# The heat equation with order-respecting absorption and particle systems with
topological interaction

Rami Atar111Viterbi Faculty of Electrical Engineering, Technion, Haifa 32000,
Israel
<EMAIL_ADDRESS>
(November 15, 2020)

###### Abstract

A PDE formulation is proposed, referred to as a heat equation with order-
respecting absorption, aimed at characterizing hydrodynamic limits of a class
of particle systems on the line with topological interaction that have so far
been described by free boundary problems. It consists of the heat equation
with measure-valued injection and absorption terms, where the absorption
measure respects the usual order on ${\mathbb{R}}$ in the sense that, for all
$r\in{\mathbb{R}}$, it charges $(-\infty,r)$ only at times when the solution
vanishes on $(r,\infty)$. The formulation is used to obtain new hydrodynamic
limit results for two models. One is a variant of the main model studied by
Carinci, De Masi, Giardinà and Presutti [21] where Brownian particles undergo
injection according to a general injection measure, and removal that is
restricted to the rightmost particle of the configuration. This partially
addresses a conjecture of [21]. Next a Brownian particle system is considered
where the $Q$-quantile member of the population is removed until extinction,
where $Q$ is a given $[0,1]$-valued continuous function of time. Here, unlike
in earlier work on the subject, the removal mechanism acts on particles that
are ‘at the boundary’ but are not rightmost or leftmost. Finally, further
potential uses of order-respecting absorption are mentioned.

##  1 Introduction

Particle systems with topological interactions are models for particle
dynamics on the real line where the so called 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 [20, 21, 22, 23, 24, 26]. A prototype of this class
of models is the following variant of the basic model studied by Carinci, De
Masi, Giardinà and Presutti in the monograph [21]. 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: 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 order-respecting
absorption (ORA) condition

$\int_{{\mathbb{R}}_{+}}u((r,\infty),t)\beta((-\infty,r)\times
dt)=0\quad\text{for all}\quad r\in{\mathbb{R}},$ (1)

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 §2 for the precise definition of
a solution and statement of convergence). This model is referred to in this
paper as 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 [21] and references therein). In
particular, the basic model of [21] (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 [21]
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
$-\frac{1}{2}\partial_{x}u(0,t)=j$, and the following conditions at the free
boundary,

$u(X_{t},t)=0,\qquad-\frac{1}{2}\partial_{x}u(X_{t},t)=j.$

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 [21], for which existence and
uniqueness are established. It is also established that this solution
describes the hydrodynamic limit of the particle configuration process.

In [21], 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 [21, §15.1] 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 [21]). 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 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 [20, 21, 22, 23, 24], 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
$\beta({\mathbb{R}}\times[0,t])=t\wedge 1$, 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 [20], where particles perform random walks on
$[0,N]\cap{\mathbb{Z}}$ rather than Brownian motion. In [22], 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 [21], 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 [23]
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 [19]. A variant of this model, in which the branching is nonlocal,
was studied in [24], 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 §6) 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 [24] 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 [18] 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 §2). 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 [21] for the history of its use; it has
been employed in particular in [20, 21, 22, 23, 24, 26]. 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 [21] 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 [21, 20, 23] in that we do not construct stochastic
barriers, but instead our arguments are based on tightness of the rescaled
processes.

#### Organization of this paper

§2.1 introduces the particle system with RAB and asserts that the hydrodynamic
limit exists (Theorem 2.1). In §2.2, a general treatment from [17] 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 (8), and a uniqueness result is stated (Theorem 2.2). The main
result regarding this model, Theorem 2.3, is given in §2.3, stating that the
scaling limit has a density and identifying this density as the unique
solution of (8). Also, a comparison principle for solutions is stated (Theorem
2.4), 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. §3 gives
the proof of Theorem 2.2, starting with a representation of $u$ in terms of
$\alpha$ and $\beta$ as a mild solution (§3.1) followed by mass transport
inequalities, some of which are borrowed from [21], stated in §3.2. Lower and
upper barriers are constructed in §3.3 and §3.4, resp. The result is finally
proved in §3.5 by showing that the upper and lower barriers cannot be
separated by more than one element. In §4.1, Theorem 2.3 is proved, and as a
corollary, Theorem 2.1. The proof is based on a tightness argument (Lemma
4.2), followed by estimates on the empirical density, used in conjunction with
a result due to Doob found in [25] on existence of a measurable density, to
construct such a density for any subsequential limit of the rescaled
configuration process $\bar{\xi}^{N}$ (Lemma 4.3). In Lemma 4.4 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 2.3 that appears at the end of the
section shows that $(u,\beta)$ constitute solutions to equation (8). The
uniqueness of such solution established in Theorem 2.2 then implies the
existence of limits. In §4.2 Theorem 2.4 is proved via a coupling argument for
the particle system. §5 treats the RAQ model starting with §5.1 that construct
the model. The heat equation (35) is presented and the main result regarding
this model, Theorem 5.1, is stated in §5.2. §5.3 lists and proves further mass
transport inequalities. Barriers are constructed in §5.4, and §5.5 completes
the proof. In §6 further examples are provided of models that can be
formulated via ORA, yet to be studied.

#### 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 càdlàg 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.

##  2 Removal at boundary: model and results

###  2.1 Brownian particles with injection and boundary removal

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

$\varepsilon_{0}:=\inf_{t\in{\mathbb{R}}_{+}}(1+I_{t}-J_{t})>0.$ (2)

In addition, it is required that $\sup_{t}I_{t}=\sup_{t}J_{t}=\infty$, 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 $\\{x^{i},1\leq i\leq N\\}$, $\\{x^{i},i\geq
N+1\\}$ 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, $x^{i}$, $1\leq i\leq N$ 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, $x^{i}$, $i\geq N$
are IID real-valued RVs distributed according to $\pi$, representing the
injection positions of particles injected after time $0$. Finally,
$\\{\tilde{B}^{i}_{t},t\geq 0\\}$, $i\in{\mathbb{N}}$ 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 $i\geq N+1$, the injection time of
particle $i$ is at $\sigma^{i}=\sigma^{i}(N)=\inf\\{t:N^{I}_{t}\geq i\\}$. Set
$\sigma^{i}=0$ for $i\leq N$. For $i\in{\mathbb{N}}$, we refer to the process
$\\{B^{i}_{t},t\in[\sigma^{i},\infty)\\}$ as the potential trajectory
associated with particle $i\in{\mathbb{N}}$, where

$B^{i}_{t}=x^{i}+\tilde{B}^{i}_{t-\sigma_{i}},\qquad t\geq\sigma^{i}.$ (3)

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
$\eta^{i}=\eta^{i}(N)=\inf\\{t:N^{J}_{t}\geq i\\}$, for $i\in{\mathbb{N}}$.

The number of particles in the system at time $t$ is $N+[NI_{t}]-[NJ_{t}]\geq
N\varepsilon_{0}-1$. Thus (2) ensures that the number of particles is always
positive, provided $N\varepsilon_{0}-1\geq 1$, 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 $\operatorname*{arg\,max}\\{g(i):i\in
C\\}=\min\\{i\in C:g_{i}=\max_{C}g\\}$. Let $[a,b)=\emptyset$ if $b\leq a$ 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
$A_{t}=\\{i:i\leq N^{I}_{t}\\}$. For $j\geq 1$, $A_{t}$ 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

$A_{\eta_{j}}=C_{j}\setminus\\{\operatorname*{arg\,max}\\{B^{i}_{\eta_{j}}:i\in
C_{j}\\}\\}.$

Finally, for $t\in(\eta_{j},\eta_{j+1})$, new injections are accounted for by
letting $A_{t}=A_{\eta_{j}}\cup\\{i:N^{I}_{\eta_{j}}<i\leq N^{I}_{t}\\}$. 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
$i\in\cup_{t}A_{t}$, 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 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

$\xi^{N}_{t}(dx)=\sum_{i=1}^{N^{I}_{t}}\boldsymbol{\delta}_{B^{i}_{t}}(dx)1_{\\{t<\tau^{i}\\}}=\sum_{i\in{\mathbb{N}}}\boldsymbol{\delta}_{B^{i}_{t}}(dx)1_{\\{\sigma^{i}\leq
t<\tau^{i}\\}}.$

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
${\cal M}_{\rm loc}({\mathbb{R}}_{+}\times{\mathbb{R}})$,

$\alpha^{N}(dx,dt)=\sum_{i\geq
N+1}\boldsymbol{\delta}_{(x_{i},\sigma_{i})}(dx,dt).$

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,

$\gamma^{N}_{t}=\xi^{N}_{t}+\int_{[0,t]}\beta^{N}(\cdot,ds)=\sum_{i\leq
N^{I}_{t}}\boldsymbol{\delta}_{B^{i}_{t\wedge\tau_{i}}},$ (4)
$\zeta^{N}_{t}=\sum_{i=1}^{N^{I}_{t}}\boldsymbol{\delta}_{B^{i}_{t}}.$ (5)

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,

$\int_{(-\infty,r)\times{\mathbb{R}}_{+}}\langle
1_{[r,\infty)},\bar{\xi}^{N}_{t}\rangle\bar{\beta}^{N}(dx,dt)=0,\qquad
r\in{\mathbb{R}}.$ (6)

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.

######  Theorem 2.1.

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.

The proof of this result is given at the end of §4, and is based on uniqueness
of solutions to the heat equation with an ORA condition, described next.

###  2.2 Heat equation with ORA

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

$\begin{cases}\partial_{t}u-\frac{1}{2}\partial_{xx}u=\mu&\text{in
}{\mathbb{R}}\times{\mathbb{R}}_{+},\\\ \\\ u(\cdot,0)=u_{0}&\text{on
}{\mathbb{R}}.\end{cases}$ (7)

Let $q\in(1,\infty)$. A (weak) $L_{q}$-solution of (7) 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 [17]. In particular, by
[17, Theorem 1 and Remark 1(b)], for $1<q<\infty$, $0\leq\sigma<\frac{q+1}{q}$
and $p\geq 1$ 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 $v[y,z)=\int_{y}^{z}v\,dx$
when $v\in L_{1}({\mathbb{R}},{\mathbb{R}})$. Consider the problem

$\begin{cases}{\it(i)}\hskip
35.00005pt\partial_{t}u-\frac{1}{2}\partial_{xx}u=\alpha-\beta&\text{in
}{\mathbb{R}}\times{\mathbb{R}}_{+},\\\ \\\ {\it(ii)}\hskip
33.00008ptu(\cdot,0)=u_{0}&\text{on }{\mathbb{R}},\\\ \\\ {\it(iii)}\hskip
28.00006pt\displaystyle\int_{{\mathbb{R}}_{+}}u([r,\infty),t)\beta((-\infty,r)\times
dt)=0&\text{for all }r\in{\mathbb{R}},\\\ \\\ {\it(iv)}\hskip
31.0001pt\beta({\mathbb{R}}\times[0,t])=J_{t}&\text{for all
}t\in{\mathbb{R}}_{+}.\end{cases}$ (8)

Define a (weak) solution to (8) 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 (8)(iii) and (8)(iv) hold, and moreover for any test function
$\varphi\in C_{c}^{\infty}({\mathbb{R}}\times{\mathbb{R}}_{+})$,

$-\int_{0}^{\infty}(\partial_{t}\varphi+\frac{1}{2}\partial_{xx}\varphi,u)dt=(\varphi(\cdot,0),u_{0})+\int_{{\mathbb{R}}\times{\mathbb{R}}_{+}}\varphi\,d\alpha-\int_{{\mathbb{R}}\times{\mathbb{R}}_{+}}\varphi\,d\beta.$
(9)

######  Theorem 2.2.

Assume there exists a solution $(u,\beta)$ to (8). Then it is unique.

This result is proved in §3.

###  2.3 Main result

The main result regarding this model summarizes Theorems 2.1 and 2.2, and
relates the limit $(\xi,\beta)$ from the former to the solution $(u,\beta)$
from the latter.

######  Theorem 2.3.

There exists a unique solution $(u,\beta)$ to (8). 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
$N\to\infty$.

See Remark 3.3 for further properties of the solution following from [17]. The
proof of Theorem 2.3 is based on §3–4 and is given at the end of §4.

Following is comparison principle for solutions of (8). 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 $u[r,\infty)\leq v[r,\infty)$ 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
$\nu[r,\infty)\leq\mu[r,\infty)$ for all $r\in{\mathbb{R}}$.

######  Theorem 2.4.

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 (8)
satisfy $\tilde{u}(\cdot,t)\preccurlyeq u(\cdot,t)$, $t\in{\mathbb{R}}_{+}$.

The proof is provided in §4.2 and uses, besides Theorem 2.3, a coupling
argument which shows a similar property for the configuration process.

######  Remark 2.1.

The condition (8)(iii) is closely related to the Skorohod map
$\mathnormal{\Phi}:D({\mathbb{R}}_{+},{\mathbb{R}})\to
D({\mathbb{R}}_{+},{\mathbb{R}})$ given by
$\mathnormal{\Phi}(\varphi)(t)=-\inf_{s\in[0,t]}(\varphi(s)\wedge 0)$. 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 (8). Denote

$\widehat{u}_{t}(r)=u([r,\infty),t),\quad\widehat{\beta}_{t}(r)=\beta((-\infty,r)\times[0,t]),\quad\widehat{\gamma}_{t}(r)=u([r,\infty),t)+\beta([r,\infty)\times[0,t])-J_{t}.$
(10)

Then by (8)(iv),

$\displaystyle\widehat{u}_{t}(r)$
$\displaystyle=\widehat{u}_{t}(r)+\beta((-\infty,r)\times[0,t])+\beta([r,\infty)\times[0,t])-J_{t}$
$\displaystyle=\widehat{\gamma}_{t}(r)+\widehat{\beta}_{t}(r).$

One also has $\widehat{u}_{t}(r)\geq 0$ and, by (8)(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

$\widehat{\beta}_{t}(r)=-\inf_{s\in[0,t]}(\widehat{\gamma}_{s}(r)\wedge
0),\qquad\widehat{u}_{t}(r)=\widehat{\gamma}_{t}(r)+\widehat{\beta}_{t}(r).$
(11)

From this one may draw two conclusions. First, an alternative way to define a
solution $(u,\beta)$ is to replace (8)(iii)–(iv) by the relation (11), where
again $\widehat{\beta}$ and $\widehat{\gamma}$ are given by (10).

Second, let $\mathnormal{\Psi}$ denote the solution map for the problem (7),
so that $u=\mathnormal{\Phi}(\mu;u_{0})$. Then a solution $(u,\beta)$ of (8)
satisfies $u=\mathnormal{\Phi}(\alpha-\beta;u_{0})$. As a result, the third
part of (10) 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 (11) 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 (8), can be
deduced from abstract fixed point theorems is not obvious to us.

##  3 Removal at boundary: uniqueness via barriers

This section proves uniqueness of solutions to (8) by constructing barriers,
which are shown to constitute upper and lower bounds to any solution, in the
sense of mass transport inequalities. In §3.1 a representation of $u$ in terms
of $\alpha$ and $\beta$ is given. In §3.2, some mass transport inequalities
are provided, used in the sequel. The lower and upper barriers are constructed
in§3.3 and §3.4. Finally, the proof of Theorem 2.2 appears in §3.5. Throughout
this section, an admissible data $(u_{0},\pi,I,J)$ is fixed.

###  3.1 Mild solutions

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

$G_{t}u(y)=\int_{\mathbb{R}}G_{t}(x,y)u(x)dx,$

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 $t\geq\tau$.

######  Lemma 3.1.

Assume that a solution $(v,\beta)$ to (8) exists. Then $v$ has a version $u$
given by

$u(y,t)=G_{t}u_{0}(y)+G*\alpha(y,t)-G*\beta(y,t).$ (12)

Moreover, for $0\leq\tau<t$,

$u(y,t)=G_{t-\tau}u(\cdot,\tau)(y)+G*\alpha(y,t;\tau)-G*\beta(y,t;\tau).$ (13)

Furthermore, $u\in L_{\infty,{\rm loc}}({\mathbb{R}}_{+},L_{1})\cap
L_{\infty,{\rm loc}}({\mathbb{R}}_{+},L_{\infty})$.

######  Remark 3.1.

If $(v,\beta)$ is a solution to (8) and $u=v$ a.e. then $(u,\beta)$ is also a
solution; specifically, Condition (8)(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 $J\in AC^{\uparrow}({\mathbb{R}}_{+},{\mathbb{R}}_{+})$.

######  Remark 3.2.

Throughout what follows, by a solution to (8) we will always mean the version
given by (12). In view of the above lemma there is no loss of generality in
doing so.

Proof. Let $(v,\beta)$ be a solution, and define $u$ by (12). First we argue
that the integrals in (12) 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 $\varepsilon\downarrow 0$. To show this, bound the above
integral by

$\displaystyle\int_{[0,t-\varepsilon]}(t-s)^{-1/2}dI_{s}$
$\displaystyle=I_{t-\varepsilon}\varepsilon^{-1/2}-\frac{1}{2}\int_{0}^{t-\varepsilon}I_{s}(t-s)^{-3/2}ds$
$\displaystyle=(I_{t-\varepsilon}-I_{t})\varepsilon^{-1/2}+\frac{1}{2}\int_{0}^{t-\varepsilon}(I_{t}-I_{s})(t-s)^{-3/2}ds+I_{t}t^{-1/2}$
$\displaystyle\leq
c\int_{0}^{t}(t-s)^{\rho_{0}-\frac{3}{2}}ds+I_{t}t^{-1/2}<\infty,$ (14)

where $\rho_{0}>1/2$ is used. A similar argument applies for the integral in
(12) by appealing to (8)(iv) and the Hölder continuity of $J$.

Next, given a test function $\varphi$, identity (9) 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 $|u(y,t)|\leq
G_{t}u_{0}(y)+G*\alpha(y,t)+G*\beta(y,t)$. Hence

$\|u(\cdot,t)\|_{1}\leq\|u_{0}\|_{1}+\|\pi\|_{1}I_{t}+J_{t}=1+I_{t}+J_{t},$
$\|u(\cdot,t)\|_{\infty}\leq\|u_{0}\|_{\infty}+\int_{[0,t]}(t-s)^{-1/2}(dI_{s}+dJ_{s}).$

The above integral is locally bounded as the calculation (14) 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 (7), 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 [17, Theorem 1], 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 (9) is satisfied
by both $u$ and $v$, they are both weak $L_{q}$-solutions to (7) and as a
result, $u=v$ a.e.

Finally, (13) follows from (12) by a direct calculation that we omit, upon
using the identity $\int G_{t-\tau}(x,y)G_{\tau}(y,z)dy=G_{t}(x,z)$. ∎

###  3.2 Mass transport inequalities

We borrow from [21] 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

$R_{\delta}(u)=\sup\\{x\in{\mathbb{R}}:u[x,\infty)\geq\delta\\}$

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 $\delta\geq 0$, 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 $\widehat{K}^{\bar{\delta}}={\tt id}-K^{\bar{\delta}}$.

######  Lemma 3.2.

Let $\delta\geq 0$ 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
$\|w\|_{1}\leq\delta$ 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 $0<\widehat{\mathnormal{\Delta}}\leq\mathnormal{\Delta}$ then
$K^{\widehat{\mathnormal{\Delta}},\delta}u\preccurlyeq
K^{\mathnormal{\Delta},\delta}u$.

Proof. i. These are well known properties. For monotonicity of $K^{\delta}$
see [21, Lemma 6.6]. As for $G_{\delta}$, the same lemma in [21] 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
$\int_{r}^{\infty}w\leq\int_{r}^{\infty}\tilde{w}$ for all $r$. First consider
$r\geq c$. In this case, noting that $w\leq u$ pointwise, we have
$\int_{r}^{\infty}w\leq\int_{c}^{\infty}u$, but
$\int_{r}^{\infty}\tilde{w}=\int_{r}^{\infty}u$.

Next, if $r<c$ we have $\int_{r}^{\infty}w\leq\delta$ but
$\int_{r}^{\infty}\tilde{w}=\delta$.

iii. We have $u(r,\infty)\leq v(r,\infty)$ for all $r$. We must show that
$\widehat{u}(r,\infty)\leq\widehat{v}(r,\infty)$ 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 $b\leq\bar{b}$ and $a\leq\bar{a}$. We split into
cases.
For $r\geq\bar{b}$ it is clear that
$\widehat{u}(r,\infty)\leq\widehat{v}(r,\infty)$.
For $r\in[\bar{a},\bar{b})$, $\widehat{v}(r,\infty)=\mathnormal{\Delta}$ but
$\widehat{u}(r,\infty)\leq\widehat{u}(a,\infty)=\mathnormal{\Delta}$.
For $r\in[a,\bar{a})$,
$\widehat{u}(r,\infty)\leq\widehat{u}(a,\infty)=\mathnormal{\Delta}$, but
$\widehat{v}(r,\infty)\geq\widehat{v}(\bar{a},\infty)=\mathnormal{\Delta}$.
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. ∎

###  3.3 Lower barriers

For $u\in L_{1}({\mathbb{R}},{\mathbb{R}}_{+})$ and $0\leq\tau<t$ 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

$P_{n,\delta}G_{\delta}u^{(\delta,-)}_{(n-1)\delta}\in H_{j_{n}(\delta)}.$
(15)

Our goal here is to show that the lower barriers form lower bounds on
solutions to (8).

######  Proposition 3.1.

Fix $\delta>0$. Then (15) holds for all $n\in{\mathbb{N}}$, and consequently
the lower barriers are well defined. Moreover, let $(u,\beta)$ be a solution
to (8). Then for $n\in{\mathbb{Z}}_{+}$,

$u^{(\delta,-)}_{n\delta}\preccurlyeq u_{n\delta}.$

Proof. We have $\|u_{0}\|_{1}=1$. Therefore for $n=1$, the $L_{1}$ norm of the
LHS of (15) is $1+I_{\delta}\geq J_{\delta}$, where (2) is used. This shows
that (15) holds for $n=1$. Next, assume that (15) holds for all
$n\in\\{1,\ldots,m\\}$. Then $u_{1},\ldots,u_{m}$ are well defined, and

$\|u_{m\delta}^{(\delta,-)}\|_{1}=\|u_{(m-1)\delta}^{(\delta,-)}\|_{1}+(I_{m\delta}-I_{(m-1)\delta})-(J_{m\delta}-J_{(m-1)\delta})$
(16)

whence $\|u_{m\delta}^{(\delta,-)}\|_{1}=1+I_{m\delta}-J_{m\delta}$. As a
result, the $L_{1}$ norm of the LHS of (15) with $n=m+1$ is

$1+I_{(m+1)\delta}-J_{m\delta}>j_{m+1}(\delta)$ (17)

by (2). This shows that (15) 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 3.1, $u_{n\delta}=PGg-h$, 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 3.2(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 $Kw\preccurlyeq w-h$. 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 $r\geq r_{0}$,

$\displaystyle w(r,\infty)-h(r,\infty)$
$\displaystyle=w(r,\infty)-w(r,\infty)$ $\displaystyle=0$ $\displaystyle\leq
w(r,\infty)-h(r,\infty).$

For $r<r_{0}$,

$\displaystyle w(r,\infty)-q(r,\infty)$
$\displaystyle=w(r,\infty)-j_{n}(\delta)$ $\displaystyle\leq
w(r,\infty)-h(r,\infty).$

This shows $w-q\preccurlyeq w-h$ and completes the proof. ∎

###  3.4 Upper barriers

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 $n\in{\mathbb{N}}$,

$\displaystyle u^{({\bar{\delta}},+)}_{n\delta}$
$\displaystyle=K_{n,{\bar{\delta}}}P_{n,\delta}G_{\delta}u^{({\bar{\delta}},+)}_{(n-1)\delta}+e_{n,{\bar{\delta}}}$
$\displaystyle e_{n,{\bar{\delta}}}(y)$
$\displaystyle=e^{-\mathnormal{\Delta}^{4}/2\delta}j_{n}(\delta)1_{[\sigma_{n},\sigma_{n}+1]}(y),\qquad
y\in{\mathbb{R}}_{+},$ (18) $\displaystyle\sigma_{n}$
$\displaystyle=\sigma_{n}({\bar{\delta}})=R_{\mathnormal{\Delta}}(P_{n,\delta}G_{\delta}u^{({\bar{\delta}},+)}_{(n-1)\delta}).$

Once again, for the definition to be valid, one must assure that for all
$n\in{\mathbb{N}}$,

$P_{n,\delta}G_{\delta}u^{({\bar{\delta}},+)}_{(n-1)\delta}\in
H_{\mathnormal{\Delta}+j_{n}(\delta)}.$ (19)

######  Proposition 3.2.

For $\mathnormal{\Delta}\in(0,\varepsilon_{0})$ and $\delta>0$, (19) holds for
all $n$ and consequently the upper barriers are well defined. Moreover, let
$(u,\beta)$ be a solution to (8). 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 $n\delta\leq T$.

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 (8) also has a removal mechanism, expressed by $\beta$ and
constrained by conditions (8)(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.

######  Lemma 3.3.

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}\geq
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 $j_{n}(\delta)>0$.}$

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 (8)(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 (12). Denoting the last term there by
$\eta(y)=\int_{{\mathbb{R}}\times[0,t]}G_{t-s}(x,y)\beta(dx,ds)$, clearly

$\int_{b}^{\infty}\eta(y)dy\leq\int_{-\infty}^{\infty}\eta(y)dy=\beta({\mathbb{R}}\times[0,t])=J_{t}\leq
J_{\delta}.$

Since $u(\cdot,t)$ vanishes on $[b,\infty)$, we have for the first term in
(12),

$\int_{b}^{\infty}\int_{\mathbb{R}}G_{t}(x,y)u_{0}(x)dx\,dy\leq\int_{b}^{\infty}\eta(y)dy\leq
J_{\delta}.$

Using $\int_{x}^{\infty}G_{t}(x,y)dy=1/2$,

$\displaystyle J_{\delta}$
$\displaystyle\geq\int_{x\in[b,\infty)}\int_{y\in[b,\infty)}G_{t}(x,y)u_{0}(x)dy\,dx$
$\displaystyle\geq\frac{1}{2}\int_{b}^{\infty}u_{0}(x)dx=\frac{3J_{\delta}}{2},$

a contradiction due to the assumption $J_{\delta}=j_{1}(\delta)>0$. This
proves the claim. ∎

Proof of Proposition 3.2. First, to prove that when
$\mathnormal{\Delta}<\varepsilon_{0}$ one has (19) for all $n$ (and therefore
the upper barriers are well defined) we argue as in the proof of Proposition
3.1. In place of equality in (16) we write an inequality $\geq$, and in place
of (17), we write
$1+I_{(m+1)\delta}-J_{m\delta}>\mathnormal{\Delta}+j_{m+1}(\delta)$, that
again follows from (2). 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 3.1 and (2) that $\|u(\cdot,t)\|_{1}\geq\varepsilon_{0}$. 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 3.3 is valid
provided merely that $j_{n}(\delta)>0$.

It follows from Lemma 3.1 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
$\|u(\cdot,t)\|_{1}\leq c_{1}$ 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 3.1, $u_{n\delta}=PGf-h$, 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

$w-h\preccurlyeq Kw+e_{n,{\bar{\delta}}}$ (20)

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 3.2(i) and
(iii)), which completes the proof. It remains to show (20).

First, if $j_{n}(\delta)=0$ then both sides of (20) 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 3.3 are thus
satisfied, and the lemma asserts that $\rho_{n,\delta}\geq b$. Write
$h=h_{1}+h_{2}$, where

$h_{1}(y)=h(y)1_{\\{y<b-\mathnormal{\Delta}^{2}\\}},\qquad
h_{2}(y)=h(y)1_{\\{y\geq b-\mathnormal{\Delta}^{2}\\}}.$

Note that $\|h\|_{1}=j$. A straightforward calculation based on

$\displaystyle\int_{{\mathbb{R}}\times[(n-1)\delta,n\delta]}G_{n\delta-s}(x,y)\beta(dx,ds)=\int_{[b,\infty)\times[(n-1)\delta,n\delta]}G_{n\delta-s}(x,y)\beta(dx,ds)$

(which follows from Lemma 3.3), 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 $G_{t}(0,[a,\infty))\leq
e^{-a^{2}/2t}$ $a>0$, $t<a^{2}$, shows that the mass of $h_{1}$ is much
smaller, specifically,

$\|h_{1}\|_{1}\leq e^{-\mathnormal{\Delta}^{4}/2\delta}j,$ (21)

provided $\delta<\mathnormal{\Delta}^{4}$. Let $q\in(0,1]$ be defined by
$\|h_{2}\|_{1}=qj$. Then $q\geq 1-e^{-\mathnormal{\Delta}^{4}/2\delta}$.

Our next step toward showing (20) is to show

$w-h\preccurlyeq K^{\mathnormal{\Delta},qj}w.$ (22)

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

$R_{\mathnormal{\Delta}}(w)\leq R_{3j}(f)-\mathnormal{\Delta}^{2},$ (23)

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 (22).

To show (23), note first, by boundedness $\|f\|_{\infty}<c_{\infty}$, that for
all $\delta$ so small that $3j<\mathnormal{\Delta}/6$ and recalling
$2c_{\infty}\mathnormal{\Delta}^{2}<\mathnormal{\Delta}/6$,

$R_{\mathnormal{\Delta}/3}(f)\leq R_{3j}(f)-2\mathnormal{\Delta}^{2}.$ (24)

Also, we argue that for all small $\delta$,

$R_{2\mathnormal{\Delta}/3}(Gf)\leq\theta:=R_{\mathnormal{\Delta}/3}(f)+\mathnormal{\Delta}^{2}.$
(25)

To show this we must to show $(Gf)[\theta,\infty)\leq 2\mathnormal{\Delta}/3$.
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

$(Gf)[\theta,\infty)=(Gf_{1})[\theta,\infty)+(Gf_{2})[\theta,\infty)\leq\mathnormal{\Delta}/3+(GA)[\theta,\infty)$

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 $(GA)[\theta,\infty)\leq
c_{1}G_{\delta}(0,[\mathnormal{\Delta}^{2},\infty))$, which of course is
smaller than $\mathnormal{\Delta}/3$ if $\delta$ is small. This shows (25).

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,

$R_{\mathnormal{\Delta}}(w)=R_{\mathnormal{\Delta}}(PGf)=R_{\mathnormal{\Delta}}(Gf+\bar{\pi})\leq
R_{2\mathnormal{\Delta}/3}(Gf).$

Combining this with (24) and (25) gives (23). Hence (22) 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
$R_{\mathnormal{\Delta}}(w)\leq R_{\mathnormal{\Delta}}(PGg)$, 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 (22),

$w-h\preccurlyeq K^{\mathnormal{\Delta},qj}w\preccurlyeq
K^{\mathnormal{\Delta},j}w+(1-q)j1_{[R_{\mathnormal{\Delta}}(PGg),R_{\mathnormal{\Delta}}(PGg)+1]}.$

Recalling that by (21), $1-q\leq e^{-\mathnormal{\Delta}^{4}/2\delta}$, it
follows that the last term is $\preccurlyeq e_{n,{\bar{\delta}}}$. This proves
(20) and completes the proof. ∎

###  3.5 Proof of uniqueness result

The last step is showing that the lower and upper barriers become close upon
taking $\delta\to 0$ then $\mathnormal{\Delta}\to 0$.

######  Proposition 3.3.

Given $T$, let $\mathnormal{\Delta}_{0}=\mathnormal{\Delta}_{0}(T)$ and
$\delta_{0}=\delta_{0}(T,\mathnormal{\Delta}_{0})$ be as in Proposition 3.2.
Then for $\mathnormal{\Delta}\in(0,\mathnormal{\Delta}_{0})$,
$\delta\in(0,\delta_{0})$ and $n\in{\mathbb{Z}}_{+}$, $n\delta\leq T$, 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

$\|\varepsilon_{n}\|_{1}\leq\mathnormal{\Delta}+e^{-\mathnormal{\Delta}^{4}/2\delta}\sum_{i=1}^{n}j_{i}(\delta)=\mathnormal{\Delta}+e^{-\mathnormal{\Delta}^{4}/2\delta}J_{n\delta}.$

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 $n\geq 1$. Assume $u^{(+)}_{(n-1)\delta}\preccurlyeq
u^{(-)}_{(n-1)\delta}+\varepsilon_{n-1}=:f_{n-1}$, and $\varepsilon_{n-1}$ is
nonnegative and
$\|\varepsilon_{n-1}\|_{1}\leq\mathnormal{\Delta}_{n-1}:=\mathnormal{\Delta}+e^{-\mathnormal{\Delta}^{4}/2\delta}\sum_{i=1}^{n-1}j_{i}(\delta)$.
Then by definition of the upper barriers,

$\displaystyle u^{(+)}_{n\delta}$
$\displaystyle=K^{\mathnormal{\Delta},j_{n}(\delta)}PGu^{(+)}_{(n-1)\delta}+e_{n}.$

Now, $\mathnormal{\Delta}\leq\mathnormal{\Delta}_{n-1}$. Hence by Lemma
3.2(iv),

$\displaystyle u^{(+)}_{n\delta}$ $\displaystyle\preccurlyeq
K^{\mathnormal{\Delta}_{n-1},j_{n}(\delta)}PGu^{(+)}_{(n-1)\delta}+e_{n}$
$\displaystyle\preccurlyeq
K^{\mathnormal{\Delta}_{n-1},j_{n}(\delta)}PGf_{n-1}+e_{n},$

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,
$\|w\|_{1}\leq\mathnormal{\Delta}_{n-1}$ and

$PGu^{(-)}_{(n-1)\delta}=PGf_{n-1}-w.$

Hence by Lemma 3.2(ii), $K^{\mathnormal{\Delta}_{n-1}}PGf_{n-1}\preccurlyeq
PGu^{(-)}_{(n-1)\delta}$. By the monotonicity of $K^{j_{n}(\delta)}$, denoting
$\varepsilon_{n}=\widehat{K}^{\mathnormal{\Delta}_{n-1}}PGf_{n-1}+e_{n}$,

$\displaystyle u^{(+)}_{n\delta}$ $\displaystyle\preccurlyeq
K^{j_{n}(\delta)}PGu^{(-)}_{(n-1)\delta}+\varepsilon_{n}$
$\displaystyle=u^{(-)}_{n\delta}+\varepsilon_{n}.$

Now, $\varepsilon_{n}$ is nonnegative. Moreover,
$\|\varepsilon_{n}\|_{1}=\|\widehat{K}^{\mathnormal{\Delta}_{n-1}}PGf_{n-1}\|_{1}+\|e_{n}\|_{1}\leq\mathnormal{\Delta}_{n-1}+e^{-\mathnormal{\Delta}^{4}/2\delta}j_{n}(\delta)=\mathnormal{\Delta}_{n}$.
This completes the proof. ∎

Proof of Theorem 2.2. Note first that once uniqueness is established for the
$u$ component of the solution $(u,\beta)$, uniqueness of the $\beta$ component
follows immediately from (9). 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 3.1 and
3.2, for every small $\mathnormal{\Delta}>0$ there exists $n_{0}$ such that
for every $n>n_{0}$,

$u^{(\delta_{n},-)}_{t}[r,\infty)\leq
u^{1}_{t}[r,\infty)<u^{2}_{t}[r,\infty)\leq
u^{(\mathnormal{\Delta},\delta_{n},+)}_{t}[r,\infty).$

By Proposition 3.3,

$u^{(\mathnormal{\Delta},\delta_{n},+)}_{t}[r,\infty)-u^{(\delta_{n},-)}_{t}[r,\infty)\leq\|\varepsilon_{n}(t,\mathnormal{\Delta},\delta_{n})\|_{1}\leq\mathnormal{\Delta}+e^{-\mathnormal{\Delta}^{4}/2\delta_{n}}J_{t}.$

On taking $n\to\infty$ and then $\mathnormal{\Delta}\downarrow 0$, the RHS
converges to zero, a contradiction. ∎

######  Remark 3.3.

Here are further properties of the solution deduced from [17, Theorem 1 and
Remark 1(b)]. For $1<q<\infty$, $0\leq\sigma<\frac{q+1}{q}$ and $p\geq 1$ 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
$\rho\in[0,1)$, $u\in L_{p,\rm loc}({\mathbb{R}}_{+},C_{0}({\mathbb{R}})\cap
C^{\rho}({\mathbb{R}}))$.

##  4 Removal at boundary: convergence

###  4.1 Proof of main result

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 4.1, the
rescaled processes are shown to satisfy equation (9) with a certain error
term. Lemma 4.2 establishes tightness of these processes. Existence of
measurable density for any subsequential limit of $\bar{\xi}^{N}$ is argued in
Lemma 4.3. Then, Lemma 4.4 shows that subsequential limits satisfy the ORA
condition. The proof of Theorem 2.3 appears at the end of the section.

A first relation to the PDE is as follows.

######  Lemma 4.1.

Fix $\varphi\in C_{c}^{\infty}({\mathbb{R}}\times{\mathbb{R}}_{+})$ and $T$ so
large that $\varphi(\cdot,t)=0$ for all $t\geq T$. 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

$\langle\bar{M}^{N}\rangle_{t}\leq\|\partial_{x}\varphi\|_{\infty}^{2}N^{-1}(1+J_{t})t.$
(26)

Proof. By the assumed property $\varphi(\cdot,T)=0$ and (4), we have

$\displaystyle 0=\int_{{\mathbb{R}}}\varphi(x,T)\bar{\gamma}_{T}^{N}(dx)$
$\displaystyle=N^{-1}\sum_{i\leq
N^{I}_{T}}\varphi(B^{i}_{T\wedge\tau^{i}},T).$

For any $t$ and $i\leq N^{I}_{t}$, we have $\sigma_{i}\leq t$, hence by Itô’s
formula,

$\displaystyle\varphi(B^{i}_{t\wedge\tau^{i}},t)$
$\displaystyle=\varphi(x^{i},\sigma_{i})+\int_{\sigma_{i}}^{t}\partial_{t}\varphi(B^{i}_{s\wedge\tau^{i}},s)ds+\int_{\sigma_{i}}^{t\wedge\tau^{i}}\partial_{x}\varphi(B^{i}_{s},s)dB^{i}_{s}$
$\displaystyle\qquad+\frac{1}{2}\int_{\sigma_{i}}^{t\wedge\tau^{i}}\partial_{xx}\varphi(B^{i}_{s},s)ds.$

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 $i\leq N^{I}_{T}$,

$\varphi(B^{i}_{T\wedge\tau^{i}},T)=\varphi(x^{i},\sigma_{i})+\int_{\sigma^{i}}^{T\wedge\tau^{i}}(\partial_{t}+\frac{1}{2}\partial_{xx})\varphi(B^{i}_{s},s)ds-\varphi(B^{i}_{T\wedge\tau^{i}},T\wedge\tau^{i})+M^{i}_{T}.$

Hence

$\displaystyle 0$
$\displaystyle=\int_{\mathbb{R}}\varphi(\cdot,0)d\bar{\xi}^{N}_{0}+\int_{{\mathbb{R}}\times[0,T]}\varphi
d\bar{\alpha}^{N}+\int_{0}^{T}\int_{\mathbb{R}}(\partial_{t}+\frac{1}{2}\partial_{xx})\varphi(x,t)\bar{\xi}^{N}_{t}(dx)dt-\int_{{\mathbb{R}}\times[0,T]}\varphi
d\bar{\beta}^{N}+\bar{M}^{N}_{T},$

where $\bar{M}^{N}_{t}=N^{-1}\sum_{i\leq N^{I}_{t}}M^{N,i}_{t}$. The
integration range ${\mathbb{R}}\times[0,T]$ can be replaced by
${\mathbb{R}}\times{\mathbb{R}}_{+}$ thanks to the property
$\varphi(\cdot,t)=0$, $t\geq T$. Finally, the quadratic variation bound (26)
is straightforward. ∎

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.

######  Lemma 4.2.

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
${\mathbb{P}}(\xi\in C({\mathbb{R}}_{+},{\cal M}_{+}({\mathbb{R}})))=1$.

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 $\lim_{\varepsilon\to
0}k(\varepsilon)=0$, let

$K(c,k)=\\{\beta:\text{for every $\varepsilon\in{\cal E}$},\
\beta({\mathbb{R}}\times[0,T)\leq c,\
\beta([-k(\varepsilon),k(\varepsilon)]^{c}\times[0,T))\leq\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
$\bar{\beta}^{N}({\mathbb{R}}\times[0,T))=N^{-1}N^{J}_{T-}\leq J_{T}$.

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 $y^{i}=\sup_{t\in[0,T]}|\tilde{B}^{i}_{t}|$,

$\displaystyle\bar{\beta}^{N}([-k,k]^{c}\times[0,T))$ $\displaystyle\leq
N^{-1}\sum_{i\leq N^{I}_{T}}1_{\\{\sup_{t\in[\sigma^{i},T]}|B^{i}_{t}|\geq
k\\}}$ $\displaystyle\leq N^{-1}\sum_{i\leq N^{I}_{T}}(1_{\\{|x^{i}|\geq
k/2\\}}+1_{\\{y^{i}\geq k/2\\}})$ $\displaystyle\to{\mathbb{P}}(x^{1}\geq
k/2)+I_{T}{\mathbb{P}}(z\geq k/2)+(1+I_{T}){\mathbb{P}}(y^{1}\geq k/2),$

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
${\mathbb{P}}(\bar{\beta}^{N}\in K)\to 0$ 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
$w(f,\delta,T)=\sup\\{|f(s)-f(t)|:s,t\in[0,T],|s-t|\leq\delta\\}$, 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
${\mathbb{P}}(\bar{\xi}^{N}_{T}\in K)\to 1$ as $N\to\infty$; and (ii) for
every $T$ and $\varepsilon>0$ there exists $\delta>0$ such that

$\limsup_{N}{\mathbb{P}}(w(\bar{\xi}^{N},\delta,T)\geq\varepsilon)\leq\varepsilon.$

Once these two properties are proved we can use [27] Corollary 3.7.4 (p. 129)
by which $\bar{\xi}^{N}$ is a relatively compact sequence. Moreover, because
we use $w$ rather than $w^{\prime}$ (cf. (3.6.2) of [27] (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 $0<t-s\leq\delta$. The cardinality of the symmetric different
between $A^{N}_{s}$ and $A^{N}_{t}$ is given by

$N^{I}_{t}-N^{I}_{s}+N^{J}_{t}-N^{J}_{s}\leq Nw_{1}(\delta),$ (27)

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
$\delta\to 0$. Therefore given $\varepsilon>0$ there exists $\delta>0$ such
that

$\limsup_{N}{\mathbb{P}}(\mathnormal{\Omega}^{N})=0,\qquad\text{where}\qquad\mathnormal{\Omega}^{N}=\mathnormal{\Omega}^{N}_{\varepsilon,\delta}=\Big{\\{}\frac{\\#\\{i\leq
N^{I}_{T}:w(\tilde{B}^{i},\delta,T)\geq\eta\\}}{N^{I}_{T}}\geq\eta\Big{\\}},$

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 $\mathnormal{\Omega}^{N}$,

$\xi^{N}_{s}(C)\leq\xi^{N}_{t}(C^{\eta})+w_{1}(\delta)N+\eta N^{I}_{T}.$

Similarly,

$\xi^{N}_{t}(C)\leq\xi^{N}_{s}(C^{\eta})+w_{1}(\delta)N+\eta N^{I}_{T}.$

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

$\limsup_{N}{\mathbb{P}}(w(\bar{\xi}^{N},\delta,T)\geq\varepsilon)=0,$

and the proof is complete. ∎

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.

######  Lemma 4.3.

Let $(\xi,\beta)$ be a subsequential limit of
$(\bar{\xi}^{N},\bar{\beta}^{N})$. Then, given $T<\infty$,

$\sup_{t\in[0,T]}\sup_{a,b\in{\mathbb{R}}:a<b}(b-a)^{-1}\xi_{t}(a,b)\leq
c\quad\text{a.s.},$ (28)

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 ${\mathbb{R}}$.

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 $a<b$,

$\xi_{t}(a,b)\leq c_{1}(b-a)\quad\text{a.s.}$ (29)

Recall that $\zeta^{N}$ denotes the particle configuration without removals.
Note that by construction, $\xi^{N}_{t}(B)\leq\zeta^{N}_{t}(B)$ for all
$t\in{\mathbb{R}}_{+}$, $B\in{\cal B}({\mathbb{R}})$,
$\omega\in\mathnormal{\Omega}$. The argument toward (29) is based on showing,
for $t$, $a$, $b$ fixed,

${\mathbb{P}}(\bar{\zeta}^{N}_{t}(a,b)>c_{1}(b-a))\to 0.$ (30)

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,
${\mathbb{P}}(\xi_{t}(a,b)>c_{1}(b-a))\leq\limsup{\mathbb{P}}(\bar{\xi}^{N}_{t}(a,b)>c_{1}(b-a))=0$.

The convergence (30) is a standard result but we have not found a reference
that covers precisely this setting. The proof is simple. We have by (5),
$\zeta^{N}_{t}(a,b)=\sum_{i=1}^{N^{I}_{t}}1_{\\{B_{t}^{i}\in(a,b)\\}}$. For a
first moment calculation, denoting $\bar{I}^{N}_{s}=N^{-1}[NI_{s}]$,

$\displaystyle{\mathbb{E}}\bar{\zeta}^{N}_{t}(a,b)$
$\displaystyle=N^{-1}\sum_{i\leq
N^{I}_{t}}{\mathbb{P}}(x^{i}+\tilde{B}^{i}_{t-\sigma^{i}}\in(a,b))$
$\displaystyle=N^{-1}\sum_{i\leq N}\int
G_{t}(x,(a,b))u_{0}(x)dx+N^{-1}\sum_{N<i\leq N^{I}_{t}}\int
G_{t-\sigma^{i}}(x,(a,b))\pi(dx)$
$\displaystyle=G_{t}u_{0}(a,b)+\int_{{\mathbb{R}}\times[0,t]}G_{t-s}(x,(a,b))\pi(dx)d\bar{I}^{N}_{s}$
$\displaystyle\leq
G_{t}u_{0}(a,b)+\int_{{\mathbb{R}}\times[0,t-\varepsilon]}G_{t-s}(x,(a,b))\pi(dx)d\bar{I}^{N}_{s}+(\bar{I}^{N}_{t}-\bar{I}^{N}_{t-\varepsilon}).$

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 $t\in[0,T]$,

$\limsup_{N}{\mathbb{E}}\bar{\zeta}^{N}_{t}(a,b)\leq
G_{t}u_{0}(a,b)+G*\alpha((a,b),t)\leq c_{2}(b-a),$

where the last inequality follows from the $L_{\infty,{\rm
loc}}({\mathbb{R}}_{+},L_{\infty})$ bound stated in Lemma 3.1, 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 ${\rm
var}(\bar{\zeta}^{N}_{t}(a,b))\leq N^{-2}N^{I}_{t}\leq N^{-1}(1+I_{t})$. Hence
(30) follows if we set $c_{1}=2c_{2}$. This shows (29).

To deduce from this the statement (28) and the existence of a measurable
density we proceed in several steps. It follows from (29) that on an event of
full measure, for fixed $t\in[0,T]$ and all rational $a,b$ one has
$\xi_{t}(a,b)\leq c_{1}(b-a)$. As a result, on the same event, for all
$a,b\in{\mathbb{R}}$ and $a<b$, one has $\xi_{t}(a,b)\leq 2c_{1}(b-a)$. 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 $a<b$, $\xi_{t}(a,b)\leq
2c_{1}(b-a)$. This proves (28) 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 [25] (see also the
remark that appears on the same page). The measurable spaces denoted in [25]
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 $dx$.
∎

The following lemma addresses condition (8)(iii).

######  Lemma 4.4.

Let $(\xi,\beta)$ be a subsequential limit of
$(\bar{\xi}^{N},\bar{\beta}^{N})$. Then a.s.,

$X(\xi,\beta,r):=\int_{{\mathbb{R}}_{+}}\xi_{t}[r,\infty)\beta((-\infty,r)\times
dt)=0,\quad\text{for all }r\in{\mathbb{R}}.$ (31)

Proof. We first prove a version of (31) for rational $r$. More precisely,
consider the collection $\mathnormal{\Sigma}$ of all (deterministic) tuples
$\sigma=(r,t_{1},t_{2},\eta,\delta)\in{\mathbb{Q}}^{5}$, $0\leq t_{1}<t_{2}$,
$\eta>0$, $\delta>0$. 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 (6), 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 4.3). 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 $\int_{A_{\eta}}a_{t}db_{t}>0$
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
$\int_{(t_{1},t_{2})}a_{t}db_{t}>0$. By taking a subset we may assume without
loss that $t_{1},t_{2}\in{\mathbb{Q}}$. Consequently, $\xi_{t}(r,\infty)=a\geq
2\eta>\eta$ 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
${\mathbb{P}}(\mathnormal{\Omega}^{0})\leq{\mathbb{P}}(\cup_{\sigma\in\mathnormal{\Sigma}}\mathnormal{\Omega}_{\sigma})=0$.
∎

Proof of Theorem 2.3. Again consider a subsequential limit $(\xi,\beta)$ of
$(\bar{\xi}^{N},\bar{\beta}^{N})$. Let $u$ be the density from Lemma 4.3. We
show that a.s., $(u,\beta)$ is a weak solution to (8).

First, to address (9), recall that $\\{x^{i},1\leq i\leq N\\}$ 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
4.1 shows that $(u,\beta)$ satisfies (9) a.s.

That condition (8)(iii) is satisfied by $(u,\beta)$ follows from Lemma 4.4.

Condition (8)(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 (8), 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
$\bar{\xi}^{N}_{t}({\mathbb{R}})=N^{-1}(N^{I}_{t}-N^{J}_{t})\leq 1+I_{t}$ and
the latter follows the Lipschitz bound (28). We have thus shown that
$(u,\beta)$ is a.s. a weak solution to (8). By Theorem (2.2), it is in fact
the unique weak solution to (8). In particular, the limit $(\xi,\beta)$ does
not depend on the subsequence. In other words, there exists a unique weak
solution $(u,\beta)$ to (8), and
$(\bar{\xi}^{N},\bar{\beta}^{N})\to(\xi,\beta)$ in probability, where, for
every $t$, $\xi_{t}(dx)=u(x,t)dx$. ∎

Proof of Theorem 2.1. This result is contained in Theorem 2.3. ∎

###  4.2 Comparison via coupling

In this section we prove the comparison principle, Theorem 2.4.

Proof of Theorem 2.4. 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 §2.1 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 4.2, where the inequality (27) 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 2.3. ∎

##  5 Removal at quantile

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 §5.1. The heat equation with a
(modified) ORA condition is described in §5.2, where the main result regarding
this model is also stated. §5.3 provides several further mass transport
inequalities used later. The barriers are constructed and analyzed in §5.4.
Finally, in §5.5 the proof of uniqueness and convergence is provided.

###  5.1 Particle system with quantile removal

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 $q(t)=(1-t)Q(t)$, $0\leq t\leq 1$. 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 $0\leq t<1$. 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 $q(t)\leq 1-t$ for $t\in[0,1)$ and vanishing on
$[1,\infty)$.

The initial number of particles is $N$. Mutually independent BMs
$\\{B^{i}_{t},t\geq 0\\}$, $1\leq i\leq N$ are given, where $B^{i}$ starts at
$x^{i}$ for $1\leq i\leq N$, 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 $t_{k}=kN^{-1}$, $0\leq k\leq N$, while
$t_{N+1}=\infty$, where at each time $t_{k}$, $1\leq k\leq N$, one particle is
removed. Given admissible data $(u_{0},q)$, let $Q(t)=(1-t)^{-1}q(t)$ for
$0\leq t<1$ (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 $1\leq k\leq N-1$, let
$i_{k}$ be the $Q(t_{k})$-quantile member of the particle configuration at
$t_{k}-$. That is,

$i_{k}=\min\\{j\in P:\\#\\{i\in P:b_{i}\geq b_{j}\\}=\lceil
nQ(t_{k})\rceil\\},$

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 $\lceil 0\rceil=1$ 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
$1\leq i\leq N$ via the relation $\tau^{i_{k}}=t_{k}$, for $1\leq k\leq N$.

The configuration process and the removal measure are defined by

$\xi^{N}_{t}(dx)=\sum_{i=1}^{N}\boldsymbol{\delta}_{B^{i}_{t}}(dx)1_{\\{t<\tau^{i}\\}},\qquad\beta^{N}(dx,dt)=\sum_{i\leq
N}\boldsymbol{\delta}_{(B^{i}_{\tau^{i}},\tau^{i})}(dx,dt),$

and the two extended configuration processes by

$\gamma^{N}_{t}(dx)=\sum_{i\leq
N}\boldsymbol{\delta}_{B^{i}_{t\wedge\tau_{i}}}(dx),\qquad\zeta^{N}_{t}(dx)=\sum_{i\leq
N}\boldsymbol{\delta}_{B^{i}_{t}}(dx).$

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
$\xi^{N}_{t_{k}}(r,\infty)=\xi^{N}_{t_{k}-}(r,\infty)\leq\lceil
nQ(t_{k})\rceil-1\leq nQ(t_{k})\leq Nq(t_{k})+1$. Similarly if a particle is
removed within $(r,\infty)$ then $\xi^{N}_{t_{k}}(-\infty,r)\leq
N(1-t_{k}-q(t_{k}))+1$.

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

$\displaystyle{\cal I}_{r}^{+}(\xi,\beta,c)$
$\displaystyle=\int_{{\mathbb{R}}_{+}}[\xi_{t}(r,\infty)-c(t)]^{+}\beta((-\infty,r)\times
dt),$ (32) $\displaystyle{\cal I}_{r}^{-}(\xi,\beta,c)$
$\displaystyle=\int_{{\mathbb{R}}_{+}}[\xi_{t}(-\infty,r)-(1-t-c(t))]^{+}\beta((r,\infty)\times
dt).$ (33)

Then

${\cal
I}_{r}^{+}\Big{(}\bar{\xi}^{N},\bar{\beta}^{N},q(\cdot)+\frac{1}{N}\Big{)}={\cal
I}_{r}^{-}\Big{(}\bar{\xi}^{N},\bar{\beta}^{N},q(\cdot)-\frac{1}{N}\Big{)}=0,\qquad
r\in{\mathbb{R}}.$ (34)

###  5.2 PDE and main result

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 (32)–(33) with $u((r,\infty),t)$ in place of $\xi_{t}(r,\infty)$, etc.
Consider the problem

$\begin{cases}{\it(i)}\hskip
35.00005pt\partial_{t}u-\frac{1}{2}\partial_{xx}u=-\beta&\text{in
}{\mathbb{R}}\times{\mathbb{R}}_{+},\\\ \\\ {\it(ii)}\hskip
33.00008ptu(\cdot,0)=u_{0}&\text{on }{\mathbb{R}},\\\ \\\ {\it(iii)}\hskip
28.00006pt{\cal I}^{\pm}_{r}(u,\beta,q)=0&\text{for all }r\in{\mathbb{R}},\\\
\\\ {\it(iv)}\hskip 31.0001pt\beta({\mathbb{R}}\times[0,t])=t\wedge
1&\text{for all }t\in{\mathbb{R}}_{+}.\end{cases}$ (35)

A solution to (35) 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 (35)(iii) and (35)(iv) hold, and for any $\varphi\in
C_{c}^{\infty}({\mathbb{R}}\times{\mathbb{R}}_{+})$,

$-\int_{0}^{\infty}(\partial_{t}\varphi+\frac{1}{2}\partial_{xx}\varphi,u)dt=(\varphi(\cdot,0),u_{0})-\int_{{\mathbb{R}}\times{\mathbb{R}}_{+}}\varphi\,d\beta.$
(36)

######  Theorem 5.1.

There exists a unique solution $(u,\beta)$ to (35). 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
$N\to\infty$.

###  5.3 More mass transport inequalities

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 $u\in H_{\delta}$,

$\tilde{R}_{\delta}(u)=\inf\\{x\in{\mathbb{R}}:u(-\infty,x]\geq\delta\\},$

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 $u\in H_{\delta}$,

$L^{\mathnormal{\Delta},\delta}=\begin{cases}K^{\delta}u&\mathnormal{\Delta}\leq
0,\\\
K^{\mathnormal{\Delta},\delta}u&\mathnormal{\Delta}>0,\,\mathnormal{\Delta}+\delta<\|u\|_{1},\\\
\tilde{K}^{\delta}u&\mathnormal{\Delta}>0,\mathnormal{\Delta}+\delta\geq\|u\|_{1}.\end{cases}$

######  Lemma 5.1.

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

v. If $-\infty<\widehat{\mathnormal{\Delta}}\leq\mathnormal{\Delta}<\infty$
and $u\in H_{\mathnormal{\Delta}-\widehat{\mathnormal{\Delta}}+\delta}$ then

$L^{\mathnormal{\Delta},\delta}u\preccurlyeq
L^{\widehat{\mathnormal{\Delta}},\delta}K^{\mathnormal{\Delta}-\widehat{\mathnormal{\Delta}}}u+\widehat{K}^{\mathnormal{\Delta}-\widehat{\mathnormal{\Delta}}}u.$

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
$\tilde{K}^{\delta}u(r,\infty)\leq\tilde{K}^{\delta}v(r,\infty)$ for all $r$.
Note first that one must have $\|u\|_{1}\leq\|v\|_{1}$. Let
$r_{u}=\tilde{R}_{\delta}(u)$ and $r_{v}=\tilde{R}_{\delta}(v)$.

Case 1: $r_{u}\leq r_{v}$. For $r\geq r_{v}$ this follows from $u\preccurlyeq
v$. For $r\leq r_{v}$,

$\tilde{K}^{\delta}u(r,\infty)\leq
u(r,\infty)\leq\|u\|_{1}-\delta\leq\|v\|_{1}-\delta=\tilde{K}^{\delta}v(r,\infty).$

Case 2: $r_{u}>r_{v}$. Again, the claim holds trivially when $r>r_{u}$. For
$r\leq r_{v}$,
$\tilde{K}^{\delta}u(r,\infty)=\|u\|_{1}-\delta\leq\|v\|_{1}-\delta=\tilde{K}^{\delta}v(r,\infty)$.
For $r_{v}\leq r\leq r_{u}$,

$\tilde{K}^{\delta}u(r,\infty)\leq u(r,\infty)\leq
v(r,\infty)=\tilde{K}^{\delta}v(r,\infty).$

ii. For $\mathnormal{\Delta}\leq 0$, the claim holds by the monotonicity of
$K^{\delta}$ (Lemma 3.2(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 3.2(iii)). If
$\mathnormal{\Delta}+\delta\geq\|u\|_{1}\vee\|v\|_{1}$, the claim follows from
the monotonicity of $\tilde{K}^{\delta}$ just proved. Because
$\|u\|_{1}\leq\|v\|_{1}$, the only remaining case is
$\|u\|_{1}\leq\mathnormal{\Delta}+\delta<\|v\|_{1}$. In this case we must show
$\tilde{K}^{\delta}u(r,\infty)\leq K^{\mathnormal{\Delta},\delta}v(r,\infty)$.
Denote $r_{u}=\tilde{R}_{\delta}(u)$ and $r_{v}=R_{\mathnormal{\Delta}}(v)$.

Case 1: $r_{u}\leq r_{v}$. For $r\geq r_{v}$ the claim clearly holds. For
$r<r_{v}$,

$\tilde{K}^{\delta}u(r,\infty)\leq\|\tilde{K}^{\delta}u\|_{1}=\|u\|_{1}-\delta\leq\mathnormal{\Delta}=K^{\mathnormal{\Delta},\delta}v(r_{v},\infty)\leq
K^{\mathnormal{\Delta},\delta}v(r,\infty).$

Case 2: $r_{u}>r_{v}$. For $r\geq r_{u}$, the claim clearly holds. For $r\leq
r_{v}$, again

$\tilde{K}^{\delta}u(r,\infty)=\|u\|_{1}-\delta\leq\mathnormal{\Delta}=K^{\mathnormal{\Delta},\delta}v(r_{v},\infty)\leq
K^{\mathnormal{\Delta},\delta}v(r,\infty).$

For $r_{v}<r<r_{u}$,

$\tilde{K}^{\delta}u(r,\infty)\leq u(r,\infty)\leq
v(r,\infty)=K^{\mathnormal{\Delta},\delta}v(r,\infty).$

iii. Because $\|u-w\|_{1}=\|\tilde{K}^{\delta}u\|_{1}$, this claim follows
from Lemma 3.2(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 3.2(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
$\mathnormal{\Delta}\geq\|u\|_{1}-\delta$ then both sides of the inequality
are equal to $\tilde{K}^{\delta}u$.

Finally, consider the case $\widehat{\mathnormal{\Delta}}\leq 0$ and denote
the RHS of the asserted inequality by $v$. Then
$v=K^{\mathnormal{\Delta}-\widehat{\mathnormal{\Delta}},\delta}u$. Since by
assumption
$0\leq\mathnormal{\Delta}-\widehat{\mathnormal{\Delta}}\leq\|u\|_{1}-\delta$,
one has $v=L^{\mathnormal{\Delta}-\widehat{\mathnormal{\Delta}},\delta}u$. But
$\mathnormal{\Delta}\leq\mathnormal{\Delta}-\widehat{\mathnormal{\Delta}}$,
hence $L^{\mathnormal{\Delta},\delta}u\preccurlyeq
L^{\mathnormal{\Delta}-\widehat{\mathnormal{\Delta}},\delta}u=v$ by part (iv)
above. ∎

###  5.4 Barriers

######  Lemma 5.2.

Assume that a solution $(v,\beta)$ to (35) exists. Then $v$ has a version $u$
given by

$u(y,t)=G_{t}u_{0}(y)-G*\beta(y,t).$ (37)

Moreover, for $0\leq\tau<t$,

$u(y,t)=G_{t-\tau}u(\cdot,\tau)(y)-G*\beta(y,t;\tau).$ (38)

Furthermore, $u\in L_{\infty,{\rm loc}}({\mathbb{R}}_{+},L_{1})\cap
L_{\infty,{\rm loc}}({\mathbb{R}}_{+},L_{\infty})$.

As was the case in §3, a solution to (35) will always mean the version given
by (37), and this results in no loss of generality.

Proof. The proof of Lemma 3.1 does not rely on the ORA condition (8)(iii) in
any way, and hence the result is a special case of Lemma 3.1. ∎

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 $n\delta<1-\mathnormal{\Delta}$,

$\displaystyle u^{({\bar{\delta}},+)}_{n\delta}$
$\displaystyle=L^{(+)}_{n,{\bar{\delta}}}G_{\delta}u^{({\bar{\delta}},+)}_{(n-1)\delta}+e^{(+)}_{n,{\bar{\delta}}}$
$\displaystyle e^{(+)}_{n,{\bar{\delta}}}(y)$
$\displaystyle=e^{-\mathnormal{\Delta}^{4}/2\delta}\delta
1_{[\sigma^{(+)}_{n},\sigma^{(+)}_{n}+1]}(y),\qquad y\in{\mathbb{R}},$ (39)
$\displaystyle\sigma^{(+)}_{n}$
$\displaystyle=\sigma^{(+)}_{n}({\bar{\delta}})=R_{\mathnormal{\Delta}}(G_{\delta}u^{({\bar{\delta}},+)}_{(n-1)\delta}).$

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
$\|u^{({\bar{\delta}},+)}_{(n-1)\delta}\|_{1}\geq
1-(n-1)\delta>\mathnormal{\Delta}>\delta$, 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 $n\in{\mathbb{N}}$,
$n\delta<1-\mathnormal{\Delta}$,

$\displaystyle u^{({\bar{\delta}},-)}_{n\delta}$
$\displaystyle=L^{(-)}_{n,{\bar{\delta}}}G_{\delta}u^{({\bar{\delta}},-)}_{(n-1)\delta}+e^{(-)}_{n,{\bar{\delta}}}$
$\displaystyle e^{(-)}_{n,{\bar{\delta}}}(y)$
$\displaystyle=e^{-\mathnormal{\Delta}^{4}/2\delta}\delta
1_{[\sigma^{(-)}_{n}-1,\sigma^{(-)}_{n}]}(y),\qquad y\in{\mathbb{R}},$ (40)
$\displaystyle\sigma^{(-)}_{n}$
$\displaystyle=\sigma^{(-)}_{n}({\bar{\delta}})=\tilde{R}_{\mathnormal{\Delta}}(G_{\delta}u^{({\bar{\delta}},-)}_{(n-1)\delta}).$

Following [21], define, for $u,v\in L_{1}({\mathbb{R}},{\mathbb{R}}_{+})$ and
$m>0$, the relation $u\preccurlyeq v$ mod $m$ as

$u(r,\infty)\leq v(r,\infty)+m,\quad\text{for all }r\in{\mathbb{R}}.$

######  Proposition 5.1.

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 (35) one has

$u^{({\bar{\delta}},-)}_{n\delta}\preccurlyeq u_{n\delta}\text{ mod
}e^{-\mathnormal{\Delta}^{4}/2\delta},\qquad\text{and}\qquad
u_{n\delta}\preccurlyeq u^{({\bar{\delta}},+)}_{n\delta}.$

By Lemma 5.2, there is a constant $c_{\infty}$ such that for any solution
$(u,\beta)$, $\|u(\cdot,t)\|_{1}\leq 1$ and $\|u(\cdot,t)\|_{\infty}\leq
c_{\infty}$, $t\in[0,1]$.

######  Lemma 5.3.

There exists $\delta_{0}>0$ such that for any solution $(u,\beta)$ and any
$(n,\delta)\in{\mathbb{N}}\times(0,\delta_{0})$ satisfying $n\delta\leq 1$,

$\rho_{*}(\beta;[(n-1)\delta,n\delta])\geq
R_{q^{(+)}_{n,\delta}+c_{\infty}\delta^{1/2}}(u_{(n-1)\delta}),$

provided that
$q^{(+)}_{n,\delta}+c_{\infty}\delta^{1/2}\leq\|u_{(n-1)\delta}\|_{1}$, and

$\rho^{*}(\beta;[(n-1)\delta,n\delta])\leq
R_{q^{(-)}_{n,\delta}-c_{\infty}\delta^{1/2}}(u_{(n-1)\delta}),$

provided that $q^{(-)}_{n,\delta}-c_{\infty}\delta^{1/2}\geq 0$.

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 (35)(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 $u((r,\infty),t)\leq
q(t)\leq q^{(+)}_{1,\delta}$. 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 (37) by
$\eta(y)=\int_{{\mathbb{R}}\times[0,t]}G_{t-s}(x,y)\beta(dx,ds)$. Then

$\int_{r}^{\infty}\eta(y)dy\leq\int_{-\infty}^{\infty}\eta(y)dy=\beta({\mathbb{R}}\times[0,t])=t\leq\delta.$

Since $u((r,\infty),t)\leq q^{(+)}_{1,\delta}$, we have by (37),

$\int_{r}^{\infty}\int_{\mathbb{R}}G_{t}(x,y)u_{0}(x)dx\,dy\leq
q^{(+)}_{1,\delta}+\int_{r}^{\infty}\eta(y)dy\leq q^{(+)}_{1,\delta}+\delta.$

Hence if $A=\int_{r}^{\infty}G_{t}(x,(-\infty,r))u_{0}(x)dx$,

$\displaystyle q^{(+)}_{1,\delta}+\delta$
$\displaystyle\geq\int_{r}^{\infty}G_{t}(x,(r,\infty))u_{0}(x)dx=-A+u_{0}(r,\infty)$
$\displaystyle=-A+q^{(+)}_{1,\delta}+c_{\infty}\delta^{1/2}.$

However,

$\displaystyle A$ $\displaystyle\leq
c_{\infty}\int_{r}^{\infty}G_{t}(x,(-\infty,r))dx$
$\displaystyle=c_{\infty}\int_{0}^{\infty}G_{t}(0,(x,\infty))dx=c_{\infty}(2\pi)^{-1/2}t^{1/2}<\frac{c_{\infty}}{2}\delta^{1/2}.$

This gives $c_{\infty}\delta^{1/2}-\delta<\frac{1}{2}c_{\infty}\delta^{1/2}$,
which is impossible when $\delta$ is small. ∎

Proof of Proposition 5.1. There is an obvious symmetry in equation (35) 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 (35). Define
$(\tilde{u},\tilde{\beta})$ by composing $(u,\beta)$ with the reflection
$x\mapsto-x$. Then $(\tilde{u},\tilde{\beta})$ is a solution of (35) 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,

$u(-\infty,r]\leq u^{({\bar{\delta}},-)}(-\infty,r]$

hence

$u^{({\bar{\delta}},-)}[r,\infty)\leq
u[r,\infty)+\|u^{({\bar{\delta}},-)}\|_{1}-\|u\|_{1}.$

Now, directly from (40),
$\|u^{({\bar{\delta}},-)}_{n\delta}\|_{1}=1-n\delta+ne^{-\mathnormal{\Delta}^{4}/2\delta}\delta$.
By Lemma 5.2 and (35)(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 3.2, 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 5.2,
$u_{n\delta}=Gf-h$,
$h(y)=\int_{{\mathbb{R}}\times[(n-1)\delta,n\delta]}G_{n\delta-s}(x,y)\beta(dx,ds)$.
Moreover, $u^{({\bar{\delta}},+)}_{n\delta}=LGg+e_{n,{\bar{\delta}}}$.
Denoting $w=Gf$, consider the inequality

$w-h\preccurlyeq Lw+e_{n,{\bar{\delta}}}.$ (41)

If it holds then, using $f\preccurlyeq g$ and monotonicity of $G$ and $L$ (see
Lemma 5.1(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 (41). Let $q^{*}=q^{(+)}_{n,\delta}$.

First, if $q^{*}+\mathnormal{\Delta}+\delta\geq\|f\|_{1}=\|w\|_{1}$ then by
the way in which the operator $L$ is defined, $Lw=\tilde{K}^{\delta}w$. Hence
by Lemma 5.1(iii), noting that $\|h\|_{1}=\delta$, we have
$w-h\preccurlyeq\tilde{K}^{\delta}w$. This shows (41).

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 5.3 $\rho_{*}(\beta;[(n-1)\delta,n\delta])\geq
r_{*}$. Let $h=h_{1}+h_{2}$, where

$h_{1}(y)=h(y)1_{\\{y<r_{*}-\mathnormal{\Delta}^{2}\\}},\qquad
h_{2}(y)=h(y)1_{\\{y\geq r_{*}-\mathnormal{\Delta}^{2}\\}}.$

Again, recall $\|h\|_{1}=\delta$. Using

$\displaystyle\int_{{\mathbb{R}}\times[(n-1)\delta,n\delta]}G_{n\delta-s}(x,y)\beta(dx,ds)=\int_{[r_{*},\infty)\times[(n-1)\delta,n\delta]}G_{n\delta-s}(x,y)\beta(dx,ds)$

and $G_{t}(0,[a,\infty))\leq e^{-a^{2}/2t}$ $a>0$, $t<a^{2}$, it is easy to
see that

$\|h_{1}\|_{1}\leq e^{-\mathnormal{\Delta}^{4}/2\delta}\delta,$ (42)

provided $\delta<\mathnormal{\Delta}^{4}$. Let $p\in(0,1]$ be defined by
$\|h_{2}\|_{1}=p\delta$. Then $p\geq 1-e^{-\mathnormal{\Delta}^{4}/2\delta}$.

In order to show (41) it will now be argued that

$w-h\preccurlyeq K^{q^{*}+\mathnormal{\Delta},p\delta}w.$ (43)

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

$R_{q^{*}+\mathnormal{\Delta}}(w)\leq
R_{q^{*}+c_{\infty}\delta^{1/2}}(f)-\mathnormal{\Delta}^{2},$ (44)

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 (43). Toward showing (44), it follows from $\|f\|_{\infty}<c_{\infty}$
that for all $\delta$ small (specifically,
$c_{\infty}\delta^{1/2}<\mathnormal{\Delta}/4$ and recalling
$2c_{\infty}\mathnormal{\Delta}^{2}<\mathnormal{\Delta}/4$),

$R_{q^{*}+\mathnormal{\Delta}/2}(f)\leq
R_{q^{*}+c_{\infty}\delta^{1/2}}(f)-2\mathnormal{\Delta}^{2}.$ (45)

Moreover, for small $\delta$,

$R_{q^{*}+\mathnormal{\Delta}}(Gf)\leq\theta:=R_{q^{*}+\mathnormal{\Delta}/2}(f)+\mathnormal{\Delta}^{2},$
(46)

as we now show. This is shown by proving that $(Gf)[\theta,\infty)\leq
q^{*}+\mathnormal{\Delta}$. 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

$(Gf)[\theta,\infty)=(Gf_{1})[\theta,\infty)+(Gf_{2})[\theta,\infty)\leq
q^{*}+\mathnormal{\Delta}/2+(GA)[\theta,\infty)$

where $A=\|f_{2}\|_{1}\boldsymbol{\delta}_{x}$ with
$x=R_{q^{*}+\mathnormal{\Delta}/2}(f)$. Because $\|f\|_{1}\leq 1$,
$(GA)[\theta,\infty)\leq
G_{\delta}(0,[\mathnormal{\Delta}^{2},\infty))<\mathnormal{\Delta}/2$ for
small $\delta$. This shows (46).

The bounds (45) and (46) imply (44). This establishes (43).

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
$\preccurlyeq(1-p)\delta
1_{[R_{\mathnormal{\Delta}}(w),R_{\mathnormal{\Delta}}(w)+1]}$. Moreover,
since $w\preccurlyeq Gg$, one has $R_{\mathnormal{\Delta}}(w)\leq
R_{\mathnormal{\Delta}}(Gg)$, hence the same term is dominated by $(1-p)\delta
1_{[R_{\mathnormal{\Delta}}(Gg),R_{\mathnormal{\Delta}}(Gg)+1]}$. Using this
in (43),

$w-h\preccurlyeq K^{q^{*}+\mathnormal{\Delta},p\delta}w\preccurlyeq
K^{q^{*}+\mathnormal{\Delta},\delta}w+(1-p)\delta
1_{[R_{\mathnormal{\Delta}}(Gg),R_{\mathnormal{\Delta}}(Gg)+1]}.$

Since $1-p\leq e^{-\mathnormal{\Delta}^{4}/2\delta}$, the last term is
$\preccurlyeq e_{n,{\bar{\delta}}}$, and (41) follows. ∎

The proof of the following result is similar to that of Proposition 3.3, but
various details differ and so the proof is presented in full.

######  Proposition 5.2.

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

$\|\varepsilon_{n}\|_{1}\leq
3\mathnormal{\Delta}+e^{-\mathnormal{\Delta}^{4}/2\delta}\delta n.$

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 $n\geq 1$. The induction assumption is $u^{(+)}_{(n-1)\delta}\preccurlyeq
u^{(-)}_{(n-1)\delta}+\varepsilon_{n-1}=:f_{n-1}$, $\varepsilon_{n-1}$ is
nonnegative,
$\|\varepsilon_{n-1}\|_{1}\leq\mathnormal{\Delta}_{n-1}:=3\mathnormal{\Delta}+e^{-\mathnormal{\Delta}^{4}/2\delta}\delta
n$. Then $u^{(+)}_{n\delta}=L^{(+)}Gu^{(+)}_{(n-1)\delta}+e^{(+)}_{n}$. Hence

$\displaystyle u^{(+)}_{n\delta}$
$\displaystyle=L^{q^{(+)}_{n\delta}+\mathnormal{\Delta},\delta}Gu^{(+)}_{(n-1)\delta}+e_{n}^{(+)}$
$\displaystyle\preccurlyeq
L^{q^{(+)}_{n\delta}+\mathnormal{\Delta},\delta}Gf_{n-1}+e_{n}^{(+)}$

where one uses the monotonicity of $L$ (Lemma 5.1(ii)) and $G$. Using Lemma
5.1(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,
$q^{(+)}_{n,\delta}+\mathnormal{\Delta}-\mathnormal{\Delta}_{n-1}\leq
q^{(-)}_{n,\delta}-\mathnormal{\Delta}-\delta$. Thus by Lemma 5.1(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
$\|\varepsilon_{n}\|_{1}\leq\mathnormal{\Delta}_{n-1}+\|e^{(+)}_{n}\|_{1}=\mathnormal{\Delta}_{n}$,
as claimed.

Hence it remains to show $z\preccurlyeq Gu^{(-)}_{(n-1)\delta}$. But
$w:=G\varepsilon_{n-1}$ is pointwise nonnegative,
$\|w\|_{1}\leq\mathnormal{\Delta}_{n-1}$ and

$Gu^{(-)}_{(n-1)\delta}=Gf_{n-1}-w.$

Hence by Lemma 3.2(ii), $z\preccurlyeq Gu^{(-)}_{(n-1)\delta}$, and the proof
is complete. ∎

###  5.5 Uniqueness and convergence

Proof of Theorem 5.1.

Step 1: The uniqueness result.

It will be shown that if there exists a solution $(u,\beta)$ to (35) 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 2.2, 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 5.2 and (35)(iv) dictate that $u(\cdot,t)=0$ for
$t\geq 1$. 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 5.1, for every small
$\mathnormal{\Delta}>0$ and every $n>n_{0}(\mathnormal{\Delta})$,

$a(n,\mathnormal{\Delta}):=u^{(\delta_{n},-)}_{t}[r,\infty)-e^{\mathnormal{\Delta}^{4}/2\delta_{n}}\leq
u^{1}_{t}[r,\infty)<u^{2}_{t}[r,\infty)\leq
b(n,\mathnormal{\Delta}):=u^{(\mathnormal{\Delta},\delta_{n},+)}_{t}[r,\infty),$

whereas by Proposition 5.2,

$b(n,\mathnormal{\Delta})-a(n,\mathnormal{\Delta})\leq\|\varepsilon_{n}(t,\mathnormal{\Delta},\delta_{n})\|_{1}+e^{-\mathnormal{\Delta}^{4}/2\delta_{n}}\leq
3\mathnormal{\Delta}+2e^{-\mathnormal{\Delta}^{4}/2\delta_{n}}.$

A contradiction is obtained upon sending $n\to\infty$ then
$\mathnormal{\Delta}\downarrow 0$. This establishes uniqueness.

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 2.3,
comprising §4. These steps are Lemmas 4.1, 4.2, 4.3 and 4.4, and finally the
proof of Theorem 2.3. The only part of §4 where the specifics of the removal
mechanism are used is the proof of Lemma 4.4 (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 4.2. However the removal form is
not used). Thus the statements of Lemmas 4.1, 4.2 and 4.3 and their proofs
cover the current model as a special case. The only part of §4 that needs to
be modified is Lemma 4.4. A modified statement and proof are provided below in
Lemma 5.4.

Likewise, the proof of Theorem 2.3 based on uniqueness and Lemmas 4.1–4.4 is
valid here, with Lemma 5.4 in place of Lemma 4.4. This completes the proof of
Theorem 5.1. ∎

######  Lemma 5.4.

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 ${\cal I}^{\pm}_{r}(\xi,\beta,q)=0$.

Proof. We provide a proof for ${\cal I}^{+}$; the one for ${\cal I}^{-}$ is
similar. Recalling (32), we must show

${\cal
I}^{+}_{r}(\xi,\beta,q)=\int_{{\mathbb{R}}_{+}}[\xi_{t}(r,\infty)-q(t)]^{+}\beta((-\infty,r)\times
dt)=0,\quad\text{for all }r\in{\mathbb{R}}.$ (47)

Let $\mathnormal{\Sigma}$ be the collection of tuples
$\sigma=(r,t_{1},t_{2},\eta,\delta)\in{\mathbb{Q}}^{5}$, $0\leq t_{1}<t_{2}$,
$\eta>0$, $\delta>0$. 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 (34). 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 4.4 (with trivial adaptations).
∎

##  6 Other models

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\. 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 (8) 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
[28] for the study of a hydrodynamic limit of a polymer pinning model.

2\. The $N$-BBM model. As mentioned in the introduction, hydrodynamic limits
of this model were identified in [23] as a solution to a FBP analyzed in [19].
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 [24].

3\. 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 (35)(iii),

${\cal J}^{\pm}_{r}(u,\alpha,a)={\cal J}^{\pm}_{r}(u,\beta,b)=0,\qquad
r\in{\mathbb{R}},$

where

$\displaystyle{\cal J}_{r}^{+}(u,\gamma,c)$
$\displaystyle=\int_{{\mathbb{R}}_{+}}[u((r,\infty),t)-c]^{+}\gamma((-\infty,r)\times
dt),$ $\displaystyle{\cal J}_{r}^{-}(u,\gamma,c)$
$\displaystyle=\int_{{\mathbb{R}}_{+}}[u((-\infty,r),t)-(1-c)]^{+}\gamma((r,\infty)\times
dt).$

4\. Other total orders. Equation (8) could be considered with condition
(8)(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.

##### 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.

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