\section{Introduction}

Particle systems with topological interactions are
models for particle dynamics on the real line where
the so called {\it boundary particles}, namely the rightmost and leftmost ones,
have a special role.
Typically in these models, all particles undergo motion, branching
or nonlocal duplication, whereas the boundary particles are
in addition subject to a removal (or injection) mechanism.
The study of their hydrodynamic limits and the characterization of these limits
in terms of free boundary problems (FBP)
has been the subject of interest in the literature
\cite{de-masi-top, de-masi-book, de-masi-excl, de-masi-nbbm,
de-masi-nbbm2,dur11}.
A prototype of this class of models is the following variant of the basic model studied by
Carinci, De Masi, Giardin\`a and Presutti in the monograph \cite{de-masi-book}.
It is a system of Brownian particles on the real line
subject to injection of new particles and removal,
where injection follows a given probability measure $\pi$
and removal is constrained to the rightmost particle.
Initially, $N$ particles are distributed according to
$u_0(x)dx$, where $u_0$ is given.
The cumulative number of injections (resp., removals)
by time $t$ is $[NI_t]$ (resp., $[NJ_t]$) where $I$ and $J$ are given
continuous nondecreasing functions starting at $0$, and $1+I-J>0$.
The particle configuration measure at time $t$ is denoted by $\xi^N_t$,
and its rescaled version by $\bar\xi^N_t=N^{-1}\xi^N_t$.
Key to a new viewpoint proposed in this paper is the fact that
the removal mechanism is order-respecting:
{\it for every $r\in{\mathbb R}$, at times when a particle is removed from $(-\infty,r)$,
no particles occupy $(r,\infty)$.}
This condition translates in a natural way to a condition about the associated heat equation,
provided it is formulated with a measure-valued RHS.
This equation, for the unknown $(u,\beta)$, takes the form
\[
{\cal L} u=\alpha-\beta,
\]
where ${\cal L}=\partial_t-\frac{1}{2}\partial_{xx}$,
$\alpha$ and $\beta$ are measures on ${\mathbb R}\times{\mathbb R}_+$
governing injection and removal, respectively,
and $\alpha$ is given by the problem data via $\alpha(dx,dt)=\pi(dx)dI_t$.
It is considered with initial condition $u(\cdot,0)=u_0$, a positivity condition of $u$,
a cumulative removal condition $\beta({\mathbb R}\times[0,t])=J_t$,
and the {\it order-respecting absorption} (ORA) condition
\begin{equation}\label{01}
\int_{{\mathbb R}_+}u((r,\infty),t)\beta((-\infty,r)\times dt)=0
\quad\text{for all}\quad r\in{\mathbb R},
\end{equation}
where one denotes $u((r,\infty),t)=\int_r^\infty u(y,t)dy$.
Under some restrictions on $(u_0,I,J)$, it is shown
in this paper that there exists a unique solution $(u,\beta)$,
and that the limit of $\bar\xi^N_t$ exists and has a density given by $u(\cdot,t)$
(see \S\ref{sec2} for the precise definition of a solution and statement of convergence).
This model is referred to in this paper as {\it removal at boundary} (RAB).

This formulation is to be contrasted with the FBP approach
that has been successfully used many times as a means of characterizing
scaling limits of this and various related models (see \cite{de-masi-book} and references therein).
In particular, the basic model of \cite{de-masi-book} (studied alongside several others)
corresponds to the above description
with $\pi=\boldsymbol{\delta}_0$ (throughout, the Dirac measure at $x\in{\mathbb R}$ or $x\in{\mathbb R}\times{\mathbb R}_+$
is denoted by $\boldsymbol{\delta}_x$), $I_t=J_t=jt$, $j>0$ a constant,
and where the Brownian particles live in ${\mathbb R}_+$, reflecting at $0$.
The FBP \cite{de-masi-book} consists of the heat
equation ${\cal L} u=0$ on a time varying domain $(0,X_t)$,
with initial condition $u(\cdot,0)=u_0$, boundary condition $-\frac12\partial_xu(0,t)=j$,
and the following conditions at the free boundary,
\[
u(X_t,t)=0,\qquad  -\frac12\partial_x u(X_t,t)=j.
\]
This problem is closely related to the Stefan problem,
for which it is well known that local existence and uniqueness
of classical solutions hold (for classical initial conditions)
but global existence fails in general. Hence a relaxed notion of solutions
is developed in \cite{de-masi-book}, for which existence and uniqueness
are established. It is also established that this solution describes
the hydrodynamic limit of the particle configuration process.

In \cite{de-masi-book}, constraining the particles to ${\mathbb R}_+$ and injecting
at the leftmost site assures that injection always occurs to the left
of the free boundary. It is conjectured in \cite[\S 15.1]{de-masi-book}
that when applying a general injection distribution,
which may result in injection to the right of the free boundary,
the hydrodynamic limit is ruled by a FBP (specifically, (15.1.1)--(15.1.3) in \cite{de-masi-book}).
Such a condition is not important when it comes to the ORA formulation,
which completely avoids the use of a free boundary. While our results
are not concerned with a FBP,
they address the conjecture in a weaker sense, as they show that
for a general injection distribution, (and further, time varying rates
and no mass conservation condition) the limit exists, and moreover
is characterized by an alternative PDE,
namely the heat equation with ORA.

Another model treated in this paper is referred to as {\it removal at quantile} (RAQ).
Initially $N$ particles are distributed on the line according to $u_0(x)dx$.
At rate $N$, the $Q$-quantile member of the population is removed
until no particles are left, where $Q$ is a given continuous function
of time with values in $[0,1]$. The motivation to study this model is
that, according to a formal argument, it gives rise to a topological interaction
of a somewhat different nature than models studied earlier.
Namely, denote the mass to the left and right of the $Q$-quantile member
by $q_L(t)=(1-t)(1-Q(t))$ and, resp., $q_R(t)=(1-t)Q(t)$.
Focus first on the case where $Q$ is $C^1$ and takes values in $(0,1)$, and moreover,
$\lambda_L(t),\lambda_R(t)\in[0,1]$ for all $t$, where $\lambda_L(t)=-\frac{dq_L(t)}{dt}$,
$\lambda_R(t)=-\frac{dq_R(t)}{dt}$.
In this special case, the FBP formally associated with the model
is given by the heat equation ${\cal L} u=0$ on the time varying domain
$(-\infty,X_t)\cup(Y_t,\infty)$, $X_t<Y_t$, for $t\in(0,1)$,
coupled with the initial condition $u(\cdot,0)=u_0$
and the conditions at the free boundary
\[
u(X_t,t)=u(Y_t,t)=0,
\qquad
-\frac{1}{2}\partial_x u(X_t,t)=\lambda_L(t),
\qquad
\frac{1}{2}\partial_x u(Y_t,t)=\lambda_R(t),
\qquad t\in(0,1).
\]
Thus the macroscopic configuration is supported on
$(-\infty,X_t]\cup[Y_t,\infty)$ and absorption takes place at $X_t$ and $Y_t$.
Unlike in earlier work
\cite{de-masi-top, de-masi-book, de-masi-excl, de-masi-nbbm, de-masi-nbbm2},
according to this description,
the removal mechanism acts on particles that are `at the boundary' but are not rightmost or leftmost.
If $Q$ is allowed to vary in $[0,1]$ then one must deal with free boundaries
taking values in $[-\infty,\infty]$, for $X_t=-\infty$
(resp., $Y_t=\infty$) when $Q(t)=1$ (resp., $Q(t)=0$).
For the more general case where $\lambda_L$ or $\lambda_R$ are not restricted to
$[0,1]$ (or when $Q$ is merely continuous)
we make no attempt to propose a FBP formulation, which seems to be even more challenging.

We propose an alternative formulation via a variant of the ORA condition.
It does not rely on the above heuristic or on its implication that
the macroscopic configuration is supported
on two connected components. Rather, it expresses an absorption rule
that again reflects the particle system's removal
mechanism. It consists of the heat equation
${\cal L} u=-\beta$ with an initial condition $u(\cdot,0)=u_0$,
a cumulative removal condition $\beta({\mathbb R}\times[0,t])=t\w1$, and the conditions
\[
\int_{{\mathbb R}_+}[u((r,\infty),t)-q_R(t)]^+\beta((-\infty,r)\times dt)=0,\qquad r\in{\mathbb R},
\]
\[
\int_{{\mathbb R}_+}[u((-\infty,r),t)-q_L(t)]^+\beta((r,\infty)\times dt)=0,\qquad r\in{\mathbb R}.
\]
It is shown that the hydrodynamic limit exists and is characterized as the unique
solution to this equation.

The first paper to study hydrodynamic limits for a model closely related
to the aforementioned
RAB was \cite{de-masi-top}, where particles perform random walks on $[0,N]\cap{\mathbb Z}$
rather than Brownian motion.
In \cite{de-masi-excl}, a one dimensional symmetric simple exclusion process
with birth of the leftmost hole and death of the rightmost particle was considered,
and convergence at the hydrodynamic scale was proved.
A formal limit gave rise to a FBP that is close to that of \cite{de-masi-book},
but a rigorous connection
was not proved because existence and uniqueness of solution to this FBP are not known.
A related class of models is the Brunet-Derrida evolution-selection
mechanisms, of which the $N$-branching Brownian motion is a special case.
In this model, Brownian particles on the line undergo
branching, and the leftmost particle is removed whenever a new one is born.
The hydrodynamic limit was proved to exist in \cite{de-masi-nbbm} and
the limit was identified as the local solution of a FBP,
but the identification on an arbitrary time interval was
conditional on existence of a global solution and on
continuity of the free boundary. These issues were then settled in \cite{ber19}.
A variant of this model, in which the branching is nonlocal,
was studied in \cite{de-masi-nbbm2},
where the hydrodynamic limit was proved to exist.
Its characterization as the solution of a FBP was also proved
conditionally on existence of a classical solution to the FBP.
However, existence is not known in general.

The results of this paper demonstrate that
ORA may provide a route to circumvent obstacles
related to FBP (existence of classical solutions, regularity
of the free boundary, etc.), and consequently may be applicable in cases
where FBP formulations are too difficult to analyze.
In the future, it is desired to justify this statement more broadly.
For example, it is straightforward to formulate
(as we do in \S\ref{sec6}) ORA for the $N$-BBM and its extensions
alluded to above, and so it would be of interest
to apply the approach in settings such as \cite{de-masi-nbbm2} where
the FBP is challenging.

ORA is closely related to (indeed, inspired by)
the Skorohod map. Particularly, the Skorohod map
in measure space was introduced in \cite{abkr}
as a means of describing priority in queueing models, especially
in settings of a continuum of priority classes, and was used
to establish law-of-large-numbers limits of such systems.
A relation between ORA and the Skorohod map can be made precise,
and moreover, a solution to the heat equation with ORA
can be seen as a fixed point of a related map (see \S\ref{sec2}).
However, we were not able to base a proof of our main results on fixed point techniques.

The proof of uniqueness of solutions to the heat equation relies heavily on
the method of barriers. We refer to \cite{de-masi-book} for
the history of its use;
it has been employed in particular in \cite{de-masi-top, de-masi-book,
de-masi-excl, de-masi-nbbm, de-masi-nbbm2, dur11}.
Barriers are discrete approximate solutions that trap the true solutions
from above and below in the sense of mass transport inequalities,
and that have a unique separating element.
The specific form of the barriers from \cite{de-masi-book}
is used, up to simple adaptations, in the construction of the lower barriers
for the RAB model. For the upper barriers of RAB and both lower and upper barriers of RAQ,
additional ideas are necessary and the constructions differ considerably.
Existence of solutions is not proved analytically, but follows as a consequence
of the convergence of the probabilistic model.
For convergence we deviate from the method employed in \cite{de-masi-book,
de-masi-top, de-masi-nbbm}
in that we do not construct stochastic barriers,
but instead our arguments are based on tightness of the rescaled processes.

\subsubsection*{Organization of this paper}
\S \ref{sec21} introduces the particle system with RAB
and asserts that the hydrodynamic limit exists (Theorem~\ref{th0}).
In \S\ref{sec22}, a general treatment from \cite{ama01}
of parabolic initial value problems involving measures is mentioned
in the special case of the heat equation. The uniqueness results from this reference
are crucially used in the sequel. Then, the heat equation with ORA is formulated \eqref{018},
and a uniqueness result is stated (Theorem \ref{th1}).
The main result regarding this model,
Theorem \ref{th2}, is given in \S\ref{sec23}, stating that the scaling
limit has a density and identifying this density as the unique solution of \eqref{018}.
Also, a comparison principle for solutions is stated (Theorem \ref{th5}),
according to which solutions depend on the problem data in such a way
that an increase in the absorption rate (i.e., the derivative of $J$) causes a decrease
of the solution $u$ w.r.t.\ the mass transport partial order.
\S \ref{sec3} gives the proof of Theorem \ref{th1}, starting with
a representation of $u$ in terms of $\alpha$ and $\beta$ as a mild solution (\S\ref{sec30})
followed by mass transport inequalities, some of which are borrowed from
\cite{de-masi-book}, stated in \S\ref{sec30+}.
Lower and upper barriers are constructed in \S\ref{sec31} and \S\ref{sec32}, resp.
The result is finally proved in \S\ref{sec33} by showing that the upper and lower
barriers cannot be separated by more than one element.
In \S\ref{sec41}, Theorem \ref{th2} is proved,
and as a corollary, Theorem \ref{th0}. The proof is based on a tightness argument
(Lemma \ref{lem6}), followed by estimates on the empirical density,
used in conjunction with a result due to Doob found in \cite{del-mey-82}
on existence of a measurable density, to construct such a density for any subsequential limit of
the rescaled configuration process $\bar\xi^N$ (Lemma \ref{lem5}).
In Lemma \ref{lem7} it is deduced from the particle system's removal
structure that these densities, $u$ (paired with $\beta$, also obtained by a scaling limits),
satisfy the ORA condition. Based on these lemmas, the proof of Theorem \ref{th2}
that appears at the end of the section shows that $(u,\beta)$
constitute solutions to equation \eqref{018}. The uniqueness of such solution
established in Theorem \ref{th1} then implies the existence of limits.
In \S\ref{sec42} Theorem \ref{th5} is proved via a coupling argument for the particle system.
\S\ref{sec5} treats the RAQ model starting with \S\ref{sec51} that
construct the model.
The heat equation \eqref{B018} is presented and the main result regarding this model,
Theorem \ref{Bth2}, is
stated in \S\ref{sec52}. \S\ref{sec53} lists and proves further mass transport inequalities.
Barriers are constructed in \S\ref{sec54}, and \S\ref{sec55} completes the proof.
In \S\ref{sec6} further examples are provided of models
that can be formulated via ORA, yet to be studied.


\subsubsection*{Notation}
The `spacial' and `temporal' domains we work with are ${\mathbb R}$ and,
respectively, ${\mathbb R}_+=[0,\infty)$.
For $X={\mathbb R}$ or ${\mathbb R}\times{\mathbb R}_+$ let $C_c^\infty(X)$ be the space
of compactly supported smooth functions on $X$. Let
$C_0(X)$ denote the space of continuous functions on $X$ vanishing at infinity,
and endow it with the maximum norm.
For $s\in{\mathbb R}$ and $p\in[0,\infty]$ denote by $W^s_p(X)$ the usual Sobolev space,
so that $W^0_p(X)=L_p(X)$.
Abbreviate $W^s_p({\mathbb R})$ and $L_p({\mathbb R})$ to $W^s_p$ and $L_p$, resp.
Denote by ${\cal M}(X)$ the space of locally finite signed Borel measures on $X$
and endow it with the topology of weak convergence on compacts.
Let ${\cal M}_+(X)\subset{\cal M}(X)$ denote the set of positive measures.
Denote by ${\cal M}_{\rm loc}({\mathbb R}\times{\mathbb R}_+)$ the
space of signed Borel measures that are finite on ${\mathbb R}\times[0,T]$
for any finite $T$.
Use similarly the notation $L_{p,\rm loc}({\mathbb R}_+,L_p)$, etc.

For $u,v\in{\cal B}({\mathbb R},{\mathbb R})$ and $\xi\in{\cal M}_+({\mathbb R})$,
denote $\langle u,\xi\rangle=\int_{\mathbb R} u\, d\xi$ and $(u,v)=\int_{\mathbb R} uv\,dx$,
provided the integrals exist.
For $\xi\in{\cal M}_+({\mathbb R})$, and an interval, say $[a,b]$,
$\xi[a,b]$ is used as shorthand for $\xi([a,b])$, and for $v\in L_1$,
$v[a,b]$ is shorthand for $\int_a^b v\,dx$.

For $X$ a Polish space, let $C({\mathbb R}_+,X)$ be the space of continuous paths,
and $D({\mathbb R}_+,X)$ the space of c\`adl\`ag  paths. Endow the latter with
the Skorohod $J_1$ topology.

Let $C^\uparrow({\mathbb R}_+,{\mathbb R}_+)$ denote the subset of $C({\mathbb R}_+,{\mathbb R}_+)$
of nondecreasing functions that vanish at zero.
For $I\in C^\uparrow({\mathbb R}_+,{\mathbb R}_+)$, denote by $dI_t$
the corresponding Stieltjes measure on ${\mathbb R}_+$.
Let $AC^\uparrow({\mathbb R}_+,{\mathbb R}_+)$ denote the subset of $C^\uparrow({\mathbb R}_+,{\mathbb R}_+)$
of absolutely continuous functions.




\section{Removal at boundary: model and results}\label{sec2}
\beginsec



\subsection{Brownian particles with
injection and boundary removal}\label{sec21}

A tuple $(u_0,\pi,I,J)$ is said to be an admissible data if
$u_0\in L_1({\mathbb R},{\mathbb R}_+)\cap L_\infty({\mathbb R},{\mathbb R}_+)$,
$\|u_0\|_1=1$, $\pi$ is a Borel probability measure on ${\mathbb R}$,
$I\in C^\uparrow({\mathbb R}_+,{\mathbb R}_+)\cap C^{\rho_0}({\mathbb R}_+,{\mathbb R}_+)$ and
$J\in AC^\uparrow({\mathbb R}_+,{\mathbb R}_+)\cap C^{\rho_0}({\mathbb R}_+,{\mathbb R}_+)$ for some
$\rho_0>1/2$, and
\begin{equation}\label{055}
\varepsilon_0:=\inf_{t\in{\mathbb R}_+}(1+I_t-J_t)>0.
\end{equation}
In addition, it is required that $\sup_tI_t=\sup_tJ_t=\infty$,
but this condition is not an important one;
it is imposed only to slightly simplify the description of the particle system.
The roles played by these four ingredients
are to determine, respectively,
the initial configuration of particles, the distribution of injected particles,
and the cumulative number of injected (resp., removed) particles.
The precise details are as follows.

The model is indexed by $N$, where $N$ is the initial number of particles.
Let an admissible data $(u_0,\pi,I,J)$ be given.
The construction uses three mutually independent tuples $\{x^i,1\le i\le N\}$,
$\{x^i,i\ge N+1\}$ and $\{\tilde B^i_t, t\in{\mathbb R}_+,i\in{\mathbb N}\}$,
defined on a probability space $(\mathnormal{\Omega},{\cal F},{\mathbb P})$.
Here, $x^i$, $1\le i\le N$ are IID real-valued RVs distributed
according to the probability measure $u_0(x)dx$,
representing the initial positions of those particles that are in the system at time $0$.
Also, $x^i$, $i\ge N$ are IID real-valued RVs distributed according to $\pi$,
representing the injection positions of particles injected after time $0$.
Finally, $\{\tilde B^i_t, t\ge0\}$, $i\in{\mathbb N}$ are mutually independent
standard Brownian motions (SBMs).

The $N$ initial particles are labeled by $\{1,\ldots,N\}$, and the remaining ones
by $\{N+1,N+2,\ldots\}$, in the order of injection.
The injections occur at deterministic times determined by the function $I$.
The number of particles introduced (present initially or injected later)
by time $t$ is given by $N^I_t:=N+[NI_t]$.
Thus
for $i\ge N+1$, the injection time of particle $i$
is at $\sigma^i=\sigma^i(N)=\inf\{t:N^I_t\ge i\}$.
Set $\sigma^i=0$ for $i\le N$.
For $i\in{\mathbb N}$, we refer to the process
$\{B^i_t,t\in[\sigma^i,\infty)\}$ as the {\it potential trajectory associated with particle $i\in{\mathbb N}$}, where
\begin{equation}\label{060}
B^i_t=x^i+\tilde B^i_{t-\sigma_i},\qquad t\ge\sigma^i.
\end{equation}

Particle removal times are dictated similarly by the function $J$.
The number of removals by $t$ is $N^J_t=[NJ_t]$,
and the $i$-th time of removal is $\eta^i=\eta^i(N)=\inf\{t:N^J_t\ge i\}$,
for $i\in{\mathbb N}$.

The number of particles in the system at time $t$ is
$N+[NI_t]-[NJ_t]\ge N\varepsilon_0-1$.
Thus \eqref{055} ensures that the number of particles
is always positive, provided $N\varepsilon_0-1\ge1$, a condition
assumed throughout.

The collection of potential trajectories and removal times
are used to construct dynamics under which, at each time $\eta_i$,
the rightmost particle is removed.
It is possible that an injection and a removal occur simultaneously;
in this case the construction obeys the rule `inject and then remove',
and therefore it is possible for an injected particle to be removed immediately.

The details are as follows.
For a finite nonempty subset $C$ of ${\mathbb N}$ and $g:C\to{\mathbb R}$,
denote $\argmax \{g(i):i\in C\}=\min\{i\in C:g_i=\max_C g\}$.
Let $[a,b)=\emptyset$ if $b\le a$ and $[a,b]=\emptyset$ if $b<a$.
A process $A_t$, representing the population of particles
alive at time $t$, taking values in the set of finite nonempty subsets of ${\mathbb N}$,
is constructed recursively.
On $[0,\eta^1)$, set $A_t=\{i:i\le N^I_t\}$.
For $j\ge1$, $A_t$ is defined on $[\eta_j,\eta_{j+1})$ based
on $A|_{[0,\eta_j)}$. Let
\[
C_j=A_{\eta_j-}\cup\{k:\sigma_k=\eta_j\}.
\]
Thus $C_j$ consists of those particles that are alive at $\eta_j-$ and, possibly one more
particle, namely the one injected exactly at $\eta_j$, if there is one. Next,
the rightmost particle is removed, by setting
\[
A_{\eta_j}=C_j\setminus\{\argmax\{B^i_{\eta_j}:i\in C_j\}\}.
\]
Finally, for $t\in(\eta_j,\eta_{j+1})$, new injections are accounted for by letting
$A_t=A_{\eta_j}\cup\{i: N^I_{\eta_j}<i\le N^I_t\}$.
This defines $A_t$ for all $t\in[0,\infty)$.

The removal time of particle $i\in{\mathbb N}$, denoted $\tau_i$,
can now be deduced from the construction of $A_t$.
If $i\in{\mathbb N}$ and $i\in\cup_tA_t$, let $\tau_i=\sup\{t:i\in A_t\}$.
Otherwise, $i$ must be a particle injected and removed at the same time,
in which case let $\tau_i=\sigma_i$.
The {\it trajectory associated with particle $i$}
is given by $\{B^i_t:t\in[\sigma_i,\tau_i)\}$.
By construction, $\tau_i$ is a stopping time on the filtration $F_t=\sigma\{B^i_s, s\in[\sigma_i,t],i\in{\mathbb N}\}$.



The configuration process, with sample paths in $D({\mathbb R}_+,{\cal M}_+({\mathbb R}))$ is defined as
\[
\xi^N_t(dx)=
\sum_{i=1}^{N^I_t}\boldsymbol{\delta}_{B^i_t}(dx)1_{\{t<\tau^i\}}
=\sum_{i\in{\mathbb N}}\boldsymbol{\delta}_{B^i_t}(dx)1_{\{\sigma^i\le t<\tau^i\}}.
\]
The removal positions and times are recorded by an ${\cal M}_{\rm loc}({\mathbb R}_+\times{\mathbb R})$-valued
RV,
\[
\beta^N(dx,dt)=\sum_{i\in{\mathbb N}:\,\tau^i<\infty}\boldsymbol{\delta}_{(B^i_{\tau^i},\tau^i)}(dx,dt).
\]
The injection positions and times are encoded in the (deterministic)
member of ${\cal M}_{\rm loc}({\mathbb R}_+\times{\mathbb R})$,
\[
\alpha^N(dx,dt)=\sum_{i\ge N+1}\boldsymbol{\delta}_{(x_i,\sigma_i)}(dx,dt).
\]
It is also useful to consider two extended configuration process.
One keeps track of the removed particles, viewed at their removal positions,
and another merely ignores the removal mechanism. That is,
\begin{equation}\label{056}
\gamma^N_t=\xi^N_t+\int_{[0,t]}\beta^N(\cdot,ds)=\sum_{i\le N^I_t}\boldsymbol{\delta}_{B^i_{t\wedge\tau_i}},
\end{equation}
\begin{equation}\label{058+}
\zeta^N_t=\sum_{i=1}^{N^I_t}\boldsymbol{\delta}_{B^i_t}.
\end{equation}
Denote normalized versions by $\bar\xi^N=N^{-1}\xi^N$, $\bar\beta^N=N^{-1}\beta^N$, etc.

Because the removal occurs at the rightmost particle,
the construction satisfies, for every $i\in{\mathbb N}$ with $\tau_i<\infty$ and every $r\in{\mathbb R}$,
\[
B_{\tau_i}< r \quad \text{ implies } \quad \xi_{\tau_i}[r,\infty)=0.
\]
As a result,
\begin{equation}\label{058}
\int_{(-\infty,r)\times{\mathbb R}_+}\langle 1_{[r,\infty)},\bar\xi^N_t\rangle\bar\beta^N(dx,dt)=0,
\qquad
r\in{\mathbb R}.
\end{equation}

View $\bar\xi^N$ and $\bar\beta^N$ as random elements taking values
in $D({\mathbb R}_+,{\cal M}({\mathbb R}))$ and ${\cal M}_{\rm loc}({\mathbb R}\times{\mathbb R}_+)$, respectively,
and endow the product space with the product topology.
\begin{theorem}\label{th0}
There exists a pair
$(\xi,\beta)\in C({\mathbb R}_+,{\cal M}_+({\mathbb R}))\times{\cal M}_{+,{\rm loc}}({\mathbb R}\times{\mathbb R}_+)$
such that, as $N\to\infty$, $(\bar\xi^N,\bar\beta^N)\to(\xi,\beta)$ in probability.
\end{theorem}

The proof of this result is given at the end of \S\ref{sec4},
and is based on uniqueness of solutions to
the heat equation with an ORA condition, described next.

\subsection{Heat equation with ORA}\label{sec22}

First, consider the heat equation with measure-valued RHS.
Namely, for
$\mu\in{\cal M}_{\rm loc}({\mathbb R}\times{\mathbb R}_+)$ and $u_0\in L_1({\mathbb R})\cap L_\infty({\mathbb R})$
consider the problem
\begin{equation}\label{017}
\begin{cases}
\partial_tu-\frac{1}{2}\partial_{xx}u=\mu & \text{in } {\mathbb R}\times{\mathbb R}_+,\\ \\
u(\cdot,0)=u_0 & \text{on } {\mathbb R}.
\end{cases}
\end{equation}
Let $q\in(1,\infty)$. A (weak) $L_q$-solution of \eqref{017} is a function
$u\in L_{1,\rm loc}({\mathbb R}_+,L_q)$ satisfying
\[
-\int_0^\infty(\partial_t\varphi+\frac{1}{2}\partial_{xx}\varphi,u)dt
=(\varphi(\cdot,0),u_0)+\int_{{\mathbb R}\times{\mathbb R}_+}\varphi\,d\mu
\]
for all $\varphi\in C_c^\infty({\mathbb R}\times{\mathbb R}_+)$.
Such problems were analyzed in far greater generality in \cite{ama01}.
In particular, by \cite[Theorem 1 and Remark 1(b)]{ama01}, for $1<q<\infty$,
$0\le\sigma<\frac{q+1}{q}$ and $p\ge1$ such that $\frac{2}{p}+\frac{1}{q}>1+\sigma$,
this problem possesses a unique $L_q$-solution $u$.
Moreover, $u\in L_{p,\rm loc}({\mathbb R}_+,W^\sigma_q)$,
and is independent of $q\in(1,\infty)$.
Furthermore, $u\in L_{p,\rm loc}({\mathbb R}_+,C_0({\mathbb R})\cap C^\rho({\mathbb R}))$,
for $p\in[1,2)$ and $\rho\in[0,1)$.

We base on this result the following problem formulation.
Let an admissible data $(u_0,\pi,I,J)$ be given.
Denote by $\alpha\in{\cal M}_{+,{\rm loc}}({\mathbb R}\times{\mathbb R}_+)$
the measure $\alpha(dx,dt)=\pi(dx)dI_t$.
Recall our notation $v[y,z)=\int_y^zv\,dx$ when $v\in L_1({\mathbb R},{\mathbb R})$.
Consider the problem
\begin{equation}\label{018}
\begin{cases}
{\it (i)}\hspace{3.5em}
\partial_tu-\frac{1}{2}\partial_{xx}u=\alpha-\beta & \text{in } {\mathbb R}\times{\mathbb R}_+,\\ \\
{\it (ii)}\hspace{3.3em}
u(\cdot,0)=u_0 & \text{on } {\mathbb R}, \\ \\
{\it (iii)}\hspace{2.8em}
\displaystyle\int_{{\mathbb R}_+}u([r,\infty),t)\beta((-\infty,r)\times dt)=0 & \text{for all } r\in{\mathbb R},\\ \\
{\it (iv)}\hspace{3.1em}
\beta({\mathbb R}\times[0,t])=J_t & \text{for all } t\in{\mathbb R}_+.
\end{cases}
\end{equation}
Define a (weak) solution to \eqref{018} to be a pair
$(u,\beta)\in L_{1,\rm loc}({\mathbb R}_+,L_1)
\cap L_{\infty,\rm loc}({\mathbb R}_+,L_\infty)
\times{\cal M}_{+,{\rm loc}}({\mathbb R}\times{\mathbb R}_+)$
such that $u$ is a.e.\ nonnegative,
conditions \eqref{018}(iii) and \eqref{018}(iv) hold,
and moreover for any test function $\varphi\in C_c^\infty({\mathbb R}\times{\mathbb R}_+)$,
\begin{equation}\label{054}
-\int_0^\infty(\partial_t\varphi+\frac{1}{2}\partial_{xx}\varphi,u)dt
=(\varphi(\cdot,0),u_0)
+\int_{{\mathbb R}\times{\mathbb R}_+}\varphi\, d\alpha - \int_{{\mathbb R}\times{\mathbb R}_+}\varphi\, d\beta.
\end{equation}

\begin{theorem}\label{th1}
Assume there exists a solution $(u,\beta)$ to \eqref{018}.
Then it is unique.
\end{theorem}
This result is proved in \S\ref{sec3}.

\subsection{Main result}\label{sec23}

The main result regarding this model summarizes Theorems \ref{th0}
and \ref{th1}, and relates the limit $(\xi,\beta)$ from the former
to the solution $(u,\beta)$ from the latter.

\begin{theorem}\label{th2}
There exists a unique solution $(u,\beta)$ to \eqref{018}.
Moreover, there exists a version of $u$ such that letting
$\xi_t$ be defined by $\xi_t(dx)=u(x,t)dx$ for every $t\in{\mathbb R}_+$, one has
$(\xi,\beta)\in C({\mathbb R}_+,{\cal M}_+({\mathbb R}))\times{\cal M}_{+,{\rm loc}}({\mathbb R}\times{\mathbb R}_+)$
and $(\bar\xi^N,\bar\beta^N)\to(\xi,\beta)$ in probability, as $N\to\infty$.
\end{theorem}

See Remark \ref{rem3} for further properties of the solution
following from \cite{ama01}. The proof of Theorem \ref{th2}
is based on \S\ref{sec3}--\ref{sec4} and is given at the end of \S\ref{sec4}.

Following is comparison principle for solutions of \eqref{018}.
It is largely a corollary of the main result.
To state it we need the notion of a mass transport inequality, used extensively throughout.

On $L_1({\mathbb R},{\mathbb R})$, the partial order $u\preccurlyeq v$ is defined
as $u[r,\infty)\le v[r,\infty)$ for all $r\in{\mathbb R}$.
The same notation is used for an analogous
partial order on the space of finite Borel measures
on ${\mathbb R}$, that is, $\nu\preccurlyeq\mu$ is defined as $\nu[r,\infty)\le\mu[r,\infty)$ for all $r\in{\mathbb R}$.

\begin{theorem}\label{th5}
Let each of the tuples $(u_0,\pi,I,J)$ and $(\tilde u_0,\tilde\pi,\tilde I,\tilde J)$
be admissible data and assume $\tilde u_0\preccurlyeq u_0$, $\tilde\pi\preccurlyeq\pi$,
$I-\tilde I\in C^\uparrow({\mathbb R}_+,{\mathbb R}_+)$, $\tilde J-J\in C^\uparrow({\mathbb R}_+,{\mathbb R}_+)$.
Then the corresponding solutions $(u,\beta)$, $(\tilde u,\tilde\beta)$ of \eqref{018}
satisfy $\tilde u(\cdot,t)\preccurlyeq u(\cdot,t)$, $t\in{\mathbb R}_+$.
\end{theorem}

The proof is provided in \S\ref{sec42} and uses, besides Theorem \ref{th2},
a coupling argument which shows a similar property for the configuration process.

\begin{remark}\label{rem4}
The condition \eqref{018}(iii) is closely related to the Skorohod map
$\mathnormal{\Phi}:D({\mathbb R}_+,{\mathbb R})\to D({\mathbb R}_+,{\mathbb R})$ given by
$\mathnormal{\Phi}(\varphi)(t)=-\inf_{s\in[0,t]}(\varphi(s)\w0)$.
It is shown that the latter can be used as an alternative
to the former in the problem formulation. Further, it is shown
that the component $\beta$ of the solution can be seen
as the fixed point of a transformation defined using $\mathnormal{\Phi}$.

Let $(u,\beta)$ be a solution to \eqref{018}.
Denote
\begin{equation}\label{090}
\widehat u_t(r)=u([r,\infty),t),
\quad
\widehat\beta_t(r)=\beta((-\infty,r)\times[0,t]),
\quad
\widehat\gamma_t(r)=u([r,\infty),t)+\beta([r,\infty)\times[0,t])-J_t.
\end{equation}
Then by \eqref{018}(iv),
\begin{align*}
\widehat u_t(r)
&=\widehat u_t(r)+\beta((-\infty,r)\times[0,t])+\beta([r,\infty)\times[0,t])-J_t
\\
&=\widehat\gamma_t(r)+\widehat\beta_t(r).
\end{align*}
One also has $\widehat u_t(r)\ge0$ and, by \eqref{018}(iii), for every $r$,
\[
\int_{{\mathbb R}_+}\widehat u_t(r)d\widehat\beta_t(r)=0.
\]
Skorohod's lemma then states that $\widehat u$ and $\widehat\beta$
can be recovered from $\widehat\gamma$ via
\begin{equation}\label{091}
\widehat\beta_t(r)=-\inf_{s\in[0,t]}(\widehat\gamma_s(r)\w0),
\qquad
\widehat u_t(r)=\widehat\gamma_t(r)+\widehat\beta_t(r).
\end{equation}
From this one may draw two conclusions.
First, an alternative way to define a solution $(u,\beta)$
is to replace \eqref{018}(iii)--(iv) by the relation \eqref{091},
where again $\widehat\beta$ and $\widehat\gamma$ are given by \eqref{090}.

Second, let $\mathnormal{\Psi}$ denote the solution map for the problem \eqref{017},
so that $u=\mathnormal{\Phi}(\mu;u_0)$.
Then a solution $(u,\beta)$ of \eqref{018} satisfies $u=\mathnormal{\Phi}(\alpha-\beta;u_0)$.
As a result, the third part of \eqref{090}
gives $\widehat\gamma$ as the image of $(\mathnormal{\Phi}(\alpha-\beta),\beta,J)$ under a map.
Considering $(u_0,\alpha,J)$ as fixed, let this relation be written as
$\widehat\gamma=\widetilde\mathnormal{\Phi}(\beta)$ for a suitable $\widetilde\mathnormal{\Phi}$.
Next, let the relation expressed in the first part of \eqref{091}
be written as $\widehat\beta=\mathnormal{\Psi}(\widehat\gamma)$.
Since $\widehat\beta$ determines $\beta$, one further has
$\beta=\widetilde\mathnormal{\Psi}(\widehat\gamma)$.
Thus $\beta$ is a fixed point
of $\widetilde\mathnormal{\Psi}\circ\widetilde\mathnormal{\Phi}$.
Whether existence and uniqueness of a fixed point,
thus of solutions to \eqref{018},
can be deduced from abstract fixed point theorems is not obvious to us.
\end{remark}






\section{Removal at boundary: uniqueness via barriers}\label{sec3}
\beginsec

This section proves uniqueness of solutions to \eqref{018}
by constructing barriers, which are shown to constitute upper
and lower bounds to any solution, in the sense of mass transport inequalities.
In \S\ref{sec30} a representation of $u$ in terms of $\alpha$ and $\beta$ is given.
In \S\ref{sec30+}, some mass transport inequalities are provided,
used in the sequel.
The lower and upper barriers are constructed in\S\ref{sec31} and \S\ref{sec32}.
Finally, the proof of Theorem \ref{th1} appears in \S\ref{sec33}.
Throughout this section, an admissible data $(u_0,\pi,I,J)$ is fixed.

\subsection{Mild solutions}\label{sec30}

Let the heat kernel be
denoted by $G_t(x,y)=(2\pi t)^{-1/2}e^{-(x-y)^2/2t}$.
Denote by $G_t$ the operator
\[
G_tu(y)=\int_{\mathbb R} G_t(x,y)u(x)dx,
\]
for $u\in L_1({\mathbb R},{\mathbb R}_+)$.
For $\gamma\in{\cal M}_{+,{\rm loc}}({\mathbb R}\times{\mathbb R}_+)$, denote
\[
G*\gamma(y,t)=\int_{{\mathbb R}\times[0,t]}G_{t-s}(x,y)\gamma(dx,ds),
\]
\[
G*\gamma(y,t;\tau)=\int_{{\mathbb R}\times[\tau,t]}G_{t-s}(x,y)\gamma(dx,ds),
\]
for $t\ge\tau$.

\begin{lemma}\label{lem1}
Assume that a solution $(v,\beta)$ to \eqref{018} exists.
Then $v$ has a version $u$ given by
\begin{equation}\label{012-}
u(y,t)=G_tu_0(y)+G*\alpha(y,t)-G*\beta(y,t).
\end{equation}
Moreover, for $0\le \tau<t$,
\begin{equation}\label{012}
u(y,t)=G_{t-\tau}u(\cdot,\tau)(y)+G*\alpha(y,t;\tau)-G*\beta(y,t;\tau).
\end{equation}
Furthermore, $u\in L_{\infty,{\rm loc}}({\mathbb R}_+,L_1)\cap L_{\infty,{\rm loc}}({\mathbb R}_+,L_\infty)$.
\end{lemma}

\begin{remark}\label{rem1}
If $(v,\beta)$ is a solution to \eqref{018}
and $u=v$ a.e.\ then $(u,\beta)$ is also a solution; specifically,
Condition \eqref{018}(iii) is inherited from the former to the latter.
Indeed, for a given $r$, $v([r,\infty),t)$ and $u([r,\infty),t)$ may differ for $t$
in a null set w.r.t.\ the Lebesgue measure $dt$.
However, this is a null set also w.r.t.\ the measure
$\beta((-\infty,r)\times dt)\ll\beta({\mathbb R}\times dt)=dJ_t\ll dt$.
The latter is a consequence of our assumption $J\in AC^\uparrow({\mathbb R}_+,{\mathbb R}_+)$.
\end{remark}
\begin{remark}\label{rem2}
Throughout what follows,
by a solution to \eqref{018} we will always mean the version given
by \eqref{012-}. In view of the above lemma there is no loss of generality in doing so.
\end{remark}

\proof
Let $(v,\beta)$ be a solution, and define $u$ by \eqref{012-}.
First we argue that the integrals in
\eqref{012-} are finite. By monotone convergence,
it suffices that $\int_{{\mathbb R}\times[0,t-\varepsilon]}G_{t-s}(x,y)\alpha(dx,ds)$
remains bounded as $\varepsilon\downarrow0$.
To show this, bound the above integral by
\begin{align}
\int_{[0,t-\varepsilon]}(t-s)^{-1/2}dI_s
&=
I_{t-\varepsilon}\varepsilon^{-1/2}-\frac{1}{2}\int_0^{t-\varepsilon}I_s(t-s)^{-3/2}ds
\notag
\\
&=(I_{t-\varepsilon}-I_t)\varepsilon^{-1/2}
+\frac{1}{2}\int_0^{t-\varepsilon}(I_t-I_s)(t-s)^{-3/2}ds+I_tt^{-1/2}
\notag
\\
&\le c\int_0^t(t-s)^{\rho_0-\frac{3}{2}}ds+I_tt^{-1/2}<\infty,
\label{023}
\end{align}
where $\rho_0>1/2$ is used.
A similar argument applies for the integral in \eqref{012-}
by appealing to \eqref{018}(iv) and the H\"older continuity of $J$.

Next, given a test function $\varphi$, identity \eqref{054}
is satisfied by $(u,\beta)$ as follows by direct calculation
that uses the fact that $G_t(x,y)$ solves the heat equation
and again the finiteness of $\int_{[0,t]}(t-s)^{-1/2}dI_s$.
We do not include this calculation because it is standard.

We now show that $u$ is a member of
$L_{\infty,{\rm loc}}({\mathbb R}_+,L_1)$ (hence $L_{1,{\rm loc}}({\mathbb R}_+,L_1)$)
and of $L_{\infty,{\rm loc}}({\mathbb R}_+,L_\infty)$. We have
$|u(y,t)|\le G_tu_0(y)+G*\alpha(y,t)+G*\beta(y,t)$.
Hence
\[
\|u(\cdot,t)\|_1\le\|u_0\|_1+\|\pi\|_1I_t+J_t=1+I_t+J_t,
\]
\[
\|u(\cdot,t)\|_\infty \le\|u_0\|_\infty+\int_{[0,t]}(t-s)^{-1/2}(dI_s+dJ_s).
\]
The above integral is locally bounded as the calculation \eqref{023} shows,
upon using one more time the fact $I,J\in C^{\rho_0}$.
This shows the claim.

We can now appeal to the uniqueness result due to Amann,
of weak $L_q$-solutions to \eqref{017}, for $q\in(1,\infty)$,
where we take $\mu=\alpha-\beta$
(thus $\beta$ in this case is taken as data).
Namely, as already mentioned, by \cite[Theorem 1]{ama01},
this problem possesses a unique weak $L_q$-solution.
Both $u$ and $v$ are in
$L_{1,\rm loc}({\mathbb R}_+,L_1)\cap L_{\infty,\rm loc}({\mathbb R}_+,L_\infty)$,
and consequently in $L_{1,\rm loc}({\mathbb R}_+,L_q)$ for all $q\in(1,\infty)$.
Since \eqref{054} is satisfied by both $u$ and $v$, they are
both weak $L_q$-solutions to \eqref{017} and as a result, $u=v$ a.e.

Finally, \eqref{012} follows from \eqref{012-} by a direct calculation
that we omit, upon using the identity
$\int G_{t-\tau}(x,y)G_\tau(y,z)dy=G_t(x,z)$.
\qed


\subsection{Mass transport inequalities}\label{sec30+}

We borrow from \cite{de-masi-book} the use of mass transport inequalities.

For $\delta>0$, $K^\delta$ acts on $H_\delta=\{u\in L_1({\mathbb R},{\mathbb R}_+):
\|u\|_1>\delta\}$
by cutting mass of size $\delta>0$ that is rightmost.
That is, for $u\in H_\delta$, let
\[
R_\delta(u)=\sup\{x\in{\mathbb R}:u[x,\infty) \ge \delta\}
\]
and
\[
K^\delta u(x)=u(x)1_{(-\infty,R_\delta(u)]}(x).
\]
When $\delta=0$ set $K^\delta={\tt id}$, the identity map.
Also, denote $\widehat K^\delta={\tt id}-K^\delta$.
We also use an operator
that removes the mass of size $\delta$ lying between $R_{\mathnormal{\Delta}+\delta}$ and $R_{\mathnormal{\Delta}}$.
More precisely, given $\mathnormal{\Delta}>0$ and $\delta\ge0$, let ${\bar\delta}$ denote
the pair $(\mathnormal{\Delta},\delta)$.
Then the operator $K^{\bar\delta}$ acts on $H_{\mathnormal{\Delta}+\delta}$
as
\[
K^{{\bar\delta}}u(x)=K^{\mathnormal{\Delta},\delta}u(x)=u(x)1_{\{(-\infty, R_{\mathnormal{\Delta}+\delta}(u)]\cup(R_\mathnormal{\Delta}(u),\infty)\}}(x).
\]
Also set $\widehat K^{\bar\delta}={\tt id}-K^{\bar\delta}$.

\begin{lemma}\label{lem3}
Let $\delta\ge0$ and assume $u,v\in H_\delta$.

i. If $u\preccurlyeq v$ then $K^\delta u\preccurlyeq K^\delta v$
and $G_\delta u\preccurlyeq G_\delta v$.

ii. If also $w\in L_1({\mathbb R},{\mathbb R}_+)$ is such that $\|w\|_1\le\delta$
and $u-w\in L_1({\mathbb R},{\mathbb R}_+)$ then $K^\delta u\preccurlyeq u-w$.

Next let $\mathnormal{\Delta}>0$ and assume $u,v\in H_{\mathnormal{\Delta}+\delta}$.

iii. If $u\preccurlyeq v$ then $K^{\mathnormal{\Delta},\delta}u\preccurlyeq K^{\mathnormal{\Delta},\delta}v$.

iv. If $0<\widehat\mathnormal{\Delta}\le\mathnormal{\Delta}$ then $K^{\widehat\mathnormal{\Delta},\delta}u\preccurlyeq K^{\mathnormal{\Delta},\delta}u$.
\end{lemma}
\proof
i.
These are well known properties. For monotonicity
of $K^\delta$ see \cite[Lemma 6.6]{de-masi-book}.
As for $G_\delta$, the same lemma in \cite{de-masi-book}
shows monotonicity of the Neumann Laplacian evolution operator on ${\mathbb R}_+$,
but the same proof holds for the one on ${\mathbb R}$.

ii.
We have $K^\delta=u-\tilde w$ where $\tilde w=u1_{(c,\infty)}$,
$\int_c^\infty u=\delta$. Hence it suffices to show that
$\int_r^\infty w\le\int_r^\infty\tilde w$ for all $r$.
First consider $r\ge c$. In this case, noting that $w\le u$ pointwise,
we have $\int_r^\infty w\le\int_c^\infty u$, but
$\int_r^\infty\tilde w=\int_r^\infty u$.

Next, if $r<c$ we have $\int_r^\infty w\le\delta$ but $\int_r^\infty\tilde w=\delta$.

iii. We have $u(r,\infty)\le v(r,\infty)$ for all $r$. We must show that
$\widehat u(r,\infty)\le\widehat v(r,\infty)$ for all $r$, where
\[
\widehat u=u1_{(-\infty,a)\cup(b,\infty)},
\text{ where $u(a,b)=\delta$, $u(b,\infty)=\mathnormal{\Delta}$},
\]
\[
\widehat v=v1_{(-\infty,\bar a)\cup(\bar b,\infty)},
\text{ where $v(\bar a,\bar b)=\delta$, $v(\bar b,\infty)=\mathnormal{\Delta}$}.
\]
To this end, note first that $b\le\bar b$ and $a\le\bar a$. We split into cases.
\\
For $r\ge\bar b$ it is clear that $\widehat u(r,\infty)\le\widehat v(r,\infty)$.
\\
For $r\in[\bar a,\bar b)$, $\widehat v(r,\infty)=\mathnormal{\Delta}$ but $\widehat u(r,\infty)\le\widehat u(a,\infty)=\mathnormal{\Delta}$.
\\
For $r\in[a,\bar a)$, $\widehat u(r,\infty)\le\widehat u(a,\infty)=\mathnormal{\Delta}$,
but $\widehat v(r,\infty)\ge\widehat v(\bar a,\infty)=\mathnormal{\Delta}$.
\\
For $r<a$, $\widehat u(r,\infty)=u(r,\infty)-\delta$ and $\widehat v(r,\infty)=v(r,\infty)-\delta$.

iv.
Note that
\[
K^{\widehat\mathnormal{\Delta},\delta}u= K^\delta v+z,\qquad
K^{\mathnormal{\Delta},\delta}u=K^{\mathnormal{\Delta}-\widehat\mathnormal{\Delta},\delta}v+z,
\]
where
\[
v=K^{\widehat\mathnormal{\Delta}}u,\qquad z=\widehat K^{\widehat\mathnormal{\Delta}}u.
\]
Hence it suffices to prove $K^\delta v\preccurlyeq K^{\mathnormal{\Delta}-\widehat\mathnormal{\Delta},\delta}v$.
To this end, note that $K^{\mathnormal{\Delta}-\widehat\mathnormal{\Delta},\delta}v=v-w\in L_1({\mathbb R},{\mathbb R}_+)$,
and $\|w\|_1=\delta$. Therefore we can use part (ii) of the lemma,
by which $K^\delta v\preccurlyeq v-w$.
This completes the proof.
\qed


\subsection{Lower barriers}\label{sec31}

For $u\in L_1({\mathbb R},{\mathbb R}_+)$ and $0\le\tau<t$ let
\[
P^{\tau,t}u=u+G*\alpha(\cdot,t;\tau).
\]
Between time $(n-1)\delta$ and $n\delta$, the amount of mass removed is given by
$j_n(\delta):=J_{n\delta}-J_{(n-1)\delta}$. Denote $K_{n,\delta}=K^{j_n(\delta)}$.
The injection of mass during the same time interval is according to
$P_{n,\delta}:=P^{(n-1)\delta,n\delta}$.
The lower barriers are defined for each $\delta>0$ and $n\in{\mathbb Z}_+$
as $u^{(\delta,-)}_0=u_0$ and
\[
u^{(\delta,-)}_{n\delta}= K_{n,\delta}P_{n,\delta}G_\delta u^{(\delta,-)}_{(n-1)\delta},\qquad n\in{\mathbb N}.
\]
Note that for the RHS to be well defined one must have
\begin{equation}\label{061}
P_{n,\delta}G_\delta u^{(\delta,-)}_{(n-1)\delta}\in H_{j_n(\delta)}.
\end{equation}
Our goal here is to show that the
lower barriers form lower bounds on solutions to \eqref{018}.

\begin{proposition}\label{prop1}
Fix $\delta>0$. Then \eqref{061} holds for all $n\in{\mathbb N}$, and consequently
the lower barriers are well defined. Moreover,
let $(u,\beta)$ be a solution to \eqref{018}.
Then for $n\in{\mathbb Z}_+$,
\[
u^{(\delta,-)}_{n\delta}\preccurlyeq u_{n\delta}.
\]
\end{proposition}

\proof
We have $\|u_0\|_1=1$. Therefore for $n=1$, the $L_1$ norm of the LHS
of \eqref{061} is $1+I_\delta\ge J_\delta$, where \eqref{055} is used. This shows
that \eqref{061} holds for $n=1$.
Next, assume that \eqref{061} holds for all $n\in\{1,\ldots,m\}$.
Then $u_1,\ldots,u_{m}$ are well defined, and
\begin{equation}\label{063}
\|u_{m\delta}^{(\delta,-)}\|_1=\|u_{(m-1)\delta}^{(\delta,-)}\|_1+(I_{m\delta}-I_{(m-1)\delta})
-(J_{m\delta}-J_{(m-1)\delta})
\end{equation}
whence $\|u_{m\delta}^{(\delta,-)}\|_1=1+I_{m\delta}-J_{m\delta}$.
As a result, the $L_1$ norm of the LHS of \eqref{061} with $n=m+1$ is
\begin{equation}\label{064}
1+I_{(m+1)\delta}-J_{m\delta}>j_{m+1}(\delta)
\end{equation}
by \eqref{055}.
This shows that \eqref{061} holds for $n=m+1$, which completes
the argument for the first assertion.

Next, to prove the partial order, again arguing by induction,
assume $f\preccurlyeq g$ where $f=u^{(\delta,-)}_{(n-1)\delta}$ and $g=u_{(n-1)\delta}$.
Write $K$, $P$ and $G$ for $K_{n,\delta}$, $P_{n,\delta}$ and $G_\delta$, resp.
We have $u^{(\delta,-)}_{n\delta}=KPGf$.
Moreover, by Lemma \ref{lem1},
$u_{n\delta}=PGg-h$, where
\[
h(y)=\int_{{\mathbb R}\times[(n-1)\delta,n\delta]}G_{n\delta-s}(x,y)\beta(dx,ds).
\]
Hence the proof will be complete provided we show that
$KPGf\preccurlyeq PGg-h$.

By Lemma \ref{lem3}(i), both operators $G$ and $K$
preserve $\preccurlyeq$. It is trivial that this is also true for $P$.
Denote $w=PGg$. Suppose one shows $Kw\preccurlyeq w-h$. Then
\[
KPGf\preccurlyeq KPGg =Kw\preccurlyeq w-h,
\]
and the proof would be complete.

It thus suffices to show $K w \preccurlyeq w-h$.
Denote $r_0=R_{j_n(\delta)}(w)$. Then $\int_{r_0}^\infty w(x)dx=j_n(\delta)$
and $Kw=w-q$ where $q(y)=w(y)1_{\{y>r_0\}}$.
Next, the function $w-h$ is nonnegative
(as required by the definition of a solution)
and $\int h(y)dy=j_n(\delta)$.
Hence for $r\ge r_0$,
\begin{align*}
w(r,\infty)-h(r,\infty)&=w(r,\infty)-w(r,\infty)\\
&=0\\
&\le w(r,\infty)-h(r,\infty).
\end{align*}
For $r<r_0$,
\begin{align*}
w(r,\infty)-q(r,\infty)&=w(r,\infty)-j_n(\delta)\\
&\le w(r,\infty)-h(r,\infty).
\end{align*}
This shows $w-q\preccurlyeq w-h$ and completes the proof.
\qed


\subsection{Upper barriers}\label{sec32}

The upper barriers are defined for ${\bar\delta}=(\mathnormal{\Delta},\delta)\in(0,\infty)^2$
and $n\in{\mathbb Z}_+$ as follows.
Let
\[
K_{n,{\bar\delta}}=K^{\mathnormal{\Delta},j_n(\delta)}.
\]
Set $u^{({\bar\delta},+)}_0=u_0$, and for $n\in{\mathbb N}$,
\begin{align}
\notag
u^{({\bar\delta},+)}_{n\delta}
&=K_{n,{\bar\delta}}P_{n,\delta}G_\delta u^{({\bar\delta},+)}_{(n-1)\delta}+e_{n,{\bar\delta}}
\\
\label{016}
e_{n,{\bar\delta}}(y)&=e^{-\mathnormal{\Delta}^4/2\delta}j_n(\delta)1_{[\sigma_n,\sigma_n+1]}(y),
\qquad y\in{\mathbb R}_+,
\\
\notag
\sigma_n&= \sigma_n({\bar\delta})=R_\mathnormal{\Delta}(P_{n,\delta}G_\delta u^{({\bar\delta},+)}_{(n-1)\delta}).
\end{align}
Once again, for the definition to be valid, one must assure that for all $n\in{\mathbb N}$,
\begin{equation}\label{062}
P_{n,\delta}G_\delta u^{({\bar\delta},+)}_{(n-1)\delta}
\in H_{\mathnormal{\Delta}+j_n(\delta)}.
\end{equation}

\begin{proposition}\label{prop2}
For $\mathnormal{\Delta}\in(0,\varepsilon_0)$ and $\delta>0$, \eqref{062} holds for all $n$
and consequently the upper barriers are well defined.
Moreover, let $(u,\beta)$ be a solution to \eqref{018}.
Then given $T\in(0,\infty)$
there exists $\mathnormal{\Delta}_0\in(0,\varepsilon_0)$ such that for every $\mathnormal{\Delta}\in(0,\mathnormal{\Delta}_0)$
there exists $\delta_0>0$ such that for $\delta\in(0,\delta_0)$
and $n\in{\mathbb Z}_+$ one has
\[
u_{n\delta}\preccurlyeq u^{({\bar\delta},+)}_{n\delta},
\]
provided $n\delta\le T$.
\end{proposition}

Given $\gamma\in{\cal M}_{+,{\rm loc}}({\mathbb R}\times{\mathbb R}_+)$ and $[t_1,t_2]\subset{\mathbb R}_+$,
the infimum and the supremum of the support of the measure
$\tilde\gamma(\cdot)=\gamma(\cdot\times[t_1,t_2])$
are denoted by
$\rho_*(\gamma;[t_1,t_2])$ and, respectively, $\rho^*(\gamma;[t_1,t_2])$.

If we ignore for a moment the `error' terms $e_{n,{\bar\delta}}$,
the recursion consists of iterates of $K_{n,{\bar\delta}}P_{n,\delta}G_\delta$.
The cutting operator removes mass at a point
that leaves $\mathnormal{\Delta}$ units of mass to its right.
Of course, the PDE \eqref{018} also has a removal
mechanism, expressed by $\beta$ and constrained by
conditions \eqref{018}(iii)--(iv).
A central part of the argument is to compare the two removal mechanisms.
That is, if $\rho_*(\beta;[(n-1)\delta,n\delta])$, which expresses
the lowest reach of $\beta$ during the interval $[(n-1)\delta,n\delta]$,
can be bounded below then cutting a mass sufficiently to the left
will result in removal mostly to the left of that
occurring in the PDE model.
For this to be useful one needs to show that removal
at the point that leaves mass of size $\mathnormal{\Delta}$ to its right,
for arbitrarily small $\mathnormal{\Delta}>0$,
is indeed ``sufficiently to the left". This requires $\delta$ to be much smaller than
$\mathnormal{\Delta}$, and is made precise by including the aforementioned
error terms.

\begin{lemma}\label{lem2}
Given a solution $(u,\beta)$, $\delta>0$ and $n\in{\mathbb N}$, denote
$\rho_{n,\delta}=\rho_*(\beta;[(n-1)\delta,n\delta])$. Then
\[
\rho_{n,\delta}\ge b(n,\delta,u_{(n-1)\delta}):=R_{3j_n(\delta)}(u_{(n-1)\delta}),
\quad \text{provided\ \ $u_{(n-1)\delta}\in H_{3j_n(\delta)}$ and $j_n(\delta)>0$.}
\]
\end{lemma}

\proof
We consider only $n=1$; the proof is similar for all $n$.
Fix $\delta$ and a solution $(u,\beta)$.
By contradiction, assume $\rho_{1,\delta}<b=b(1,\delta,u_0)=R_{3J_\delta}(u_0)$.
(Note that $R_{3J_\delta}(u_0)$ is well defined and finite
by the assumption $u_0\in H_{3J_\delta}$).
Then there exists $x<b$ such that $\beta[(-\infty,x]\times[0,\delta])>0$, hence
$\beta((-\infty,b)\times[0,\delta])>0$.
Using \eqref{018}(iii), $\int_{[0,\delta]}u([b,\infty),t)\beta((-\infty,b)\times dt)=0$,
we deduce that
there exists $t\in[0,\delta]$ such that $u([b,\infty),t)=0$.
It is impossible that $t=0$ since $u_0[b,\infty)=3J_\delta>0$.
Fix such $t$.
We now appeal to identity \eqref{012-}.
Denoting the last term there by
$\eta(y)=\int_{{\mathbb R}\times[0,t]}G_{t-s}(x,y)\beta(dx,ds)$, clearly
\[
\int_b^\infty\eta(y)dy\le\int_{-\infty}^\infty\eta(y)dy=\beta({\mathbb R}\times[0,t])=J_t\le J_\delta.
\]
Since $u(\cdot,t)$ vanishes on $[b,\infty)$, we have for the first term in \eqref{012-},
\[
\int_b^\infty\int_{\mathbb R} G_t(x,y)u_0(x)dx\,dy\le\int_b^\infty\eta(y)dy\le J_\delta.
\]
Using $\int_x^\infty G_t(x,y)dy=1/2$,
\begin{align*}
J_\delta&\ge\int_{x\in[b,\infty)}\int_{y\in[b,\infty)}G_t(x,y)u_0(x)dy\,dx\\
&\ge\frac{1}{2}\int_b^\infty u_0(x)dx=\frac{3J_\delta}{2},
\end{align*}
a contradiction due to the assumption $J_\delta=j_1(\delta)>0$.
This proves the claim.
\qed

\vspace{\baselineskip}

\noindent{\bf Proof of Proposition \ref{prop2}.}
First, to prove that when $\mathnormal{\Delta}<\varepsilon_0$ one has \eqref{062} for all $n$
(and therefore the upper barriers are well defined)
we argue as in the proof of Proposition \ref{prop1}.
In place of equality in \eqref{063} we write an inequality $\ge$,
and in place of \eqref{064}, we write $1+I_{(m+1)\delta}-J_{m\delta}>\mathnormal{\Delta}+j_{m+1}(\delta)$,
that again follows from \eqref{055}. This shows the first claim.

We turn to the main assertion.
This result is stated for $n\delta\in[0,T]$. Fix $T$.
Assume that $\mathnormal{\Delta}<\varepsilon_0/2$.
It follows from Lemma \ref{lem1} and \eqref{055} that
$\|u(\cdot,t)\|_1\ge\varepsilon_0$. Since $J$ is continuous on $[0,T]$,
we may and will assume also that $\delta$ is so small that
$\mathnormal{\Delta}+3j_n(\delta)<\varepsilon_0$ whenever $n\delta\in[0,T]$. As a result, the bound
asserted in Lemma \ref{lem2} is valid provided merely
that $j_n(\delta)>0$.

It follows from Lemma \ref{lem1} that there exist constants $c_1$, $c_\infty$,
depending only on $T$ and $I|_{[0,T]},J|_{[0,T]}$ such that
any solution $(u,\beta)$ (corresponding to the given admissible data)
satisfies $\|u(\cdot,t)\|_1\le c_1$ and $\|u(\cdot,t)\|_\infty<c_\infty$, $t\in[0,T]$.
In what follows we consider only $\mathnormal{\Delta}$ so small that $2c_\infty\mathnormal{\Delta}^2<\mathnormal{\Delta}/6$
and, as already mentioned, $\mathnormal{\Delta}<\varepsilon_0/2$.

Arguing by induction, assume that $f\preccurlyeq g$
where $f=u_{(n-1)\delta}$ and $g=u^{({\bar\delta},+)}_{(n-1)\delta}$.
Write $K$, $P$ and $G$ for $K_{n,{\bar\delta}}$, $P_{n,\delta}$ and $G_\delta$,
respectively.
Then using Lemma \ref{lem1}, $u_{n\delta}=PGf-h$, where
\[
h(y)=\int_{{\mathbb R}\times[(n-1)\delta,n\delta]}G_{n\delta-s}(x,y)\beta(dx,ds),
\]
whereas $u^{({\bar\delta},+)}_{n\delta}=KPGg+e_{n,{\bar\delta}}$.
Denote $w=PGf$. If one shows
\begin{equation}\label{013}
w-h\preccurlyeq Kw+e_{n,{\bar\delta}}
\end{equation}
then $u_{n\delta}=w-h\preccurlyeq KPGf+e_{n,{\bar\delta}}
\preccurlyeq KPGg+e_{n,{\bar\delta}}
=u^{({\bar\delta},+)}_{n\delta}$, by the monotonicity
of $G$ and of $K^{\mathnormal{\Delta},\delta}$ (Lemma \ref{lem3}(i) and (iii)),
which completes the proof.
It remains to show \eqref{013}.

First, if $j_n(\delta)=0$ then both sides of \eqref{013} coincide with $w$,
because $h=0$, $e_{n,{\bar\delta}}=0$ and $K={\tt id}$.

Next consider $j_n(\delta)>0$.
Denote $j=j_n(\delta)$ and $b=R_{3j_n(\delta)}(f)=R_{3j}(f)$.
The assumptions of Lemma \ref{lem2} are thus satisfied, and the lemma asserts
that $\rho_{n,\delta}\ge b$.
Write $h=h_1+h_2$, where
\[
h_1(y)=h(y) 1_{\{y<b-\mathnormal{\Delta}^2\}},
\qquad
h_2(y)=h(y) 1_{\{y\ge b-\mathnormal{\Delta}^2\}}.
\]
Note that $\|h\|_1=j$. A straightforward calculation
based on
\begin{align*}
\int_{{\mathbb R}\times[(n-1)\delta,n\delta]}G_{n\delta-s}(x,y)\beta(dx,ds)
=\int_{[b,\infty)\times[(n-1)\delta,n\delta]}G_{n\delta-s}(x,y)\beta(dx,ds)
\end{align*}
(which follows from Lemma \ref{lem2}),
the identity $\beta({\mathbb R}_+\times[n\delta,t])=J_t-J_{(n-1)\delta}$, $t\in[(n-1)\delta,n\delta]$
and the estimate $G_t(0,[a,\infty))\le e^{-a^2/2t}$ $a>0$, $t<a^2$,
shows that the mass of $h_1$ is much smaller, specifically,
\begin{equation}\label{053}
\|h_1\|_1\le e^{-\mathnormal{\Delta}^4/2\delta}j,
\end{equation}
provided $\delta<\mathnormal{\Delta}^4$.
Let $q\in(0,1]$ be defined by
$\|h_2\|_1=qj$. Then $q\ge1-e^{-\mathnormal{\Delta}^4/2\delta}$.

Our next step toward showing \eqref{013} is to show
\begin{equation}\label{051}
w-h\preccurlyeq K^{\mathnormal{\Delta},qj}w.
\end{equation}
By definition, $h_2$ is supported to the right of
$b-\mathnormal{\Delta}^2$.
On the other hand, $K^{\mathnormal{\Delta},qj}w=w-\tilde h$, where $\|\tilde h\|_1=qj$
and $\tilde h$ is supported to the left of $R_\mathnormal{\Delta}(w)$.
Thus if we can show
\begin{equation}\label{020}
R_\mathnormal{\Delta}(w)\le R_{3j}(f)-\mathnormal{\Delta}^2,
\end{equation}
then using $\|\tilde h\|_1=\|h_2\|_1$ we would have $w-h_2\preccurlyeq K^{\mathnormal{\Delta},qj}w$,
hence
\[
w-h=w-h_1-h_2\preccurlyeq w-h_2\preccurlyeq K^{\mathnormal{\Delta},qj}w,
\]
giving \eqref{051}.

To show \eqref{020}, note first, by boundedness $\|f\|_\infty<c_\infty$,
that for all $\delta$ so small that $3j<\mathnormal{\Delta}/6$ and recalling $2c_\infty\mathnormal{\Delta}^2<\mathnormal{\Delta}/6$,
\begin{equation}
\label{052}
R_{\mathnormal{\Delta}/3}(f)\le R_{3j}(f)-2\mathnormal{\Delta}^2.
\end{equation}
Also, we argue that for all small $\delta$,
\begin{equation}\label{021}
R_{2\mathnormal{\Delta}/3}(Gf)\le \theta:=R_{\mathnormal{\Delta}/3}(f)+\mathnormal{\Delta}^2.
\end{equation}
To show this we must to show $(Gf)[\theta,\infty)\le2\mathnormal{\Delta}/3$.
Let $f=f_1+f_2=\widehat K^{\mathnormal{\Delta}/3}f+K^{\mathnormal{\Delta}/3}f$. Because $\|f_1\|_1=\mathnormal{\Delta}/3$,
we have $\|Gf_1\|_1=\mathnormal{\Delta}/3$, and
\[
(Gf)[\theta,\infty)=(Gf_1)[\theta,\infty)+(Gf_2)[\theta,\infty)
\le \mathnormal{\Delta}/3+(G A)[\theta,\infty)
\]
where $A=\|f_2\|_1\boldsymbol{\delta}_x$ is an atom of mass equal to that of $f_2$
at the point $x=R_{\mathnormal{\Delta}/3}(f)$.
Recalling that $\|f\|_1<c_1$, we have
$(GA)[\theta,\infty)\le c_1G_\delta(0,[\mathnormal{\Delta}^2,\infty))$,
which of course is smaller than
$\mathnormal{\Delta}/3$ if $\delta$ is small. This shows \eqref{021}.

For $\bar\pi:=G*\alpha(\cdot,n\delta;(n-1)\delta)$
we have $\|\bar\pi\|_1=I_{n\delta}-I_{(n-1)\delta}$.
Hence for all small $\delta$, $\|\bar\pi\|_1<\mathnormal{\Delta}/3$. As a result,
\[
R_\mathnormal{\Delta}(w)=R_\mathnormal{\Delta}(PGf)=R_\mathnormal{\Delta}(Gf+\bar\pi)
\le R_{2\mathnormal{\Delta}/3}(Gf).
\]
Combining this with \eqref{052} and \eqref{021} gives \eqref{020}.
Hence \eqref{051} follows.

Finally, note that
\[
K^{\mathnormal{\Delta},qj}w=K^{\mathnormal{\Delta},j}w+\widehat K^{\mathnormal{\Delta}+qj,(1-q)j}w.
\]
The last term is a nonnegative function supported to the left of $R_\mathnormal{\Delta}(w)$,
hence $\preccurlyeq (1-q)j1_{[R_\mathnormal{\Delta}(w),R_\mathnormal{\Delta}(w)+1]}$.
Now, $f\preccurlyeq g$ hence $w=PGf\preccurlyeq PGg$.
Hence $R_\mathnormal{\Delta}(w)\le R_\mathnormal{\Delta}(PGg)$, and
\[
\widehat K^{\mathnormal{\Delta}+qj,(1-q)j}w\preccurlyeq  (1-q)j1_{[R_\mathnormal{\Delta}(PGg),R_\mathnormal{\Delta}(PGg)+1]}.
\]
Combining this with \eqref{051},
\[
w-h\preccurlyeq K^{\mathnormal{\Delta},qj}w
\preccurlyeq K^{\mathnormal{\Delta},j}w+(1-q)j1_{[R_\mathnormal{\Delta}(PGg),R_\mathnormal{\Delta}(PGg)+1]}.
\]
Recalling that by \eqref{053}, $1-q\le e^{-\mathnormal{\Delta}^4/2\delta}$,
it follows that the last term is
$\preccurlyeq e_{n,{\bar\delta}}$. This proves \eqref{013} and completes the proof.
\qed

\subsection{Proof of uniqueness result}\label{sec33}

The last step is showing that the lower and upper barriers
become close upon taking $\delta\to0$ then $\mathnormal{\Delta}\to0$.

\begin{proposition}\label{prop3}
Given $T$, let $\mathnormal{\Delta}_0=\mathnormal{\Delta}_0(T)$ and $\delta_0=\delta_0(T,\mathnormal{\Delta}_0)$ be as in
Proposition \ref{prop2}. Then
for $\mathnormal{\Delta}\in(0,\mathnormal{\Delta}_0)$, $\delta\in(0,\delta_0)$ and $n\in{\mathbb Z}_+$, $n\delta\le T$,
one has
\[
u^{({\bar\delta},+)}_{n\delta}\preccurlyeq u^{(\delta,-)}_{n\delta}+\varepsilon_n,
\]
where $\varepsilon_n=\varepsilon_n(T,{\bar\delta})\in L_1({\mathbb R},{\mathbb R}_+)$ and
\[
\|\varepsilon_n\|_1\le\mathnormal{\Delta}+e^{-\mathnormal{\Delta}^4/2\delta}\sum_{i=1}^nj_i(\delta)=\mathnormal{\Delta}+e^{-\mathnormal{\Delta}^4/2\delta}J_{n\delta}.
\]
\end{proposition}

\proof
By induction.
Abbreviate $u^{(\delta,-)}$,  $u^{({\bar\delta},+)}$, $e_{n,{\bar\delta}}$, $P_{n,\delta}$
and $G_\delta$ to $u^{(-)}$, $u^{(+)}$, $e_n$, $P$ and $G$, respectively.
For $n=0$ the claim holds because $u^{(\pm)}_0=u_0$.

Next, consider $n\ge1$.
Assume $u^{(+)}_{(n-1)\delta}\preccurlyeq u^{(-)}_{(n-1)\delta}+\varepsilon_{n-1} =:f_{n-1}$,
and $\varepsilon_{n-1}$ is nonnegative and
$\|\varepsilon_{n-1}\|_1\le \mathnormal{\Delta}_{n-1}:=\mathnormal{\Delta}+e^{-\mathnormal{\Delta}^4/2\delta}\sum_{i=1}^{n-1}j_i(\delta)$.
Then by definition of the upper barriers,
\begin{align*}
u^{(+)}_{n\delta} &=K^{\mathnormal{\Delta},j_n(\delta)}PGu^{(+)}_{(n-1)\delta}+e_n.
\end{align*}
Now, $\mathnormal{\Delta}\le\mathnormal{\Delta}_{n-1}$. Hence by Lemma \ref{lem3}(iv),
\begin{align*}
u^{(+)}_{n\delta}&\preccurlyeq K^{\mathnormal{\Delta}_{n-1},j_n(\delta)}PGu^{(+)}_{(n-1)\delta}+e_n
\\
&\preccurlyeq K^{\mathnormal{\Delta}_{n-1},j_n(\delta)}PGf_{n-1}+e_n,
\end{align*}
where monotonicity of $K$, $P$ and $G$ is used.
We can rewrite the above as
\[
K^{j_n(\delta)}K^{\mathnormal{\Delta}_{n-1}} PGf_{n-1}+\widehat K^{\mathnormal{\Delta}_{n-1}} PGf_{n-1}+e_n.
\]
Note that $w:=G\varepsilon_{n-1}$ is pointwise nonnegative,
$\|w\|_1\le\mathnormal{\Delta}_{n-1}$ and
\[
PGu^{(-)}_{(n-1)\delta}=PGf_{n-1}-w.
\]
Hence by Lemma \ref{lem3}(ii),
$K^{\mathnormal{\Delta}_{n-1}} PGf_{n-1}\preccurlyeq PGu^{(-)}_{(n-1)\delta}$.
By the monotonicity of $K^{j_n(\delta)}$, denoting
$\varepsilon_n=\widehat K^{\mathnormal{\Delta}_{n-1}}PGf_{n-1}+e_n$,
\begin{align*}
u^{(+)}_{n\delta} &\preccurlyeq K^{j_n(\delta)}PGu^{(-)}_{(n-1)\delta}+\varepsilon_n\\
&=u^{(-)}_{n\delta}+\varepsilon_n.
\end{align*}
Now, $\varepsilon_n$ is nonnegative. Moreover,
$\|\varepsilon_n\|_1=\|\widehat K^{\mathnormal{\Delta}_{n-1}}PGf_{n-1}\|_1+\|e_n\|_1
\le\mathnormal{\Delta}_{n-1}+e^{-\mathnormal{\Delta}^4/2\delta}j_n(\delta)=\mathnormal{\Delta}_n$.
This completes the proof.
\qed

\vspace{\baselineskip}

\noindent{\bf Proof of Theorem \ref{th1}.}
Note first that once uniqueness is established for the $u$ component
of the solution $(u,\beta)$, uniqueness of the $\beta$ component
follows immediately from \eqref{054}. To show the former,
argue by contradiction and assume that $(u^i,\beta^i)$, $i=1,2$ are two
solutions where $u^1$ and $u^2$ are distinct.
Then there exist $t>0$ and $r\in{\mathbb R}$ such that,
say, $u^1_t[r,\infty)<u^2_t[r,\infty)$. Fix such $t$ and $r$.
Given $n\in{\mathbb N}$ denote $\delta_n=tn^{-1}$.
Then by Propositions \ref{prop1} and \ref{prop2}, for every small $\mathnormal{\Delta}>0$
there exists $n_0$ such that for every $n>n_0$,
\[
u^{(\delta_n,-)}_t[r,\infty)\le u^1_t[r,\infty)< u^2_t[r,\infty)\le u^{(\mathnormal{\Delta},\delta_n,+)}_t[r,\infty).
\]
By Proposition \ref{prop3},
\[
u^{(\mathnormal{\Delta},\delta_n,+)}_t[r,\infty)-u^{(\delta_n,-)}_t[r,\infty)\le\|\varepsilon_n(t,\mathnormal{\Delta},\delta_n)\|_1
\le\mathnormal{\Delta}+e^{-\mathnormal{\Delta}^4/2\delta_n}J_t.
\]
On taking $n\to\infty$ and then $\mathnormal{\Delta}\downarrow0$, the RHS converges to zero,
a contradiction.
\qed

\begin{remark}\label{rem3}
Here are further properties of the solution deduced from
\cite[Theorem 1 and Remark 1(b)]{ama01}.
For $1<q<\infty$,
$0\le\sigma<\frac{q+1}{q}$ and $p\ge1$ such that $\frac{2}{p}+\frac{1}{q}>1+\sigma$,
$u\in L_{p,\rm loc}({\mathbb R}_+,W^\sigma_q)$, and for $p\in[1,2)$ and $\rho\in[0,1)$,
$u\in L_{p,\rm loc}({\mathbb R}_+,C_0({\mathbb R})\cap C^\rho({\mathbb R}))$.
\end{remark}



\section{Removal at boundary: convergence}\label{sec4}
\beginsec

\subsection{Proof of main result}\label{sec41}

In this section the convergence result is proved based on the uniqueness
of solutions to the PDE. It proceeds in four steps.
First, in Lemma \ref{lem4}, the rescaled processes are shown
to satisfy equation \eqref{054} with a certain error term.
Lemma \ref{lem6} establishes tightness of these processes.
Existence of measurable density for any subsequential limit of $\bar\xi^N$
is argued in Lemma \ref{lem5}.
Then, Lemma \ref{lem7} shows that
subsequential limits satisfy the ORA condition.
The proof of Theorem \ref{th2} appears at the end of the section.


A first relation to the PDE is as follows.
\begin{lemma}\label{lem4}
Fix $\varphi\in C_c^\infty({\mathbb R}\times{\mathbb R}_+)$ and $T$ so large that $\varphi(\cdot,t)=0$ for all $t\ge T$.
Then
\[
-\int_0^\infty\langle(\partial_t+\frac{1}{2}\partial_{xx})\varphi(\cdot,t),\bar\xi^N_t\rangle dt
=\langle\varphi(\cdot,0),\bar\xi^N_0\rangle
+\int_{{\mathbb R}\times{\mathbb R}_+}\varphi d\bar\alpha^N
-\int_{{\mathbb R}\times{\mathbb R}_+}\varphi d\bar\beta^N+\bar M^N_T.
\]
Here, $\bar M^N_t$ is an $\{F_t\}$-martingale that starts at zero
with quadratic variation bounded by
\begin{equation}\label{019}
\langle\bar M^N\rangle_t
\le\|\partial_x\varphi\|_\infty^2N^{-1}(1+J_t)t.
\end{equation}
\end{lemma}

\proof
By the assumed property $\varphi(\cdot,T)=0$ and \eqref{056}, we have
\begin{align*}
0=\int_{{\mathbb R}}\varphi(x,T)\bar\gamma_T^N(dx)
&=N^{-1}\sum_{i\le N^I_T}\varphi(B^i_{T\wedge\tau^i},T).
\end{align*}
For any $t$ and $i\le N^I_t$, we have $\sigma_i\le t$, hence by It\^o's formula,
\begin{align*}
\varphi(B^i_{t\wedge\tau^i},t)&=\varphi(x^i,\sigma_i)+\int_{\sigma_i}^t\partial_t\varphi(B^i_{s\wedge\tau^i},s)ds
+\int_{\sigma_i}^{t\wedge\tau^i}\partial_x\varphi(B^i_s,s)dB^i_s
\\
&\qquad
+\frac{1}{2}\int_{\sigma_i}^{t\wedge\tau^i}\partial_{xx}\varphi(B^i_s,s)ds.
\end{align*}
For $i\in{\mathbb N}$ denote $M^i_t=M^{N,i}_t
=\int_{\sigma_i}^{(t\vee\sigma_i)\wedge\tau^i}\partial_x\varphi(B^i_s,s)dB^i_s$.
Thus for $i\le N^I_T$,
\[
\varphi(B^i_{T\wedge\tau^i},T)
=\varphi(x^i,\sigma_i)+\int_{\sigma^i}^{T\wedge\tau^i}(\partial_t+\frac{1}{2}\partial_{xx})\varphi(B^i_s,s)ds
-\varphi(B^i_{T\wedge\tau^i},T\wedge\tau^i) +M^i_T.
\]
Hence
\begin{align*}
0&=\int_{\mathbb R}\varphi(\cdot,0)d\bar\xi^N_0+\int_{{\mathbb R}\times[0,T]}\varphi d\bar\alpha^N
+\int_0^T\int_{\mathbb R}(\partial_t+\frac12\partial_{xx})\varphi(x,t)\bar\xi^N_t(dx)dt
-\int_{{\mathbb R}\times[0,T]}\varphi d\bar\beta^N
+\bar M^N_T,
\end{align*}
where $\bar M^N_t=N^{-1}\sum_{i\le N^I_t}M^{N,i}_t$.
The integration range ${\mathbb R}\times[0,T]$ can be replaced by ${\mathbb R}\times{\mathbb R}_+$
thanks to the property $\varphi(\cdot,t)=0$, $t\ge T$.
Finally, the quadratic variation bound \eqref{019} is straightforward.
\qed

Recall that ${\cal M}({\mathbb R})$ is equipped with
the weak convergence topology and
$D({\mathbb R}_+,{\cal M}({\mathbb R}))$ with the corresponding $J_1$ topology.
${\cal M}_{\rm loc}({\mathbb R}\times{\mathbb R}_+)$ is equipped with
topology of weak convergence on compacts. Together they are equipped with
the product topology.

\begin{lemma}\label{lem6}
The sequence of laws of $(\bar\xi^N,\bar\beta^N)$, $N\in{\mathbb N}$,
is tight. For every subsequential limit $(\xi,\beta)$,
one has ${\mathbb P}(\xi\in C({\mathbb R}_+,{\cal M}_+({\mathbb R})))=1$.
\end{lemma}

\proof
Tightness may be argued separately for each component.
For $\bar\beta^N$, it will be shown that for every $T$,
the sequence of law of $\bar\beta^N|_{{\mathbb R}\times[0,T)}$ is tight
with respect to the topology of weak convergence. This is sufficient.
Fix $T<\infty$ and write $\bar\beta^N$ for the restriction
$\bar\beta^N|_{{\mathbb R}\times[0,T)}$.
Consider the following subset of ${\cal M}_+({\mathbb R}\times[0,T))$.
Given a constant $c$ and a collection of constants
$k(\varepsilon)$, $\varepsilon\in{\cal E}:=\{n^{-1}:n\in{\mathbb N}\}$
satisfying $\lim_{\varepsilon\to0}k(\varepsilon)=0$, let
\[
K(c,k)=\{\beta: \text{for every $\varepsilon\in{\cal E}$},
\ \beta({\mathbb R}\times[0,T)\le c,\ \beta([-k(\varepsilon),k(\varepsilon)]^c\times[0,T))\le\varepsilon\}.
\]
By Prohorov's theorem this set has a compact closure,
and by Portmanteau's theorem it is closed. Hence $K(c,k)$ is compact.

Now, the total mass of $\bar\beta^N$ is deterministic, and given by
$\bar\beta^N({\mathbb R}\times[0,T))=N^{-1}N^J_{T-}\le J_T$.

The position of each particle that has lived between time $0$ and $T$
(which includes the position where it was removed) can be bounded
in terms of its initial position and the maximal displacement of the driving BM.
Hence denoting $y^i=\sup_{t\in[0,T]}|\tilde B^i_t|$,
\begin{align*}
\bar\beta^N([-k,k]^c\times[0,T))
&\le N^{-1}\sum_{i\le N^I_T}1_{\{\sup_{t\in[\sigma^i,T]}|B^i_t|\ge k\}}
\\
&\le N^{-1}\sum_{i\le N^I_T}(1_{\{|x^i|\ge k/2\}}
+1_{\{y^i\ge k/2\}})\\
&\to {\mathbb P}(x^1\ge k/2)+I_T{\mathbb P}(z\ge k/2)+(1+I_T){\mathbb P}(y^1\ge k/2),
\end{align*}
in probability, where we recall that $x^1$ is distributed according to
$\pi$, let $z$ be distributed according to $u_0(x)dx$, and use the LLN.
For every $\varepsilon$ let $k=k(\varepsilon)$ be so large that the RHS is less than
$\varepsilon/2$. Then setting $K=K(J_T,k(\cdot))$ one has
${\mathbb P}(\bar\beta^N\in K)\to0$ as $N\to\infty$. This shows tightness
of the laws of $\bar\beta^N$.

Denote by $d_L(\cdot,\cdot)$ the Levy distance
on ${\cal M}_+({\mathbb R})$ and recall that it induces weak convergence.
For $f:{\mathbb R}_+\to{\mathbb R}$ denote $w(f,\delta,T)=\sup\{|f(s)-f(t)|:s,t\in[0,T],|s-t|\le\delta\}$,
and for $f:{\mathbb R}_+\to{\cal M}_+({\mathbb R})$ use the same notation
for the modulus of continuity with respect to $d_L$.
The argument for $\bar\xi^N$ is based on showing
(i) for every $T$ there exists a compact $K=K(T)\subset{\cal M}_+({\mathbb R})$
such that ${\mathbb P}(\bar\xi^N_T\in K)\to1$ as $N\to\infty$;
and (ii)
for every $T$ and $\varepsilon>0$ there exists $\delta>0$ such that
\[
\limsup_N {\mathbb P}(w(\bar\xi^N,\delta,T)\ge\varepsilon)\le\varepsilon.
\]
Once these two properties are proved we can use
\cite{ethkur} Corollary 3.7.4 (p.\ 129) by which $\bar\xi^N$
is a relatively compact
sequence. Moreover, because we use $w$ rather than $w'$
(cf.\ (3.6.2) of \cite{ethkur} (p.\ 122)), this in fact establishes $C$-tightness,
which proves the second statement.
The argument for (i) follows by similar considerations
to the one given above for $\bar\beta^N$, and is thus omitted.

It remains to show (ii).
Recall that $A^N_t$ denotes the index set for the particles existing
in the configuration at time $t$.
Consider $s,t\in[0,T]$ such that $0<t-s\le\delta$.
The cardinality of the symmetric different between $A^N_s$ and $A^N_t$
is given by
\begin{equation}\label{02}
N^I_t-N^I_s+N^J_t-N^J_s\le Nw_1(\delta),
\end{equation}
where $w_1(\delta)=w(I,\delta,T)+w(J,\delta,T)$.

Now,
$w(\tilde B^i,\delta,T)$ are IID and each converges to zero as $\delta\to0$.
Therefore given $\varepsilon>0$ there exists $\delta>0$ such that
\[
\limsup_N{\mathbb P}(\mathnormal{\Omega}^N)=0,
\qquad \text{where}
\qquad
\mathnormal{\Omega}^N=\mathnormal{\Omega}^N_{\varepsilon,\delta}
=\Big\{\frac{\#\{i\le N^I_T:w(\tilde B^i,\delta,T)\ge\eta\}}{N^I_T}\ge\eta\Big\},
\]
and we have denoted $\eta=c\varepsilon$, $c=(2+I_T)^{-1}$.
Given a set $C\subset{\mathbb R}$ let $C^\eta$ denote the $\eta$-neighborhood.
On the event $\mathnormal{\Omega}^N$, the following holds:
except for at most $w_1(\delta)N$ particles (removed between $s$ and $t$),
and at most $\eta N^I_T$ particles (whose displacement exceeds $\eta$),
each particle $i\in A^N_s$ travels less than $\eta$ between $s$ and $t$.
Hence, for any Borel set $C$, on $\mathnormal{\Omega}^N$,
\[
\xi^N_s(C) \le \xi^N_t(C^\eta)+w_1(\delta)N+\eta N^I_T.
\]
Similarly,
\[
\xi^N_t(C) \le \xi^N_s(C^\eta)+w_1(\delta)N+\eta N^I_T.
\]
By making $\delta$ smaller if necessary, we may assume that $w_1(\delta)<\eta$.
Consequently, $w_1(\delta)N+\eta N^I_T<\eta N+\eta N(1+I_T)=\varepsilon$.
We obtain that on $\mathnormal{\Omega}^N$,
\[
d_L(\bar\xi^N_s,\bar\xi^N_t)<\varepsilon.
\]
This shows that
\[
\limsup_N {\mathbb P}(w(\bar\xi^N,\delta,T)\ge\varepsilon)=0,
\]
and the proof is complete.
\qed


The next two lemmas address limit points $(\xi,\beta)$
of $(\bar\xi^N,\bar\beta^N)$. The first states that for every $t$,
$\xi_t(dx)$ has a density.

\begin{lemma}\label{lem5}
Let $(\xi,\beta)$ be a subsequential limit of $(\bar\xi^N,\bar\beta^N)$.
Then, given $T<\infty$,
\begin{equation}\label{070}
\sup_{t\in[0,T]}\sup_{a,b\in{\mathbb R}:a<b}(b-a)^{-1}\xi_t(a,b)\le c\quad\text{a.s.},
\end{equation}
where $c<\infty$ is a deterministic constant that depends only on $T$
and the data.
Moreover, there exists a full measure event $\mathnormal{\Omega}^1$ and
a ${\cal B}({\mathbb R})\otimes{\cal B}({\mathbb R}_+)\otimes{\cal F}$-measurable
function $u(x,t,\omega)$ such that for every $(t,\omega)\in{\mathbb R}_+\times\mathnormal{\Omega}^1$,
$u(\cdot,t,\omega)$ is a density of $\xi_t(\cdot,\omega)$ with respect to
the Lebesgue measure on ${\mathbb R}$.
\end{lemma}

\proof
Fix $T$. The first step is to show that there exists a constant $c_1$ such that
for every $t\in[0,T]$ and every $a<b$,
\begin{equation}\label{050}
\xi_t(a,b)\le c_1(b-a) \quad \text{a.s.}
\end{equation}
Recall that $\zeta^N$ denotes the particle configuration
without removals. Note that by construction,
$\xi^N_t(B)\le\zeta^N_t(B)$ for all $t\in{\mathbb R}_+$, $B\in{\cal B}({\mathbb R})$, $\omega\in\mathnormal{\Omega}$.
The argument toward \eqref{050} is based on showing,
for $t$, $a$, $b$ fixed,
\begin{equation}\label{050+}
{\mathbb P}(\bar\zeta^N_t(a,b)>c_1(b-a))\to0.
\end{equation}
If the above is true, the same is true for $\bar\xi^N_t(a,b)$.
Since along a subsequence one has
$\bar\xi^N\Rightarrow\xi$ and the latter has continuous sample paths,
one also has $\bar\xi^N_t\Rightarrow\xi_t$.
As a result, ${\mathbb P}(\xi_t(a,b)>c_1(b-a))\le\limsup{\mathbb P}(\bar\xi^N_t(a,b)>c_1(b-a))=0$.

The convergence \eqref{050+} is a standard result
but we have not found a reference that covers precisely this setting.
The proof is simple.
We have by \eqref{058+},
$\zeta^N_t(a,b)=\sum_{i=1}^{N^I_t}1_{\{B_t^i\in(a,b)\}}$.
For a first moment calculation, denoting $\bar I^N_s=N^{-1}[NI_s]$,
\begin{align*}
{\mathbb E}\bar\zeta^N_t(a,b)&=
N^{-1}\sum_{i\le N^I_t}{\mathbb P}(x^i+\tilde B^i_{t-\sigma^i}\in(a,b))
\\
&=
N^{-1}\sum_{i\le N}\int G_t(x,(a,b))u_0(x)dx
+N^{-1}\sum_{N<i\le N^I_t}\int G_{t-\sigma^i}(x,(a,b))\pi(dx)
\\
&=
G_tu_0(a,b)+\int_{{\mathbb R}\times[0,t]}G_{t-s}(x,(a,b)) \pi(dx) d\bar I^N_s
\\
&\le
G_tu_0(a,b)+\int_{{\mathbb R}\times[0,t-\varepsilon]}G_{t-s}(x,(a,b)) \pi(dx) d\bar I^N_s
+(\bar I^N_t-\bar I^N_{t-\varepsilon}).
\end{align*}
On $[0,t-\varepsilon]$, $s\mapsto\int G_{t-s}(x,(a,b))\pi(dx)$
is continuous and bounded.
Hence the above integral converges to
$\int_{{\mathbb R}\times[0,t-\varepsilon]}G_{t-s}(x,(a,b))\pi(dx)dI_s$. Thus for $t\in[0,T]$,
\[
\limsup_N{\mathbb E}\bar\zeta^N_t(a,b)\le G_tu_0(a,b)
+G*\alpha((a,b),t)\le c_2(b-a),
\]
where the last inequality follows from the $L_{\infty,{\rm loc}}({\mathbb R}_+,L_\infty)$
bound stated in Lemma \ref{lem1}, upon applying this lemma with
data $J=0$ on $[0,T]$.
Next, a straightforward second moment calculation that uses the independence
of the particles gives 
${\rm var}(\bar\zeta^N_t(a,b))\le N^{-2}N^I_t\le N^{-1}(1+I_t)$.
Hence \eqref{050+} follows if we set $c_1=2c_2$.
This shows \eqref{050}.

To deduce from this the statement \eqref{070}
and the existence of a measurable density we proceed in several steps.
It follows from \eqref{050} that on an event of full measure,
for fixed $t\in[0,T]$ and all rational $a,b$ one has $\xi_t(a,b)\le c_1(b-a)$.
As a result, on the same event, for all $a,b\in{\mathbb R}$ and $a<b$, one has
$\xi_t(a,b)\le 2c_1(b-a)$.
Next, the same can be stated for all rational $t$ at once.
Now using the continuity of $t\mapsto\xi_t$ and the fact that $(a,b)$
is an open set we get that on an event of full measure,
for all $t\in[0,T]$ and $a<b$, $\xi_t(a,b)\le2c_1(b-a)$.
This proves \eqref{070} with $c=2c_1$.

We may now assume without loss of generality that $\xi=0$ away from
the full measure event under consideration, obtaining
that for every $t\in[0,T]$ and $\omega\in\mathnormal{\Omega}$, the measure $\xi_t(dx,\omega)$
is absolutely continuous with respect to $dx$.
By repeating the process for all $T\in{\mathbb N}$, the same conclusion holds
for $(t,\omega)\in{\mathbb R}_+\times\mathnormal{\Omega}$.

We can now appeal to Theorem 58 in Chapter V (page 52) of \cite{del-mey-82}
(see also the remark that appears on the same page).
The measurable spaces denoted in \cite{del-mey-82}
by $(\mathnormal{\Omega},{\cal F})$ and $(T,{\cal T})$ are taken to be, in our notation,
$({\mathbb R},{\cal B}({\mathbb R}))$ and $({\mathbb R}_+\times\mathnormal{\Omega},{\cal B}({\mathbb R}_+)\otimes{\cal F})$, respectively.
According to this result there exists
a ${\cal B}({\mathbb R})\otimes{\cal B}({\mathbb R}_+)\otimes{\cal F}$-measurable function
$u(x,t,\omega)$, such that for every $(t,\omega)\in{\mathbb R}_+\times\mathnormal{\Omega}$,
$u(\cdot,t,\omega)$ is a density of $\xi_t(dx,\omega)$ with respect to $dx$.
\qed

The following lemma addresses condition \eqref{018}(iii).

\begin{lemma}\label{lem7}
Let $(\xi,\beta)$ be a subsequential limit of $(\bar\xi^N,\bar\beta^N)$.
Then a.s.,
\begin{equation}\label{025}
X(\xi,\beta,r):=\int_{{\mathbb R}_+}\xi_t[r,\infty)\beta((-\infty,r)\times dt)=0,\quad \text{for all } r\in{\mathbb R}.
\end{equation}
\end{lemma}

\proof
We first prove a version of \eqref{025} for rational $r$.
More precisely, consider the collection $\mathnormal{\Sigma}$
of all (deterministic) tuples $\sigma=(r,t_1,t_2,\eta,\delta)\in{\mathbb Q}^5$,
$0\le t_1<t_2$, $\eta>0$, $\delta>0$. We show that for every
$\sigma\in\mathnormal{\Sigma}$, ${\mathbb P}(\mathnormal{\Omega}_\sigma)=0$ where
\[
\mathnormal{\Omega}_\sigma=\{\inf_{t\in[t_1,t_2]}\xi_t[r,\infty)>\eta,\, \beta((-\infty,r)\times(t_1,t_2))>\delta\}.
\]
Fix $\sigma=(r,t_1,t_2,\eta,\delta)$. If ${\mathbb P}(\mathnormal{\Omega}_\sigma)>0$ then by
the weak convergence $(\bar\xi^N,\bar\beta^N)\Rightarrow(\xi,\beta)$
and the a.s.\ continuity of the limit $\xi$, one must have for all large $N$,
\[
{\mathbb P}\Big(\inf_{t\in[t_1,t_2]}\bar\xi^N_t[r,\infty)>\eta/2,
\,\bar\beta^N(-\infty,r)\times(t_1,t_2)>\delta/2\Big)>0.
\]
However, by \eqref{058}, the above probability is zero for all $N$.
This shows ${\mathbb P}(\mathnormal{\Omega}_\sigma)=0$.
Consequently, ${\mathbb P}(\cup\mathnormal{\Omega}_\sigma)=0$.

Next consider the event
\[
\mathnormal{\Omega}^0=\{\text{that there exists $r\in{\mathbb R}$ such that
$X(\xi,\beta,r)>0$}\}.
\]
Given any $r\in{\mathbb R}$, consider $r_n\downarrow r$, $r_n\in{\mathbb Q}$.
Then $\beta((-\infty,r),dt)\ll\beta((-\infty,r_n),dt)$ for every $n$.
Moreover, $\xi_t[r_n,\infty)\to\xi_t[r,\infty)$ as $n\to\infty$ uniformly for $t$
in any compact set (as follows from the Lipschitz property
$x\to\xi_t[x,\infty)$ stated in Lemma \ref{lem5}).
Consequently, if $X(\xi,\beta,r)>0$ then
there exists $n$ such that $X(\xi,\beta,r_n)>0$. Hence
on $\mathnormal{\Omega}^0$ there exists $r\in{\mathbb Q}$ such that $X(\xi,\beta,r)>0$.
Next, the condition $X(\xi,\beta,r)>0$ (with $r\in{\mathbb Q}$) implies that
there exists $\eta\in{\mathbb Q}\cap(0,\infty)$ such that
$\int_{A_\eta}a_tdb_t>0$ where $a_t=\xi_t[r,\infty)=\xi_t(r,\infty)$,
$b_t=\beta(-\infty,r)\times[0,t]$ and $A_\eta=\{t:a_t>2\eta\}$.
The trajectory $t\mapsto a_t$ is continuous on $(0,\infty)$
(using the fact that $\xi\in C({\mathbb R}_+,{\cal M}_+({\mathbb R}))$ and
that for each $t$, $\xi_t$ has no atoms).
Hence $A_\eta$ is a countable union of disjoint open intervals.
For one of those intervals, say $(t_1,t_2)$, we have
$\int_{(t_1,t_2)}a_tdb_t>0$. By taking a subset we may assume without loss
that $t_1, t_2\in{\mathbb Q}$.
Consequently, $\xi_t(r,\infty)=a\ge2\eta>\eta$ on the closed interval $[t_1,t_2]$,
while $b_{t_2}>b_{t_1}$.
Moreover, notice that by construction, for every $a$,
$\bar\beta^N({\mathbb R}\times[0,s])=N^{-1}[NJ_s]$. Hence by
the continuity of $J$, $\beta$ does not charge sets ${\mathbb R}\times\{t\}$.
As a result, there exists $\delta\in{\mathbb Q}\cap(0,\infty)$ such that
\[
\beta((-\infty,r)\times(t_1,t_2))>\delta.
\]
This shows that ${\mathbb P}(\mathnormal{\Omega}^0)\le{\mathbb P}(\cup_{\sigma\in\mathnormal{\Sigma}}\mathnormal{\Omega}_\sigma)=0$.
\qed


\noindent{\bf Proof of Theorem \ref{th2}.}
Again consider a subsequential limit $(\xi,\beta)$ of
$(\bar\xi^N,\bar\beta^N)$. Let $u$ be the density from Lemma \ref{lem5}.
We show that a.s., $(u,\beta)$ is a weak solution to \eqref{018}.

First, to address \eqref{054},
recall that $\{x^i,1\le i\le N\}$ are IID $u_0(dx)$, hence
$\langle\varphi(\cdot,0),\bar\xi^N_0\rangle\to(\varphi(\cdot,0),u_0)$ in probability.
Using this and the relation $\xi_t(dx)=u(x,t)dx$
in Lemma \ref{lem4} shows that $(u,\beta)$ satisfies \eqref{054} a.s.

That condition \eqref{018}(iii) is satisfied by $(u,\beta)$
follows from Lemma \ref{lem7}.

Condition \eqref{018}(iv) is an immediate
consequence of the particle system construction where the removal count
is given by $N^J_t=[NJ_t]$.

To show that $(u,\beta)$ is a.s.\ a weak solution to \eqref{018},
it remains to prove that $u\in L_{1,{\rm loc}}({\mathbb R}_+,L_1)\cap L_{\infty,{\rm loc}}({\mathbb R}_+,L_\infty)$.
The former condition is a consequence of
$\bar\xi^N_t({\mathbb R})=N^{-1}(N^I_t-N^J_t)\le 1+I_t$ and the latter follows
the Lipschitz bound \eqref{070}.
We have thus shown that $(u,\beta)$ is a.s.\ a weak solution to \eqref{018}.
By Theorem \eqref{th1}, it is in fact the unique weak solution
to \eqref{018}. In particular, the limit $(\xi,\beta)$ does not depend on
the subsequence. In other words, there exists
a unique weak solution $(u,\beta)$ to \eqref{018},
and $(\bar\xi^N,\bar\beta^N)\to(\xi,\beta)$ in probability, where,
for every $t$, $\xi_t(dx)=u(x,t)dx$.
\qed

\vspace{\baselineskip}

\noindent{\bf Proof of Theorem \ref{th0}.}
This result is contained in Theorem \ref{th2}.
\qed


\subsection{Comparison via coupling}\label{sec42}

In this section we prove the comparison principle, Theorem \ref{th5}.

\noindent{\bf Proof of Theorem \ref{th5}.}
The proof is based on a coupling. Because it is
a straightforward coupling construction, we only give a sketch.
Before we can apply coupling, we need two preparatory steps.

Step 1. A right-continuous nondecreasing function
from ${\mathbb R}_+$ to ${\mathbb Z}_+$ starting at $0$, for which every jump is of size $1$,
is said to be a simple step function.
The construction from \S \ref{sec21} uses $N^I_t=N+[NI_t]$
and $N^J_t=[NJ_t]$ as the injection and removal counts, but we claim that
the main result still holds if these are replaced by any sequence
$N^{I,N}_t=N+\widehat N^{I,N}_t$ and $N^{J,N}_t$ such that
$\widehat N^{I,N}$ and $N^{J,N}$ are simple step functions,
as long as $N^{-1}\widehat N^{I,N}\to I$ and $N^{-1}N^{J,N}\to J$,
as $N\to\infty$, uniformly on compacts. The only place in the proof
that needs some attention is the tightness argument, Lemma \ref{lem6},
where the inequality \eqref{02} should be modified to add
an $o(N)$ term on the RHS. The proof carries over with this change.

Step 2.
By performing discretization (such as $x\mapsto[x]$) to
$N(\tilde J_t-J_t)$ and $NJ_t$ it is seen that
simple step functions $N^{\tilde J,N}$ and $N^{J,N}$ can be
constructed in such a way that the jumps of the former dominate
those of the latter (every time the latter function jumps, so does the former),
and moreover, their normalized versions converge uniformly
on compacts to $\tilde J$ and $J$.
(If a jump of size $2$ occurs for $N^{\tilde J,N}$ via the above discretization,
as it may, one splits it to two nearby jumps).
Similarly $\widehat N^{\tilde I,N}$ and $\widehat N^{I,N}$ can be
constructed in such a way that the jumps of the latter dominate
those of the former, with convergence of their normalized versions
to $\tilde I$ and $I$.

Using the domination of the jumps of $N^{I,N}$ over those of $N^{\tilde I,N}$,
it follows by an elementary argument that a coupled pair of particle systems
can be constructed in such a way that the counting measures
$\tilde\zeta^N$ and $\zeta^N$ (that, we recall, account
merely for the potential trajectories) satisfy a.s., for all $t$,
\[
\tilde\zeta^N_t \preccurlyeq \zeta^N_t.
\]
To deduce a similar statement about the counting measures
of the true trajectories, argue by induction on the times of removal.
At times when a joint removal occurs, the rightmost particle is removed
at both systems. Since by the induction assumption the counting measures
are ordered as $\tilde\xi^N\preccurlyeq\xi^N$ right before the removal,
the removal of rightmost particle from both configurations preserves this order.
At times of a jump of $N^{\tilde J,N}$ alone, a particle is removed only
from the configuration $\tilde\xi^N$, again keeping the order.
As a consequence, $\tilde\xi^N_t \preccurlyeq \xi^N_t$ holds a.s.\ at all times.
The result is now a simple consequence of Theorem \ref{th2}.
\qed




\section{Removal at quantile}\label{sec5}
\beginsec

In this section the approach is applied to RAQ, a model that follows a far more
general removal mechanism, where, for a given continuous function $Q$,
the $Q(t)$-quantile member of the population is removed at time $t$.
On the other hand, the model is simpler than the RAB model
in two respects: the removal rate is constant, and there are
no injections, and therefore the population dies out after $N$ removals.

Whereas the construction and analysis of the barriers, as well as
some further details of the uniqueness proof are different,
the convergence proof is almost identical to the one provided for RAB.

The particle system is constructed in \S\ref{sec51}.
The heat equation with a (modified) ORA condition is described in
\S\ref{sec52}, where the main result regarding this model is also stated.
\S\ref{sec53} provides several further mass transport inequalities
used later.
The barriers are constructed and analyzed in \S\ref{sec54}.
Finally, in \S\ref{sec55} the proof of uniqueness and convergence is provided.


\subsection{Particle system with quantile removal}\label{sec51}

Simply put, the system consists of Brownian particles whose
initial number is $N$, subjected to removal
defined in terms of a given $Q:[0,1]\to[0,1]$.
At a fixed rate $N$, the particle that is the
$Q(t)$-quantile member of the current population
is removed, until there are no particles left.
The precise details of the construction are as follows.

Let $u_0\in L_1({\mathbb R},{\mathbb R}_+)\cap L_\infty({\mathbb R},{\mathbb R}_+)$, $\|u_0\|_1=1$ be given,
and let $q(t)=(1-t)Q(t)$, $0\le t\le 1$. It turns out to be more natural to
regard $q$, rather that $Q$, as data;
note that $Q(t)$ can be recovered from $q(t)$ for all $0\le t<1$.
Thus an admissible data for this problem is a pair $(u_0,q)$, where
$u_0$ is as above, and
$q:{\mathbb R}_+\to[0,1]$ is a continuous function satisfying $q(t)\le1-t$
for $t\in[0,1)$ and vanishing on $[1,\infty)$.

The initial number of particles is $N$.
Mutually independent BMs $\{B^i_t,t\ge0\}$, $1\le i\le N$ are given,
where $B^i$ starts at $x^i$ for $1\le i\le N$,
and $x^i$ are IID, distributed
according to the probability measure $u_0(x)dx$.

The configuration is constructed recursively on intervals ${\rm INT}_k$
given by $[t_k,t_{k+1})$ where $t_k=kN^{-1}$, $0\le k\le N$, while
$t_{N+1}=\infty$, where at each time $t_k$, $1\le k\le N$, one particle is removed.
Given admissible data $(u_0,q)$, let $Q(t)=(1-t)^{-1}q(t)$ for $0\le t<1$
(it is not necessary to define $Q(1)$ because at time $1-$
there is only one particle to remove).
The collection of particles present in the system during
${\rm INT}_k$ is denoted by ${\rm POP}_k$; in particular, $|{\rm POP}_k|=N-k$.
For $t\in{\rm INT}_0$, the $N$ particle positions are
$B^i_t$, and ${\rm POP}_0=\{1,\ldots,N\}$.
For $1\le k\le N-1$,
let $i_k$ be the $Q(t_k)$-quantile member of the particle
configuration at $t_k-$. That is,
\[
i_k=\min\{j\in P:\#\{i\in P:b_i\ge b_j\} = \lceil nQ(t_k)\rceil\},
\]
where $P={\rm POP}_{k-1}$, $n=N-k+1=|{\rm POP}_{k-1}|$,
and for $i\in P$, $b_i=B^i_{t_k-}$.
Here, the convention $\lceil0\rceil=1$ is used.
Set ${\rm POP}_k={\rm POP}_{k-1}\setminus\{i_k\}$.
At the final removal time, $t_N=1$, the last
remaining particle is removed. Thus $i_N$ is the unique
member of ${\rm POP}_{N-1}$, and ${\rm POP}_N$ is
the empty set.
Finally, $\tau^i$, the removal time of particle $i$,
is well defined for all $1\le i\le N$ via the relation $\tau^{i_k}=t_k$,
for $1\le k\le N$.

The configuration process and the removal measure are defined by
\[
\xi^N_t(dx)=
\sum_{i=1}^N\boldsymbol{\delta}_{B^i_t}(dx)1_{\{t<\tau^i\}},
\qquad
\beta^N(dx,dt)=\sum_{i\le N}\boldsymbol{\delta}_{(B^i_{\tau^i},\tau^i)}(dx,dt),
\]
and the two extended configuration processes by
\[
\gamma^N_t(dx)=\sum_{i\le N}\boldsymbol{\delta}_{B^i_{t\wedge\tau_i}}(dx),
\qquad
\zeta^N_t(dx)=\sum_{i\le N}\boldsymbol{\delta}_{B^i_t}(dx).
\]
The rescaled versions are denoted by $\bar\xi^N=N^{-1}\xi^N$, etc.

Fix $r\in{\mathbb R}$.
If at time $t=t_k=kN^{-1}$ a particle is removed at a position within $(-\infty,r)$
then, again denoting $n=N-k+1$, one has
$\xi^N_{t_k}(r,\infty)=\xi^N_{t_k-}(r,\infty)\le \lceil nQ(t_k)\rceil-1\le
nQ(t_k)\le Nq(t_k)+1$.
Similarly if a particle is removed within $(r,\infty)$
then $\xi^N_{t_k}(-\infty,r)\le N(1-t_k-q(t_k))+1$.

To express this in terms of $\bar\xi^N$ and $\bar\beta^N$
use the following notation. For $(\xi,\beta,c)\in D({\mathbb R}_+,{\cal M}_+({\mathbb R}))
\times{\cal M}_{+,{\rm loc}}({\mathbb R}\times{\mathbb R}_+)\times C({\mathbb R}_+,[0,1])$, let
\begin{align}
\label{B120}
{\cal I}_r^+(\xi,\beta,c)
&=\int_{{\mathbb R}_+}[\xi_t(r,\infty)-c(t)]^+\beta((-\infty,r)\times dt),
\\
\label{B121}
{\cal I}_r^-(\xi,\beta,c)
&=\int_{{\mathbb R}_+}[\xi_t(-\infty,r)-(1-t-c(t))]^+\beta((r,\infty)\times dt).
\end{align}
Then
\begin{equation}\label{B058}
{\cal I}_r^+\Big(\bar\xi^N,\bar\beta^N,q(\cdot)+\frac{1}{N}\Big)
={\cal I}_r^-\Big(\bar\xi^N,\bar\beta^N,q(\cdot)-\frac{1}{N}\Big)
=0,\qquad
r\in{\mathbb R}.
\end{equation}



\subsection{PDE and main result}\label{sec52}

With a slight abuse of notation, for $u\in L_{1,{\rm loc}}({\mathbb R}_+,L_1)$,
let ${\cal I}_r^\pm(u,\beta,c)$ be defined as in \eqref{B120}--\eqref{B121}
with $u((r,\infty),t)$ in place of $\xi_t(r,\infty)$, etc.
Consider the problem
\begin{equation}\label{B018}
\begin{cases}
{\it (i)}\hspace{3.5em}
\partial_tu-\frac{1}{2}\partial_{xx}u=-\beta & \text{in } {\mathbb R}\times{\mathbb R}_+,\\ \\
{\it (ii)}\hspace{3.3em}
u(\cdot,0)=u_0 & \text{on } {\mathbb R}, \\ \\
{\it (iii)}\hspace{2.8em}
{\cal I}^\pm_r(u,\beta,q)=0 & \text{for all } r\in{\mathbb R},\\ \\
{\it (iv)}\hspace{3.1em}
\beta({\mathbb R}\times[0,t])=t\w1 & \text{for all } t\in{\mathbb R}_+.
\end{cases}
\end{equation}
A solution to \eqref{B018} is a pair
$(u,\beta)\in L_{1,\rm loc}({\mathbb R}_+,L_1)
\cap L_{\infty,\rm loc}({\mathbb R}_+,L_\infty)
\times{\cal M}_{+,{\rm loc}}({\mathbb R}\times{\mathbb R}_+)$
such that $u$ is a.e.\ nonnegative,
conditions \eqref{B018}(iii) and \eqref{B018}(iv) hold,
and for any $\varphi\in C_c^\infty({\mathbb R}\times{\mathbb R}_+)$,
\begin{equation}\label{B054}
-\int_0^\infty(\partial_t\varphi+\frac{1}{2}\partial_{xx}\varphi,u)dt
=(\varphi(\cdot,0),u_0) - \int_{{\mathbb R}\times{\mathbb R}_+}\varphi\, d\beta.
\end{equation}


\begin{theorem}\label{Bth2}
There exists a unique solution $(u,\beta)$ to \eqref{B018}.
Moreover, if for every $t$ we set
$\xi_t(dx)=u(x,t)dx$ then $(\xi,\beta)\in C({\mathbb R}_+,{\cal M}_+({\mathbb R}))\times{\cal M}_{+,{\rm loc}}({\mathbb R}\times{\mathbb R}_+)$
and $(\bar\xi^N,\bar\beta^N)\to(\xi,\beta)$ in probability, as $N\to\infty$.
\end{theorem}


\subsection{More mass transport inequalities}\label{sec53}


In this section it is convenient to work with an extention
of $K^{\mathnormal{\Delta},\delta}$, denoted by $L^{\mathnormal{\Delta},\delta}$, to $\mathnormal{\Delta}\in{\mathbb R}$
(but still $\delta>0$).
Introduce, analogously to $R_\delta$, for $\delta>0$ and $u\in H_\delta$,
\[
\tilde R_\delta(u)=\inf\{x\in{\mathbb R}:u(-\infty,x]\ge\delta\},
\]
and analogously to $K^\delta$,
\[
\tilde K^\delta u(x)=u(x)1_{[\tilde R_\delta(u),\infty)}(x).
\]
This operator cuts off the leftmost mass of size $\delta$.
For $\mathnormal{\Delta}\in{\mathbb R}$, $\delta>0$ and $u\in H_\delta$,
\[
L^{\mathnormal{\Delta},\delta}=
\begin{cases}
K^\delta u
& \mathnormal{\Delta}\le0,\\
K^{\mathnormal{\Delta},\delta}u
& \mathnormal{\Delta}>0,\, \mathnormal{\Delta}+\delta<\|u\|_1,\\
\tilde K^\delta u
& \mathnormal{\Delta}>0, \mathnormal{\Delta}+\delta\ge\|u\|_1.
\end{cases}
\]

\begin{lemma}\label{Blem3+}
Let $\mathnormal{\Delta}\in{\mathbb R}$, $\delta>0$, $u,v\in H_\delta$ be given, where $u\preccurlyeq v$. Then

i. $\tilde K^\delta u\preccurlyeq\tilde K^\delta v$.

ii. $L^{\mathnormal{\Delta},\delta}u\preccurlyeq L^{\mathnormal{\Delta},\delta}v$.

iii. If in addition $w\in L_1({\mathbb R},{\mathbb R}_+)$ is such that $\|w\|_1=\delta$,
and $u-w\in L_1({\mathbb R},{\mathbb R}_+)$ then $u-w\preccurlyeq\tilde K^\delta u$.

iv. If $-\infty<\widehat\mathnormal{\Delta}\le\mathnormal{\Delta}<\infty$ then
$L^{\widehat\mathnormal{\Delta},\delta}u\preccurlyeq L^{\mathnormal{\Delta},\delta}u$.

v. If $-\infty<\widehat\mathnormal{\Delta}\le\mathnormal{\Delta}<\infty$ and $u\in H_{\mathnormal{\Delta}-\widehat\mathnormal{\Delta}+\delta}$ then
\[
L^{\mathnormal{\Delta},\delta}u\preccurlyeq L^{\widehat\mathnormal{\Delta},\delta}K^{\mathnormal{\Delta}-\widehat\mathnormal{\Delta}}u+\widehat K^{\mathnormal{\Delta}-\widehat\mathnormal{\Delta}}u.
\]
\end{lemma}

\proof
i. Although in the special case when $\|u\|_1=\|v\|_1$,
the monotonicity of $\tilde K^\delta$ follows from that of
$K^\delta$ (via reflection $x\mapsto-x$), this is not the case in general.
We must show $\tilde K^\delta u(r,\infty)\le\tilde K^\delta v(r,\infty)$ for all $r$.
Note first that one must have $\|u\|_1\le\|v\|_1$. Let $r_u=\tilde R_\delta(u)$
and $r_v=\tilde R_\delta(v)$.

Case 1: $r_u\le r_v$. For $r\ge r_v$ this follows from $u\preccurlyeq v$.
For $r\le r_v$,
\[
\tilde K^\delta u(r,\infty)\le u(r,\infty)\le\|u\|_1-\delta\le\|v\|_1-\delta=\tilde K^\delta v(r,\infty).
\]

Case 2: $r_u>r_v$. Again, the claim holds trivially when $r>r_u$.
For $r\le r_v$, $\tilde K^\delta u(r,\infty)=\|u\|_1-\delta\le\|v\|_1-\delta=\tilde K^\delta v(r,\infty)$.
For $r_v\le r\le r_u$,
\[
\tilde K^\delta u(r,\infty)\le u(r,\infty)\le v(r,\infty)=\tilde K^\delta v(r,\infty).
\]

ii. For $\mathnormal{\Delta}\le0$, the claim holds by the monotonicity of $K^\delta$ (Lemma \ref{lem3}(i)).
Next assume $\mathnormal{\Delta}>0$. If $\mathnormal{\Delta}+\delta<\|u\|_1\wedge \|v\|_1$, the claim follows from
the monotonicity of $K^{\mathnormal{\Delta},\delta}$ (Lemma \ref{lem3}(iii)). If $\mathnormal{\Delta}+\delta\ge\|u\|_1\vee\|v\|_1$,
the claim follows from the monotonicity of $\tilde K^\delta$ just proved.
Because $\|u\|_1\le\|v\|_1$, the only remaining case is $\|u\|_1\le\mathnormal{\Delta}+\delta<\|v\|_1$.
In this case we must show $\tilde K^\delta u(r,\infty)\le K^{\mathnormal{\Delta},\delta}v(r,\infty)$.
Denote $r_u=\tilde R_\delta(u)$ and $r_v=R_\mathnormal{\Delta}(v)$.

Case 1: $r_u\le r_v$. For $r\ge r_v$ the claim clearly holds. For $r<r_v$,
\[
\tilde K^\delta u(r,\infty)\le \|\tilde K^\delta u\|_1=\|u\|_1-\delta\le\mathnormal{\Delta}
=K^{\mathnormal{\Delta},\delta}v(r_v,\infty)\le K^{\mathnormal{\Delta},\delta}v(r,\infty).
\]

Case 2: $r_u>r_v$. For $r\ge r_u$, the claim clearly holds.
For $r\le r_v$, again
\[
\tilde K^\delta u(r,\infty)=\|u\|_1-\delta\le\mathnormal{\Delta}=K^{\mathnormal{\Delta},\delta}v(r_v,\infty)\le K^{\mathnormal{\Delta},\delta}v(r,\infty).
\]
For $r_v<r<r_u$,
\[
\tilde K^\delta u(r,\infty)\le u(r,\infty)\le v(r,\infty)=K^{\mathnormal{\Delta},\delta}v(r,\infty).
\]

iii. Because $\|u-w\|_1=\|\tilde K^\delta u\|_1$, this claim follows from
Lemma \ref{lem3}(ii) by reflection.

iv. One can write $L^{a,\delta}u$ as $K^{g(a),\delta}u$ where
$g(a)=(a\vee 0)\wedge(\|u\|_1-\delta)$. Since $g$ is a nondecreasing
function, the result follows from Lemma \ref{lem3}(iv).

v. If $\widehat\mathnormal{\Delta}>0$ and $\mathnormal{\Delta}<\|u\|_1-\delta$ then both sides of the
asserted inequality are equal to $K^{\mathnormal{\Delta},\delta}u$.
If $\widehat\mathnormal{\Delta}>0$ and $\mathnormal{\Delta}\ge\|u\|_1-\delta$
then both sides of the inequality are equal
to $\tilde K^\delta u$.

Finally, consider the case $\widehat\mathnormal{\Delta}\le0$ and denote the
RHS of the asserted inequality by $v$.
Then $v=K^{\mathnormal{\Delta}-\widehat\mathnormal{\Delta},\delta}u$.
Since by assumption $0\le\mathnormal{\Delta}-\widehat\mathnormal{\Delta}\le\|u\|_1-\delta$,
one has $v=L^{\mathnormal{\Delta}-\widehat\mathnormal{\Delta},\delta}u$. But $\mathnormal{\Delta}\le \mathnormal{\Delta}-\widehat\mathnormal{\Delta}$,
hence $L^{\mathnormal{\Delta},\delta}u\preccurlyeq L^{\mathnormal{\Delta}-\widehat\mathnormal{\Delta},\delta}u=v$ by
part (iv) above.
\qed



\subsection{Barriers}\label{sec54}


\begin{lemma}\label{Blem1}
Assume that a solution $(v,\beta)$ to \eqref{B018} exists.
Then $v$ has a version $u$ given by
\begin{equation}\label{B012-}
u(y,t)=G_tu_0(y)-G*\beta(y,t).
\end{equation}
Moreover, for $0\le \tau<t$,
\begin{equation}\label{B012}
u(y,t)=G_{t-\tau}u(\cdot,\tau)(y)-G*\beta(y,t;\tau).
\end{equation}
Furthermore, $u\in L_{\infty,{\rm loc}}({\mathbb R}_+,L_1)\cap L_{\infty,{\rm loc}}({\mathbb R}_+,L_\infty)$.
\end{lemma}

As was the case in \S\ref{sec3},
a solution to \eqref{B018} will always mean the version given
by \eqref{B012-}, and this results in no loss of generality.

\proof The proof of Lemma \ref{lem1} does not rely on the ORA condition
\eqref{018}(iii) in any way,
and hence the result is a special case of Lemma \ref{lem1}.
\qed

The barriers are defined for parameters
$0<\delta<\mathnormal{\Delta}$. Denote again ${\bar\delta}=(\mathnormal{\Delta},\delta)$.
Let
\[
q^{(-)}_{n,\delta}=\min_{[(n-1)\delta,n\delta]}q,
\qquad
q^{(+)}_{n,\delta}=\max_{[(n-1)\delta,n,\delta]}q
\]
and
\[
L_{n,{\bar\delta}}^{(+)}=L^{q^{(+)}_{n,\delta}+\mathnormal{\Delta},\delta}.
\]
Then the upper barriers are defined by $u^{({\bar\delta},+)}_0=u_0$, and for $n\in{\mathbb N}$
such that $n\delta<1-\mathnormal{\Delta}$,
\begin{align}
\notag
u^{({\bar\delta},+)}_{n\delta}
&=L^{(+)}_{n,{\bar\delta}}G_\delta u^{({\bar\delta},+)}_{(n-1)\delta}
+e^{(+)}_{n,{\bar\delta}}
\\
\label{B016-}
e^{(+)}_{n,{\bar\delta}}(y)
&=e^{-\mathnormal{\Delta}^4/2\delta}\del1_{[\sigma^{(+)}_n,\sigma^{(+)}_n+1]}(y),
\qquad y\in{\mathbb R},
\\
\notag
\sigma^{(+)}_n&= \sigma^{(+)}_n({\bar\delta})=R_\mathnormal{\Delta}(G_\delta u^{({\bar\delta},+)}_{(n-1)\delta}).
\end{align}
For the first and third lines to be well defined one must have
$\|u^{({\bar\delta},+)}_{(n-1)\delta}\|_1>\delta$ and, respectively, $>\mathnormal{\Delta}$.
Because every iteration cuts off a mass of size $\delta$ (and adds
the mass of the error term), the condition $n\delta<1-\mathnormal{\Delta}$ does
assure $\|u^{({\bar\delta},+)}_{(n-1)\delta}\|_1\ge 1-(n-1)\delta>\mathnormal{\Delta}>\delta$,
hence the upper barriers are well defined.

The lower barriers are constructed analogously (and are again
well defined). Namely, let
\[
L^{(-)}_{n,{\bar\delta}}=L^{q^{(-)}_{n,\delta}-\mathnormal{\Delta}-\delta,\delta}.
\]
Then $u^{({\bar\delta},-)}_0=u_0$, and for $n\in{\mathbb N}$, $n\delta<1-\mathnormal{\Delta}$,
\begin{align}
\notag
u^{({\bar\delta},-)}_{n\delta}
&=L^{(-)}_{n,{\bar\delta}}G_\delta u^{({\bar\delta},-)}_{(n-1)\delta}+e^{(-)}_{n,{\bar\delta}}
\\
\label{B016}
e^{(-)}_{n,{\bar\delta}}(y)
&=e^{-\mathnormal{\Delta}^4/2\delta}\del1_{[\sigma^{(-)}_n-1,\sigma^{(-)}_n]}(y),
\qquad y\in{\mathbb R},
\\
\notag
\sigma^{(-)}_n&= \sigma^{(-)}_n({\bar\delta})=\tilde R_\mathnormal{\Delta}(G_\delta u^{({\bar\delta},-)}_{(n-1)\delta}).
\end{align}

Following \cite{de-masi-book}, define, for $u,v\in L_1({\mathbb R},{\mathbb R}_+)$ and $m>0$,
the relation $u\preccurlyeq v$ mod $m$ as
\[
u(r,\infty)\le v(r,\infty)+m, \quad \text{for all } r\in{\mathbb R}.
\]
\begin{proposition}\label{Bprop2}
For every $\mathnormal{\Delta}>0$ sufficiently small
and every $0<\delta<\delta_0=\delta_0(\mathnormal{\Delta})$ and $n\in{\mathbb Z}_+$ such that $n\delta<1-\mathnormal{\Delta}$,
for any solution $(u,\beta)$ to \eqref{B018} one has
\[
u^{({\bar\delta},-)}_{n\delta}\preccurlyeq u_{n\delta} \text{ mod } e^{-\mathnormal{\Delta}^4/2\delta},
\qquad \text{and} \qquad
u_{n\delta}\preccurlyeq u^{({\bar\delta},+)}_{n\delta}.
\]
\end{proposition}

By Lemma \ref{Blem1}, there is a constant $c_\infty$ such that
for any solution $(u,\beta)$,
$\|u(\cdot,t)\|_1\le1$ and $\|u(\cdot,t)\|_\infty\le c_\infty$, $t\in[0,1]$.

\begin{lemma}\label{Blem2}
There exists $\delta_0>0$ such that
for any solution $(u,\beta)$ and any $(n,\delta)\in{\mathbb N}\times(0,\delta_0)$
satisfying $n\delta\le 1$,
\[
\rho_*(\beta;[(n-1)\delta,n\delta])\ge
R_{q^{(+)}_{n,\delta}+c_\infty\delta^{1/2}}(u_{(n-1)\delta}),
\]
provided that $q^{(+)}_{n,\delta}+c_\infty\delta^{1/2}\le\|u_{(n-1)\delta}\|_1$,
and
\[
\rho^*(\beta;[(n-1)\delta,n\delta])
\le R_{q^{(-)}_{n,\delta}-c_\infty\delta^{1/2}}(u_{(n-1)\delta}),
\]
provided that $q^{(-)}_{n,\delta}-c_\infty\delta^{1/2}\ge0$.
\end{lemma}

\proof
It suffices to prove the result for $n=1$, and to consider only the lower bound.
Fix $\delta$ and a solution $(u,\beta)$.
Denote $r=R_{q^{(+)}_{1,\delta}+c_\infty\delta^{1/2}}(u_0)$.
First, in the special case where 
$q^{(+)}_{1,\delta}+c_\infty\delta^{1/2}=\|u_0\|_1$
and $u_0(-\infty,x)>0$ for all $x$, there is nothing to prove because
$r=-\infty$. In all other cases, $r$ is finite. Therefore it is assumed
in what follows that $r$ is finite.

Arguing by contradiction, assume
$\rho_*(\beta;[0,\delta])<r$.
Then $\beta((-\infty,r)\times[0,\delta])>0$.
Using \eqref{B018}(iii),\[
\int_{[0,\delta]}[u((r,\infty),t)-q(t)]^+\beta((-\infty,r)\times dt)=0,
\]
it follows that
there exists $t\in[0,\delta]$ such that $u((r,\infty),t)\le q(t)\le q^{(+)}_{1,\delta}$.
It is impossible that $t=0$ since $u_0(r,\infty)=q^{(+)}_{1,\delta}+c_\infty\delta^{1/2}$.
Fix such $t$.
Denote the last term in \eqref{B012-} by
$\eta(y)=\int_{{\mathbb R}\times[0,t]}G_{t-s}(x,y)\beta(dx,ds)$. Then
\[
\int_r^\infty\eta(y)dy\le\int_{-\infty}^\infty\eta(y)dy=\beta({\mathbb R}\times[0,t])=t\le\delta.
\]
Since $u((r,\infty),t)\le q^{(+)}_{1,\delta}$, we have by \eqref{B012-},
\[
\int_r^\infty\int_{\mathbb R} G_t(x,y)u_0(x)dx\,dy\le q^{(+)}_{1,\delta}
+\int_r^\infty\eta(y)dy\le q^{(+)}_{1,\delta}+\delta.
\]
Hence if $A=\int_r^\infty G_t(x,(-\infty,r))u_0(x)dx$,
\begin{align*}
q^{(+)}_{1,\delta}+\delta &\ge\int_r^\infty G_t(x,(r,\infty))u_0(x)dx
=-A+u_0(r,\infty)\\
&=-A +q^{(+)}_{1,\delta}+c_\infty\delta^{1/2}.
\end{align*}
However,
\begin{align*}
A&\le c_\infty\int_r^\infty G_t(x,(-\infty,r))dx
\\
&=c_\infty\int_0^\infty G_t(0,(x,\infty))dx
=c_\infty(2\pi)^{-1/2}t^{1/2}
<\frac{c_\infty}{2}\delta^{1/2}.
\end{align*}
This gives $c_\infty\delta^{1/2}-\delta<\frac{1}{2}c_\infty\delta^{1/2}$,
which is impossible when $\delta$ is small.
\qed

\vspace{\baselineskip}

\noindent{\bf Proof of Proposition \ref{Bprop2}.}
There is an obvious symmetry in equation \eqref{B018} w.r.t.\ the reflection
$x\mapsto-x$, thanks to which the lower bound
follows from the upper bound. More precisely,
let $(u,\beta)$ be a solution to \eqref{B018}.
Define $(\tilde u,\tilde\beta)$ by composing $(u,\beta)$
with the reflection $x\mapsto-x$. Then
$(\tilde u,\tilde\beta)$ is a solution of \eqref{B018}
with data $(u_0(-\cdot),1-t-q(t))$ in place of $(u_0,q)$.
Further, there is symmetry in the construction of the upper and lower barriers:
If $u^{({\bar\delta},-)}$ are the lower barriers for the data $(u_0,q)$ and
$U^{({\bar\delta},+)}$ are the upper barriers for the data $(u_0(-\cdot),1-t-q(t))$,
then $U^{({\bar\delta},+)}=u^{({\bar\delta},-)}(-\cdot)$.
Assume now that the statement regarding the upper barriers is true.
Apply the upper bound $\tilde u\preccurlyeq U^{({\bar\delta},+)}$. Then for all $r$,
suppressing the time variable,
\[
u(-\infty,r]\le u^{({\bar\delta},-)}(-\infty,r]
\]
hence
\[
u^{({\bar\delta},-)}[r,\infty)\le u[r,\infty)+\|u^{({\bar\delta},-)}\|_1-\|u\|_1.
\]
Now, directly from \eqref{B016}, $\|u^{({\bar\delta},-)}_{n\delta}\|_1=1-n\delta+ne^{-\mathnormal{\Delta}^4/2\delta}\delta$.
By Lemma \ref{Blem1} and \eqref{B018}(iv), $\|u_{n\delta}\|_1=1-n\delta$.
This shows that $u^{({\bar\delta},-)}_{n\delta}\preccurlyeq u_{n\delta}$ mod $e^{-\mathnormal{\Delta}^4/2\delta}$,
which gives the lower bound.
It thus suffices to prove the upper bound.

The proof strategy is similar to that of Lemma \ref{prop2},
but various details differ, hence we present the proof in full.

In what follows, the superscript `$+$'
is suppressed from all notation except the upper barriers themselves
that we continue to denote as $u^{({\bar\delta},+)}$.
The parameter $\mathnormal{\Delta}>0$ is assumed to satisfy $2c_\infty\mathnormal{\Delta}^2<\mathnormal{\Delta}/4$.

We argue by induction. Assume that $f\preccurlyeq g$
where $f=u_{(n-1)\delta}$ and $g=u^{({\bar\delta},+)}_{(n-1)\delta}$.
Write $L$ and $G$ for $L^{(+)}_{n,{\bar\delta}}$ and $G_\delta$, respectively.
By Lemma \ref{Blem1}, $u_{n\delta}=Gf-h$,
$h(y)=\int_{{\mathbb R}\times[(n-1)\delta,n\delta]}G_{n\delta-s}(x,y)\beta(dx,ds)$.
Moreover, $u^{({\bar\delta},+)}_{n\delta}=LGg+e_{n,{\bar\delta}}$.
Denoting $w=Gf$, consider the inequality
\begin{equation}\label{B013}
w-h\preccurlyeq Lw+e_{n,{\bar\delta}}.
\end{equation}
If it holds then, using $f\preccurlyeq g$ and monotonicity
of $G$ and $L$ (see Lemma \ref{Blem3+}(ii)), it will follow that
$Gf-h\preccurlyeq LGg+e_{n,{\bar\delta}}$, hence
$u_{n\delta}=Gf-h\preccurlyeq LGg+e_{n,{\bar\delta}}=u^{({\bar\delta},+)}_{n\delta}$,
and the proof will be complete.
It thus remains to show \eqref{B013}.
Let $q^*=q^{(+)}_{n,\delta}$.

First, if $q^*+\mathnormal{\Delta}+\delta\ge\|f\|_1=\|w\|_1$ then by the way in which the operator
$L$ is defined, $Lw=\tilde K^\delta w$.
Hence by Lemma \ref{Blem3+}(iii),
noting that $\|h\|_1=\delta$, we have $w-h\preccurlyeq\tilde K^\delta w$.
This shows \eqref{B013}.

In what follows we address the case $q^*+\mathnormal{\Delta}+\delta<\|f\|_1$. In this case
$Lw=K^{q^*+\mathnormal{\Delta},\delta}w$.
Let $r_*=R_{q^*+c_\infty\delta^{1/2}}(f)$, which is a finite number
provided $\delta$ is small.
By Lemma \ref{Blem2} $\rho_*(\beta;[(n-1)\delta,n\delta])\ge r_*$.
Let $h=h_1+h_2$, where
\[
h_1(y)=h(y) 1_{\{y<r_*-\mathnormal{\Delta}^2\}},
\qquad
h_2(y)=h(y) 1_{\{y\ge r_*-\mathnormal{\Delta}^2\}}.
\]
Again, recall $\|h\|_1=\delta$.
Using
\begin{align*}
\int_{{\mathbb R}\times[(n-1)\delta,n\delta]}G_{n\delta-s}(x,y)\beta(dx,ds)
=\int_{[r_*,\infty)\times[(n-1)\delta,n\delta]}G_{n\delta-s}(x,y)\beta(dx,ds)
\end{align*}
and $G_t(0,[a,\infty))\le e^{-a^2/2t}$ $a>0$, $t<a^2$,
it is easy to see that
\begin{equation}\label{B053}
\|h_1\|_1\le e^{-\mathnormal{\Delta}^4/2\delta}\delta,
\end{equation}
provided $\delta<\mathnormal{\Delta}^4$.
Let $p\in(0,1]$ be defined by
$\|h_2\|_1=p\delta$. Then $p\ge1-e^{-\mathnormal{\Delta}^4/2\delta}$.

In order to show \eqref{B013} it will now be argued that
\begin{equation}\label{B051}
w-h\preccurlyeq K^{q^*+\mathnormal{\Delta},p\delta}w.
\end{equation}
Because $h_2$ is supported to the right of $r_*-\mathnormal{\Delta}^2$
and $K^{q^*+\mathnormal{\Delta},p\delta}w=w-\tilde h$, where $\|\tilde h\|_1=p\delta$
and $\tilde h$ is supported to the left of $R_{q^*+\mathnormal{\Delta}}(w)$, if one has
\begin{equation}\label{B020}
R_{q^*+\mathnormal{\Delta}}(w)\le R_{q^*+c_\infty\delta^{1/2}}(f)-\mathnormal{\Delta}^2,
\end{equation}
then one could deduce $w-h_2\preccurlyeq K^{q^*+\mathnormal{\Delta},p\delta}w$, hence
\[
w-h=w-h_1-h_2\preccurlyeq w-h_2\preccurlyeq K^{q^*+\mathnormal{\Delta},p\delta}w,
\]
giving \eqref{B051}.
Toward showing \eqref{B020}, it follows from $\|f\|_\infty<c_\infty$
that for all $\delta$ small (specifically, $c_\infty\delta^{1/2}<\mathnormal{\Delta}/4$
and recalling $2c_\infty\mathnormal{\Delta}^2<\mathnormal{\Delta}/4$),
\begin{equation}
\label{B052}
R_{q^*+\mathnormal{\Delta}/2}(f)\le R_{q^*+c_\infty\delta^{1/2}}(f)-2\mathnormal{\Delta}^2.
\end{equation}
Moreover, for small $\delta$,
\begin{equation}\label{B021}
R_{q^*+\mathnormal{\Delta}}(Gf)\le \theta:=R_{q^*+\mathnormal{\Delta}/2}(f)+\mathnormal{\Delta}^2,
\end{equation}
as we now show.
This is shown by proving that $(Gf)[\theta,\infty)\le q^*+\mathnormal{\Delta}$.
Let $f=f_1+f_2=\widehat K^{q^*+\mathnormal{\Delta}/2}f+K^{q^*+\mathnormal{\Delta}/2}f$.
Then $\|f_1\|_1=q^*+\mathnormal{\Delta}/2$, $\|Gf_1\|_1=q^*+\mathnormal{\Delta}/2$, and
\[
(Gf)[\theta,\infty)=(Gf_1)[\theta,\infty)+(Gf_2)[\theta,\infty)
\le q^*+\mathnormal{\Delta}/2+(G A)[\theta,\infty)
\]
where $A=\|f_2\|_1\boldsymbol{\delta}_x$ with $x=R_{q^*+\mathnormal{\Delta}/2}(f)$.
Because $\|f\|_1\le1$,
$(GA)[\theta,\infty)\le G_\delta(0,[\mathnormal{\Delta}^2,\infty))<\mathnormal{\Delta}/2$
for small $\delta$. This shows \eqref{B021}.

The bounds \eqref{B052} and \eqref{B021} imply \eqref{B020}.
This establishes \eqref{B051}.

Next,
\[
K^{q^*+\mathnormal{\Delta},p\delta}w=K^{q^*+\mathnormal{\Delta},\delta}w
+\widehat K^{q^*+\mathnormal{\Delta}+p\delta,(1-p)\delta}w.
\]
The last term on the RHS is nonnegative and supported
to the left of $R_{\mathnormal{\Delta}}(w)$, with total mass $(1-p)\delta$,
hence it is $\preccurlyeq (1-p)\del1_{[R_{\mathnormal{\Delta}}(w),R_{\mathnormal{\Delta}}(w)+1]}$.
Moreover, since $w\preccurlyeq Gg$, one has
$R_\mathnormal{\Delta}(w)\le R_\mathnormal{\Delta}(Gg)$, hence the same term is dominated by
$(1-p)\del1_{[R_{\mathnormal{\Delta}}(Gg),R_{\mathnormal{\Delta}}(Gg)+1]}$.
Using this in \eqref{B051},
\[
w-h\preccurlyeq K^{q^*+\mathnormal{\Delta},p\delta}w
\preccurlyeq K^{q^*+\mathnormal{\Delta},\delta}w+(1-p)\del1_{[R_\mathnormal{\Delta}(Gg),R_\mathnormal{\Delta}(Gg)+1]}.
\]
Since $1-p\le e^{-\mathnormal{\Delta}^4/2\delta}$, the last term is
$\preccurlyeq e_{n,{\bar\delta}}$, and \eqref{B013} follows.
\qed

The proof of the following result is similar to that of Proposition \ref{prop3},
but various details differ and so the proof is presented in full.

\begin{proposition}\label{Bprop3}
For all $\mathnormal{\Delta}$ sufficiently small and $0<\delta<\delta_1(\mathnormal{\Delta})$
and all $n\in{\mathbb Z}_+$ such that $n\delta<1-3\mathnormal{\Delta}-e^{-\mathnormal{\Delta}^4/2\delta}$, one has
\[
u^{({\bar\delta},+)}_{n\delta}\preccurlyeq u^{({\bar\delta},-)}_{n\delta}+\varepsilon_n,
\]
where $\varepsilon_n=\varepsilon_n({\bar\delta})\in L_1({\mathbb R},{\mathbb R}_+)$ and
\[
\|\varepsilon_n\|_1\le3\mathnormal{\Delta}+e^{-\mathnormal{\Delta}^4/2\delta}\delta n.
\]
\end{proposition}

\proof
Abbreviate $u^{({\bar\delta},\pm)}$, $e^{({\bar\delta},\pm)}_{n,{\bar\delta}}$,
$L^{(\pm)}_{n,{\bar\delta}}$ and $G_\delta$
to $u^{(\pm)}$, $e^{(\pm)}_n$, $L^{(\pm)}$ and $G$.
For $n=0$ one has $u^{(\pm)}_0=u_0$.

Let $n\ge1$.
The induction assumption is
$u^{(+)}_{(n-1)\delta}\preccurlyeq u^{(-)}_{(n-1)\delta}+\varepsilon_{n-1} =:f_{n-1}$,
$\varepsilon_{n-1}$ is nonnegative,
$\|\varepsilon_{n-1}\|_1\le \mathnormal{\Delta}_{n-1}:=3\mathnormal{\Delta}+e^{-\mathnormal{\Delta}^4/2\delta}\delta n$.
Then $u^{(+)}_{n\delta}=L^{(+)}Gu^{(+)}_{(n-1)\delta}+e^{(+)}_n$. Hence
\begin{align*}
u^{(+)}_{n\delta}&=L^{q^{(+)}_{n\delta}+\mathnormal{\Delta},\delta}Gu^{(+)}_{(n-1)\delta}+e_n^{(+)}
\\
&\preccurlyeq L^{q^{(+)}_{n\delta}+\mathnormal{\Delta},\delta}Gf_{n-1}+e_n^{(+)}
\end{align*}
where one uses the monotonicity of $L$ (Lemma \ref{Blem3+}(ii)) and $G$.
Using Lemma \ref{Blem3+}(v),
\[
u^{(+)}_{n\delta}\preccurlyeq
L^{q^{(+)}_{n,\delta}+\mathnormal{\Delta}-\mathnormal{\Delta}_{n-1},\delta}
K^{\mathnormal{\Delta}_{n-1}} Gf_{n-1}+\widehat K^{\mathnormal{\Delta}_{n-1}} Gf_{n-1}+e_n^{(+)}.
\]
Because $q$ is continuous on $[0,1]$,
one has $\max_{n:n\delta<1}(q^{(+)}_{n,\delta}-q^{(-)}_{n,\delta})<\mathnormal{\Delta}$
for all small $\delta$. Hence for $\delta$ small,
$q^{(+)}_{n,\delta}+\mathnormal{\Delta}-\mathnormal{\Delta}_{n-1}\le q^{(-)}_{n,\delta}-\mathnormal{\Delta}-\delta$.
Thus by Lemma \ref{Blem3+}(iv), denoting
$\varepsilon_n=\widehat K^{\mathnormal{\Delta}_{n-1}} Gf_{n-1}+e_n^{(+)}$
and $z=K^{\mathnormal{\Delta}_{n-1}} Gf_{n-1}$,
\[
u^{(+)}_{n\delta}\preccurlyeq L^{q^{(-)}_{n,\delta}-\mathnormal{\Delta}-\delta,\delta}z+e^{(-)}_{n,\delta}+\varepsilon_n.
\]
Suppose $z\preccurlyeq Gu^{(-)}_{(n-1)\delta}$. Then
$u^{(+)}_{n\delta}\preccurlyeq u^{(-)}_{n\delta}+\varepsilon_n$,
and $\|\varepsilon_n\|_1\le\mathnormal{\Delta}_{n-1}+\|e^{(+)}_n\|_1=\mathnormal{\Delta}_n$, as claimed.

Hence it remains to show $z\preccurlyeq Gu^{(-)}_{(n-1)\delta}$.
But $w:=G\varepsilon_{n-1}$ is pointwise nonnegative,
$\|w\|_1\le\mathnormal{\Delta}_{n-1}$ and
\[
Gu^{(-)}_{(n-1)\delta}=Gf_{n-1}-w.
\]
Hence by Lemma \ref{lem3}(ii), $z\preccurlyeq Gu^{(-)}_{(n-1)\delta}$,
and the proof is complete.
\qed

\subsection{Uniqueness and convergence}\label{sec55}

\noindent{\bf Proof of Theorem \ref{Bth2}.}

{\it Step 1: The uniqueness result.}

It will be shown that if there exists a solution $(u,\beta)$ to \eqref{B018}
then it is unique (the existence, as in the case of RAB,
follows from the convergence result proved afterwards).
As in the proof of Theorem \ref{th1},
if uniqueness is established for the $u$ component,
uniqueness of the $\beta$ component follows.
To show uniqueness of $u$,
argue by contradiction assuming $(u^i,\beta^i)$, $i=1,2$ are two
solutions with distinct $u^1$ and $u^2$.
Note that Lemma \ref{Blem1}
and \eqref{B018}(iv) dictate that $u(\cdot,t)=0$ for $t\ge1$.
Hence there exist $0<t<1$ and $r\in{\mathbb R}$ such that,
say, $u^1_t[r,\infty)<u^2_t[r,\infty)$. For $n\in{\mathbb N}$ let $\delta_n=tn^{-1}$.
Then by Proposition \ref{Bprop2}, for every small $\mathnormal{\Delta}>0$
and every $n>n_0(\mathnormal{\Delta})$,
\[
a(n,\mathnormal{\Delta}):=u^{(\delta_n,-)}_t[r,\infty)-e^{\mathnormal{\Delta}^4/2\delta_n}
\le u^1_t[r,\infty)< u^2_t[r,\infty)\le b(n,\mathnormal{\Delta}):=u^{(\mathnormal{\Delta},\delta_n,+)}_t[r,\infty),
\]
whereas by Proposition \ref{Bprop3},
\[
b(n,\mathnormal{\Delta})-a(n,\mathnormal{\Delta})\le\|\varepsilon_n(t,\mathnormal{\Delta},\delta_n)\|_1+e^{-\mathnormal{\Delta}^4/2\delta_n}
\le3\mathnormal{\Delta}+2e^{-\mathnormal{\Delta}^4/2\delta_n}.
\]
A contradiction is obtained upon sending $n\to\infty$ then $\mathnormal{\Delta}\downarrow0$.
This establishes uniqueness.

{\it Step 2: The convergence result.}

Given step 1,
the proof of existence of a solution and the convergence assertion
follow very closely the steps of the proof of Theorem \ref{th2},
comprising \S\ref{sec4}.
These steps are Lemmas \ref{lem4}, \ref{lem6}, \ref{lem5}
and \ref{lem7}, and finally the proof of Theorem \ref{th2}.
The only part of \S\ref{sec4} where
the specifics of the removal mechanism are used is the proof
of Lemma \ref{lem7} (that is not to say that the properties
of the function $J$, governing the cumulative removal count,
are not used; they are used for example in
Lemma \ref{lem6}. However the removal form is not used).
Thus the statements
of Lemmas \ref{lem4}, \ref{lem6} and \ref{lem5} and their proofs
cover the current model as a special case.
The only part of \S\ref{sec4} that needs to be modified is
Lemma \ref{lem7}. A modified statement and proof
are provided below in Lemma \ref{Blem7}.

Likewise, the proof of Theorem \ref{th2} based on uniqueness
and Lemmas \ref{lem4}--\ref{lem7} is valid here,
with Lemma \ref{Blem7} in place of Lemma \ref{lem7}.
This completes the proof of Theorem \ref{Bth2}.
\qed

\vspace{\baselineskip}

\begin{lemma}\label{Blem7}
Let $(\xi,\beta)$ be a subsequential limit of $(\bar\xi^N,\bar\beta^N)$.
Then a.s., for all $r\in{\mathbb R}$ one has ${\cal I}^\pm_r(\xi,\beta,q)=0$.
\end{lemma}

\proof
We provide a proof for ${\cal I}^+$; the one for ${\cal I}^-$ is similar.
Recalling \eqref{B120}, we must show
\begin{equation}\label{B025}
{\cal I}^+_r(\xi,\beta,q)
=\int_{{\mathbb R}_+}[\xi_t(r,\infty)-q(t)]^+\beta((-\infty,r)\times dt)
=0,\quad \text{for all } r\in{\mathbb R}.
\end{equation}
Let $\mathnormal{\Sigma}$ be the collection of tuples $\sigma=(r,t_1,t_2,\eta,\delta)\in{\mathbb Q}^5$,
$0\le t_1<t_2$, $\eta>0$, $\delta>0$. It is shown that for every
$\sigma\in\mathnormal{\Sigma}$, ${\mathbb P}(\mathnormal{\Omega}_\sigma)=0$ where
\[
\mathnormal{\Omega}_\sigma=\{\inf_{t\in[t_1,t_2]}[\xi_t(r,\infty)-q(t)]>\eta,\, \beta((-\infty,r)\times(t_1,t_2))>\delta\}.
\]
Fix $\sigma$. If ${\mathbb P}(\mathnormal{\Omega}_\sigma)>0$ then in view of
the convergence $(\bar\xi^N,\bar\beta^N)\Rightarrow(\xi,\beta)$
and the a.s.\ continuity of the sample paths of the limit $\xi$,
one has for all large $N$,
\[
{\mathbb P}\Big(\inf_{t\in[t_1,t_2]}[\bar\xi^N_t(r,\infty)-q(t)]>\eta/2,
\,\bar\beta^N((-\infty,r)\times(t_1,t_2))>\delta/2\Big)>0.
\]
However, the above probability is zero for all $N$, in view of \eqref{B058}.
This shows ${\mathbb P}(\mathnormal{\Omega}_\sigma)=0$, and consequently, ${\mathbb P}(\cup\mathnormal{\Omega}_\sigma)=0$.

The remainder of the proof is as that of Lemma \ref{lem7}
(with trivial adaptations).
\qed


\section{Other models}\label{sec6}

Additional models formulated via ORA are presented
with no rigorous claims or proofs. The goal is twofold:
to propose alternatives to additional macroscopic models that have previously
been formulated via FBP
(items 1 and 2), and to demonstrate that the approach gives rise to new models
that are quite far from FBP (items 3 and 4).

1. {\it Removal on both extremes.}
This is a straightforward extension
of the basic model where both the leftmost
and the rightmost particles are removed according to
prescribed functions $J^L$ and $J^R$.
The equation \eqref{018} is then modified as
${\cal L} u=\alpha-\beta^L-\beta^R$ with $u(\cdot,0)=u_0$,
$\beta^L({\mathbb R}\times[0,t])=J^L_t$, $\beta^R({\mathbb R}\times[0,t])=J^R_t$,
and two ORA conditions of opposite directions,
\[
\int_{{\mathbb R}_+}u((-\infty,r],t)\beta^L((r,\infty)\times dt)=0,
\qquad
\int_{{\mathbb R}_+}u([r,\infty),t)\beta^R((-\infty,r)\times dt)=0,
\qquad r\in{\mathbb R}.
\]
A FBP for this model (without injection, until extinction) was instrumental
in \cite{lac14} for the study of a hydrodynamic limit of a polymer pinning model.


2.
{\it The $N$-BBM model.} As mentioned in the introduction, hydrodynamic limits
of this model were identified in \cite{de-masi-nbbm} as a solution to
a FBP analyzed in \cite{ber19}. An alternative macroscopic model
consists of the equation ${\cal L} u=u-\beta$
coupled with an initial condition, a mass conservation condition
$\int_{\mathbb R} u(x,t)dx=1$, and the condition
\[
\int_{{\mathbb R}_+}u((-\infty,r],t)\beta((r,\infty)\times dt)=0, \qquad r\in{\mathbb R}.
\]
It is straightforward to extend this equation to the case of
nonlocal branching studied in \cite{de-masi-nbbm2}.

3. {\it Injection at quantile.} Similarly to removal at quantile,
one can formulate an equation with injection at a specified
quantile of the population. For example, consider a system
of $N$ Brownian particles on the line with rate-$N$ injection
and removal, injection occurring at the location of the
$a$-quantile member of the population and removal is restricted to the $b$-quantile
member. Here, $a\in(0,1)$ and $b\in[0,1]$ are parameters.
It is of interest to relate the hydrodynamic limit to an equation,
for the unknown $(u,\alpha,\beta)$, given by
${\cal L} u=\alpha-\beta$ coupled with an initial condition,
the condition $\alpha({\mathbb R}\times[0,t])=\beta({\mathbb R}\times[0,t])=t$, and
a variant of \eqref{B018}(iii),
\[
{\cal J}^\pm_r(u,\alpha,a)={\cal J}^\pm_r(u,\beta,b)=0,
\qquad
r\in{\mathbb R},
\]
where
\begin{align*}
{\cal J}_r^+(u,\gamma,c)
&=\int_{{\mathbb R}_+}[u((r,\infty),t)-c]^+\gamma((-\infty,r)\times dt),
\\
{\cal J}_r^-(u,\gamma,c)
&=\int_{{\mathbb R}_+}[u((-\infty,r),t)-(1-c)]^+\gamma((r,\infty)\times dt).
\end{align*}

4. {\it Other total orders.}
Equation \eqref{018} could be considered with
condition \eqref{018}(iii) replaced by
\[
\int_{x:x\prec r, t\in{\mathbb R}_+} \int_{y:r\prec y}u(y,t)dy\,\beta(dx,dt)=0,
\qquad r\in{\mathbb R},
\]
for a given (measurable) total order $\prec$ on ${\mathbb R}$.
This formulation is meaningful also for the heat equation in higher dimension,
in which case $\prec$ is a given total order of ${\mathbb R}^d$.
Of course, these equations are no longer alternatives to FBP,
especially in higher
dimension. A serious challenge in implementing barrier techniques, even in dimension $1$,
is that the monotonicity properties of the heat kernel extensively used in the proofs
fail to hold w.r.t.\ other orders.



\paragraph{Acknowledgment}
The author would like to thank Kavita Ramanan for valuable discussions and comments.
This research was supported in part by the Israel Science Foundation grant 1035/20.




\footnotesize

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