Note that the sequence
$\bigl{\\{}\bigl{(}\overline{{\cal X}}_{n}(\tau_{1}),\cdots,\overline{{\cal
X}}_{n}(\tau_{k})\bigr{)}:n\in{\mathbb{N}}\bigr{\\}}$ is a tight sequence of
${\cal H}^{k}$-valued random variables. Consider any sub-sequential limit
$({\cal Z}_{1},\cdots,{\cal Z}_{k})$ of the above sequence. The work of Fontes
et al. [FINR04] ensures that

${\cal Z}_{i}\stackrel{{\scriptstyle d}}{{=}}{\mathcal{W}}\text{ for all
}1\leq i\leq k.$

Because of Remark 3.6, in order to prove Proposition 3.3 it suffices to show
that $({\cal Z}_{1},\cdots,{\cal Z}_{k})$ satisfies conditions (b) and (c) as
well. Condition (a) follows from the fact that, as long as the scaled discrete
web paths are disjoint, they evolve independently.

To show condition (a) we fix $1\leq i,j\leq k$. Set
$(x_{i},t_{i})=(x_{j},t_{j})=(0,0)$, i.e., $i$-th and $j$-th path both start
from origin. The DW paths evolve independently as long as they are supported
on disjoint sets of lattice points and hence, taking two different starting
points does not pose any additional challenge. For each $n\geq 1$, the
collection
$\\{N^{n}_{(x,t)}(\tau):\tau\in[0,1]\\}_{(x,t)\in{\mathbb{Z}}^{2}_{\text{even}}}$
denotes a collection of i.i.d. Poisson processes with intensity $1/\sqrt{n}$
and these processes are taken to be independent. Corresponding to this,
$\pi^{\mathbf{0},n}_{\tau_{i}}$ and $\pi^{\mathbf{0},n}_{\tau_{j}}$ denote the
unscaled DW paths starting from the origin at dynamic times $\tau_{i}$ and
$\tau_{j}$ respectively. We comment here that the marginal distribution of
$\pi^{\mathbf{0},n}_{\tau_{i}}$ does not depend on $n$ but the joint
distribution of
$(\pi^{\mathbf{0},n}_{\tau_{i}},\pi^{\mathbf{0},n}_{\tau_{j}})$ depends on
$n$. To simplify our notation, we denote these two paths simply as $S^{i,n}$
and $S^{j,n}$ respectively. W.l.o.g. we assume that $\tau_{i}<\tau_{j}$.
Corresponding pair of $n$-th order diffusivelly scaled paths are given by

$(S^{i}_{n}(t),S^{j}_{n}(t))_{t\geq
0}:=(S^{i,n}(nt)/\sqrt{n},S^{j,n}(nt)/\sqrt{n})_{t\geq 0}.$

For this part of the proof we follow the ideas in [NRS09] and in [NRS09A]. We
observe that the pair of ‘scaled’ DW paths $S^{i}_{n}$ and $S^{j}_{n}$
alternates between times at which these two paths are equal (i.e., they “stick
together”) and times at which they move independently. As soon as the paths
$S^{i}_{n}$ and $S^{j}_{n}$ meet at time $t=k/n$ for some $k\in{\mathbb{N}}$,
they continue to coincide and move together as long as the clock at
$(\sqrt{n}S^{i}_{n}(k^{\prime}/n),k^{\prime})=(\sqrt{n}S^{j}_{n}(k^{\prime}/n),k^{\prime})$
for some $k^{\prime}\in\\{k+1,k+2,\cdots\\}$ does not ring during the interval
$(\tau_{i},\tau_{j}]$. After such a clock ticks, the two random walk paths are
independent till the time they meet again. This suggests the following time
decomposition. Set $T^{n}_{0}=0$ and for $k\geq 0$ we define

$\displaystyle T^{n}_{2k+1}$ $\displaystyle:=\inf\\{m\geq
T^{n}_{2k},m\in{\mathbb{N}}:\text{ the Poisson clock
}N^{n}_{(\sqrt{n}S^{i}_{n}(m/n),m)}\text{ ticks in
}(\tau_{i},\tau_{j}]\\}\text{ and }$ $\displaystyle T^{n}_{2k+2}$
$\displaystyle:=\inf\\{m>T^{n}_{2k+1},m\in{\mathbb{N}}:S^{i}_{n}(m/n)=S^{j}_{n}(m/n)\\}.$

As discussed before, on the interval $[T^{n}_{2k}/n,T^{n}_{2k+1}/n]$, both the
scaled paths $S^{i}_{n}$ and $S^{j}_{n}$ coincide and at time $T^{n}_{2k+1}/n$
they start moving independently until meeting at $T^{n}_{2k+2}/n$. In other
words, if we skip the intervals $[T^{n}_{2k}/n,T^{n}_{2k+1}/n)_{k\geq 0}$, the
pair $(S^{i}_{n},S^{j}_{n})$ behave as two independent diffusively scaled
random walks $(\tilde{S}^{i}_{n},\tilde{S}^{j}_{n})$, while if we skip
intervals $[T^{n}_{2k+1}/n,T^{n}_{2k+2}/n]_{k\geq 0}$, the two walks coincide
to form a single diffusively scaled random walk $\tilde{S}^{s}_{n}$ which
evolves independent of $(\tilde{S}^{i}_{n},\tilde{S}^{j}_{n})$. We define
$\Delta T^{n}_{k}:=T^{n}_{2k+1}-T^{n}_{2k}$ and we have

${\mathbb{P}}(\Delta
T^{n}_{k}\geq\ell)\leq\exp{(-\ell\frac{|\tau_{j}-\tau_{i}|}{\sqrt{n}})}.$

Further, the collection $\\{\Delta T^{n}_{k}:k\geq 0\\}$ is independent of the
rescaled random walk paths. This analysis, which is a scaled version of Lemma
3.2 of [NRS09], is summarised as follows:

Distributionally this scaled pair is equivalent to

$\displaystyle S^{i}_{n}(t)$
$\displaystyle=\tilde{S}^{i}_{n}(C_{n}(t))+\tilde{S}^{s}_{n}(t-C_{n}(t))\text{
and }$ $\displaystyle S^{j}_{n}(t)$
$\displaystyle=\tilde{S}^{j}_{n}(C_{n}(t))+\tilde{S}^{s}_{n}(t-C_{n}(t)),$

where $\tilde{S}^{i}_{n},\tilde{S}^{j}_{n},\tilde{S}^{s}_{n}$ are three
independent rescaled random walks, and $C_{n}$ is the right continuous inverse
of $\hat{L}_{n}(t)+t$ with,

  * (i)

$\hat{L}^{n}(t):=\frac{1}{n}\sum_{k=1}^{\lfloor\hat{l}^{n}(t)\sqrt{n}\rfloor}T^{n}_{k}$;

  * (ii)

$\hat{l}^{n}(t):=\frac{1}{\sqrt{n}}\\#\\{k\leq
nt:\tilde{S}^{i}_{n}(k/n)=\tilde{S}^{j}_{n}(k/n)\\}$,

where $T^{n}_{k}$’s are i.i.d. non-negative random variables with
${\mathbb{P}}(T^{n}_{k}\geq\ell)=\exp{(-\ell\frac{|\tau_{i}-\tau_{j}|}{\sqrt{n}})}$.

Then we have

${\mathbb{E}}(T^{n}_{k})=\frac{\exp{(-\frac{|\tau_{i}-\tau_{j}|}{\sqrt{n}})}}{1-\exp{(-\frac{|\tau_{i}-\tau_{j}|}{\sqrt{n}})}}\text{
and
}Var(T^{n}_{k})=\frac{\exp{(-\frac{|\tau_{i}-\tau_{j}|}{\sqrt{n}})}}{\bigl{(}1-\exp{(-\frac{|\tau_{i}-\tau_{j}|}{\sqrt{n}})}\bigr{)}^{2}}.$

We know that $\hat{l}^{n}(t)$ converges in distribution to $L(t)$ where $L(t)$
is the local time at zero of $(B_{1}-B_{2})$ where $B_{1}$ and $B_{2}$ are two
independent Brownian motions (see Theorem 1.1 of [B82] and Theorem 4.1 of
[K63]).

By Skorohod’s representation theorem we assume that we are working on a
probability space $(\Omega_{l},{\cal B}_{l},{\mathbb{P}}_{l})$ such that
$\hat{l}^{n}(t)$ converges almost surely to $L(t)$. Note that the scaled
independent random walks
$\\{(\bar{S}^{i}_{n}(s),\bar{S}^{j}_{n}(s),\bar{S}^{s}_{n}(s)):s\geq 0\\}$
${\cal B}_{l}$ are measurable. Let us denote the probability space for the
i.i.d. collections $\\{T^{n}_{k}:k\geq 1\\}_{n\geq 1}$ as $(\Omega_{d},{\cal
B}_{d},{\mathbb{P}}_{d})$ and let
${\mathbb{P}}={\mathbb{P}}_{l}\times{\mathbb{P}}_{d}$. Then we have

$\displaystyle Var(\hat{L}^{n}(t)\mid{\cal B}_{l})$
$\displaystyle=Var\bigl{(}\frac{1}{n}\sum_{k=1}^{\lfloor\hat{l}^{n}_{t}\sqrt{n}\rfloor}T^{n}_{k}\bigr{)}$
$\displaystyle=\frac{1}{n^{2}}\sum_{k=1}^{\lfloor\hat{l}^{n}_{t}\sqrt{n}\rfloor}Var(T^{n}_{k})$
$\displaystyle=\frac{1}{n^{2}}\lfloor\hat{l}^{n}_{t}\sqrt{n}\rfloor\frac{\exp{(-\frac{|\tau_{i}-\tau_{j}|}{\sqrt{n}})}}{\bigl{(}1-\exp{(-\frac{|\tau_{i}-\tau_{j}|}{\sqrt{n}})}\bigr{)}^{2}}$
$\displaystyle\equiv O(1/\sqrt{n}).$

Therefore we have ${\mathbb{P}}_{l}$ a.s. $Var(\hat{L}^{n}(t)\mid{\cal
B}_{l})\to 0$ as $n\to\infty$. Thus we have
$|\hat{L}^{n}(t)-{\mathbb{E}}(\hat{L}^{n}(t)\mid{\cal B}_{l})|$ converges to
$0$ in Probability. Since,

$\lim_{n\to\infty}{\mathbb{E}}(\hat{L}^{n}(t)\mid{\cal
B}_{l})=\lim_{n\to\infty}\frac{\lfloor\hat{l}^{n}_{t}\sqrt{n}\rfloor}{n(|\tau_{i}-\tau_{j}|/\sqrt{n})}=\frac{\lim_{n\to\infty}\hat{l}^{n}_{t}}{|\tau_{i}-\tau_{j}|}=\frac{{L}_{t}}{|\tau_{i}-\tau_{j}|},{\mathbb{P}}_{l}\text{
a.s.}$

Therefore, we have $\hat{L}^{n}(t)$ converges to $L(t)/|\tau_{i}-\tau_{j}|$ in
probability which implies convergence in distribution for local times. This
completes the proof for Proposition 3.3.

##  4 Tightness part

In this section we finally prove that the scaled DyDW processes
$\\{\overline{{\cal X}}_{n}(\tau):\tau\in[0,1]\\}$ form a tight sequence as
${\cal D}^{{\cal H}}([0,1])$ valued random variables and thereby completes the
proof of Theorem 3.2.

####  4.0.1 Description of notations

Recall that the $n$-th scaled discrete DyDW process is denoted as
$\\{\overline{{\cal X}}_{n}(\tau):\tau\in[0,1]\\}$, a ${\cal H}$-valued
stochastic process evolving over the dynamic time interval $[0,1]$.

Fix $n\in{\mathbb{N}}$. We consider the scaled DyDW process
$\\{\overline{{\cal X}}_{n}(\tau):\tau\in[0,1]\\}$. We denote the random
configuration of all outgoing edges observed over the dynamic time interval
$[0,1]$ as

$\displaystyle{\cal A}_{n}$
$\displaystyle:=\\{\langle(x,t),h_{\tau,n}(x,t)\rangle:(x,t)\in{\mathbb{Z}}^{2}_{\text{even}},\tau\in[0,1]\\}$
$\displaystyle=\\{\langle(x,t),(x+1,t+1)\rangle:(x,t)\in{\mathbb{Z}}^{2}_{\text{even}},h_{\tau,n}(x,t)=(x+1,t+1)\text{
for some }\tau\in[0,1]\\}$
$\displaystyle\bigcup\\{\langle(x,t),(x-1,t+1)\rangle:(x,t)\in{\mathbb{Z}}^{2}_{\text{even}},h_{\tau,n}(x,t)=(x-1,t+1)\text{
for some }\tau\in[0,1]\\}.$

The random quantity ${\cal A}_{n}$ basically denotes the random configuration
of all ‘arrow’s observed over dynamic time domain $[0,1]$ as described in
[SS08]. An ${\cal A}_{n}$-path, is the graph of a function
$\pi:[\sigma_{\pi},\infty]\mapsto{\mathbb{R}}\cup\\{\star\\}$, with
$\sigma_{\pi}\in{\mathbb{Z}}\cup\\{\pm\infty\\}$, such that
$\langle(\pi(t),t),(\pi(t+1),t+1)\rangle\in{\cal A}_{n}$ and $\pi$ is linear
on the interval $[t,t+1]$ for all $t\in[\sigma_{\pi},\infty]\cap{\mathbb{Z}}$,
while $\pi(\pm\infty)=\times$ whenever $\pm\infty\in[\sigma_{\pi},\infty]$.

The closure of collection of all ${\cal A}_{n}$-paths, i.e., paths along the
arrows in ${\cal A}_{n}$, is denoted as ${\mathcal{N}}_{n}$. We observe that
${\mathcal{N}}_{n}$ denotes the $n$-th scaled discrete net considered in
[SS08] and is a ${\cal H}$-valued random variable.

More generally, for any dynamic time interval $I\subseteq[0,1]$ collection of
all outgoing edges observed over $I$ is denoted as

${\cal
A}_{I,n}:=\\{\langle(x,t),h_{\tau,n}(x,t)\rangle:(x,t)\in{\mathbb{Z}}^{2}_{\text{even}},\tau\in
I\\}$

and the closure of collection of all ${\cal A}_{I,n}$-paths is denoted as
${\mathcal{N}}_{I,n}$.

Sun et. al showed that the discrete net ${\mathcal{N}}_{n}$ converges in
distribution to the standard Brownian net (Theorem 1.1 of [SS08]). Their
argument also proves that for any deterministic $0\leq\tau_{1}<\tau_{2}\leq 1$
we have

${\mathcal{N}}_{[\tau_{1},\tau_{2}],n}\Rightarrow{\mathcal{N}}_{[\tau_{1},\tau_{2}]}\text{
as }n\to\infty.$

Fix $\epsilon>0$. For $n\in{\mathbb{N}}$ and for any interval
$I\subseteq[0,1]$ considering ${\cal H}$-valued random variable
${\mathcal{N}}_{I,n}$ we define the following quantities:

$\displaystyle{\cal S}_{I,n}$ $\displaystyle:={\cal S}({\mathcal{N}}_{I,n}),$
$\displaystyle D^{\epsilon}_{I,n}$
$\displaystyle:=D^{\epsilon}({\mathcal{N}}_{I,n}),$ $\displaystyle
T^{\epsilon}_{I,n}$ $\displaystyle:=T^{\epsilon}({\mathcal{N}}_{I,n})\text{
and }$ $\displaystyle A^{\epsilon}_{I,n}$
$\displaystyle:=A^{\epsilon}({\mathcal{N}}_{I,n}).$

For $(x,t)\in{\mathbb{Z}}^{2}_{\text{even}}$ a scaled lattice point
$(x,t)_{n}:=S_{n}(x,t)$ may belong to the set ${\cal S}_{I,n}$ only if the
associated Markov process $\\{N^{n}_{(x,t)}(\tau):\tau\in[0,1]\\}$ ticks at
least once during the dynamic time interval $I$. In fact, for
$I=[\tau_{1},\tau_{2})$, to be in the set ${\cal S}_{[\tau_{1},\tau_{2}),n}$,
the associated Markov process must tick at least once during the interval
$(\tau_{1},\tau_{2})$. We define the (random) set $\Xi^{\epsilon}_{I,n}$ as

$\Xi^{\epsilon}_{I,n}:=D^{\epsilon}_{I,n}\cap A^{\tilde{\epsilon}}_{I,n},$

where $\tilde{\epsilon}$ is defined as in (2.3). The notation
$\Lambda^{\epsilon}_{I,n}$ denotes the random set of Poisson clock rings
attached to points in the set $\Xi^{\epsilon}_{I,n}$ over dynamic time
interval $I$. For simplicity of notation we set

$\displaystyle\Xi^{\epsilon}_{[0,1],n}$
$\displaystyle:=\Xi^{\epsilon}_{n}\text{ and }$
$\displaystyle\Lambda^{\epsilon}_{[0,1],n}$
$\displaystyle:=\Lambda^{\epsilon}_{n}.$

As observed earlier, for all $I\subseteq[0,1]$ and any $n\in{\mathbb{N}}$ we
have $\Xi^{\epsilon}_{I,n}\subseteq\Xi^{\epsilon}_{n}$ and
$\Lambda^{\epsilon}_{I,n}\subseteq\Lambda^{\epsilon}_{n}$ a.s.

####  4.0.2 Proof of tightness

First we need to show that the scaled DyDW process $\\{\overline{{\cal
X}}_{n}(\tau):\tau\in[0,1]\\}$ has RCLL ${\cal H}$-valued paths a.s. We will
prove a weaker version that for all large $n$, the process $\\{\overline{{\cal
X}}_{n}(\tau):\tau\in[0,1]\\}$ has RCLL paths a.s. For $\delta>0$ with a
slight abuse of notation let $\omega_{\overline{{\cal X}}_{n}}(\delta)$ denote
the (generalised) modulus of continuity for the DyDW process
$\\{\overline{{\cal X}}_{n}(\tau):\tau\in[0,1]\\}$. We first show that the
same argument of Lemma 2.12 holds for the scaled DyDW process as well and
gives the following lemma.

######  Lemma 4.1.

Fix $\epsilon>0$ and $0\leq\tau_{1}<\tau_{2}\leq 1$. For all large $n$ on the
event
$\Lambda^{\epsilon}_{[\tau_{1},\tau_{2}),n}\cap(\tau_{1},\tau_{2})=\emptyset$
we must have $\omega_{\overline{{\cal
X}}_{n}}([\tau_{1},\tau_{2}))\leq\epsilon$.

###### Proof.

The argument is essentially same as that of Lemma 2.12 and we only give a
sketch of the proof here. As argued earlier, in order to have
$\omega_{\overline{{\cal X}}_{n}}([\tau_{1},\tau_{2}))>\epsilon$ there must be
a scaled path $\pi^{n}_{s_{1}}\in{\cal X}_{n}(s_{1})$ for some
$s_{1}\in[\tau_{1},\tau_{2})$ with a scaled separating (branching) point
$(x,t)$ on it’s trajectory such that $(x,t)\in
D^{\epsilon}_{[\tau_{1},\tau_{2}),n}$.

We choose a scaled lattice point $(y^{n},s^{n})\in
S_{n}({\mathbb{Z}}^{2}_{\text{even}})$ on the trajectory of the path
$\pi^{n}_{s_{1}}$ and above $t$ such that the difference between $s^{n}$ and
starting time of $\pi^{n}_{s_{1}}$ in $\tanh(\cdot)$ metric lies in
$(\tilde{\epsilon},2\tilde{\epsilon})$. For $n$ large enough, we can always
make such a choice and only for this part of the argument we require $n$ to be
large. Under the assumption
$\Lambda^{\epsilon}_{[\tau_{1},\tau_{2}),n}\cap(\tau_{1},\tau_{2})=\emptyset$
we have $\Xi^{\epsilon}_{[\tau_{1},\tau_{2}),n}=\emptyset$ as well and
consequently it follows that for all $s\in[\tau_{1},\tau_{2})$, the scaled
path starting from $(y^{n},s^{n})$ at dynamic time $s$ does not have a member
of $D^{\epsilon}_{[\tau_{1},\tau_{2}),n}$ on it’s trajectory. Rest of the
argument is exactly same as Lemma 2.12. ∎

######  Lemma 4.2.

For each $n$, the scaled DyDW process $\\{{\cal X}_{n}(\tau):\tau\in[0,1]\\}$
is RCLL a.s.

###### Proof.

The DyDW process is constructed on the discrete set of scaled lattice points.
Because of Lemma 4.1, for each $n$ to show that the process $\\{{\cal
X}_{n}(\tau):\tau\in[0,1]\\}$ is RCLL, it is enough to show that the set
$\Xi^{\epsilon}_{n}$ is contained in a compact box a.s. Fix
$n\in{\mathbb{N}}$. Properties of the metric space ${\mathbb{R}}^{2}_{c}$
ensures that there exists $M\in{\mathbb{N}}$ such that

  * (i)

any scaled lattice point $(y^{n},s^{n})$ in the set $\Xi^{\epsilon}_{n}$ must
have $s^{n}\in[-M,M]$ and

  * (ii)

at least one of the two outgoing paths starting from
$(y^{n},s^{n})\in\Xi^{\epsilon}_{n}$ must intersect with the box $[-M,M]^{2}$.

Since, the scaled DyDW paths start from scaled lattice points and for any
$\tau\in[0,1]$ the modulus of continuity of all the paths in ${\cal
X}_{n}(\tau)$ is uniformly bounded by $\sqrt{n}$, condition (i) and (ii)
ensure that the set of points in $\Xi^{\epsilon}_{n}$ must be contained in a
compact box. This completes the proof. ∎

In order to show that the sequence $\\{\overline{{\cal
X}_{n}}(\tau):\tau\in[0,1]\\}_{n\in{\mathbb{N}}}$ is tight we need to show the
following proposition.

######  Proposition 4.3.

Fix $\epsilon>0$ and $\gamma\in(0,1)$. There exist
$n_{0}=n_{0}(\epsilon,\gamma)\in{\mathbb{N}}$ and
$\delta=\delta(\epsilon,\gamma)$ such that

${\mathbb{P}}(\omega_{\overline{{\cal X}_{n}}}(\delta)>\epsilon\text{ for some
}n\geq n_{0})<\gamma.$ (10)

We first describe a strategy to prove the above proposition. Because of Lemma
4.1, in order to prove Proposition 4.3 it suffices to show that there exist
$n_{0}=n_{0}(\epsilon,\gamma)\in{\mathbb{N}}$ and
$\delta=\delta(\epsilon,\gamma)$ such that with probability bigger than
$1-\gamma$, for all $n\geq n_{0}$ there exists a partition (possibly random)
$\\{[\tau_{i},\tau_{i+1}):0\leq i\leq\ell-1\\}$ of $[0,1]$ with

  * (i)

$\min\\{\tau_{i+1}-\tau_{i}:0\leq i\leq\ell-1\\}>\delta$ and

  * (ii)

$\Lambda^{\epsilon}_{(\tau_{i},\tau_{i+1}),n}=\emptyset$ for all $0\leq
i\leq\ell-1$.

Towards this we choose $m\in{\mathbb{N}}$ (we will specify the choice of $m$
later) and partition the interval $[0,1]$ into $m$ many intervals of length
$1/m$ given by $\\{[j/m,(j+1)/m):0\leq j\leq m-1\\}$. For simplicity of
notation we denote $\Lambda^{\epsilon/2}_{[j/m,(j+1)/m),n}$ simply as
$\Lambda^{\epsilon/2}_{j,n}$. We note that we have considered $\epsilon/2$
instead of $\epsilon$ and the reason for considering $\epsilon/2$ will be
explained shortly. For any set $A$ let $\\#A$ denote the cardinality of $A$.
We define the event

$\displaystyle
F^{m}_{n}:=(\Lambda^{\epsilon/2}_{0,n}=\Lambda^{\epsilon/2}_{m-1,n}=\emptyset)\bigcap\bigl{(}\cap_{j=1}^{m-2}(\\#(\cup_{i=j-1}^{j+1}\Lambda^{\epsilon/2}_{i,n})\leq
1)\bigr{)}.$ (11)

In other words, the event $F^{m}_{n}$ ensures that for any $1\leq j\leq m-2$
the set $\Lambda^{\epsilon/2}_{j,n}$ can have at most one element. Further, if
the set $\Lambda^{\epsilon/2}_{j,n}$ is non-empty, then both the
‘neighbouring’ sets $\Lambda^{\epsilon/2}_{(j-1),n}$ and
$\Lambda^{\epsilon/2}_{(j+1),n}$ must be empty.

On the event $F^{m}_{n}$ for $n\geq n_{0}$ let $I\subset\\{0,1,\cdots,m-1\\}$
be such that $\Lambda^{\epsilon/2}_{j,n}$ is non-empty if and only if $j\in I$
and set

$\displaystyle\Lambda^{\epsilon/2}_{j,n}=\begin{cases}\\{s_{j}\\}&\text{ if
}j\in I\\\ \emptyset&\text{ if }j\notin I.\end{cases}$

Precisely, for each $j\in I$, the set $\Lambda^{\epsilon/2}_{j,n}$ is a
singleton set denoted by $\\{s_{j}\\}$. This allows us to choose a partition
of the unit interval as

$\Bigl{(}\bigcup_{0\leq j\leq m-1:j-1,j,j+1\notin
I}[j/m,(j+1)/m)\Bigr{)}\bigcup\Bigl{(}\bigcup_{j\in
I}[(j-1)/m,s_{j})\cup[s_{j},(j+2)/m)\Bigr{)}.$

In other words, for $j\in I$ we modify the partition
$\cup_{\ell=j-1}^{j+1}[\ell/m,(\ell+1)/m)$ as
$[(j-1)/m,s_{j})\cup[s_{j},(j+2)/m)$. Choose $\delta<1/m$ and we observe that
each interval in the above partition has length strictly bigger than $\delta$.
For $0\leq j\leq m-1$ with $j-1,j,j+1\notin I$ we have
$\Lambda^{\epsilon/2}_{j,n}=\emptyset$ implying $\omega_{\overline{{\cal
X}_{n}}}([j/m,(j+1)/m))<\epsilon/2<\epsilon$. On the other hand for $j\in I$,
a.s. $j/m$ is not a jump point and Remark 2.6 ensures that we have

$\omega_{\overline{{\cal X}}_{n}}([(j-1)/m,s_{j}))\leq\omega_{\overline{{\cal
X}}_{n}}\bigl{(}[(j-1)/m,j/m)\bigr{)}+\omega_{\overline{{\cal
X}}_{n}}\bigl{(}[j/m,s_{j})\bigr{)}.$ (12)

For both the intervals on the R.H.S., we have that the corresponding
$\Lambda^{\epsilon/2}$ sets are empty giving us that $\omega_{\overline{{\cal
X}}_{n}}([(j-1)/m,s_{j}))\leq\epsilon/2+\epsilon/2\leq\epsilon$. Similar logic
applies for the part $\omega_{{\cal X}_{n}}([s_{j},(j+2)/m))$ too, as we have
$\Lambda^{\epsilon/2}_{[s_{j},(j+2)/m),n}\cap(s_{j},(j+2)/m)=\emptyset$. This
shows that on the event $F^{m}_{n}$ for any $0<\delta<1/m$ we have
$\omega_{\overline{{\cal X}}_{n}}(\delta)<\epsilon$. Hence, in order to prove
Proposition 4.3, it is enough to prove the following proposition.

######  Proposition 4.4.

Fix $\epsilon>0$ and $\gamma\in(0,1)$. There exist
$n_{0}=n_{0}(\epsilon,\gamma),m_{0}=m_{0}(\epsilon,\gamma)\in{\mathbb{N}}$
such that

${\mathbb{P}}(F^{m_{0}}_{n}\text{ for all }n\geq n_{0})\geq 1-\gamma.$ (13)

This proposition will be proved through a sequence of lemmas. We first
describe our heuristics in words. The exact value of $m_{0}\in{\mathbb{N}}$
will be specified later (see (18)). We consider a partition
$\\{[j/m_{0},(j+1)/m_{0}):0\leq j\leq m_{0}-1\\}$ of the unit interval into
sub intervals of width $1/m_{0}$. Corresponding to the partition
$\\{[j/m_{0},(j+1)/m_{0}):0\leq j\leq m_{0}-1\\}$ of $[0,1]$ we consider the
set $\Lambda^{\epsilon/2}_{j}=\Lambda^{\epsilon/2}_{[j/m_{0},(j+1)/m_{0})}$
for $0\leq j\leq m_{0}-1$. We choose $m_{0}\in{\mathbb{N}}$ large enough so
that for any $j$ the set $\Lambda^{\epsilon/2}_{j}$ contains at most one
element and for most $j$’s, the set $\Lambda^{\epsilon/2}_{j}$ is empty.
Precisely we define an event $B_{m_{0}}$ such that

$\displaystyle B_{m_{0}}:=$
$\displaystyle(\Lambda^{\epsilon/2}_{0}=\Lambda^{\epsilon/2}_{m_{0}-1}=\emptyset)\bigcap\bigl{(}\\#(\Lambda^{\epsilon/2}_{j}\cup\Lambda^{\epsilon/2}_{j+1})\leq
1\text{ for all }1\leq j\leq m_{0}-2\bigr{)},$

and choose $m_{0}$ large enough so that ${\mathbb{P}}(B_{m_{0}})$ is close to
one. Here is a brief justification about why such a choice of $m_{0}$ is
always possible. As the set $\Lambda^{\epsilon/2}$ is a.s. finite, the r.v.
$Z:=\min\\{s_{1},|s_{1}-s_{2}|,1-s_{2}:s_{1},s_{2}\in\Lambda^{\epsilon/2}\\}$
is strictly positive with probability $1$. We observe that on the event
$\\{Z>\delta\\}$, the event $B_{m_{0}}$ holds for any $m_{0}\in{\mathbb{N}}$
with $m_{0}<\delta/3$. As ${\mathbb{P}}(Z\leq\delta)$ decreases to zero as
$\delta\downarrow 0$, we can choose $m_{0}$ large enough to make
${\mathbb{P}}(B_{m_{0}})$ arbitrarily close to one.

Based on the above choice of $m_{0}$ we show that with high probability for
all large $n$ the set $\Lambda^{\epsilon/2}_{j,n}$ is non-empty only if the
set $\Lambda^{\epsilon/2}_{j}$ is non-empty and a non-empty
$\Lambda^{\epsilon/2}_{j,n}$ contains exactly one element. This implies (13).
In other words, to prove (13) it suffices to show that there exist
$n_{0}=n_{0}(\epsilon,\gamma),m_{0}=m_{0}(\epsilon,\gamma)\in{\mathbb{N}}$
such that

${\mathbb{P}}\bigl{(}B_{m_{0}}\bigcap(\cap_{j=0}^{m_{0}-1}(\\#\Lambda^{\epsilon/2}_{j,n}\leq\\#\Lambda^{\epsilon/2}_{j}\text{
for all }n\geq n_{0}))\bigr{)}\geq 1-\gamma.$ (14)

In the following, through a sequence of lemmas we prove (14) and thereby prove
Proposition 4.4 to obtain tightness. Towards this we work on a specific
probability space to use properties of almost sure convergence. Below we will
construct our required probability space of almost sure convergence.

We consider a partition $\\{[j/m_{0},(j+1)/m_{0}):0\leq j\leq m_{0}-1\\}$ of
the unit interval into sub intervals of width $1/m_{0}$. Corresponding to this
partition of the dynamic time interval, the $m_{0}+1$ dimensional vector of
‘corresponding’ nets is given by

$({\mathcal{N}},{\mathcal{N}}_{0},\cdots,{\mathcal{N}}_{m_{0}-1}).$

We observe that for any $0\leq j\leq m_{0}-1$ we have
${\mathcal{N}}_{j}\stackrel{{\scriptstyle d}}{{=}}{\mathcal{N}}(1/m_{0})$.
Further, we observe that

$\pi\in{\mathcal{W}}(s)\text{ for some }s\in[j/m,(j+1)/m)\text{ only if
}\pi\in{\mathcal{N}}_{j}.$

The next proposition proves that as $n\to\infty$, the vector of scaled
discrete nets
$\bigl{(}{\mathcal{N}}_{n},{\mathcal{N}}_{0,n},\cdots,{\mathcal{N}}_{m_{0}-1,n}\bigr{)}$
as ${\cal H}^{m_{0}+1}$ valued random variable converges in distribution to
the vector of corresponding nets
$\bigl{(}{\mathcal{N}},{\mathcal{N}}_{0},\cdots,{\mathcal{N}}_{m_{0}-1}\bigr{)}$.

######  Proposition 4.5.

As $n\to\infty$, we have

$\bigl{(}{\mathcal{N}}_{n},{\mathcal{N}}_{0,n},\cdots,{\mathcal{N}}_{m_{0}-1,n})\Rightarrow\bigl{(}{\mathcal{N}},{\mathcal{N}}_{0},\cdots,{\mathcal{N}}_{m_{0}-1}\bigr{)},$

as ${\cal H}^{m_{0}+1}$ valued random variables.

We postpone proving Proposition 4.5 for the moment and proceed. In fact, as
the DyDW paths are independent till the time they meet, Sun et. al. [SS08]
showed that the scaled rightmost and leftmost paths jointly with their first
meeting times converge to left right Brownian paths and their first meeting
times. The same holds for each of the dynamic time interval
$[j/m_{0},(j+1)/m_{0})$ for $0\leq j\leq m_{0}-1$. We need to introduce some
notations. For $(x,t)\in{\mathbb{Q}}^{2}$ let $l^{(x,t)}$ and $r^{(x,t)}$
respectively denote the leftmost and rightmost path in the Brownian net
${\mathcal{N}}$ starting from $(x,t)$. Let $l^{(x,t)}_{j}$ and $r^{(x,t)}_{j}$
respectively denote the same for the Brownian net ${\mathcal{N}}_{j}$
constructed over dynamic time interval $[j/m_{0},(j+1)/m_{0})$. For
$(x,t),(y,s)\in{\mathbb{Q}}^{2}$ let $\tau^{(x,t),(y,s)}$ denote the first
meeting time of $l^{(x,t)}$ and $r^{(y,s)}$ and let $\tau^{(x,t),(y,s)}_{j}$
denote the same for paths $l^{(x,t)}_{j}$ and $r^{(y,s)}_{j}$. For
$(x,t)\in{\mathbb{Q}}^{2}$ let
$(x^{n},t^{n})\in{\mathbb{Z}}^{2}_{\text{even}}$ be such that
$S_{n}(x^{n},t^{n})\to(x,t)$ as $n\to\infty$. For $(x,t)\in{\mathbb{Q}}^{2}$
let $l^{(x^{n},t^{n})}_{n}$ and $r^{(x^{n},t^{n})}_{n}$ respectively denote
the scaled leftmost and rightmost DW path in ${\mathcal{N}}_{n}$ starting from
$S_{n}(x^{n},t^{n})$. $l^{(x^{n},t^{n})}_{j,n}$ and $r^{(x^{n},t^{n})}_{j,n}$
denote the same in ${\mathcal{N}}_{j,n}$. For $(x,t),(y,s)\in{\mathbb{Q}}^{2}$
the first meeting times $\tau^{(x,t),(y,s)}_{n}$ and
$\tau^{(x,t),(y,s)}_{j,n}$ are similarly defined. Assuming Proposition 4.5, we
have

$\displaystyle(l^{(x^{n},t^{n})}_{n},r^{(y^{n},s^{n})}_{n},\tau^{(x,t),(y,s)}_{n},l^{(x^{n},t^{n})}_{0,n},r^{(y^{n},s^{n})}_{0,n},\tau^{(x,t),(y,s)}_{0,n},\cdots,l^{(x^{n},t^{n})}_{m_{0}-1,n},r^{(y^{n},s^{n})}_{m_{0}-1,n},\tau^{(x,t),(y,s)}_{m_{0}-1,n}\bigr{)}_{(x,t),(y,s)\in{\mathbb{Q}}^{2}}$
$\displaystyle\Rightarrow(l^{(x,t)},r^{(y,s)},\tau^{(x,t),(y,s)},l^{(x,t)}_{0},r^{(y,s)}_{0},\tau^{(x,t),(y,s)}_{0},\cdots,l^{(x,t)}_{m_{0}-1},r^{(y,s)}_{m_{0}-1},\tau^{(x,t),(y,s)}_{m_{0}-1}\bigr{)}_{(x,t),(y,s)\in{\mathbb{Q}}^{2}}.$
(15)

Here we give a brief justification of Equation (4.0.2). From Proposition 4.5
we have joint convergence of the vector of scaled discrete nets to the vector
of Brownian nets. This guarantees that the vector of left right scaled DyDW
paths in (4.0.2) converges to the vector of respective left right coalescing
Brownian motions. Since, scaled DyDW paths are independent till the time they
meet (at a fixed dynamic time point), respective first intersection times
jointly converge too.

By Skorohod’s embedding, we assume that we are working on a probability space
so that the convergence as in (4.0.2) happens almost surely. In what follows,
we will continue to work on this probability space and we will use
consequences of this almost sure convergence extensively. It should be
mentioned here that the exact choice of $m_{0}\in{\mathbb{N}}$ will be
mentioned later in Equation (18).

The next lemma follows from consequences of almost sure convergence in path
space.

######  Lemma 4.6.

For any $0\leq j\leq m_{0}-1$, we have equality of the following events

$\\{\Lambda^{\epsilon/2}_{j}\neq\emptyset\\}=\\{\Lambda^{\epsilon/2}_{j,n}\neq\emptyset\text{
for infinitely many }n\\}.$

###### Proof.

Because of stationary nature of our model, it suffices to prove Lemma 4.6 for
$j=0$. On the event $\\{\Lambda^{\epsilon/2}_{0}\neq\emptyset\\}$, we must
have $\\{\Xi^{\epsilon/2}_{0}\neq\emptyset\\}$ as the set
$\Lambda^{\epsilon/2}_{0}$ consists of Poisson clock rings in the dynamic time
interval $[0,1/m_{0})$ associated to points in the set $\Xi^{\epsilon/2}_{0}$.
On the other hand, it is not difficult to see that we must have

$\\{\Xi^{\epsilon/2}_{0}\neq\emptyset\\}\subseteq\\{\Lambda^{\epsilon/2}_{0}\neq\emptyset\\},$

as there must be some switching event in the interval $[0,1/m_{0})$ (actually
$(0,1/m_{0})$) in order to have the set $\Xi^{\epsilon/2}_{0}$ non-empty.
Hence, the two events are equal. Similar reasoning holds for discrete scaled
nets as well. Hence, to prove Lemma 4.6 it is enough to show equality of the
following events

$\\{\Xi^{\epsilon/2}_{0,n}\neq\emptyset\text{ for infinitely many
}n\\}=\\{\Xi^{\epsilon/2}_{0}\neq\emptyset\\}.$

On the event $\\{\Xi^{\epsilon/2}_{0,n}\neq\emptyset\text{ for infinitely many
}n\text{'s}\\}$, almost sure convergence of path families ensures that the
limiting Brownian net ${\mathcal{N}}_{0}$ (constructed over dynamic time
interval $[0,1/m_{0})$) must have a separation point $(x,t)$ with an incoming
forward path of age at least $\tilde{\epsilon}=\tilde{\epsilon}(\epsilon/2)$.
Further, diameter of the maximal excursion set generated at $(x,t)$ is bigger
than $\epsilon$. This implies that the set $\Xi^{\epsilon/2}_{[0,1/m_{0})}$
must be non-empty. Hence, we have

$\\{\Xi^{\epsilon/2}_{0,n}\neq\emptyset\text{ for infinitely many
}n\\}\subseteq\\{\Xi^{\epsilon/2}_{0}\neq\emptyset\\}.$

On the other hand, on the event $\\{\Xi^{\epsilon/2}_{0}\neq\emptyset\\}$
consider a separating point $(x,t)\in\Xi^{\epsilon/2}_{0}$. There must exist
skeletal rightmost and leftmost Brownian paths
$\pi^{r},\pi^{l}\in{\mathcal{N}}$ passing through $(x,t)$ and forming the
right boundary and left boundary of the maximal excursion set generated at
$(x,t)$ respectively. Further, their first meeting time must be smaller than
$t$. Almost sure convergence of rightmost and leftmost path families together
with their first meeting times ensure that there must be sequences of paths
$\\{\pi^{n}_{l}:n\in{\mathbb{N}}\\}$ and $\\{\pi^{n}_{r}:n\in{\mathbb{N}}\\}$
approximating paths $\pi^{r}$ and $\pi^{l}$ respectively and their first
meeting time converges to that of $\pi^{r}$ and $\pi^{l}$ which is smaller
than $t$.

After intersection, scaled discrete web paths (rightmost and leftmost)
continue to move together till they encounter a separating point (for path
family ${\mathcal{N}}_{0,n}$). As $\\{\pi^{n}_{l}:n\in{\mathbb{N}}\\}$ and
$\\{\pi^{n}_{r}:n\in{\mathbb{N}}\\}$ approximate paths $\pi^{r}$ and $\pi^{l}$
with an $\epsilon$ fat maximal excursion set in between them, after the first
meeting time there must be a separating point $(x_{n},t_{n})\in
S_{n}({\mathbb{Z}}^{2}_{\text{even}})$ on their trajectory with maximal
excursion set of diameter $\epsilon$ at least. More precisely, for infinitely
many $n$’s there exist $(x_{n},t_{n})$ such that

$\pi^{n}_{r}(t_{n})=\pi^{n}_{l}(t_{n})=x_{n}\text{ and }(x_{n},t_{n})\in
D^{\epsilon/2}_{0,n}.$

To complete the proof we need to ensure that $(x_{n},t_{n})\in
A^{\tilde{\epsilon}/2}_{0,n}$ as well. This follows from the fact that the
limiting net has incoming path(s) at $x,t$ of age
$\tilde{\epsilon}(\epsilon/2)$ at least. For large $n$, there must be
approximating incoming path(s) in ${\cal N}_{0,n}$ passing through
$(x_{n},t_{n})$. This completes the proof. ∎

Lemma 4.6 suggests that for all $0\leq j\leq m_{0}-1$, on the event
$\\{\Lambda^{\epsilon/2}_{j}=\emptyset\\}$, we must have that the r.v.
$\sup\\{n\in{\mathbb{N}}:\Lambda^{\epsilon/2}_{j,n}\neq\emptyset\\}$ is a.s.
finite. This observation is used to prove the next lemma.

######  Lemma 4.7.

For each $\gamma\in(0,1)$, there exist
$m_{0}=m_{0}(\epsilon,\gamma),n^{\prime}=n^{\prime}(\epsilon,\gamma)\in{\mathbb{N}}$
such that

${\mathbb{P}}\bigl{(}\Lambda^{\epsilon/2}_{[0,1/m_{0}),n}\neq\emptyset\text{
for some }n\geq n^{\prime}\bigr{)}<\gamma.$ (16)

###### Proof.

We note that the random set $\Lambda^{\epsilon/2}$ is a.s. finite. Further as
$m\to\infty$, the set $\Lambda^{\epsilon/2}_{[0,1/m)}$ monotonically decreases
to empty set. Hence, we can choose $m_{0}\in{\mathbb{N}}$ such that

${\mathbb{P}}(\Lambda^{\epsilon/2}_{[0,1/m_{0})}\neq\emptyset)<\gamma/2,$

where $\Lambda^{\epsilon/2}_{[0,1/m_{0})}$ is defined considering the DyBW
process over dynamic time interval $[0,1/m_{0})$. We define $X$ as the non-
negative integer valued random variable

$\displaystyle
X:=\sup\\{n\in{\mathbb{N}}:\Lambda^{\epsilon/2}_{[0,1/m_{0}),n}\neq\emptyset\\}.$

By the previous lemma on the event $\\{X=\infty\\}$, the set
$\Lambda^{\epsilon/2}_{[0,1/m_{0})}$ must be non-empty. This gives us that

$\displaystyle{\mathbb{P}}(X=\infty)\leq{\mathbb{P}}(\Lambda^{\epsilon/2}_{[0,1/m_{0})}\neq\emptyset)<\gamma/2.$

Hence, we can choose $n^{\prime}\in{\mathbb{N}}$ such that ${\mathbb{P}}(X\leq
n^{\prime})>1-\gamma$. This completes the proof. ∎

Now we are ready to prove Proposition 4.4.

Proof of Proposition 4.4: As discussed earlier, it suffices to prove Equation
(14). Fix $\epsilon>0$ and $\gamma\in(0,1)$. Remember that we need to find
$m_{0}=m_{0}(\epsilon,\gamma)>0$ and
$n_{0}=n_{0}(\epsilon,\gamma)\in{\mathbb{N}}$ such that Equation (14) holds.

For $\delta>0$ we define the event $A(\delta)$ as

$\displaystyle A(\delta)$ $\displaystyle:=\bigl{\\{}\min\\{|s_{1}-s_{2}|\wedge
s_{1}\wedge(1-s_{2}):s_{1},s_{2}\in\Lambda^{\epsilon/2}\\}>\delta\bigr{\\}}.$
(17)

We first choose $m_{1}=m_{1}(\epsilon,\gamma)\in{\mathbb{N}}$ such that
${\mathbb{P}}(A(2/m_{1})^{c})<\gamma/8$. Recall that the random set
$\Lambda^{\epsilon/2}$ consists of finitely many distinct points a.s. and
hence, such a choice of $m_{1}$ is always possible. Further, on the event
$A(2/m_{1})$ we must have

$\\#\Lambda^{\epsilon/2}\leq\lfloor m_{1}/2\rfloor+1<m_{1}.$

Next, using Lemma 4.7 we choose $m_{0}\geq m_{1}$ and
$n_{3}=n^{\prime}(\epsilon/2,\gamma/(8m_{1}))$ such that

$\displaystyle{\mathbb{P}}\bigl{(}\\#\Lambda^{\epsilon/2}_{0,n}=\\#\Lambda^{\epsilon/2}_{[0,1/m_{0}),n}\geq
1\text{ for some }n\geq n_{3}\bigr{)}<\gamma/(8m_{1}).$ (18)

As $m_{0}\geq m_{1}$, we observe that on the event $A(2/m_{1})$, any interval
of the form $[j/m_{0},(j+1)/m_{0})$ can have at most one element from the set
$\Lambda^{\epsilon/2}$ and for any such interval, the neighbouring interval(s)
cannot have any element from the set $\Lambda^{\epsilon/2}$.

For $0\leq j\leq m_{0}-1$, let $N^{j}=N^{j}(\epsilon,m_{0})$ denote a non-
negative integer valued random variable defined as

$\displaystyle
N^{j}:=\mathbf{1}_{\\{\Lambda^{\epsilon/2}_{j}=\emptyset\\}}\sup\\{n\in{\mathbb{N}}:\Lambda^{\epsilon/2}_{j,n}\neq\emptyset\\}.$

Lemma 4.6 ensures that the non-negative r.v. $N^{j}$ is a.s. finite. Using the
collection $\\{N^{j}:0\leq j\leq m_{0}-1\\}$ we define

$\displaystyle N=N(\epsilon,m_{0}):=\max\\{N^{j}:0\leq j\leq m_{0}-1\\},$

which is an a.s. finite non-negative integer valued r.v. as well. We choose
$n_{4}\geq n_{3}$ so that

$\displaystyle{\mathbb{P}}(N\geq n_{4})<\gamma/8.$ (19)

We will show that with the above choice of $m_{0}$ and with $n_{0}=n_{4}$
Equation (13) holds. We define the event $E$ as

$\displaystyle E:=\bigcap_{n\geq
n_{0}}\\{\\#\Lambda^{\epsilon/2}_{j,n}\leq\\#\Lambda^{\epsilon/2}_{j}\text{
for all }0\leq j\leq m_{0}-1\\}.$

With the choice of $m_{0}$ and $n_{0}$ are as before, we observe the following
inclusion of events:

$\\{F^{m_{0}}_{n}\text{ for all }n\geq n_{0}\\}\subseteq A(2/m_{1})\cap E.$

Hence, in order to prove (13), it is enough to show that

$\displaystyle 2{\mathbb{P}}(A(2/m_{1})^{c})+{\mathbb{P}}(N\geq
n_{0})+{\mathbb{P}}(E^{c}\cap(N<n_{0})\cap A(2/m_{1}))<\gamma.$ (20)

The choice of $m_{1}$ and $n_{0}$ ensure that it suffices to show that

${\mathbb{P}}(E^{c}\cap(N<n_{0})\cap A(2/m_{1}))<\gamma/2.$

Let $\Gamma$ denote the collection of all possible subsets of the set
$\\{1,\cdots,m_{0}-2\\}$ with at most $m_{1}$ many elements such that these
subsets do not have consecutive elements. In other words,

$\displaystyle\Gamma:=\bigl{\\{}$
$\displaystyle\lambda\subseteq\\{1,\cdots,m_{0}-2\\}:\\#\lambda\leq
m_{1}\text{ and }\nexists j\in\\{1,\cdots,m_{0}-2\\}\text{ such that both }$
$\displaystyle j\text{ and }j+1\text{ are in }\lambda\bigr{\\}}.$

For $\lambda\in\Gamma$ we define the event $G_{\lambda}$ as

$\displaystyle G_{\lambda}:=$
$\displaystyle\bigl{(}\cap_{j\in\lambda}(\Lambda^{\epsilon/2}_{j,n}\neq\emptyset\text{
for infinitely many
}n)\bigr{)}\bigcap\bigl{(}\cap_{j\notin\lambda}(\Lambda^{\epsilon/2}_{j,n}=\emptyset\text{
for all }n\geq n_{0})\bigr{)}.$

It is not difficult to see that for $\lambda,\lambda^{\prime}\in\Lambda$ with
$\lambda\neq\lambda^{\prime}$, the events $G_{\lambda}$ and
$G_{\lambda^{\prime}}$ are disjoint. By definition, on the event $A(2/m_{1})$
we have $\Lambda^{\epsilon/2}_{0}=\Lambda^{\epsilon/2}_{m_{0}-1}=\emptyset$.
Hence, Lemma 4.6 gives us the following inclusion relation

$\bigl{(}A(2/m_{1})\cap(N<n_{0})\bigr{)}\subseteq\cup_{\lambda\in\Gamma}G_{\lambda}.$

Therefore, we can express ${\mathbb{P}}\bigl{(}E^{c}\cap(N<n_{0})\cap
A(2/m_{1})\bigr{)}$ as

$\displaystyle{\mathbb{P}}(E^{c}\cap(N<n_{0})\cap A(2/m_{1}))$
$\displaystyle\leq{\mathbb{P}}\bigl{(}E^{c}\cap(\cup_{\lambda\in\Lambda}G_{\lambda})\bigr{)}$
$\displaystyle={\mathbb{P}}\bigl{(}\cup_{\lambda\in\Lambda}(E^{c}\cap
G_{\lambda})\bigr{)}$
$\displaystyle=\sum_{\lambda\in\Gamma}{\mathbb{P}}\bigl{(}E^{c}\mid
G_{\lambda}\bigr{)}{\mathbb{P}}(G_{\lambda})$
$\displaystyle\leq\sum_{\lambda\in\Gamma}{\mathbb{P}}\bigl{(}\cup_{j\in\lambda}\bigl{(}\\#\Lambda^{\epsilon/2}_{j,n}\geq
2\text{ for some }n\geq n_{0}\bigr{)}\mid
G_{\lambda}\bigr{)}{\mathbb{P}}(G_{\lambda})$ (21)

The last step follows from the fact that given that the event $G_{\lambda}$
has occurred for some $\lambda\in\Lambda$, the only way that the complement
event $E^{c}$ can occur is due to the set $\Lambda^{\epsilon/2}_{j,n}$ for
some $j\in\lambda$ and for some $n\geq n_{0}$. Further, Lemma 4.6 ensures that
on the event $G_{\lambda}$, the set $\Lambda^{\epsilon/2}_{j}$ must be non-
empty. Hence, for any $\lambda\in\Lambda$ conditional that the event
$G_{\lambda}$ has occurred, we have the following event inclusion

$E^{c}\subseteq\bigl{(}\\#\Lambda^{\epsilon/2}_{j,n}\geq 2\text{ for some
}n\geq n_{0}\bigr{)}.$

This justifies the last inequality in (4.0.2). We can now write (4.0.2) as

$\displaystyle\sum_{\lambda\in\Gamma}{\mathbb{P}}\Bigl{(}\cup_{j\in\lambda}\bigl{(}\\#\Lambda^{\epsilon/2}_{j,n}\geq
2\text{ for some }n\geq n_{0}\bigr{)}\mid
G_{\lambda}\Bigr{)}{\mathbb{P}}(G_{\lambda})$ $\displaystyle\leq$
$\displaystyle\sum_{\lambda\in\Gamma}m_{1}{\mathbb{P}}\Bigl{(}\bigl{(}\\#\Lambda^{\epsilon/2}_{j,n}\geq
2\text{ for some }n\geq n_{0}\bigr{)}\mid
G_{\lambda}\Bigr{)}{\mathbb{P}}(G_{\lambda}).$ (22)

The last inequality follows from application of union bound and the fact that
$\\#\lambda\leq m_{1}$ for all $\lambda\in\Lambda$. We observe that the
collection of random variables
$\\{\Lambda^{\epsilon/2}_{j,n}:n\in{\mathbb{N}}\\}$ depends on the evolution
of the i.i.d. Markov processes
$\\{I^{n}_{(x,t)}(\cdot):(x,t)\in{\mathbb{Z}}^{2}_{\text{even}}\\}_{n\in{\mathbb{N}}}$
over dynamic time domain $[j/m_{0},(j+1)/m_{0})$.

Fix $n\in{\mathbb{N}}$. For $0\leq j\leq m_{0}-1$ let $X^{j}_{n}$ denote the
non-negative integer valued random variable defined as
$X^{j}_{n}:=\\#\Lambda^{\epsilon/2}_{j,n}$. For $0\leq j\leq m_{0}-1$ we
define

$Z^{j}:=(X^{j}_{1},X^{j}_{2},\cdots)$

which takes vales in $({\mathbb{N}}\cup\\{0\\})^{\mathbb{N}}$. The next lemma
shows that the set $\\{Z^{j}:0\leq j\leq m_{0}-1\\}$ forms an i.i.d.
collection of random variables taking vales in
$({\mathbb{N}}\cup\\{0\\})^{\mathbb{N}}$.

######  Lemma 4.8.

$\\{Z^{j}:0\leq j\leq m_{0}-1\\}$ forms an i.i.d. collection of random
variables.

###### Proof.

Because of stationarity of our process, it follows that for fixed $n$ the
random variables $X^{j}_{n}$’s for $0\leq j\leq m_{0}-1$ are identically
distributed. Fix Borel subsets $B^{0},...,B^{m_{0}-1}$ in appropriate space.
Let $I_{j}(B^{j})$ be the indicator random variable of the event $\\{Z^{j}\in
B^{j}\\}$. For $0\leq j\leq m_{0}-1$, set the $\sigma$-field

${\cal
G}_{j}:=\sigma\bigl{(}(N^{n}_{(x,t)}(\tau),I^{\ell}_{(x,t)}):(x,t)\in{\mathbb{Z}}^{2}_{\text{even}},\tau\in[0,j/m_{0}),n\geq
1,0\leq\ell\leq N^{n}_{(x,t)}(j/m_{0})\bigr{)}.$

Now we have

$\displaystyle{\mathbb{P}}(Z^{j}\in B^{j}\text{ for }0\leq j\leq m_{0}-1)$
$\displaystyle={\mathbb{E}}(\Pi_{j=0}^{m_{0}-1}I_{j}(B^{j}))$
$\displaystyle={\mathbb{E}}\Bigl{(}{\mathbb{E}}(\Pi_{j=0}^{m_{0}-1}I_{j}(B^{j})\mid{\cal
G}_{m_{0}-1})\Bigr{)}$
$\displaystyle={\mathbb{E}}\Bigl{(}\Pi_{j=0}^{m_{0}-2}I_{j}(B^{j}){\mathbb{E}}(I_{m_{0}-1}(B^{m_{0}-1})\mid{\cal
G}_{m_{0}-1})\Bigr{)}$
$\displaystyle={\mathbb{E}}\Bigl{(}\Pi_{j=0}^{m_{0}-2}I_{j}(B^{j})\Bigr{)}{\mathbb{P}}(Z^{0}\in
B^{m_{0}-1}),$

where the penultimate step follows from the fact that the r.v. $I_{j}(B^{j})$
is measurable w.r.t. the $\sigma$-field ${\cal G}^{m_{0}-1}$. The last
equality follows from stationarity nature of our model which ensures equality
of distribution of $Z^{j}$’s. Applying the same argument repetitively
completes the proof. ∎

Lemma 4.8 gives us that the event $(\\#\Lambda^{\epsilon/2}_{j,n}\geq 2\text{
for some }n\geq n_{0})$ does not depend on the collection $\\{Z^{i}:0\leq
i\leq m_{0}-1,i\neq j\\}$. Note that the event
$(\\#\Lambda^{\epsilon/2}_{i,n}\geq 1\text{ for infinitely }n)$ is measurable
w.r.t. the $\sigma$-field $\sigma(Z^{i})$. Hence, Equation (4.0.2) becomes

$\displaystyle\sum_{\lambda\in\Gamma}m_{1}{\mathbb{P}}\Bigl{(}\bigl{(}\\#\Lambda^{\epsilon/2}_{j,n}\geq
2\text{ for some }n\geq n_{0}\bigr{)}\mid
G_{\lambda}\Bigr{)}{\mathbb{P}}(G_{\lambda})$ $\displaystyle\leq$
$\displaystyle\sum_{\lambda\in\Gamma}m_{1}{\mathbb{P}}\Bigl{(}\bigl{(}\\#\Lambda^{\epsilon/2}_{j,n}\geq
2\text{ for some }n\geq n_{0}\bigr{)}\mid(\\#\Lambda^{\epsilon/2}_{j,n}\geq
1\text{ for infinitely }n)\Bigr{)}{\mathbb{P}}(G_{\lambda})$
$\displaystyle\leq$
$\displaystyle\sum_{\lambda\in\Gamma}m_{1}{\mathbb{P}}\Bigl{(}\bigl{(}\\#\Lambda^{\epsilon/2}_{0,n}\geq
2\text{ for some }n\geq n_{0}\bigr{)}\mid(\\#\Lambda^{\epsilon/2}_{0,n}\geq
1\text{ for infinitely }n)\Bigr{)}{\mathbb{P}}(G_{\lambda}).$ (23)

The last step follows from stationarity of our model. We recall that for any
$s>0$ the set $\Lambda^{\epsilon/2}_{[0,s],n}$ is defined based on the scaled
DyDW (dynamic discrete web) restricted over dynamic time interval $[0,s]$
only. We define,

$\displaystyle\sigma^{n}:=(\inf\\{s\in[0,1/m_{0}):\\#\Lambda^{\epsilon/2}_{[0,s],n}\geq
1\\})\wedge 1/m_{0}.$

We observe that for every $n$, the r.v. $\sigma^{n}$ is a stopping time w.r.t.
the filtration

$\Bigl{\\{}{\cal
G}^{s}_{0,n}:=\sigma\bigl{(}(N^{n}_{(x,t)}(\tau),I^{\ell}_{(x,t)}):(x,t)\in{\mathbb{Z}}^{2}_{\text{even}},\tau\in[0,s],0\leq\ell\leq
N^{n}_{(x,t)}(s)\bigr{)}:s\in[0,1/m_{0})\Bigr{\\}}.$

The strong Markov property of the scaled process (w.r.t. dynamic time) allows
us to obtain

$\displaystyle{\mathbb{P}}\bigl{(}(\\#\Lambda^{\epsilon/2}_{0,n}\geq 2\text{
for some }n\geq n_{0})\mid(\\#\Lambda^{\epsilon/2}_{0,n}\geq 1\text{ for
infinitely many }n)\bigr{)}$ $\displaystyle=$
$\displaystyle{\mathbb{P}}\bigl{(}(\\#\Lambda^{\epsilon/2}_{(\sigma^{n},1/m_{0}),n}\geq
1\text{ for some }n\geq n_{0})\mid(\sigma^{n}<1/m_{0}\text{ for infinitely
many }n)\bigr{)}$ $\displaystyle=$
$\displaystyle{\mathbb{P}}\Bigl{(}\bigcup_{n\geq
n_{0}}\bigl{(}\\#\Lambda^{\epsilon/2}_{(0,1/m_{0}-\sigma^{n}),n}\geq
1\mid{\cal G}^{\sigma^{n}}_{0,n}\bigr{)}\Bigr{)}$ (24)

As $\sigma^{n}$ is a stopping time for the scaled DyDW process, applying Lemma
4.7 to (4.0.2) we obtain

${\mathbb{P}}\Bigl{(}\bigcup_{n\geq
n_{0}}\bigl{(}\\#\Lambda^{\epsilon/2}_{(0,1/m_{0}-\sigma^{n}),n}\geq
1\mid{\cal
G}^{\sigma^{n}}_{0,n}\bigr{)}\Bigr{)}\leq{\mathbb{P}}\bigl{(}\\#\Lambda^{\epsilon/2}_{[0,1/m_{0})}\geq
1\text{ for some }n\geq n_{0}\bigr{)}<\gamma/(8m_{1}).$

Finally, putting this estimate in (4.0.2) gives us that

${\mathbb{P}}(E^{c}\cap(N<n_{0})\cap
A(2/m_{1}))<\gamma/8\sum_{\lambda\in\Gamma}{\mathbb{P}}(G_{\lambda})=\gamma/8{\mathbb{P}}(\cup_{\lambda\in\Gamma}G_{\lambda})\leq\gamma/8.$

As discussed earlier, this completes the proof of (13) and thereby proves
Proposition 4.4. ∎

In order to complete our proof we need to prove Proposition 4.5. To prove
Proposition 4.5 we need the following notion of hopping at intersection times
which is taken from [SS08].

For any collection of paths ${\cal K}\subset\Pi$ we let ${\cal
H}_{\text{int}}({\cal K})$ denote the smallest set of paths containing ${\cal
K}$ that is closed under hopping at intersection times, that is, ${\cal
H}_{\text{int}}({\cal K})$ is the set of all paths $\pi\in\Pi$ of the form

$\pi=\cup^{m}_{i=1}\\{(\pi_{i}(t),t):t\in[t_{k-1},t_{k}]\\},$ (25)

where $\pi_{1},\cdots,\pi_{m}\in{\cal K}$,
$\sigma_{\pi_{1}}=t_{0}<\cdots<t_{m}=\infty$, and $t_{i}$ is an intersection
time of $\pi_{i}$ and $\pi_{i+1}$ for each $i=1,\cdots,m-1$.

The following corollary gives us a way to identify the corresponding Brownian
net corresponding to the DyBW $\\{{\mathcal{W}}(\tau):\tau\in[0,1]\\}$.

######  Corollary 4.9.

Let $\\{{\mathcal{W}}(\tau):\tau\in[0,1]\\}$ denote the DyBW process with
${\mathcal{N}}$ denoting the corresponding net. Let ${\cal K}={\cal
K}({\mathcal{W}})$ denote the collection of paths

$\displaystyle{\cal K}:=\\{\pi:\pi\in{\mathcal{W}}(\tau)\text{ for some
}\tau\in[0,1]\cap{\mathbb{Q}}\\}\subseteq\Pi,$ (26)

consisting of DyBW paths at rational dynamic time points. Let
${\mathcal{N}}^{\prime}$ be a standard Brownian net defined on the same
probability space such that ${\cal
K}({\mathcal{W}})\subset{\mathcal{N}}^{\prime}$ a.s. Then we must have
${\mathcal{N}}^{\prime}={\mathcal{N}}$ a.s., i.e., ${\mathcal{N}}^{\prime}$
must be the corresponding net only.

###### Proof.

We proved that the DyBW has ${\cal H}$-valued RCLL paths a.s. Hence,
$\overline{{\cal K}}$ contains the set $\\{\pi:\pi\in{\mathcal{W}}(\tau)\text{
for some }\tau\in[0,1]\\}$. As ${\mathcal{N}}^{\prime}$ gives a compact
collection collection of paths, our assumption

${\cal K}\subseteq{\mathcal{N}}^{\prime}\text{ implies }\overline{{\cal
K}}\subseteq{\mathcal{N}}^{\prime}.$

This implies that

$\\{\pi:\pi\in{\mathcal{W}}(\tau)\text{ for some
}\tau\in[0,1]\\}\subseteq{\mathcal{N}}^{\prime}.$ (27)

Next for any $n\in{\mathbb{N}}$, paths in the ‘corresponding’ partial net
${\mathcal{N}}_{n}$ belong to ${\cal
H}_{\text{int}}(\\{{\mathcal{W}}(\tau):\tau\in[0,1]\\})$. By definition for
any ${\cal K}_{1}\subseteq{\cal K}_{2}\subseteq\Pi$ we have ${\cal
H}_{\text{int}}({\cal K}_{1})\subseteq{\cal H}_{\text{int}}({\cal K}_{2})$.
Since, Brownian net is closed under hopping (see Proposition 1.4. of [SS08]),
for every $n\in{\mathbb{N}}$ Equation (27) allows us to obtain

${\mathcal{N}}_{n}\subseteq{\cal
H}_{\text{int}}(\\{{\mathcal{W}}(\tau):\tau\in[0,1]\\})\subseteq{\cal
H}_{\text{int}}({\mathcal{N}}^{\prime})={\mathcal{N}}^{\prime}.$

On the other hand, [NRS10] showed (see Theorem of [NRS10]) for the
corresponding net ${\mathcal{N}}$ that

${\mathcal{N}}=\lim_{n\to\infty}{\mathcal{N}}_{n}\text{ a.s}.$

As ${\mathcal{N}}^{\prime}$ is a compact collection of paths a.s., this
implies that ${\mathcal{N}}\subseteq{\mathcal{N}}^{\prime}$ as well. Since
they both have the same distribution, ${\mathcal{N}}^{\prime}$ can not have
more paths and we must have ${\mathcal{N}}^{\prime}={\mathcal{N}}$ a.s. This
completes the proof. ∎

Next we present an extension of the above corollary. Recall the $m_{0}+1$
dimensional random vector of the “corresponding” Brownian nets
$({\mathcal{N}},{\mathcal{N}}_{0},{\mathcal{N}}_{1},\cdots,{\mathcal{N}}_{m_{0}-1})$
where ${\mathcal{N}}_{j}$ is constructed using markings in the dynamic time
interval $[j/m_{0},(j+1)/m_{0})$ only.

######  Corollary 4.10.

Let
$({\mathcal{N}}^{\prime},{\mathcal{N}}^{\prime}_{0},{\mathcal{N}}^{\prime}_{1},\cdots,{\mathcal{N}}^{\prime}_{m_{0}-1})$
denote a $(m_{0}+1)$ dimensional random vector of Brownian nets such that
${\mathcal{N}}^{\prime}\stackrel{{\scriptstyle d}}{{=}}{\mathcal{N}}(1)$ and
for any $0\leq j\leq m_{0}-1$, we have
${\mathcal{N}}^{\prime}_{j}\stackrel{{\scriptstyle
d}}{{=}}{\mathcal{N}}(1/m_{0})$. Suppose there exists a DyBW process
$\\{{\mathcal{W}}(\tau):\tau\in[0,1]\\}$ define on the same probability space
such that we have

$\\{\pi:\pi\in{\mathcal{W}}(\tau)\text{ for some
}\tau\in[j/m_{0},(j+1)/m_{0})\cap{\mathbb{Q}}\\}\subset{\mathcal{N}}^{\prime}_{j}.$

Then we must have

$\bigl{(}{\mathcal{N}},{\mathcal{N}}_{0},\cdots,{\mathcal{N}}_{m_{0}-1}\bigr{)}=\bigl{(}{\mathcal{N}}^{\prime},{\mathcal{N}}^{\prime}_{0},\cdots,{\mathcal{N}}^{\prime}_{m_{0}-1}\bigr{)}\text{
a.s. }$

where the l.h.s. denotes the random vector of ‘corresponding’ Brownian nets
for the DyBW process $\\{{\mathcal{W}}(\tau):\tau\in[0,1]\\}$.

The proof follows from the same argument as the earlier one. Now we are ready
to prove Proposition 4.5.

Proof of Proposition 4.5 : In Section 3.2 we proved finite dimensional
convergence for the scaled DyDW to the DyBW. This implies following
distributional convergence

$\\{{\cal
X}_{n}(\tau):\tau\in[0,1]\cap{\mathbb{Q}}\\}\Rightarrow\\{{\mathcal{W}}(\tau):\tau\in[0,1]\cap{\mathbb{Q}}\\}\text{
as }n\to\infty,$

as a sequence of ${\cal H}^{\mathbb{N}}$ valued random variables.

We consider the sequence of ${\cal H}^{m_{0}+1}\times{\cal H}^{\mathbb{N}}$
valued random variables

$\bigl{\\{}\bigl{(}{\mathcal{N}}_{n},{\mathcal{N}}_{0,n},\cdots,{\mathcal{N}}_{m_{0}-1,n}\bigr{)},\\{{\cal
X}_{n}(\tau):\tau\in[0,1]\cap{\mathbb{Q}}\\}:n\in{\mathbb{N}}\bigr{\\}}.$

It is not difficult to see that the above sequence is tight.

Let $\Bigl{(}\bigl{(}{\cal Z},{\cal Z}_{0},\cdots,{\cal
Z}_{m_{0}-1}\bigr{)},\\{{\mathcal{W}}(\tau):\tau\in[0,1]\cap{\mathbb{Q}}\\}\Bigr{)}$
be a subsequential limit of of the above sequence. By Skorohod’s embedding
theorem we assume that we are working on a probability space where we have
almost sure convergence. By the work of [SS08] for all $0\leq j\leq m_{0}-1$
we have that

${\cal Z}\stackrel{{\scriptstyle d}}{{=}}{\mathcal{N}}(1)\text{ and }{\cal
Z}_{j}\stackrel{{\scriptstyle d}}{{=}}{\mathcal{N}}(1/m_{0}).$

Further our construction ensures that for any $0\leq j\leq m_{0}-1$ we have,

$\pi^{n}\in{\cal X}_{n}(\tau)\text{ for some
}\tau\in[j/m_{0},(j+1)/m_{0})\text{ only if }\pi^{n}\in{\mathcal{N}}_{j,n}.$

In other words, for all $0\leq j\leq m_{0}-1$ we have

$\\{\pi^{n}:\pi^{n}\in{\cal X}_{n}(\tau)\text{ for some
}\tau\in[j/m_{0},(j+1)/m_{0})\cap{\mathbb{Q}}\\}\subseteq{\mathcal{N}}_{j,n}.$

Hence, the limiting random variable ${\cal Z}^{j}$ must contain the set of
paths $\\{\pi\in{\mathcal{W}}(\tau)\text{ for some
}\tau\in[j/m_{0},(j+1)/m_{0})\cap{\mathbb{Q}}\\}$. By Corollary 4.10 we have
that

$({\cal Z},{\cal Z}_{0},\cdots,{\cal Z}_{m-1})\stackrel{{\scriptstyle
d}}{{=}}({\mathcal{N}},{\mathcal{N}}_{0},\cdots,{\mathcal{N}}_{m_{0}-1}).$

This completes the proof. ∎

Acknowledgement: Part of the work on this paper was done when K.S. visited K.R
at NYU Abu Dhabi. Both authors wish to thank NYU Abu Dhabi for hospitality and
support.

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Department of Mathematics, 1 Hawk Drive, New Paltz, NY 12561

E-mail address, K. Ravishankar<EMAIL_ADDRESS>
Department of Mathematics, Ashoka University, Sonipat, India

E-mail address, K. Saha<EMAIL_ADDRESS>