text:
\section{Introduction}

We consider spatial birth-and-death processes
with time dependent rates.
At each moment of time
the system is
represented as a finite collection
of motionless points in $\mathbb{R} ^\d$. We interpret
the points as particles, or individuals.
Existing particles
may die and new particles may appear.
Each particle is characterized by its
location.

 The state space of a spatial birth-and-death
 Markov process on $\mathbb{R} ^\d$ with finite number of points
 is the space of finite subsets of $\mathbb{R} ^\d$,
 \[
 \Gamma _0(\mathbb{R} ^\d)=\{ \eta \subset \mathbb{R} ^\d : |\eta| < \infty \},
\]
where $|\eta|$ is the number of points of $\eta$.

Denote by $\mathscr{B}(\mathbb{R} ^\d)$ the Borel $\sigma$-algebra on $\mathbb{R} ^\d$.
 The evolution of the spatial birth-and-death process on $\mathbb{R}^\d$
 admits the following description. Two measurable functions characterize the
 development in time,
 the birth rate
 $b: \mathbb{R} ^\d \times \mathbb{R} _+ \times \Gamma _0 (\mathbb{R}^\d) \rightarrow [0,\infty)$
 and the death rate
 $d: \mathbb{R} ^\d \times \mathbb{R} _+ \times \Gamma _0 (\mathbb{R}^\d) \rightarrow [0,\infty)$.
 If the system is in state $\eta \in \Gamma _0$
 at time $t$, then the probability
 that a new particle appears (a ``birth'') in a bounded set $B\in \mathscr{B}(\mathbb{R} ^\d)$
 over time interval $[t;t+ \Delta t]$ is
 \[
   \Delta t \int\limits _{ B}b(x,t, \eta)dx + o(\Delta t),
 \]
 the probability that
a particle $x \in \eta$ is deleted from the configuration (a ``death'') over
 time interval $[t;t+ \Delta t]$ is
 \[
  d(x,t, \eta) \Delta t + o(\Delta t),
 \]
 and no two events happen simultaneously.
 By an event we mean a birth or a death.
Using a slightly different terminology, we can say that
the rate at which a birth
 occurs in $B$ is $\int_B b(x, t, \eta)dx$,
 the rate at which a particle $x\in \eta$
dies is $d(x,t,\eta)$, and no two events happen at the same time.

 Such processes, in which the birth and death rates
depend on the spatial structure of the system as opposed to
 classical $\mathbb{Z} _+$-valued birth-and-death processes (see e.g.
 \cite[Page 116]{Branch2},
 \cite[Page 109]{Branch1}, and references therein),
 were first
 studied by Preston \cite{Preston}.
 A heuristic description
 similar to that above
 appeared already there. Our description
 resembles the one in \cite{GarciaKurtz}.

We say that the rates $b$ and $d$, or
the corresponding birth-and-death process,
are time-homogeneous if $b$ and $d$ do not depend on time.
By abuse of notation we write in this case $b(x,s,\eta) = b (x , \eta)$,
$d(x,s,\eta) = d (x , \eta)$.
The (heuristic) generator of a time-homogeneous spatial
birth-and-death process should be of the form

\begin{align} \label{intr generator}
L F (\eta) = \int\limits _{x \in \mathbb{R}^\d} b(x, \eta) [F(\eta \cup {x}) -F(\eta)] dx +
\sum\limits _{x \in \eta} d(x, \eta ) (F(\eta \setminus {x}) - F(\eta)),
\end{align}
for
 $F$ in an appropriate domain, where $\eta \cup {x}$
 and $\eta \setminus {x}$ are shorthands for
 $\eta \cup \{ x \}$ and
 $\eta \setminus \{ x \}$, respectively.

 Garcia and Kurtz \cite{GarciaKurtz} obtained
birth-and-death processes as solutions to
certain
stochastic integral equations.
The systems treated there involves
an infinite number of particles.
In
the earlier work \cite{Garcia} of Garcia  another approach was used:
birth-and-death processes were obtained
as projections of Poisson point processes. A
further development of the projection method
appears in \cite{GarciaKurtz2}.
Fournier and M{\'e}l{\'e}ard \cite{FournierMeleard} used
a similar equation
 for the construction of the Bolker--Pacala--Dieckmann--Law process
with finitely many particles.
Following ideas of \cite{GarciaKurtz}
and \cite{FournierMeleard},
we construct the  birth-and-death process described above
as solution to a stochastic equation.

Holley and Stroock \cite{HolleyStroock} constructed the
spatial birth-and-death process
as a Markov family of  unique solutions to the corresponding martingale problem.
For the most part, they consider a process
contained in a bounded volume, with bounded
birth and death rates.  They also
 proved the corresponding result
 for the nearest neighbor model in $\mathbb{R} ^1$
 with an infinite number of particles.

 Belavkin and Kolokoltsov \cite{BelKol03} discuss,
 among other things,
 a general structure of a Feller semigroup
 on disjoint unions of Euclidean spaces (see also references
 therein for the construction of the Markov processes
 with a given generator).
 We note that
 time-homogeneous birth-and-death processes
 need not have the Feller property.
 Eibeck and Wagner \cite{EiWag03}
 discuss convergence of particle systems
 to limiting kinetic equations.
 In particular, they construct
 the stochastic process corresponding
 to the particle system as a minimal jump process,
 or pure jump type Markov process in
 the terminology of Kallenberg \cite{KallenbergFound}.
 The jump kernel is assumed to be locally bounded.

 Kondratiev and Skorokhod \cite{KondSkor}
 constructed a
  contact process in continuum with an infinite  number of particles.
  The contact process in continuum can be described
  as the spatial birth-and-death process with
  \[
   b(x,\eta)  = \lambda \sum\limits _{y \in \eta} a(x-y),
   \ \ \  d(x,\eta) \equiv 1,
  \]
where $\lambda >0$ and $0\leq a \in L ^1 (\mathbb{R} ^\d)$.
  Under some additional assumptions,
  they showed existence of
  the process for a broad class of
  initial conditions.

  The scheme proposed by
Etheridge and Kurtz \cite{EK14} covers
a wide range of interactions
and applies to discrete and continuous models.
Their approach is based on, among other things,
assigning a certain mark ('level')
to each particle and letting this mark evolve
according to some law.
A critical event, such as birth or death,
occurs when the level hit some threshold.
  Shcherbakov and Volkov \cite{ShcherbakovVolkol} consider
  the long term behavior of birth-and-death
  processes on a finite graph with constant death rate and the
  birth rate of a special form. Bezborodov \cite{Bez15}
  obtains the spatial time-homogeneous birth-and-death process
  as a unique solution to the equation slightly different
  from the one we use here. Various questions
  not treated in this paper are considered there,
  for example the possibility of an explosion,
  continuous dependence on initial conditions
  and related semigroup of operators.

 In the aforementioned references as well as in the present work
  the system is represented by a Markov process.
  An alternative approach consists in using the concept
  of statistical dynamics that substitutes the notion of a Markov
stochastic process. This approach is based on considering evolutions of measures
and their correlation functions. For details see e.g.
\cite{semigroupapproach}, \cite{Aggreg}, and references therein.

Finkelshtein et al.
\cite{Aggreg} consider different aspects of statistical dynamics
for the aggregation model. In this model the death rate
is given by
\[
 d(x,\eta) = \exp \left( - \sum\limits _{y \in \eta \setminus x} \phi (x-y) \right),
\]
where $\phi$ is a positive measurable function. For more
details see \cite{Aggreg} and references therein.
In this paper we present a detailed long time microscopic
description of the process. Also,
we estimate the probability of extinction and
the speed of growth of the average number of points.

The paper is organized as follows.
Notation, definitions and results
are given in Section 2.
Proofs are given in Sections 3 and 4,
with two auxiliary results located to Section 5.

 \section{The set-up and main results}

 \subsection{Construction and basic properties}

 Let $X$ be a locally compact separable metric space (typically $X$
 will be a subset of $\mathbb{R} ^\d$).
 Denote by $\Gamma (X)$ the space of locally finite subsets
 of $X$
 \[
  \Gamma (X) = \{ \gamma \subset X \mid |\gamma \cap K| < \infty \},
 \]
 also called \emph{the space of configurations over} $X$.
 The space $\Gamma (X)$ can be endowed with
 a $\sigma$-field generated by the projection
 maps
 \[
   \Gamma (X) \ni \gamma \mapsto |\gamma \cap B| \in \mathbb{Z} _+
 \]
where $B$ is an arbitrary Borel subset of $X$.
For more details about the notions introduced here see e.g. \cite{DaleyVere},
\cite[Chapter 12]{KallenbergFound} or \cite{KondKuna},
and references therein.
 Throughout this paper $\Gamma _2$ stands for
 $\Gamma(\mathbb{R} _+ \times \mathbb{R} _+)$.
Let $\pi$ be the distribution of a Poisson random measure
on $(\Gamma _2, \mathscr{B}(\Gamma _2))$, with the intensity measure being
the Lebesgue measure on $\mathbb{R}_+ \times \mathbb{R} _+$.
We denote by $\mathscr{B} _t (\Gamma _2)$
the smallest sub-$\sigma$-algebra of $\mathscr{B}(\Gamma _2)$
such that for every $A _1 \in \mathscr{B} ([0,t])$,
$A _2 \in \mathscr{B} (\mathbb{R} _+)$ the map
\[
   \Gamma _2 \ni \gamma \mapsto \gamma (A_1 \times A _2 ) \in \mathbb{R} _+
\]
is $\mathscr{B} _t(\Gamma _2)$-measurable.
Similarly, define $\mathscr{B}_{>t} (\Gamma _2)$
as
the smallest sub-$\sigma$-algebra of $\mathscr{B}(\Gamma _2)$
such that for every $A _1 \in \mathscr{B} ((t, \infty))$,
$A _2 \in \mathscr{B} (\mathbb{R} _+)$ the map
\[
   \Gamma _2 \ni \gamma \mapsto \gamma (A_1 \times A _2 ) \in \mathbb{R} _+
\]
is $\mathscr{B}_{>t} (\Gamma _2)$-measurable.

Let $\eta _0$ be a (random)
initial configuration,
and let $\hat {\eta} _0$ be the point process on
$\mathbb{R} ^\d \times \Gamma _2 $ obtained by associating
to each point in $\eta _0$ an independent
Poisson point process on $\mathbb{R} _+ \times \mathbb{R} _+$,
with the distribution $\pi$. That is,
for $\eta _0 = \sum _{i=1} ^{\infty} \delta _{ x_i}$, set
\[
 \hat {\eta} _0 = \sum _{i=1} ^{\infty} \delta _{( x_i,\gamma _i)}.
\]

 Consider the stochastic equation with Poisson noise
\begin{equation} \label{se}
\begin{aligned}
\eta _t (B) = \int\limits _{(0,t] \times B \times [0, \infty ) \times \Gamma _2 }
& I _{ [0,b(x,s,\eta _{s-} )] } (u) I \{ \int _s ^t I_{[0,d(x,r, \eta _{r-})]}(v)\gamma (dr,dv) =0 \}
N(ds,dx,du,d\gamma) \\
+ \int\limits _{B \times  \Gamma _2 } &
I \{ \int _0 ^t I_{[0, d(x,r, \eta _{r-})]}(v)\gamma (dr,dv) =0 \} \hat {\eta} _0 (dx, d \gamma ) ,
\end{aligned}
\end{equation}
 where
$(\eta _t)_{t \geq 0}$ is a cadlag $\Gamma _0$-valued solution
 process,
 $N$ is a  Poisson point process on
$\mathbb{R}_+  \times \mathbb{R}^\d \times \mathbb{R}_+ \times \Gamma _2 $,
the mean measure of $N$ is $ds \times dx \times du \times \pi$.
We require the
processes $N$ and $\eta _0$ to be independent of each other.
Equation \eqref{se} is understood in the sense that the equality
holds a.s. for every bounded
$B \in \mathscr{B} (\mathbb{R} ^\d) $ and $t \geq 0$.

\textbf{Remark}.
In the first integral on the right-hand side of \eqref{se}
$x$ is the place and
$s$
is the time of birth of a new particle.
This particle is alive as long as
$\int _s ^t I_{[0,d(x,r, \eta _{r-})]}(v)\gamma (dr,dv) =0$,
where $(x,s,u, \gamma) \in N$. Thus, $\gamma$ is the process
'responsible' for death.
In the death term lies the main difference to the equation
considered by
Garcia and Kurtz \cite{GarciaKurtz}.
Adapted to our notation,
 the equation there is of the form

\begin{equation} \label{se GK}
\begin{aligned}
\eta _t (B) = \int\limits _{(0,t] \times B \times [0, \infty ) \times [0, \infty ) }
& I _{ [0,b(x,\eta _{s-} )] } (u) I \{ \int _s ^t d(x, \eta _{r-})dv  < r \}
\tilde N(ds,dx,du,dr) \\
+ \int\limits _{B \times  [0, \infty ) } &
I \{ \int _s ^t d(x, \eta _{r-})dv  < r \} \tilde {\eta} _0 (dx, d r ),
\end{aligned}
\end{equation}
where $\tilde N$ is a Poisson point process on
$\mathbb{R}_+  \times \mathbb{R}^\d \times \mathbb{R}_+ \times \mathbb{R} _+ $ with
 mean measure $ds \times dx \times du \times e ^{-r} dr$,
 and $\tilde {\eta} _0$ is obtained from $\eta _0$
 by attaching an independent unit exponential to each point.
 At first glance, \eqref{se} is more complicated than \eqref{se GK},
 since the death mechanism requires a whole Poisson random measure
 on $[0;\infty) ^2$ instead of just one exponential
 random variable. However, it is not clear
 how to a priori define a filtration $\{  \tilde {\mathscr {F}} _t  \} _ {t\geq 0}$
 compatible with $\tilde N$
 and such that
 a solution to \eqref{se GK}, if unique, should possess the Markov property
 with respect to $\{  \tilde {\mathscr {F}} _t  \} _ {t\geq 0}$.

\emph{Conditions on} $b$, $d$ and $\eta _0$.
Unless stated otherwise,
 we assume  that the birth rate $b$ satisfies
the following conditions: sublinear growth on
the second variable in the sense
that
\begin{equation} \label{sublinear growth for b}
\int\limits _{\mathbb{R} ^\d} \sup\limits _{s > 0} {b}(x, s, \eta ) dx \leq c_1|\eta| +c_2,
\end{equation}
for some constants $c_1, c_2 >0$, and
that $b(x, \cdot, \eta)$ and $d(x, \cdot, \eta)$
are right-continuous for any $x \in \mathbb{R} ^\d$
and $\eta \in \Gamma _0$.

\begin{comment}
and let $d$ satisfy (\textbf{is this condition really needed?})
\begin{equation} \label{condition on d}
\forall m \in \mathbb{N} : \sup\limits _{
\substack{ x\in \mathbb{R}^\d,|\eta| \leq m  \\ s \geq 0} }d(x, s, \eta) < \infty.
\end{equation}
\end{comment}

 We also assume that

\begin{equation} \label{condition on eta _0}
 E |\eta _0| < \infty.
\end{equation}

 We say that $N$ is \emph{compatible} with a
 right-continuous complete filtration  $\{ \mathscr{F} _t \}$
  if
 \[
 {N([0,q] \times B \times C \times \Xi)}
 \]
  is $\mathscr{F} _t $-measurable for any
 $ q \in [0,t],
B \in \mathscr{B} (\mathbb{R}^\d), C \in \mathscr{B} (\mathbb{R}_+)$, and $ {\Xi \in \mathscr{B}_t (\Gamma _2)}$,
and if
  \[
  N((t,t+q'] \times B' \times C' \times \Xi')
  \]
  is independent of $\mathscr{F} _t$
  for any
  $ q' \geq 0,
B' \in \mathscr{B} (\mathbb{R}^\d)$, ${C' \in \mathscr{B} (\mathbb{R}_+)}$, and ${ \Xi' \in \mathscr{B}_{>t} (\Gamma _2)}$.

  Sometimes we will use the representations
  \[
   N = \sum\limits _{q \in \mathcal{I}} \delta _{(s_q, x_q, u_q, \gamma _q)}, \ \ \
  \hat \eta _0 = \sum\limits _{q \in \mathcal{J}} \delta _{(x_q, \gamma _q)},
  \]
where $\mathcal{I}$ and $\mathcal{J}$ are some countable disjoint sets.

 \begin{defi} \label{weak solution}
 A \emph{(weak) solution} of equation \eqref{se} is a triple
 $(( \eta _t )_{t\geq 0} , N )$, $(\Omega , \mathscr{F} , P) $,
 $(\{ \mathscr {F} _t  \} _ {t\geq 0}) $, where
\end{defi}

  (i) $(\Omega , \mathscr{F} , P)$ is a probability space,
and $\{ \mathscr {F} _t  \} _ {t\geq 0}$ is an increasing, right-continuous
 and complete filtration of sub-$\sigma$-algebras of $\mathscr {F}$,

  (ii)   $N$ is a
 Poisson point process on $\mathbb{R}_+  \times \mathbb{R}^\d \times \mathbb{R}_+ \times \Gamma _2 $  with
 intensity  $ds \times dx \times du \times \pi$,

  (iii) $ \eta _0  $ is a random  $\mathscr {F} _0$-measurable
  element in $\Gamma _0$
  satisfying \eqref{condition on eta _0},

  (iv) the processes $N$ and $\eta _0$ are independent,
 $N$ is
  compatible with $\{ \mathscr {F} _t  \} _ {t\geq 0} $,

  (v) $( \eta _t )_{t\geq 0} $ is a cadlag $\Gamma _0$-valued process
adapted to $\{ \mathscr {F} _t  \} _ {t\geq 0} $, $\eta _t \big| _{t=0} = \eta _0$,

  (vi) all integrals in \eqref{se} are well-defined,
\[
E \int\limits _0 ^t ds \Big[ \int\limits _{\mathbb{R} ^\d} b(x,s, \eta _{s-})
  + \sum\limits _{x \in \eta _{s-}} d(x,s,\eta _{s-}) \Big] < \infty, \ \ \ t > 0
  \]
   and

  (vii) equality \eqref{se} holds a.s. for all $t\in [0,\infty]$
  and all Borel sets $B$.

  Let
 \begin{align*}
   \mathscr{S} ^{0} _t =  \sigma \bigl\{ & \eta_0 ,
 N([0,q] \times B \times C \times \Xi) ,   \\ &
 q \in [0,t],
B \in \mathscr{B} (\mathbb{R}^\d), C \in \mathscr{B} (\mathbb{R}_+),  \Xi \in \mathscr{B}_t (\Gamma _2) \bigr\},
 \end{align*}
 and
 let $\mathscr{S} _t$ be the completion of $\mathscr{S} ^{0} _t$ under $P$.
 Note that $\{ \mathscr{S} _t \}_{t\geq 0} $
 is a right-continuous filtration.

  \begin{defi} \label{def strong solution}
    A solution  of \eqref{se} is called \emph{strong}
 if $( \eta _t )_{t\geq 0} $ is adapted to
$(\mathscr{S} _t, t\geq 0)$.
  \end{defi}

\begin{rmk}
 In the definition above we considered solutions as processes indexed by
 $ t\ \in[0,\infty)$.
 The reformulations for
the case  $ t \in [0,T]$, $0<T<\infty$, are straightforward.
This remark also applies to many of the results below.
\end{rmk}

 \begin{defi}
 We say that pathwise uniqueness holds for equation \eqref{se} and an
initial distribution $ \nu $ if, whenever the triples
$(( \eta _t )_{t\geq 0} , N)$, $(\Omega , \mathscr{F} , P) $,
 $(\{ \mathscr {F} _t  \} _ {t\geq 0}) $ and
 $(( \bar \eta _t )_{t\geq 0} , N)$, $(\Omega , \mathscr{F} , P) $,
 $(\{ \bar{\mathscr {F}} _t  \} _ {t\geq 0}) $ are weak solutions of \eqref{se} with
$P \{ \eta _0 = \bar{\eta} _0 \} = 1 $ and $Law (\eta _0) = \nu $,
we have $P \{ \eta _t = \bar{\eta} _t , t \in [0,\infty ) \} = 1$
(that is, the processes $\eta , \bar{\eta} $ are indistinguishable).

\end{defi}

\begin{defi} \label{joint uniqueness in law}
 We say that \textit{joint uniqueness in law} holds for equation \eqref{se} with an initial
distribution $\nu$ if any two (weak) solutions $((\eta_t) , N)$ and
$((\eta_t  ^{ \prime }) , N  ^{\prime} )$ of \eqref{se},
$Law(\eta _0)= Law(  \eta _0  ^{\prime})=\nu$, have the same joint distribution:

$$Law ((\eta_t) , N)
= Law ((\eta_t  ^{\prime} ), N ^{\prime}  ).$$

\end{defi}

\begin{thm}\label{core thm}
Pathwise uniqueness, strong existence and joint uniqueness
in law hold for equation \eqref{se}.
 The unique solution is a  Markov process. If $b$ and $d$ are time-homogeneous,
 then the family of
 push-forward measures $\{P_{\alpha}, \alpha \in \Gamma _0 \}$ defined
 in Remark \ref{encompass}
 constitutes a Markov process, or a Markov family of probability measures,
 on $D_{\Gamma _0}[0,\infty)$.
\end{thm}
\textbf{Proof}. The statement is a
consequence of Proposition \ref{ex un}, Remark \ref{encompass} and
Proposition \ref{Markov property}. In particular,
the Markov property of $\{P_{\alpha}, \alpha \in \Gamma _0 \}$
follows from Corollary \ref{flabbergast}.

We call the unique solution of \eqref{se} (or, sometimes, the corresponding
family of measures on $D_{\Gamma _0}[0,\infty)$) a
\textit{(spatial) birth-and-death Markov process}.

 \textbf{Remark}.
 For time-homogeneous $b$ and $d$,
 the transition probabilities of
 the embedded Markov chain (see e.g. \cite[Chapter 12]{KallenbergFound})
 of the birth-and-death process
 are completely described by
 \begin{align}\label{transition probabilities}
  Q(\eta, \{ \eta \setminus \{ x \} \}) &=
  \frac{d(x,\eta)}{(B+D)(\eta)} , \ \ \ \ \ \ \ x \in \eta, \ \ \eta \in \Gamma _0,  \\
  Q(\eta, \{ \eta \cup \{ x \}, x \in U\} )
  &= \frac{\int _{x\in U} b(x, \eta) dx}{(B+D)(\eta)}, \ \ \
  U \in \mathscr{B}(\mathbb{R} ^\d), \eta \in \Gamma _0, \notag
 \end{align}
 where $(B+D)(\eta) = \int _{\mathbb{R} ^\d} b(x,\eta) dx + \sum _{x \in  \eta } d(x, \eta) $.

The following two propositions establish a rigorous relation
 between the unique solution to \eqref{se}
 and $L$ defined by \eqref{intr generator}.
 To formulate the first of them,
 let us consider the class $\mathscr{C} _b$ of cylindrical functions $F :
\Gamma _0 \to \mathbb{R} _+$ with bounded increments.
We say that $F$ has bounded increments if
\[
\sup\limits _{\eta \in \Gamma _0, x \in \mathbb{R} ^\d}
\big( F(\eta \cup \{x\} ) - F(\eta ) \big) < \infty.
\]
We say that $F$ is cylindrical
if for some $R=R_F>0$
 \begin{equation*}
   F (\eta) = F(\zeta) \text{ whenever } \eta \cap \mathbf{B}(\mathbf{o}_\d , R)
   = \zeta \cap \mathbf{B}(\mathbf{o}_\d , R),
 \end{equation*}
where $\mathbf{B}(x , R)$ is
the closed ball of radius $R$ around $x$,
and $\mathbf{o}_\d$ is the origin in $\mathbb{R} ^\d$.
 We recall that the filtration $\{ \mathscr{S} _t, t \geq 0\}$
 appeared before Definition \ref{def strong solution}.

 \begin{prop} \label{poignant}
 Let $(\eta _t) _{t \geq 0}$ be a weak solution to \eqref{se}. Then
 for any $F \in \mathscr{C} _b$
 the process
 \begin{equation} \label{hooky}
 \begin{split}
  F(\eta _t) - \int\limits _0 ^t
  \big\{ \int\limits _{\mathbb{R} ^\d} b(x,s,\eta _{s-})
  [ F(\eta _{s-} \cup \{x\}) -  F(\eta _{s-}) ] dx
  \\ -
   \sum\limits _{x \in \eta _{s-}} d(x, s, \eta _{s-})
   [ F(\eta _{s-} \setminus \{x\}) -  F(\eta _{s-}) ] \big\}
  ds
  \end{split}
 \end{equation}
is an $\{ \mathscr{S} _t, t \geq 0\}$-martingale.
In particular, the integral in \eqref{hooky} is well-defined a.s.
\end{prop}

 \subsection{Aggregation model}

 Here we consider a specific time-homogeneous model,
 which we call an aggregation model.
 This model has a property that
the death rate decreases as the number
of neighbors grows. We treat here
the death rate given below in
 \eqref{d aggr}, and,
 in addition to previous assumptions,
we require the birth rate to
grow linearly on the number of
points in configuration in
the sense \eqref{b aggr}.
We prove in Proposition \ref{extinction}
that the probability of extinction is
 small if the
initial configuration has many points in some fixed Borel set
$\Lambda \subset \mathbb{R} ^\d$.
Propositions \ref{verfemen}, \ref{finite death}
and Theorem \ref{trajectory-wise asymptotics}
describe the
pathwise behavior
of the process.

Let
\begin{equation} \label{d aggr}
 d(x,\eta) = \exp \{ - \sum\limits _{y \in \eta} \varphi (x-y) \},
\end{equation}
 where
 $\varphi $ is a nonnegative measurable function.
Proposition \ref{ex un} ensures existence
 and uniqueness of solutions,
 and that the unique solution
 is a pure jump type Markov process.

 We want to show that,
 if the initial configuration has $m$ points in some bounded region, then,
 under some  assumption on $b$ and $\varphi$,
 the probability of extinction declines at least exponentially fast by $m$.
 Also, we will give a few statements
 describing the pace of growth of the number of points of the system.

 More specifically, let $\Lambda$ be a measurable subset of $\mathbb{R}^d$, the birth rate
 and the initial condition $\eta _0$ satisfy
 the same condition as in Proposition \ref{ex un},
 and, besides that, the
 inequalities
 \begin{equation}\label{b aggr}
  \int\limits _{\Lambda} b(x,\eta) dx \geq c|\eta \cap \Lambda|, \ \ \ \eta \in \Gamma _0,
 \end{equation}
 and
  \begin{equation}\label{b monotonne}
  \ \ \ \ \ \ \
  \ \ \ \ \ \  b(x,\eta^1) \leq  b(x,\eta^2), \ \ \ \eta^1, \eta^2 \in \Gamma _0, \eta^1 \subset \eta^2,
 \end{equation}
hold
for some positive $c$. We assume also that
\begin{equation}
 \inf\limits _{x,y \in \Lambda} \varphi (x-y) \geq \log a,
\end{equation}
where $a>1$.

 We say that the process $(\eta _t)_{t\geq 0}$  \textit {extincts}
if $\inf \{ t \geq 0: \eta _t = \varnothing \} < \infty $. This infimum is called
the \textit{time of extinction}.

\begin{prop} \label{extinction}

Let $\tilde{C} >0$. Then there exists $m _0 = m _0 (\tilde{C}) \in \mathbb{N}$
such that, whenever $m \geq m_0 $,
\[
 P _\alpha \big\{ (\eta _t)_{t\geq 0} \text{ extincts }  \big\} \leq \tilde{C}^{-m}
\]
for all $\alpha$ satisfying $|\alpha \cap \Lambda| = m$

\end{prop}

\begin{prop}\label{verfemen}
 For all $\alpha \in \Gamma _0$,
\begin{equation} \label{innately}
   P_{\alpha} \bigg( \{|\eta _t \cap \Lambda| \to \infty \}
   \cup \{ \exists t': \forall t\geq t', |\eta _t \cap \Lambda| = \varnothing \} \bigg) =1.
\end{equation}

\end{prop}

\textbf{Remark}. Note that we do not require $b(\cdot, \varnothing) \equiv 0$;
if $\int _{\Lambda} b(x,\varnothing) dx >0$, then
\eqref{innately} implies
 \[
   P_{\alpha}  \{|\eta _t \cap \Lambda| \to \infty \}  =1.
 \]

 The next proposition is a consequence of the exponentially fast decay
of the death rate.

\begin{prop} \label{finite death}
 With probability $1$ only a finite number of deaths inside $\Lambda $ occur:
\[
P _{\alpha}\bigg\{ |\eta _t \cap \Lambda| - |\eta _{t-} \cap \Lambda| = -1
\textrm{\ for infinitely many different \ } t \geq 0   \bigg\} =0, \ \alpha \in \Gamma _0.
\]
\end{prop}

 \begin{thm} \label{trajectory-wise asymptotics}

Let $\alpha \in \Gamma _0$.
 For $P_{\alpha}$-almost all $\omega \in F:=
 \{ \lim\limits _{t \to \infty} |\eta _t \cap \Lambda|=\infty \}$ we have
\begin{equation} \label{implore}
 \liminf\limits _{t\to \infty} \frac{|\eta _t \cap \Lambda|}{e^{ct}} > 0 .
\end{equation}

\end{thm}

 \begin{cor} \label{growth in average}
 For all configurations $ \alpha $ with $ \alpha \cap \Lambda \ne \varnothing $,

 \begin{equation} \label{growth in aver}
\inf\limits _{t > 0}  \frac{E _{\alpha} |\eta _t \cap \Lambda|}{e^{ct}} > 0.
  \end{equation}
\end{cor}

 \textbf{Remark}. If $\Lambda$ has a finite volume and
  the birth rate is given
 constant within $\Lambda$, that is
 \[
 b(x, \eta) = c_3 >0, \quad x \in \Lambda,
 \]
 then from the proofs we can conclude that Theorem \ref{trajectory-wise asymptotics}
 and Corollary \ref{growth in average} still hold provided that
 we replace \eqref{implore} by
 \begin{equation}
 \liminf\limits _{t\to \infty} \frac{|\eta _t \cap \Lambda|}{ c _3\textrm{Vol}_ \d (\Lambda)} > 0 .
\end{equation}
 and \eqref{growth in aver} by
  \begin{equation}
\inf\limits _{t > 0}  \frac{E _{\alpha} |\eta _t \cap \Lambda|}{c _3\textrm{Vol}_ \d (\Lambda)} > 0.
  \end{equation}
 Here $c _3\textrm{Vol}_ \d (\Lambda)$ is the $\d$-dimensional volume of $ \Lambda $.
 These two growth estimates stand in contrast to the mesoscopic behavior
 of the system \cite{Aggreg}. Theorem 5.3 in \cite{Aggreg} says
 that for some values of parameters the solution to the
 mesoscopic equation started from sufficiently small initial
 condition stays bounded. On the contrary,
 the microscopic system grows whenever it survives,
 and the density always grows.

 \section{Proof of Theorem \ref{core thm} and Proposition \ref{poignant}}

 Let us start with the equation

\begin{equation} \label{se pure birth}
\overline{\eta} _t (B) = \int\limits _{(0,t] \times B \times [0, \infty ) \times \Gamma _2 }
I _{ [0, \overline {b}(x,s,\eta _{s-} )] } (u) N(ds,dx,du,d\gamma) + \eta _0 (B),
\end{equation}
where
$\overline{b}(x, \eta ) := \sup\limits _{s >0, \xi \subset \eta} b(x, s, \xi) $.
Note that $\overline{b}$ satisfies sublinear growth condition
\eqref{sublinear growth for b} if $b$ does.

 This equation
is of the type \eqref{se},
with $\overline b$ being the birth rate,
and the zero function being the death rate, and all definitions
of existence and uniqueness of solutions
are applicable here. Later a unique solution
of \eqref{se pure birth} will be
used as
 a majorant of a solution to \eqref{se}.

 \begin{prop} \label{pure birth prop}
Under assumptions \eqref{sublinear growth for b} and \eqref{condition on eta _0},
strong existence and pathwise uniqueness hold for equation \eqref{se pure birth}.
In particular, the unique solution $(\bar \eta _t)_{t\geq 0}$ satisfies
\begin{equation}\label{beschwoeren}
 E|\bar \eta _t | < \infty, \ \ \  t \geq 0.
\end{equation}
\end{prop}

 \textbf{Proof}.
For $\omega \in  \{ \int\limits _{\mathbb{R}^\d }
\overline{b}(x,  \eta _0) dx =0 \} $, set $\zeta _t \equiv \eta _0$,
$\sigma _n = \infty$, $n \in \mathbb{N}$.

 For $\omega \in F:= \{ \int\limits _{\mathbb{R}^\d }
\overline{b}(x,  \eta _0) dx >0 \}$,
 we define the sequence of random pairs $\{(\sigma _n, \zeta _{\sigma _n}) \}$, where
\[
\sigma _{n+1}= \inf \{ t>0 : \int\limits _{(\sigma _n,\sigma _n+t] \times B \times [0, \infty ) \times \Gamma _2 }
I_{[0, \overline b(x,  \zeta _{\sigma _n})]} (u) N(ds,dx,du,d\gamma) >0 \}+ \sigma _n, \ \ \sigma _0 = 0,
\]
and
\[
 \zeta _{0} = \eta _0, \ \ \
\zeta _{\sigma _{n+1}} = \zeta _{\sigma _n} \cup \{ z_{n+1} \}
 \]
for $z_{n+1} = \{x\in \mathbb{R}^\d: N (\{\sigma _{n+1}\} \times \{x \} \times [0, \overline b(x,  \zeta _{\sigma _n})] \times \Gamma _2
) >0  \}$.
The points
$z_n$ are uniquely determined almost surely on $F$.
Furthermore,
 $\sigma _{n+1} > \sigma _n$ a.s., and $\sigma _n$
 are finite a.s. on $F$
 (in particular because $\overline{b}(x,\zeta _{\sigma _n}) \geq \overline{b}(x,\eta _0)$).
For $\omega \in F$, we define $\zeta _t = \zeta _{\sigma _n}$
for $t\in [\sigma _n, \sigma _{n+1})$. Then by induction on $n$
it follows that $\sigma _n $ is a stopping time for each $n \in \mathbb{N}$, and
$\zeta _{\sigma _n}$ is $\mathscr{F} _{\sigma _n} \cap F $-measurable.
By direct substitution we see that
 $(\zeta _t)_{t\geq 0}$ is a strong
 solution to \eqref{se pure birth} on the time interval
$t\in [0, \lim\limits _{n\to \infty} \sigma _n)$.
Although we have not defined what is a solution,
or a strong solution, on a
random time interval, we do not discuss it here.
Instead we are going to show that
\begin{equation}\label{staunch}
\lim\limits _{n\to \infty} \sigma _n = \infty \ \ \ \textrm{a.s.}
\end{equation}
This relation is evidently true on the complement of $F$.
If $P(F)=0$, then \eqref{staunch} is proven.

If $P(F)>0$, define a probability measure on $F$,
$Q(A) = \frac{P(A)}{P(F)}$, $A \in \mathscr{I} := \mathscr{F} \cap F$,
and define $ \mathscr{I}_t =\mathscr{F}_t \cap F $.

The process $N$ is independent of $F$, therefore it is a
Poisson point process on the probability space $(F,\mathscr{I}, Q)$
with the same intensity,
compatible with $\{ \mathscr{I}_t \}_{t\geq 0}$.
From now on and until other is specified, we work on the filtered
probability space $(F,\mathscr{I},\{ \mathscr{I}_t \}_{t\geq 0}, Q)$.
We use the same symbols for random processes and random variables,
having in mind that
we consider their restrictions to $F$.

The process $(\zeta _t)_{t\in [0, \lim\limits _{n\to \infty} \sigma _n)}$
has the Markov property, because
the process $N$ has the strong Markov property and independent increments, see Appendix.
Indeed, conditioning on $ \mathscr{I}_{\sigma_n}$,
\[
 E \bigl[  I_{\{\zeta _{\sigma _{n+1}} = \zeta _{\sigma _{n}} \cup x
 \text{ for some } x \in B \}} \mid \mathscr{I}_{\sigma_n}\bigr]=
 \frac{\int\limits _{ B} \overline b(x, \zeta _{\sigma _{n}}) dx}{\int\limits _{\mathbb{R}^\d }
\overline{b}(x,\zeta _{\sigma _n}) dx },
\]
thus the chain $\{ \zeta _{\sigma _{n}}\}_{n \in Z_+}$ is a Markov chain,
and, given $\{ \zeta _{\sigma _{n}}\}_{n \in Z_+}$,
$\sigma _{n+1} - \sigma _n $ are distributed exponentially:
\[
E\{ I_{\{\sigma _{n+1} - \sigma _n >a\}}  \mid \{ \zeta _{\sigma _{n}}\}_{n \in Z_+}\}
= \exp \{ - a \int\limits _{\mathbb{R}^\d }
\overline{b}(x,\zeta _{\sigma _n}) dx \}.
\]
Therefore, the random variables $\gamma _n = (\sigma _{n} - \sigma _{n-1}){(\int\limits _{\mathbb{R}^\d }
\overline{b}(x,\zeta _{\sigma _n}) dx)}$
constitute a
sequence of independent
 random variables exponentially distributed
with parameter $1$, independent
of $\{ \zeta _{\sigma _{n}}\}_{n \in Z_+}$.
 Theorem 12.18 in \cite{KallenbergFound} implies  that
$(\zeta _t)_{t\in [0, \lim\limits _{n\to \infty} \sigma _n)}$
is a pure jump type Markov process.

The jump rate of $(\zeta _t)_{t\in [0, \lim\limits _{n\to \infty} \sigma _n)}$
is given by
\[
 c(\alpha) = \int\limits _{\mathbb{R}^\d }
\overline{b}(x,\alpha) dx .
\]
Condition \eqref{sublinear growth for b} implies that
$c(\alpha) \leq c_1 |\alpha| + c_2$. Consequently,
\[
c (\zeta _{\sigma _n} ) \leq c_1 |\zeta _{\sigma _n}| + c_2 =
c_1 |\zeta _0| +c_1 n  + c_2.
\]

We see that $\sum _n \frac{1}{c (\zeta _{\sigma _n} )} = \infty$ a.s.,
hence Proposition 12.19 in \cite{KallenbergFound}
implies that $\sigma _n \to \infty$.

 Now we return again to our initial probability space
$(\Omega,\mathscr{F},\{ \mathscr{F}_t \}_{t\geq 0}, P)$.
We have proved the existence of a strong solution.
The uniqueness follows by induction on jumps of the process.
Namely, let $( \tilde{\zeta} _t )_{t\geq 0} $ be another solution of \eqref{se pure birth}. From (vii)
 of Definition \ref{weak solution} and the equality

$$
\int\limits _{ (0,\sigma _1) \times \mathbb{R} ^\d  \times [0, \infty ] \times \Gamma _2 }
I _{ [0,\overline{b}(x, {\eta} _0 )] }(u) N(ds,dx,du,d \gamma) = 0 ,
$$
it follows
that $P \{ \tilde{\zeta} \text{\ has a birth before \ }
\sigma _1  \} = 0$. At the same time, the equality

$$
\int\limits _{ \{\sigma _1\} \times \mathbb{R} ^\d  \times [0, \infty ] \times \Gamma _2 }
I _{ [0,\overline{b}(x, {\eta} _0 )] }(u) N(ds,dx,du,d \gamma) = 1 ,
$$
which holds a.s.,
yields that $\tilde{\zeta}$ has a birth at the moment $\sigma _1$, and in the same point
of space at that. Therefore, $\tilde{\zeta}$ coincides with $\zeta$ up to $\sigma _1$ a.s. Similar reasoning shows
that they coincide up to $\sigma _n$ a.s., and, since $\sigma _n \to \infty$ a.s.,

$$P \{ \tilde{\zeta} _t = {\zeta} _t \text{\ for all \ } t\geq 0 \} = 1.$$

 Thus, pathwise uniqueness holds.

Now we turn our attention to \eqref{beschwoeren}.
We can write

\begin{gather}\label{gibberish}
|{\zeta} _t| =
 |{\eta} _0|+  \sum\limits _{n=1}^{\infty} I \{ |\zeta _t| - |\eta _0| \geq n \} \notag
 \\  =
  |{\eta} _0|+  \sum\limits _{n=1}^{\infty} I \{ \sigma _n \leq t \}.
\end{gather}

Since $\sigma _n = \sum\limits _{i=1}^n \frac{\gamma _i}{\int\limits _{\mathbb{R}^\d }
\overline{b}(x,\zeta _{\sigma _i}) dx}$, we have

\[
 \{ \sigma _n \leq t \} = \{ \sum\limits _{i=1}^n \frac{\gamma _i}{\int\limits _{\mathbb{R}^\d }
\overline{b}(x,\zeta _{\sigma _i}) dx} \leq t \} \subset
\{ \sum\limits _{i=1}^n \frac{\gamma _i}{c_1|\zeta _{\sigma _i}| + c_2} \leq t \}
\]
\[
\subset \{ \sum\limits _{i=1}^n \frac{\gamma _i}{(c_1+c_2)(|\eta _0|+i)} \leq t \} =
\{Z _t -Z_0 \geq n \},
\]
where
$(Z_t)$ is the Yule process,
i.e. the birth process on $\mathbb{Z} _+$
with transition rates
$p_{k, k+1} =(c_1 + c_2)k$,
$p_{k, l} = 0$, $l \ne k +1$,
see, e.g., \cite[Chapter 3, Section 5]{Branch1}.
Here $(Z_t)$ is defined as follows:
$Z_t - Z _0 = n$ when
\[
\sum\limits _{i=1}^n \frac{\gamma _i}{(c_1+c_2)(|\eta _0|+i)} \leq t
< \sum\limits _{i=1}^{n+1} \frac{\gamma _i}{(c_1+c_2)(|\eta _0|+i)},
\]
and
$Z_0 = |\eta _0|$. Thus, we have $| \zeta _t| \leq Z _t$ a.s.,
hence $E| \zeta _t| \leq  E Z _t < \infty$.
 The constructed
 solution is strong.
\qed

\begin{prop}\label{ex un}
Under assumptions \eqref{sublinear growth for b}-\eqref{condition on eta _0},
 pathwise uniqueness and strong existence hold for equation \eqref{se}.
 The unique solution $(\eta _t)$ satisfies
 \begin{equation}\label{unfathomable}
 E| \eta _t | < \infty, \ \ \  t \geq 0.
\end{equation}
\end{prop}

\textbf{Proof}. Let us define stopping times with respect to
 ${\{ \mathscr{F} _t , t\geq 0 \}} $,
$0= \theta _0 \leq \theta _1 \leq \theta _2 \leq \theta _3 \leq ...$,
and the sequence of (random) configurations
$\{ \eta _{\theta _j} \} _{j \in \mathbb{N}}$
as follows: as long
as
\[
\theta _{n+1} = \theta ^b _{n+1} \wedge \theta ^\d _{n+1} + \theta _n < \infty,
\]
where
\[
\theta ^b _{n+1}= \inf \{ t>0 : \int\limits _{ (\theta _n, \theta _n +t] \times \mathbb{R}^\d \times [0,\infty) \times \Gamma _2}
I_{[0, b(x,s,\eta _{\theta _n})]} (u) N(ds,dx,du, d \gamma) >0 \},
\]
\[
\theta ^\d _{n+1}= \inf \{ t>0 :\sum\limits _{
\substack{ q \in \mathcal{I} \cup \mathcal {J},  \\ x_q \in \eta _{\theta _n}}
}
\int  _{(\theta _n, \theta _n +t]
 \times [0, \infty ) }
I _{ [0,d(x_i,r,\eta _{\theta _n} )] } (v) \gamma _q (dr,dv) > 0 \},
\]
we set
${\eta} _{\theta _{n+1}}
=  {\eta} _{\theta _n} \cup \{ z_{n+1} \}$ if $\theta ^b _{n+1} \leq \theta ^\d _{n+1}$,
where
$\{z_{n+1}\} = \{z\in \mathbb{R}^\d: N ( \{\theta _n + \theta ^b _{n+1}\} \times \{ z \} \times \mathbb{R}_+ \times \Gamma _2) >0  \}$;
${\eta} _{\theta _{n+1}} =  {\eta} _{\theta _n} \setminus \{ z_{n+1} \}$
if $\theta ^b _{n+1} > \theta ^\d _{n+1}$, where
$\{z_{n+1}\} = \{x_q \in {\eta} _{\theta _n}: \gamma _q ( \{\theta _n + \theta ^\d _{n+1}\} \times \mathbb{R}_+) >0  \}$;
the configuration $\eta _{\theta _0} = \eta _0$
is the initial condition of \eqref{se}, ${\eta} _t = {\eta} _{\theta _n} $ for
$t \in [\theta _n , \theta _{n+1} )$.
 Note that
\[
P\{  \theta ^b _{n+1} = \theta ^\d _{n+1}  \mid
 \min \{ \theta ^b _{n+1} , \theta ^\d _{n+1} \} < \infty \} = 0,
\]
 the points $z_n$ are a.s. uniquely determined, and
 \[
 P\{  z_{n+1} \in {\eta} _{\theta _n}\mid \theta ^b _{n+1} \leq \theta ^\d _{n+1} \} =0.
 \]
If for some $n$
\[
 \theta _{n+1} = \infty,
\]
 we set $\theta _{n+k} =\infty$, $k\in \mathbb{N}$, and
$\eta _{t} =  {\eta} _{\theta _{n}}$, $t \geq {\theta _{n}}$.

Random variables $\theta _n, n \in \mathbb{N}$, are stopping times with respect to
the filtration ${\{ \mathscr{F} _t , t\geq 0 \}} $.
Using the strong
Markov property of a Poisson point process, we see that,
on $\{ \theta _n < \infty \} $, the conditional distribution of $\theta ^b _{n+1}$ given
$\mathscr {F} _{\theta _n}$ is

\[
 P\{ \theta ^b _{n+1} > p \} = \exp \{ - \int_{\theta _n} ^{\theta _n + p} b(x,s,\eta _{\theta _n}) ds \},
\]
and the conditional distribution of $\theta ^\d _{n+1}$ given
$\mathscr {F} _{\theta _n}$ is
\[
 P\{ \theta ^\d _{n+1} > p \} = \exp \{ - \int_{\theta _n} ^{\theta _n + p} d(x,s,\eta _{\theta _n}) ds \}.
\]
In particular,
$\theta ^b _n, \theta ^\d_n > 0$, $n \in \mathbb{N}$.

 We are going to show that a.s.
 \begin{equation}\label{kowtow}
  \theta _n \to \infty, \quad n \to \infty.
 \end{equation}
Denote by $\theta _k '$ the moment of the $k$-th birth. It is sufficient to show that
$\theta _k ' \to \infty$, $k\to \infty$,
because only finitely many deaths may occur between any two births, since there
are only finitely particles. By induction
on $k '$ we can see that
$\{ \theta _k' \} _{k' \in \mathbb{N}} \subset \{ \sigma _i \} _{i \in \mathbb{N}}$, where
$\sigma _i$ are the moments of births of $(\overline{\eta} _t)_{t\geq 0}$,
the solution of \eqref{se pure birth},
 and $\eta _t \subset \overline{\eta} _t$
 for all $t \in [0,\lim_n \theta _n)$.
For instance, let us show that $(\overline{\eta} _t)_{t\geq 0}$ has a birth at
 $\theta _1 '  $. We have  $\overline{\eta} _{\theta _1 ' -} \supset \overline{\eta} _{0} = \eta _0$, and
$\eta _{\theta _1 ' -} \subset \eta _{t} \mid _{t=0} = \eta _0 $,
hence for all $x \in \mathbb{R}^\d$

$$\overline{b} (x, \overline{\eta} _{\theta _1 ' -} ) \geq \overline{b} (x, \eta _{\theta _1 ' -} ) \geq b (x, \eta _{\theta _1 ' -} )$$

The latter implies that at time moment $\theta _1 '  $ a birth
 occurs for the process $(\overline{\eta} _t)_{t\geq 0}$ in the same point.
 Hence, $\eta _{\theta _1 '} \subset \overline{\eta} _{\theta _1 ' } $,
 and we can go on.
Since $\sigma _k \to \infty$ as
$k \to \infty$, we also have $\theta _k ' \to \infty$, and therefore $ \theta _n \to \infty$, $n\to \infty$.

Let us now prove the inequality from item (vi) of Definition \ref{weak solution},
\begin{equation}\label{surreptitious}
E \int\limits _0 ^t ds \Big[ \int\limits _{\mathbb{R} ^\d} b(x,s, \eta _{s-})
  + \sum\limits _{x \in \eta _{s-}} d(x,s,\eta _{s-}) \Big] < \infty, \ \ \ t > 0.
\end{equation}
  Denote the number of births and deaths
before $t$ by $b_t $ and $d_t$ respectively, i.e.
\[
 b_t  = \# \{ s:| \eta _{s}| - |\eta _{s-}| = 1 \}=
\int\limits _{(0,t] \times \mathbb{R} ^\d \times [0, \infty ) \times \Gamma _2 }
 I _{ [0,b(x,s,\eta _{s-} )] } (u)
N(ds,dx,du,d\gamma)
\]
and
\[
 d_t  = \# \{ s:| \eta _{s}| - |\eta _{s-}| = - 1 \}=
\int  _{(0, t]
 \times [0, \infty ) }
 \sum\limits _{
\substack{ q \in \mathcal{I} \cup \mathcal {J},  \\ x_q \in \eta _{r-}}
}
I _{ [0,d(x_i,r,\eta _{r-} )] } (v) \gamma _q (dr,dv).
\]

Note that $|\eta _t| = b_t - d_t + |\eta _0|$ and $\theta _k$
are the moments of jumps for $c_t : = b_t + d_t$,
so that
 \[
   c_t= \sum\limits _{k \in \mathbb{N}} I \{ \theta _k \leq t \}, \ \ \ \ t \geq 0.
  \]
For $n \in \mathbb{N}$ define
\[
c ^{(n)} _t =
\int\limits _{(0,t] \times \mathbb{R} ^\d \times [0, \infty ) \times \Gamma _2 }
 I _{ [0,b(x,s,\eta _{s-} ) \wedge n] }  (u) I \{ |x| \leq n \}
N(ds,dx,du,d\gamma)
\]
\[
 +
\int  _{0, t]
 \times [0, \infty ) }
 \sum\limits _{
\substack{ q \in \mathcal{I} \cup \mathcal {J}  \\ x_q \in \eta _{r-}}
}
I _{ [0,d(x_i,r,\eta _{r-} ) \wedge n ] } (v) I \{ |x| \leq n \} \gamma _q (dr,dv).
\]
  Then
  \[
   M^{(n)}_t (x) = c ^{(n)} _t (x) - \int\limits _0 ^t \int _{x:  |x| \leq n }
\big(b(x,s,\eta _{s-} ) \wedge n \big) ds - \int\limits _0 ^t \int _{x:  |x| \leq n }
\big(d(x,s,\eta _{s-} ) \wedge n \big) ds
  \]
is a martingale with respect to
$\{ \mathscr{S} _t \}$.
By the optional stopping theorem $EM^{(n)}_{\theta _1 \wedge t} = 0$,
hence
\[
\int\limits _0 ^{\theta _1 \wedge t } \int _{x:  |x| \leq n }
\big(b(x,s,\eta _{s-} ) \wedge n +
d(x,s,\eta _{s-} ) \wedge n \big) ds \leq  P\{\theta _{1} < t  \} \leq 1.
\]
Similarly,
\[
\int\limits _{\theta _m \wedge t} ^{\theta _{m+1} \wedge t} \int _{x:  |x| \leq n }
\big(b(x,s,\eta _{s-} ) \wedge n +
d(x,s,\eta _{s-} ) \wedge n \big) ds \leq  P\{\theta _{m} < t  \}.
\]
Consequently,
\[
\int\limits _{0} ^{t} \int _{x:  |x| \leq n }
\big(b(x,s,\eta _{s-} ) \wedge n +
d(x,s,\eta _{s-} ) \wedge n \big) ds
\]
\[
 \leq \sum\limits _{m=1} ^{\infty} E
 \int\limits _{\theta _m} ^{\theta _{m+1}} \int _{x:  |x| \leq n }
\big(b(x,s,\eta _{s-} ) \wedge n  +
d(x,s,\eta _{s-} ) \wedge n \big) ds
\]
\[
 \leq \sum\limits _{m = 1} ^ \infty  P\{\theta _{m} \leq t  \} =
 \sum\limits _{m = 1} ^ \infty  P\{c_t  \geq m  \} = E c_t .
\]
Letting
$n \to \infty$, we get by the monotone convergence theorem
\[
 \int\limits _{0} ^{t} \int _{\mathbb{R} ^\d }
\big(b(x,s,\eta _{s-} )  +
d(x,s,\eta _{s-} )  \big) ds \leq E c_t
\]
Only existing particles may disappear, hence the number of deaths $d_t $
satisfies
\[
 d_t  \leq b_t  + |\eta _0 |.
\]
Thus,
\begin{equation} \label{on the skids}
 E c_t  \leq 2 E b_t + E |\eta _0|
\leq
2 E |\bar \eta _t| + E |\eta _0| < \infty.
\end{equation}

 Since $\eta _t \subset \overline{\eta} _t$ a.s., Proposition \ref{pure birth prop}
 implies \eqref{unfathomable}.

If follows from the above construction, \eqref{kowtow} and \eqref{surreptitious}
that $(\eta _t)$ is a strong solution to \eqref{se}.
Similarly to the proof of Proposition \ref{pure birth prop},
we can show by induction on $n$ that equation \eqref{se}
has a unique solution on $[0, \theta _n]$. Namely, each two
solutions coincide on $[0, \theta _n]$ a.s.
Thus, any solution coincides with $({\eta} _t)$ a.s. for all
$t  \in [0, \theta _n] $.

 \qed

 \begin{rmk}\label{encompass}

 Assume that $b$ and $d$ are time-homogeneous.
Let $\eta _0$ be a non-random initial condition, $\eta _0 \equiv \alpha$,
$\alpha \in \Gamma _0$.
The solution of \eqref{se} with $\eta _0 \equiv \alpha$
will be denoted as $(\eta(\alpha, t))_{t\geq 0}$.
Let $P_\alpha$ be
the push-forward of $P$ under the mapping

\begin{equation}
  \Omega \ni \omega \mapsto (\eta(\alpha, \cdot)) \in D_{\Gamma _0}[0,\infty ).
\end{equation}

 It can be derived from the proof of Proposition \ref{ex un}
 that, for fixed $\omega \in \Omega$,
the unique solution is jointly measurable in $(t, \alpha)$.
Thus, the family $\{ P_\alpha \}$
 of probability measures on $D_{\Gamma _0}[0,\infty )$
is measurable in $\alpha$.
 We will often use formulations
related to the probability space
$(D_{\Gamma _0}[0,\infty ),\mathscr{B}(D_{\Gamma _0}[0,\infty )), P_{\alpha})$;
in this case, coordinate mappings will be denoted
by $\eta_t$,

\[
 \eta _t (x) = x(t), \ \ \   x \in D_{\Gamma _0}[0,\infty ).
\]

The processes $(\eta _t) _{t \in  [0,\infty )}$
and $(\eta(\alpha, \cdot))_{t \in  [0,\infty )} $
have the same law (under $P_{\alpha} $ and $P$, respectively).
As one would expect, the family
of measures $\{P_{\alpha}, \alpha \in \Gamma _0 \}$
is a Markov process, or a Markov family of probability measures;
see Proposition \ref{Markov property}
below. For a measure $\mu$ on $\Gamma _0$,
 we define

 \[
  P_{\mu} = \int P_{\alpha} \mu (d \alpha).
 \]

We denote by $E_{\mu}$ the expectation under $P_{\mu}$.

\end{rmk}

 \begin{rmk}\label{3rmk2}
Let $b_1,d_1$ be another pair of birth and death rates,
satisfying all conditions imposed on
$b$ and $d$.
Consider a unique solution $(\tilde{\eta} _t)$ of \eqref{se} with rates
$b_1, d_1$ instead of
$b,d$, but with the same initial condition $\eta _0$ and all the
other underlying structures. \textit{If for all }
$\zeta \in D$, \textit{where}
$D \in \mathscr{B}( \Gamma _0 (\mathbb{R}^\d))$ ,
$b_1 (\cdot ,\zeta) \equiv b (\cdot ,\zeta) $, $d_1 (\cdot ,\zeta) \equiv d (\cdot ,\zeta) $,
 \textit{then}
$\tilde{\eta} _t = \eta _t$ \textit{\ for all } $t\leq \inf \{s \geq 0: \eta_s \notin D\} =
\inf \{s \geq 0: \tilde{\eta}_s \notin D\} $.
This may be proven in the same way as the theorem above.

\end{rmk}

\begin{rmk}
Assume that all the conditions
of Proposition \ref{ex un} are fulfilled
except  \eqref{condition on eta _0}.
Then we could not claim that
 \eqref{unfathomable} holds.
However, we would still get a unique solution on $[0,\infty)$.
We are mostly interested in the case of a non-random
initial condition, therefore we do not discuss
the case when \eqref{unfathomable} is not satisfied.
\end{rmk}

\begin{rmk}\label{Functional dependence}
 We solved equation \eqref{se}
 $\omega$-wisely.
  We can deduce from the proof
 of Proposition \ref{ex un} that $\theta _n$ and $z_n$ are measurable
 functions of $\eta _0$ and $N$
 in the sense that, for example,
 $\theta_1 = F _1 (\eta _0, N)$ a.s.
 for a measurable  function
 $F_1: \Gamma _0
 \times \Gamma( \mathbb{R} _+ \times \mathbb{R} ^\d \times \mathbb{R} _+ \times \Gamma _2) \to \mathbb{R} _+$.
 As a consequence,
 there is a functional
 dependence of the solution
 process and the ``input'':
 the process $(\eta _t)_{t\geq 0}$
 is some function of
 $\eta _0$ and $N$.

\end{rmk}

The following corollary is a consequence of Proposition
\ref{ex un} and Remark \ref{Functional dependence}.

\begin{cor} \label{flabbergast}
Joint uniqueness in law holds for equation \eqref{se} with initial
distribution $\nu$ satisfying

\[
 \int _{\Gamma _0} |\gamma| \nu (d \gamma) < \infty.
\]

\end{cor}

As usually, the Markov property of a solution follows from uniqueness.

 \begin{prop}\label{Markov property}
 The unique solution $(\eta _t) _{t\in [0,\infty )}$ of \eqref{se} is a Markov process.

\end{prop}

\textbf{Proof}.
Take arbitrary $t > t_0 >0$. Consider
the equation

\begin{align} \label{se t_0<t}
\xi _t (B) = \int\limits _{(t_0,t] \times B \times [0, \infty ) \times \Gamma _2 }
& I _{ [0,b(x,s,\eta _{s-} )] } (u) I \{ \int _s ^t I_{[0,d(x,r, \eta _{r-})]}(v)\gamma (dr,dv) =0 \}
N(ds,dx,du,d\gamma) \notag \\
+ \int\limits _{B \times  \Gamma _2 }
I \{ \int _{t_0} ^t & I_{[0, d(x,r, \eta _{r-})]}(v)\gamma (dr,dv) =0  \} \hat {\eta} _{t_0} (dx, d \gamma )
+ \eta _{t_0} (B),  \quad
t \geq t _0.
\end{align}

The unique
solution of \eqref{se t_0<t}
is $(\eta _s) _{s\in [t_0, \infty )}$.
As in the proof of Proposition \ref{ex un}
we can see that $(\eta _s) _{s\in [t_0,\infty )}$
is measurable with respect to the filtration
generated by  random variables of the form
 $N( [s,q] \times B \times U \times \Xi)$,
and $\eta _{t_0}(B)$, where
$B \in \mathscr{B} (\mathbb{R} ^\d)$, $i \in \mathbb{Z}$,
$t_0\leq s \leq q $, $U \in \mathscr{B} (\mathbb{R} _+)$
$\Xi \in \mathscr{B} _{>t}(\Gamma _2)$.
Poisson point process have independent increments,
hence for any $\mathcal{U} \in \mathscr{B} (D _{\Gamma _0}[t_0, \infty))$
\[
 P \{ (\eta _t) _{t\in [t_0,\infty )} \in \mathcal{U} \mid \mathscr{F}_{t_0} \} =
 P \{ (\eta _t) _{t\in [t_0,\infty )} \in \mathcal{U} \mid \eta _{t_0} \}
\]
almost surely.
\qed

\textbf{Proof of Proposition} \ref{poignant}.
Let $N_1$ be the image of $N$ under the projection
\[
  (s,x,u ,\gamma) \mapsto (s,x,u).
\]
The process $N_1$ is a Poisson point process on $\mathbb{R} _+ \times \mathbb{R} ^\d \times \mathbb{R} _+$ with
intensity measure $dsdxdu$.
We have
\[
 \eta _t (B) = \int\limits _{(0,t] \times B \times [0, \infty ) \times \Gamma _2 }
 I _{ [0,b(x,s,\eta _{s-} )] } (u) I \{ \int _s ^t I_{[0,d(x,r, \eta _{r-})]}(v)\gamma (dr,dv) =0 \}
N(ds,dx,du,d\gamma)
\]
\[
+ \int\limits _{B \times  \Gamma _2 }
I \{ \int _0 ^t I_{[0, d(x,r, \eta _{r-})]}(v)\gamma (dr,dv) =0 \} \hat {\eta} _0 (dx, d \gamma )
\]
\[
 = \int\limits _{(0,t] \times B \times [0, \infty )  }
 I _{ [0,b(x,s,\eta _{s-} )] } (u)
N_1(ds,dx,du) + \eta _0 (B)
\]
\[
 - \sum\limits _{q \in \mathcal{I}\cup \mathcal{J}} \int\limits _{(0,t] \times [0, \infty ) }
 I\{ x_q \in \eta _{r-} \}
 I_{[0, d(x,r, \eta _{r-})]}(v) \gamma _q (dr,dv).
\]
  By Ito's formula for $F \in \mathscr{C} _b$
\[
 F(\eta _t) - F(\eta _0)
 =
 \int\limits _{(0,t] \times \mathbf{B}(\mathbf{o}_d, R _F ) \times [0, \infty ) }
 I _{ [0,b(x,s,\eta _{s-} )] } (u)
 \big\{ F(\eta _{s-} \cup x  ) - F(\eta _{s-}) \big\}
N_1(ds,dx,du)
\]
\[
 + \sum\limits _{q \in \mathcal{I}\cup \mathcal{J}} \int\limits _{(0,t] \times [0, \infty ) }
 I\{ x_q \in \eta _{r-} \}
 I_{[0, d(x,r, \eta _{r-})]}(v)
 \big\{ F(\eta _{s-} \setminus x  ) - F(\eta _{s-}) \big\}
 \gamma _q (dr,dv).
\]

We can write
\[
 \int\limits _{(0,t] \times \mathbf{B}(\mathbf{o}_d, R _F ) \times [0, \infty ) }
 I _{ [0,b(x,s,\eta _{s-} )] } (u)
 \big\{ F(\eta _{s-} \cup x  ) - F(\eta _{s-}) \big\}
N_1(ds,dx,du)
\]
\[
 = \int\limits _{(0,t] \times \mathbf{B}(\mathbf{o}_d, R _F )  } b(x,s,\eta _{s-} )
  \big\{ F(\eta _{s-} \cup x  ) - F(\eta _{s-}) \big\} dx ds
\]
\[
 +  \int\limits _{(0,t] \times \mathbf{B}(\mathbf{o}_d, R _F ) \times [0, \infty ) }
 I _{ [0,b(x,s,\eta _{s-} )] } (u)
  \big\{ F(\eta _{s-} \cup x  ) - F(\eta _{s-}) \big\}
\tilde N _1 (ds,dx,du),
\]
where $\tilde N = N - dsdxdu$. Since $F \in \mathscr{C}_b$,
the process
\[
 \int\limits _{(0,t] \times \mathbf{B}(\mathbf{o}_d, R _F ) \times [0, \infty ) }
 I _{ [0,b(x,s,\eta _{s-} )] } (u)
  \big\{ F(\eta _{s-} \cup x  ) - F(\eta _{s-}) \big\}
\tilde N _1 (ds,dx,du)
\]
is a martingale by item (vi) of Definition \ref{weak solution},
see e.g.  \cite[Section 3 of Chapter 2]{IkedaWat}.
Similarly,
\[
\sum\limits _{q \in \mathcal{I}\cup \mathcal{J}} \int\limits _{(0,t] \times [0, \infty ) }
 I\{ x_q \in \eta _{r-} \}
 I_{[0, d(x,r, \eta _{r-})]}(v)
 \big\{ F(\eta _{s-} \setminus x  ) - F(\eta _{s-}) \big\}
 \gamma _q (dr,dv)
\]
can be decomposed into a sum of
\[
 + \int\limits _{(0,t] }
 \sum\limits _{x \in \eta _{r-}}
d(x,r, \eta _{r-})]
 \big\{ F(\eta _{s-} \setminus x  ) - F(\eta _{s-}) \big\}
 dr
\]
and a martingale. The desired statement follows. \qed

 \subsection{Coupling}

Here we discuss the coupling
of two birth-and-death processes.
As a matter of fact,
we have already used the coupling technique in
the proof of Proposition \ref{ex un}.

Consider two
equations of the form \eqref{se},

\begin{equation} \label{2se}
\begin{split}
\xi ^{(k)} _t (B) = \int\limits _{(0,t] \times B \times [0, \infty ) \times \Gamma _2 }
& I _{ [0,b _k(x,s,\xi ^{(k)} _{s-} )] } (u) I \{ \int _s ^t I_{[0,d _k(x,r, \xi ^{(k)} _{r-})]}(v)\gamma (dr,dv) =0 \}
\\
\times
N(ds,dx,du,d\gamma)
+ \int\limits _{B \times  \Gamma _2 } &
I \{ \int _0 ^t I_{[0, d_k(x,r, \xi ^{(k)} _{r-})]}(v)\gamma (dr,dv) =0 \} \hat {\eta} _0 (dx, d \gamma ) , \ \  k=1,2.
\end{split}
\end{equation}

Assume that the initial conditions $\xi ^{(k)}_0$ and the rates
$b_k$, $d_k$ satisfy the conditions of Proposition \ref{ex un}. Let
$(\xi ^{(k)}_t)_{t \in [0,\infty )}$
be the unique strong solutions.

\begin{prop} \label{couple}
 Assume that almost surely $\xi ^{(1)}_0 \subset \xi ^{(2)}_0$, and
for any two finite configurations $\eta ^1 \subset \eta ^2$,
\begin{equation} \label{culpable}
b_1(x,s,\eta ^1) \leq b_2 (x,s, \eta ^2), \ \ \ x\in \mathbb{R}^\d, s \geq 0
\end{equation}
and
$$
d_1(x,s,\eta ^1) \geq d_2 (x,s, \eta ^2), \ \ \ x\in \eta ^1, s \geq 0
$$

Then

\begin{equation} \label{culprit}
 \xi ^{(1)}_t \subset \xi ^{(2)}_t, \ \ \ t \in [0,\infty ).
\end{equation}

\end{prop}

 We will show by induction
that each moment of birth for $(\xi ^{(1)}_t)_{t \in [0,\infty )}$ is
a moment of birth for $(\xi ^{(2)}_t)_{t \in [0,\infty )}$ too, and
each moment of death for $(\xi ^{(2)}_t)_{t \in [0,\infty )}$
is a moment of death for $(\xi ^{(1)}_t)_{t \in [0,\infty )}$
if the dying point is in $(\xi ^{(1)}_t)_{t \in [0,\infty )}$.
Moreover, in both cases
the birth or the death occurs at exactly the same point. Here a moment
of birth is a random time at which a new point appears, a
moment of death is a random time at which a point disappears
from the configuration. The statement formulated
above is in fact equivalent
to \eqref{culprit}.

 Here we deal only with
the base case, the induction step is done in the same way.
We have nothing to show
if $\tau _1$ is a moment of a birth of $(\xi ^{(2)}_t)_{t \in [0,\infty )}$
or a moment of death of $(\xi ^{(1)}_t)_{t \in [0,\infty )}$.
Assume that a new point is born for  $(\xi ^{(1)}_t)_{t \in [0,\infty )}$
at $\tau _1$,

$$
\xi ^{(1)}_{\tau _1} \setminus \xi ^{(1)}_{\tau _1 - } =\{x_1 \}.
$$
The process $(\xi ^{(1)})_{t \in [0,\infty )}$ satisfies \eqref{se}, therefore
$N_1(\{x \} \times \{ \tau _1 \} \times [0,b _k(x_1, {\tau _1}, \xi ^{(1)} _{\tau _1-} )]) = 1$.
Since
$$
\xi ^{(1)} _{\tau _1-} = \xi ^{(1)} _0 \subset \xi ^{(2)} _0 = \xi ^{(2)} _{\tau _1-},
$$
by \eqref{culpable}
$$
N_1(\{x \} \times \{ \tau _1 \} \times [0,b _k(x_1, {\tau _1}, \xi ^{(2)} _{\tau _1-} )]) = 1,
$$
hence
$$
\xi ^{(2)}_{\tau _1} \setminus \xi ^{(2)}_{\tau _1 - } =\{x_1 \}.
$$
The case when $\tau _2$ is a moment of
death for $(\xi ^{(2)}_t)_{t \in [0,\infty )}$ is analyzed
analogously.
\qed

 \section{Aggregation model: proofs}

The main idea behind our analysis in this section
is to couple the process $(\eta _t)_{t\geq 0}$
with another
birth-and-death process,
to which we can apply Lemma \ref{lumpability}.

To do so,
let us introduce another pair of the birth and death rates,
$b_1, d_1 $, and an initial condition
$\xi _0 = \eta _0 \cap \Lambda$, such that
$b_1(x, \eta) = d_1 (x, \eta) = 0 $ for $x \notin \Lambda$, $d_1 (x, \eta) =  a^{-|\eta|} $ for
$x \in \Lambda$, $b_1(x, \eta) \leq b(x, \eta)$ for all $x, \eta$, and
for some constant $c>0$
$$
\int\limits _{\Lambda} b_1(x,\eta) dx = c|\eta \cap \Lambda|, \quad \eta \in \Gamma _0.
$$
There exists a function $b_1$ satisfying these assumptions.

Functions $b_1$, $d_1$ satisfy conditions of
Theorem \ref{core thm}.
Furthermore, the conditions
of
Proposition \ref{couple} are fulfilled here:
for $\eta^1, \eta^2 \in \Gamma _0$, $\eta^1 \subset \eta^2$
we have
\[
 b_1(x,\eta^1)\leq b(x,\eta^1) \leq b(x,\eta^2)
\]
as well as
\[
 d_1(x,\eta^1)\geq d(x, \eta^1) \geq d(x,\eta^2).
\]
 Denote by $(\xi _t)_{t \geq 0}$ the unique solution
 of \eqref{se} with the birth and death rates $b_1, d_1 $
and initial condition $\xi _0$.
By Proposition \ref{couple}, $\xi _t \subset \eta _t$ hold a.s. for all
$t \geq 0$.

  In this section
we will work on the canonical probability space
 $$
 \big(D_{\Gamma _0}[0,\infty) \times D_{\Gamma _0}[0,\infty), \mathscr{B}(D_{\Gamma _0}[0,\infty) \times D_{\Gamma _0}[0,\infty)),
 P_{\alpha} \big),
 $$
where
$P_{\alpha}$ is the push-forward of
the measure $P$ under
 \[
 \Omega \ni \omega \mapsto (\eta(\alpha, \cdot),(\xi(\alpha, \cdot))
 \in D_{\Gamma _0}[0,\infty) \times D_{\Gamma _0}[0,\infty).
\]

 Consider the embedded Markov chain of the process $(\xi_t)_{t \geq 0}$,
$Y_k := \xi_{\tau _k}$, where $\tau _k$ are the moments of jumps of $(\xi _t)$.
It turns out that the process
$u = \{ u_k \}_{k \in \mathbb{N}} $, where $u_k := |Y_k|$, is a Markov chain too.
Indeed,
the equality
\[
P_{\alpha _1}\{ |Y_1| = k  \} =
P_{\alpha _2}\{ |Y_1| =k  \} , \ \ \ k\in \mathbb{N} , \alpha \in \Gamma _0.
\]
holds when $|\alpha _1 \cap \Lambda|=|\alpha _2 \cap \Lambda|$, since both sides are equal to
\[
 \left\{
  \begin{array}{l l}
    \frac{c}{c+a^{-|\alpha _1 \cap \Lambda|}} & \quad \text{if} \quad k=|\alpha _1 \cap \Lambda|+1, \\
    \frac{a^{-|\alpha _1 \cap \Lambda|}}{c+a^{-|\alpha _1 \cap \Lambda|}} & \quad \text{if} \quad k=|\alpha _1 \cap \Lambda|-1 ,\\
     0 & \quad \text{in other cases.}
  \end{array} \right.
\]
Therefore,
Lemma \ref{lumpability} is applicable here,
with $f(\cdot) = |\cdot |$.

\textbf{Proof of Proposition} \ref{extinction}. Having in mind the inclusion $\xi _t \subset \eta _t $ ($P_{\alpha}$-a.s.),
we will prove this lemma for $(\xi _t)$.

The transition probabilities for the Markov chain $\{ u_k \}_{k \in \mathbb{Z} _+} $
are given by
\begin{equation} \label{p_ij}
 p_{i,j} =  P_{\alpha}\{ u_k =j \mid |u_{k-1}| = i \} = \left\{
  \begin{array}{l l}
    \frac{c}{c+a^{-i}} & \quad \text{if} \quad j=i+1, \\
    \frac{a^{-i}}{c+a^{-i}} & \quad \text{if} \quad j=i-1 ,\\
     0 & \quad \text{in other cases,}
  \end{array} \right.
\end{equation}
for $i \in \mathbb{N} , j \in \mathbb{Z} _+$,
  and $p_{0,j} = I _ {\{ j =0 \} } $,
  see Remark \ref{finagle} and \eqref{transition probabilities}.

Since the zero is a trap and it is accessible
from all other states,
there are no recurrent states except zero,
and the process $u$ has only two possible
types of behavior on infinity:
\[
P _{\alpha} \{ \exists l \in \mathbb{N} \text{\ s.t.\ } u_l= \varnothing \text{\ or\ }
\lim\limits _{m \to \infty} u_m=\infty \} = 1.
\]

We will now use
properties of countable state space Markov chains,
see, e.g.,
 \cite[$\mathsection$ 12,
chapter 1]{Chung}. Chung considers there Markov chain with a reflecting barrier at 0, but we may still
apply those results,
adapting them correspondingly. Denote
$\varrho _m = \prod\limits _{k=1}^m \frac{p_{k,k-1}}{p_{k,k+1}}$. Then the probability
$P_{\alpha}\{ \exists k \in \mathbb{N} \text{\ s.t.\ } u_k=0  \} $ equals to 1 if and only if
$\sum\limits _{j=1}^{\infty} \varrho _j = \nobreak \infty$,
whichever initial condition ${\alpha}$, $|{\alpha} \cap \Lambda|>0$, we have.
Moreover, if $\sum\limits _{j=1}^{\infty} \varrho _j < \nobreak \infty$
and $P _{\alpha} \{ u_0=q \} =1$ (or, equivalently,
$|\alpha \cap \Lambda| =q$), then
$p_q := P _{\alpha} \{ \exists k \in \mathbb{N} \text{\ s.t.\ } u_k=0  \} =
\frac {\sum\limits _{j=q} ^{\infty} \varrho _j } {1+ \sum\limits _{j=1}^{\infty} \varrho _j}$. From \eqref{p_ij}
we see that in our case  $\varrho _j =c^{-j} a^{-\frac {j(j+1)}{2}} $, and

\begin{equation} \label{probability of extinction}
p_q = \frac {\sum\limits _{j=q} ^{\infty} c^{-j} a^{-\frac {j(j+1)}{2}} }
{1+ \sum\limits _{j=1}^{\infty} c^{-j} a^{-\frac {j(j+1)}{2}}} \leq
\frac {\sum\limits _{j=q} ^{\infty} c^{-j} a^{-\frac {j^2}{2}} }
{1+ \sum\limits _{j=1}^{\infty} c^{-j} a^{-\frac {j^2}{2}}} .
\end{equation}

Now, for arbitrary $C >1$ choose $q \in \mathbb{N}$ for which
$c^{-1}a^{-\frac{q}{2}} < C^{-1}$.
For $j>q$ we have $c^{-j} a^{-\frac {j^2}{2}} < c^{-j} a^{-\frac {jq}{2}}=(c^{-1} a^{-\frac {-q}{2}})^j<C^{-j}$,
and
\[
\sum\limits _{j=q} ^{\infty} c^{-j} a^{-\frac {j^2}{2}}<
\sum\limits _{j=q} ^{\infty} C^{-j} = \frac{C^{-q}}{1-C^{-1}},
\]
so
that the statement of the lemma for $(\xi _t)_{t \geq 0}$
follows from \eqref{probability of extinction}.
\qed

Note that
the number of particles of the process will go to
infinity with probability $1$ even though the probability of extinction is positive,
unless
$b(\cdot, \varnothing) = 0 $ almost everywhere with respect to the Lebesgue measure.
 However,
if $b(\cdot, \varnothing) \equiv 0 $, then
$$
P\bigg( \{ |\xi _t| = 0 \text{\ for large $t$  } \} \cup \{
|\xi _t| \to \infty \ , t \to \infty    \} \bigg) =1
$$
and
\[
 P\bigg( \{ |\xi _t| = 0 \text{\ for large $t$  } \} \cap \{
|\xi _t| \to \infty \ , t \to \infty    \} \bigg) =0.
\]

The following equality is also taken from
\cite[$\mathsection$ 12,
chapter 1]{Chung}; for $q>s$
and all $\beta$ with $|\beta \cap \Lambda| = q$,

$$P_{\beta} \{ \exists k\in \mathbb{N}: |u_k| = s  \} =
\frac {\sum\limits _{j=q} ^{\infty} \varrho _j (s)} {1+ \sum\limits _{j=s+1}^{\infty} \varrho  _j (s)},
$$
where $\varrho _m (s) = \prod\limits _{k=s+1}^m \frac{p_{k,k-1}}{p_{k,k+1}} =
c^{-(m-s)}a^{ -\frac12 (m-s)(m+s+1)}$;
in our case
\begin{equation}\label{meadow}
 P_{\beta} \{ \exists k\in \mathbb{N}: |u_k| = s  \} =
\frac{\sum\limits _{j=q} ^{\infty} c^{-(j-s)}a^{ -\frac12 (j-s)(j+s+1)}}
{1+ \sum\limits _{j=s+1}^{\infty} c^{-(j-s)}a^{ -\frac12 (j-s)(j+s+1)}} :=c_{q,s}  <1.
\end{equation}

Note that

\begin{equation}\label{gepraegt}
 c_{q+1,1} \to 0, \ \ \ \ q \to \infty
\end{equation}

\textbf{Proof of Proposition} \ref{verfemen}. Let $(X_k)_{k \in \mathbb{Z} _+}$ be the
embedded chain of $(\eta _t)_{t\geq 0}$. First we will show
that for all $m \in \mathbb{N}$ and $\alpha \in \Gamma _0$,

\begin{equation}\label{feisty}
 P_{\alpha}  \{|X_k \cap \Lambda| =m \text{ infinitely often } \}  =0.
\end{equation}

Let $\beta \in \Gamma _0$, $|\beta \cap \Lambda| = m$, $m \in \mathbb{N}$
(the case of $m =0$ is similar, and we do not
write it down).
Denote $\tilde k = \min \{k\in \mathbb{N} : X_k \cap \Lambda \ne X_0 \cap \Lambda \}$.
Since $\xi _t \subset \eta _t $ holds $P_{\beta}$ - a.s.,

\begin{equation}\label{damp air}
\begin{split}
 P_{\beta} \big\{ |X_k \cap \Lambda| > m, \forall k \geq \tilde k \big\} \geq & \
  P_{\beta} \big\{ |Y_k \cap \Lambda| > m, \forall k \geq 1 \big\}
 \\
  = P_{\beta} \big\{ u_k > m, & \ \forall k \geq 1  \big\}.
\end{split}
\end{equation}

 By \eqref{meadow}, the probability $P_{\beta} \{ u_k > m,  \forall k \geq 1  \}$
 is positive and does
not depend on $\beta$, $|\beta \cap \Lambda| = m$:
\begin{equation}\label{impertinent}
s_m :=
  P_{\beta} \{ u_k > m,  \forall k \geq 1  \} \geq p_{m,m+1} (1- c_{m+1,m}) >0.
\end{equation}

Define $k^m_{i}$, $i \in \mathbb{N}$, subsequently by
$k^m_{j+1} = \min \{k > k^m_{j} : |X_k \cap \Lambda| =m \text{ and }
\exists \bar k < k : |X_{\bar k} \cap \Lambda|  \ne m \}$, $k^m _0 =0$.
Note that for all $\beta$
\[
 P _{\beta} \bigg\{ \exists n_0: |X_n \cap \Lambda|=m  \text{ for all } n \geq n_0 \bigg\} =0.
\]

 By the strong Markov property,
\[
P_{\alpha}  \bigg\{|X_k \cap \Lambda| = m \text{ infinitely often } \bigg\} \leq
 P_{\alpha}  \bigg\{k^m_{j} < \infty, \forall j \in \mathbb{N} \bigg\}
\]
\[
=\prod\limits _{j=1}^{\infty} P_{\alpha}  \big\{k^m_{j+1} < \infty \mid k^m_{j} < \infty \big\} = 0,
\]
by  \eqref{damp air} and \eqref{impertinent}.
 Indeed,
if $P_{\alpha}  \{ k^m_{j} < \infty \} >0$,
then

\[
 P_{\alpha}  \{k^m_{j+1} < \infty \mid k^m_{j} < \infty \} =
 \frac{E_{\alpha} I_{ \{ k^m_{j} < \infty \} } P_{X_{k^m_{j}}}\{
 k ^m _1 < \infty
 \} }
 {E_{\alpha} I_{ \{ k^m_{j} < \infty \} }}
\]
\[
 \leq\frac{E_{\alpha} I_{ \{ k^m_{j} < \infty \} }\big(1- P_{X_{k^m_{j}}}\{ |X_k \cap \Lambda| > m,
 \forall k \geq \tilde k  \}\big) }
 {E_{\alpha} I_{ \{ k^m_{j} < \infty \} }}
\]
\[
  \leq \frac{E_{\alpha} I_{ \{ k^m_{j} < \infty \} }
  \big(1- P_{X_{k^m_{j}}}\{ u_k > m,  \forall k \geq 1  \}\big) }
 {E_{\alpha} I_{ \{ k^m_{j} < \infty \} }} = 1 - s_m <1.
\]

Having proved \eqref{feisty}, we observe that

\begin{equation}\label{cruise}
\begin{split}
 \big\{|\eta _t \cap \Lambda| \to \infty \big\}
   \cup \big\{ \exists & t': \forall t\geq t', |\eta _t \cap \Lambda| = \varnothing \big\} \\
   =
   \bigg( \bigcup _{m=1 } ^{\infty} \{|X_k \cap \Lambda| =&m \text{ infinitely often} \} \bigg)^c.
   \end{split}
\end{equation}
Note that if for some element
of probability space $\omega \in \Omega $
the process $(\eta _t)_{t \geq 0}$ is stuck in a trap $\gamma$,
$\gamma \cap \Lambda = \varnothing$,
then $\omega$ belongs to the set on the left-hand side of \eqref{cruise}
and does not belong to the set $\big\{|X_k \cap \Lambda| =m \text{ infinitely often} \big\}$,
$m \in \mathbb{N}$.

The statement of the proposition follows from \eqref{feisty} and
\eqref{cruise}.
\qed

\textbf{Proof of Proposition} \ref{finite death}.
Define $\widetilde{\eta} _t := \eta _t \cap \Lambda$
and let $\widetilde{X} _k = \widetilde{\eta} _{\varsigma _k}$, where $\varsigma _k$
is the ordered sequence of jumps of $(\widetilde{\eta} _t)_{t\geq 0}$.
Of course, the process $\{ \widetilde{\eta} _t \} _{t \geq 0}$
is not Markov in general, and neither is
$\{ \widetilde{X} _k \} _{k \in \mathbb{N}}$. However, for all $\alpha \in \Gamma _0 (\mathbb{R}^\d) $ the inequality

\[
P _{\alpha} \{ |\widetilde{X}_{1}| - |\widetilde{X}_0| = 1   \} \geq
p_ {|\alpha \cap \Lambda| , |\alpha \cap \Lambda| +1}
\]
holds,
because for every $\zeta \in \Gamma _0$, $\zeta \cap \Lambda = m$,
the integral of the birth rate $b(\cdot,\zeta)$
over $\Lambda$ is larger than $c m$,
and the cumulative death rate in $\Lambda$, $\sum\limits _{x \in \zeta \cap \Lambda}d(x,\zeta)$,
 is less than $ m a^{-m}$.

The probability of the event that absolutely no death occurs is positive, even when the initial configuration
contains only one point inside $\Lambda $:
\[
P _{\alpha}  \bigg\{ |\widetilde{\eta} _t| - |\widetilde{\eta} _{t-}| \geq 0 \text{\ for all \ } t \geq 0   \bigg\} =
P _{\alpha} \bigg\{ |\widetilde{X}_{k+1}| - |\widetilde{X}_k| = 1 \text{\ for all \ } k\in \mathbb{N}  \bigg\}
\]
\[
= \prod\limits _{k \in \mathbb{N}} P _{\alpha} \left\{ |\widetilde{X}_{k+1}| - |\widetilde{X}_k| = 1 \middle |
|\widetilde{X}_{k}| - |\widetilde{X}_{k-1}| =1 , ..., |\widetilde{X}_1| - |\widetilde{X}_0| =1  \right\}
\]
\[
\geq \prod\limits _{k \in \mathbb{N}}
\inf\limits _{\substack{ \zeta \in \Gamma ^0 (\mathbb{R}^\d), \\ |\zeta \cap \Lambda|=|\alpha \cap \Lambda| +k }}
P _{\zeta} \{ |\widetilde{X}_{1}| - |\widetilde{X}_0| = 1   \}
\]
\[
\geq \prod\limits _{i=|\alpha|} ^{\infty} p_ {i, i+1} = \prod\limits _{i=|\alpha|} ^{\infty} \frac{c}{c+a^{-i}} =
\prod\limits _{i=|\alpha|} ^{\infty} \big(1 - \frac{a^{-i}}{c+a^{-i}} \big) >0,
\]
because
the series $\sum\limits _{i=|\alpha|} ^{\infty}  \frac{a^{-i}}{c+a^{-i}} $ converges. In particular,
$\prod\limits _{i=m} ^{\infty} p_ {i, i+1} \to 1$ as $ m$ goes to $ \infty $.
Also,
\begin{equation}\label{zugunsten}
 P _{\alpha _n}  \big\{ |\widetilde{\eta} _t| - |\widetilde{\eta} _{t-}|
\geq 0 \text{\ for all \ } t \geq 0   \big\} \to 1, \ \ \ |\alpha _n \cap \Lambda| \to \infty.
\end{equation}

It is clear only an a.s. finite  number of deaths inside $\Lambda$ occurs  on
$\{ \exists t': \forall t\geq t', |\eta _t \cap \Lambda| = \varnothing \}$.

By Proposition \ref{verfemen}, it remains to show
that only an a.s. finite  number of deaths inside
$\Lambda$ occurs on $\{|\eta _t \cap \Lambda| \to \infty \} = \{|\widetilde{\eta} _t| \to \infty \} $.
Let us introduce the stopping times $\sigma _n = \inf \{s \in \mathbb{R}: |\widetilde{\eta} _s| \geq n  \}$,
which are finite on $\{|\widetilde{\eta} _t| \to \infty \} $.
Only a finite number of events (births and deaths)
occur until arbitrary finite time $P _{\beta}$-a.s.
for all $\beta \in \Gamma _0$, hence for  $n \in \mathbb{N}$

\[
P _{\alpha} \Bigl( \{ |\widetilde{\eta} _t| - |\widetilde{\eta} _{t-}| \geq 0 \text{\ for all but finitely many \ } t \geq 0   \}
\cap \{|\widetilde{\eta} _t| \to \infty \} \Bigr)
\]
\[
\geq P _{\alpha} \Bigl( \{ |\widetilde{\eta} _t| - |\widetilde{\eta} _{t-}| \geq 0 \text{\ for all \ } t \geq \sigma _n  \}
\cap \{|\widetilde{\eta} _t| \to \infty \} \Bigr)
\]
\[
=P _{\alpha} I_{\{|\widetilde{\eta} _t| \to \infty \}}
P _{\eta _{\sigma _n} }  \big\{ |\widetilde{\eta} _t| - |\widetilde{\eta} _{t-}|
\geq 0 \text{\ for all \ } t \geq 0  \big\}.
\]
From
$ |\eta _{\sigma _n}| \geq n $
we have by \eqref{zugunsten}
\[
P _{\eta _{\sigma _n} }  \big\{ |\widetilde{\eta} _t| - |\widetilde{\eta} _{t-}|
\geq 0 \text{\ for all \ } t \geq 0  \big\} \to 1,
\ \ \ n \to \infty.
\]
Therefore,
\[
P _{\alpha} \Bigl( \{ |\widetilde{\eta} _t| - |\widetilde{\eta} _{t-}| \geq 0 \text{\ for all but finitely many \ } t \geq 0   \}
\cap \{|\widetilde{\eta} _t| \to \infty \} \Bigr)= P _{\alpha} \{|\widetilde{\eta} _t| \to \infty \}.
\]
\qed

Proposition \ref{finite death} is also applicable to $(\xi)_{t \geq 0}$, since
$b_1, d_1 $ satisfy all the conditions imposed on $b,d $.

\textbf{Proof of Theorem} \ref{trajectory-wise asymptotics}. First we prove the Lemma for $(\xi)_{t \geq 0}$: we prove that
for $P_{\alpha}$-almost all $\omega \in F_1:=
 \{ \lim\limits _{t \to \infty} |\xi _t \cap \Lambda|=\infty \}$,

\begin{equation} \label{scum}
  \liminf\limits _{t\to \infty} \frac{|\xi _t \cap \Lambda|}{e^{ct}} > 0.
\end{equation}

 There is no loss in generality in assuming  $u_0 = |\alpha \cap \Lambda| >0 $.
Let $0=\tau _0 < \tau _1 < \tau _2 < ...$ be the moments of jumps
of $(\xi _t)_{t\geq 0}$, so that $\xi _{\tau _k} = Y _{k}$.
We recall that the random variables $u_n = |Y_n|$
constitute a Markov chain.
Denote $\psi (n) = cn + n a^{-n}$.
Then
\[
\int\limits _{ \Lambda } b_1(x,Y_k) dx +
\sum\limits _{x \in Y_k }d_1(x,Y_k) =  c|Y_k|+ |Y_k|a^{-|Y_k|} =\psi (u_k).
\]

By Theorem 12.17 in \cite{KallenbergFound}
 the random variables
$\gamma _k := \psi (u_k)(\tau _{k+1} - \tau _k)$, $k \in \mathbb{Z} _+$
are independent and exponentially
distributed with parameter $1$.
Furthermore, the sequence $\{\gamma _k\}$
is independent of $Y$. In particular, it
is independent of $\{ u_k\}_{k \in Z_ +}$.

From Proposition \ref{finite death} we know
that only a finite number of deaths inside $\Lambda$ occur a.s. on $ F_1$.
Particularly,
there exists a positive finite random variable $\mathbf{m}$ such that
the inequalities
\begin{equation}\label{penibel}
 u_0+ n \geq u_n \geq u_0+n-\mathbf{m}(\omega ) ,  \ \ \ n\in \mathbb{N}
\end{equation}
hold
with probability $1$.

We can write

\[
 \tau _n  = \sum\limits_{k=1}^{n-1}(\tau _{k+1} - \tau _k) =
 \sum\limits_{k=1}^{n-1}\frac{\gamma _k}{\psi (u_k)} \geq
 \sum\limits_{k=1}^{n-1}\frac{\gamma _k}{u_0+ck}.
\]

Due to Kolmogorov's two-series theorem, the series $\sum\limits_{k=1}^{\infty}\frac{\gamma _k}{u_0+ck}$
is divergent (we recall that $E\gamma _k = D \gamma _k =1$). Hence $\tau _n \to \infty$ a.s.

We will show below that
\begin{equation}\label{bigot}
 c \tau _n \leq   \ln n  +c \tilde \gamma, \ \ \ n \in \mathbb{N},
\end{equation}
where
$\tilde \gamma $ is some finite random variable. Using \eqref{bigot},
we obtain
\[
 P_{\alpha} \big\{ |\xi _t| \geq \frac{e^{ct}}{(\mathbf{m}+1)e^{c\tilde \gamma}}, t\geq 0 \big\} =
 P_{\alpha} \big\{ |\xi _{\tau _n}| \geq \frac{e^{c\tau _{n+1}}}{(\mathbf{m}+1)e^{c\tilde \gamma}}, n \in \mathbb{N} \big\}
\]
\[
 =P_{\alpha} \big\{ u_n \geq \frac {1}{\mathbf{m}+1} e^{c\tau _{n+1} - c\tilde \gamma}, n \in \mathbb{N} \big\}
 = P_{\alpha} \big\{ \ln (u_n) + \ln (\mathbf{m}+1) \geq
 c\tau _{n+1} - c \tilde \gamma, n \in \mathbb{N} \big\} =1.
\]
Therefore, \eqref{scum} holds.

Inequality \eqref{bigot} follows from the convergence
of the series

\begin{equation}\label{scam}
 \sum\limits_{k=1}^{\infty} \bigg(\frac{\gamma _k}{\psi (u_k)} - \frac{1}{ck}\bigg).
\end{equation}

To establish the convergence of \eqref{scam}, we note that

\begin{equation}\label{scam1}
 \sum\limits_{k=1}^{\infty} \bigg(\frac{\gamma _k}{\psi (u_k)} - \frac{\gamma _k}{c u_k}\bigg)
\end{equation}
converges by Kolmogorov's theorem:
\[
 -\sum\limits_{k=1}^{\infty} \bigg(\frac{\gamma _k}{\psi (u_k)} - \frac{\gamma _k}{c u_k}\bigg) =
 \sum\limits_{k=1}^{\infty} \gamma _k \frac{u_k a ^{-u_k}}{cu_k\psi (u_k)}\leq
\frac{1}{c^2} \sum\limits_{k=1}^{\infty} \gamma _k \frac{ a ^{-u_k}}{u_k}
\]
\[
=\frac{1}{c^2} \sum\limits_{k=1}^{\mathbf{m}} + \frac{1}{c^2} \sum\limits_{k=\mathbf{m}+1}^{\infty}
\leq \frac{1}{c^2} \sum\limits_{k=1}^{\mathbf{m}} \gamma _k \frac{ a ^{-u_k}}{u_k} +
\frac{1}{c^2} \sum\limits_{j=1}^{\infty} \gamma _k \frac{ a ^{-j}}{j},
\]
and
\begin{equation}\label{scam2}
 \sum\limits_{k=1}^{\infty} \bigg(\frac{1}{ck} - \frac{1}{cu_k}\bigg)
\end{equation}
converges by \eqref{penibel}.

The convergence of the series in \eqref{scam} follows from
the fact that \eqref{scam1} and \eqref{scam2}
converge.

We have thus proved \eqref{scum}.
To establish the statement of the theorem,
note that $\tilde \sigma _n = \inf \{t>0: |\eta _t|\geq n  \}$
is finite on $F$ and
\[
\big\{ \liminf \frac{|\eta _t \cap \Lambda|}{e^{ct}} =0, |\eta _t | \to \infty \big\}
\subset \big\{ \liminf \frac{|\xi _t |}{e^{ct}} =0  \big\}.
\]
It follows from what we have already proved that
\[
 P_{\beta} \big\{\liminf \frac{|\xi _t |}{e^{ct}} =0  \big\} =
 P_{\beta} \big\{(\xi _t)_{t \geq 0} \ \text{ extincts} \big\},
\ \ \ \ \beta \in \Gamma _0.
\]

Therefore, by Proposition \ref{extinction} and the
strong Markov property (for the latter see e.g. \cite[Theorem 12.14]{KallenbergFound})

\[
 P_{\alpha} \big\{\liminf \frac{|\eta _t \cap \Lambda|}{e^{ct}} =0, |\eta _t | \to \infty  \big\}
 = P_{\alpha}P_{\eta_{ \tilde \sigma_n}} \big\{ \liminf \frac{|\eta _t \cap \Lambda|}{e^{ct}} =0, |\eta _t | \to \infty  \big\}
\]
\[
 \leq P_{\alpha}P_{\eta_{\tilde \sigma_n}} \big\{ \liminf \frac{|\xi _t |}{e^{ct}} =0  \big\} \leq \tilde C ^{-n},
\]
where
$\tilde C$ is the constant that appeared in Proposition \ref{extinction}.
Since $n$ is arbitrary,
\[
 P_{\alpha}\big\{\liminf \frac{|\eta _t \cap \Lambda|}{e^{ct}} =0, |\eta _t | \to \infty  \big\} =0.
\]
\qed

\textbf{Proof of Corollary} \ref{growth in average}.
Let us fix a configuration $\alpha$, $ \alpha \cap \Lambda \ne \varnothing $.
We saw in the proof of Theorem \ref{trajectory-wise asymptotics} that
for almost all $\omega \in F = \{ \omega : \liminf\limits _{t\to \infty} \frac{|\eta _t \cap \Lambda|}{e^{ct}}
> 0 \} $ we have
\[
  P_{\alpha} \{ |\xi _t| \geq \frac{1}{(\mathbf{m} + 1) e^{c\tilde \gamma}}e^{ct}, t\geq 0 \} =1.
\]
Let $F_k$ be the set $\{\omega : \frac{1}{(\mathbf{m} + 1)e^{c\tilde \gamma}} \geq \frac 1k \}$.
Then $\bigcup\limits _{k \in \mathbb{N}} F_k = F$, and, since $P _{\alpha} (F) >0$,
$$
P _{\alpha} ( F_k ) >0
$$
for some $k \in\mathbb{N}$. Hence
\[
E _{\alpha} |\eta _t \cap \Lambda| \geq E _{\alpha} |\eta _t \cap \Lambda|I_{F_k}
\geq \frac 1k e^{ct} P_{\alpha} ( F_k ). \ \ \ \Box
\]

 \section{Appendix}

 \subsection{Markovian functions of a Markov chain}

  Let $(S, \mathscr{B}(S))$ be a Polish (state) space. Consider a
  (time-homogeneous) Markov
  chain on $(S, \mathscr{B}(S))$ as a family of probability measures
  on $S^ \infty$. Namely,  on the measurable space
  $
  {(\Omega,\mathscr{F}) = (S^ \infty , \mathscr{B}(S ^\infty ))}
 $
  consider a family of probability measures $\{P_s \}_{s \in S}$
  such that for the coordinate mappings

  \begin{align*}
 X_n: \Omega &\rightarrow S, \\
 X_n (s_1,s_2,&...)  = s_n
\end{align*}

  the process $X = \{X_n \}_{n \in \mathbb{Z} _+ }$ is a Markov chain,
  and for all $s \in S$

  $$
  P_s \{X_0 =s \} =1,
  $$
  $$
  P_s \{ X_{n+m_j}\in A_j, j=1,...,k_1 \mid \mathscr{F} _n \}
  = P_{X_n} \{ X_{m_j} \in A_j, j=1,...,k_1  \}.
  $$

  Here $A_j \in \mathscr{B} (S)$,
  $m_j \in \mathbb{N}$, $ k_1 \in \mathbb{N}$, $\mathscr{F} _n = \sigma \{ X_1,...,X_n \}$.

 The space $S$ is separable, hence
 there exists a transition probability
 kernel
 $Q: S \times \mathscr{B} (S) \rightarrow [0,1]$ such that

 $$
 Q(s,A) = P_s \{ X_1 \in A \}, \ \ \ s\in S, \ A \in \mathscr{B} (S).
 $$

  Consider a transformation of the chain $X$, $Y_n = f(X_n)$, where
  $f:S\to \mathbb{Z} _+$ is a Borel-measurable function,
  with convention $\mathscr{B}(\mathbb{Z} _+) = 2^{\mathbb{Z} _+}$.
  In this section we will give sufficient
  conditions for
 $Y = \{Y_n \}_{n \in \mathbb{Z} _+ }$  to be
  a Markov chain.
  A similar question was discussed in \cite{BRosenblatt}.

 \begin{lem} \label{lumpability}
  Assume that for any bounded Borel function $h: S\rightarrow S$

  \begin{equation}\label{insurgent}
   E_s h (X_1) =  E_q h (X_1) \text{\ whenever \ } f(s)= f(q),
  \end{equation}
  Then $Y$ is a Markov chain.
 \end{lem}

 \textbf{Remark}. Condition \eqref{insurgent} is the
 equality of distributions of $X_1$
 under two different measures, $P_s$ and $ P_q$.

 \textbf{Proof}. For the natural filtrations of the
 processes $X$ and $Y$ we have an inclusion

 \begin{equation} \label{penultimate}
 \mathscr{F} ^X _n \supset \mathscr{F} ^Y _n, \ \ \  n \in \mathbb{N},
 \end{equation}
since
$Y$ is a function of $X$. For $k\in \mathbb{N}$
 and bounded Borel functions $h_j: \mathbb{Z} _+ \rightarrow \mathbb{R}$,
 $j=1,2,...,k$
 (any function on $\mathbb{Z} _+$ is a Borel function),

 \begin{equation} \label{prance}
 \begin{split}
     E_s \left[ \prod\limits _{j=1} ^k
 h_j(Y_{n+j}) \mid \mathscr{F} ^ X _n  \right] =
 E_{X_n}  &\prod\limits _{j=1} ^k
 h_j(f(X_{j})) = \\
  \int _S Q(x_0 , dx_1) h_1(f(x_1)) \int _S Q(x_1 , dx_2) h_2(f(x_2))...
 &\int _S Q(x_{n-1} , dx_n) h_n(f(x_n)) \Bigg| _{x_0 = X _n}
 \end{split}
 \end{equation}

To transform the last integral, we introduce a new kernel:
for $y \in f(S)$ chose $x \in S$ with $f(x) = y$,
and then for $B \subset \mathbb{Z} _+$
define
\begin{equation}\label{secretive}
 \overline Q (y, B) = Q (x, f^{-1}(B)).
\end{equation}
The expression on the right-hand side
 does not depend on the choice of $x$ because of
 \eqref{insurgent}.
 To make the kernel $\overline Q$ defined on
 $\mathbb{Z} _+\times \mathscr {B} (\mathbb{Z} _+) $, we set

 \[
\overline Q (y, B) = I_{\{0 \in B \}}, \ y \notin f(S).
\]

Then, setting $z_{n} = f(x_{n})$,
it follows
from the change of variables formula for the Lebesgue integral
that

$$
\int _S Q(x_{n-1} , dx_n) h_n(f(x_n)) = \
\int _{\mathbb{Z} _+} \overline Q (f(x_{n-1}) , dz_n) h_n(z_n).
$$

Likewise, setting $z_{n-1} = f(x_{n-1})$, we get

\[
\int _S Q(x_{n-2} , dx_{n-1}) h_n(f(x_{n-1}))
\int _S Q(x_{n-1} , dx_n) h_n(f(x_n)) =
\]
\[
 \int _S Q(x_{n-2} , dx_{n-1}) h_n(f(x_{n-1}))
\int _{\mathbb{Z} _+} \overline Q (f(x_{n-1}) , dz_n) h_n(z_n) =
\]
$$
\int _{\mathbb{Z} _+} \overline Q(f(x_{n-2}) , dz_{n-1}) h_n(z_{n-1})
\int _{\mathbb{Z} _+} \overline Q (z_{n-1} , dz_n) h_n(z_n).
$$
Further proceeding, we obtain
\[
\int _S Q(x_0 , dx_1) h_1(f(x_1)) \int _S Q(x_1 , dx_2) h_2(f(x_2))...
 \int _S Q(x_{n-1} , dx_n) h_n(f(x_n)) =
\]
\[
\int _{\mathbb{Z} _+} \overline Q(z_0 , dz_1) h_1(z_1) \int _{\mathbb{Z} _+} \overline Q(z_1 , dz_2) h_2(z_2)...
 \int _{\mathbb{Z} _+} \overline Q(z_{n-1} , dz_n) h_n(z_n),
\]
where
$z_0 = f (x _0 )$.

Thus,

\[
E_s \left[ \prod\limits _{j=1} ^k
 h_j(Y_{n+j}) \mid \mathscr{F} ^ X _n  \right] =
 \]
 \[
\int _{\mathbb{Z} _+} \overline Q(f(X_0) , dz_1) h_1(z_1)
\int _{\mathbb{Z} _+} \overline Q(z_1 , dz_2) h_2(z_2)...
 \int _{\mathbb{Z} _+} \overline Q(z_{n-1} , dz_n) h_n(z_n).
\]

  This equality and  \eqref{penultimate} imply that
  $Y$ is a Markov chain.

  \begin{rmk} \label{finagle}
   The kernel $\overline Q$ and the chain $f(X_n)$ are related:
   for all $s\in S$, $n,m \in \mathbb{N}$ and $M \subset \mathbb{N}$,

   \[
    P_s \{ f(X_{n+1}) \in M \mid f(X_{n}) = m \} = \overline Q(m,M)
   \]
whenever $P_s \{  f(X_{n+1}) = m \} >0$.
   Informally, we can say that
   $\overline Q$ is the transition probability kernel for
   the chain $\{f(X_n)\}_{n\in \mathbb{Z} _+}$.
  \end{rmk}

  \begin{rmk}
   Clearly, this result holds for a Markov chain which is
   not necessarily defined on a canonical state space,
   because the property of a process to be a Markov chain
   depends on its distribution only.
  \end{rmk}

\subsection{Strong Markov property of the driving process}

 Let $N$ be compatible with a right-continuous complete filtration
$\{ \mathscr{F}_t \}_{t \geq 0}$, and $\tau$ be a finite a.s.
$\{ \mathscr{F}_t \}_{t \geq 0}$-stopping time.
For $\gamma \in \Gamma _2$, $\gamma = \sum\limits _{i} \delta _{(s_i,u_i)}$, let
$\theta _{\tau} \gamma =  \sum\limits _{i: s_i > \tau } \delta _{(s_i - \tau,u_i)}$.
Also, for $\Xi \in \mathscr{B} (\Gamma _2)$ we define the shift
\[
 \theta _{\tau} \Xi = \{ \gamma \in \Gamma _2 \mid \theta _{\tau} \gamma \in \Xi \}
\]

 Introduce another
point process $\overline N $ on $\mathbb{R} _+ \times \mathbb{R} ^\d \times \mathbb{R} _+ \times \Gamma _2 $,
\[
 \overline N ([0,s] \times U \times \Xi) =
 N ((\tau,\tau + s] \times U \times \theta _{\tau}  \Xi),
 \ \ \ s>0,  \ U \in \mathscr{B}(\mathbb{R} ^\d \times \mathbb{R} _+), \ \Xi \in \mathscr{B} (\Gamma _2).
\]

 \begin{prop}\label{Strong MP}
 The process $\overline N$ is a Poisson point process with
 intensity measure $ds \times dx \times du \times \pi$, independent of
 $\mathscr{F}_{\tau}$.
\end{prop}

 \textbf{Proof}. To prove the proposition, it is enough to show that

 (i) for any $b>a>0$, open bounded $U \subset \mathbb{R} ^\d \times \mathbb{R} _+$
 and open $\Upsilon \subset \Gamma _2$,
$\overline N ((a,b) \times U \times \Xi)$
 is a Poisson random variable
with mean $(b-a) \times l(U) \times \pi (\Xi)$,
where $l$ is the Lebesgue measure
on the corresponding space,
and

 (ii) for any $b_k>a_k>0$, $k=1,...,m$, open bounded
$U_k \subset \mathbb{R} ^\d$ and open
$ \Xi _k \subset \Gamma _2$ such that
$((a_i,b_i) \times U_i \times \Xi _i ) \cap( (a_j,b_j) \times U_j \times \Xi _j ) = \varnothing $,
$i \ne j$,
 the collection $\{\overline N ((a_k,b_k) \times U_k \times \Xi _k)\}_{k=1,m}$
 is a finite sequence of
 independent random variables, independent of $\mathscr{F}_{\tau}$.

 Let $\tau _n$ be the sequence of $\{ \mathscr{F}_t \}_{t \geq 0}$-stopping times,
$\tau _n = \frac{k}{2^n}$ on $\{ \tau \in (\frac{k-1}{2^n},\frac{k}{2^n}] \}$,
$k \in \mathbb{N}$. Then
 $\tau _n \downarrow \tau$ and
$\tau _n - \tau \leq \frac{1}{2^n}$. The
stopping times $\tau _n$ take only countably many values.
Therefore
the process $N$ satisfies the strong Markov property for $\tau _n$:
 the processes $\overline N _n$, defined by
 $$
\overline N _n ([0,s] \times U \times \Xi) :=  N ((\tau _n,\tau _n + s] \times U \times \theta _{\tau _n}  \Xi),
$$
 are Poisson point processes, independent of $\mathscr{F}_{\tau _n}$.

 To prove (i),
note that $\overline N _n ((a,b) \times U \times \Xi ) \to \overline N ((a,b) \times U \times \Xi) $ a.s.
and all random variables $\overline N _n ((a,b) \times U \times \Xi)$ have the same distribution,
therefore $\overline N  ((a,b) \times U \times \Xi)$ is a Poisson random variable with mean
$(b-a) l (U) \pi (\Xi) $.
The random variables $\overline N _n ((a,b) \times U \times \Xi)$
are independent of $\mathscr{F}_{\tau}$,
hence $\overline N  ((a,b) \times U \times \Xi)$ is independent
of $\mathscr{F}_{\tau}$, too. Similarly, (ii) follows. \qed

 \section*{Acknowledgement}

 I wish to thank my thesis advisor Yuri Kondratiev for suggesting the problem
 and superb guidance during the progress of the work. I also thank
 Vladimir Bogachev for many critical and stimulating comments.
 The author acknowledges the financial support of the DFG through the SFB 701
 (Bielefeld University) and the IRTG (IGK) 1132 ``Stochastics and Real World Models''.

\bibliographystyle{alpha}

redpajama_set_name: RedPajamaArXiv


text:
YOUR CHANCE TO COMMENT: Public Meetings February 6 and February 28 for Bozeman's review of a proposed apartment complex south of The Lakes. The West Park Neighborhood will be located south of Vaughn Drive and east of Laurel Parkway. Phase 1 of West Park, located immediately to the south of Lakes Phase 6, is proposed to include seven three-story apartment buildings with a total of 102 units. The public comment period extends from January 20 through February 28, 2023, and a public meeting of the Community Development Board will take place February 6 at 6:00 PM at Bozeman City Hall (also available via WebEx). The Bozeman City Commission is scheduled to make a decision on the preliminary plat application at its February 28 public meeting (6:00 PM at Bozeman City Hall and via WebEx). Detailed information about two linked development applications is available online: Application 22205 and Application 22178.
SNOW PLOWING: We Need Your Help! (December 2022) For residents who live on alleys in Valley West, our wonderful snow plow crew has let us know there are many trash cans encroaching on the alleys, making it difficult to pass through with a plow. We realize space is tight with all the snow piling up, but please do your best to keep obstacles out of the way so the plowing crews can do their job. Thank you!
Snow plowing on city streets: The city of Bozeman is responsible for snow plowing in city streets in Valley West and The Lakes (streets that are indicated by green street signs). The city plows snow when at least four inches have fallen. The city is scheduled to remove snow in Valley West on Wednesdays and in The Lakes on Fridays. The city also manages snow removal around the city parks (see pages 6 and 17 in this document). Valley West HOA received many complaints in winter 2022-23 that the city had fallen behind on snow plowing. Unfortunately, the HOA has no power to influence the city's plowing schedule. If needed, contact the Bozeman city Streets department here.
Snow plowing on HOA property: Valley West HOA removes snow in the alleys in Valley West, woonerfs in The Lakes (identified by blue street signs), and sidewalks throughout the community. The HOA's plowing vendor removes snow when at least two inches have fallen. When we experience a large snowfall, our HOA's plowing crews will keep up as much as possible while the snow is falling, and will come back the following day to do a cleanup if needed. We thank everyone for your patience and understanding as our valued partners work around the clock to dig us out.
Don't release nonnative fish in the ponds: Reece Buecker, a Bozeman Health School student, has been educating community members about the problems Meyers Lake and the Lakes may face when folks introduce nonnative fish into the ponds. Reece provided this helpful information: Releasing old aquatic pets into ponds is extremely detrimental to the pond's ecosystem. If you or a friend have unwanted fish, putting them in local waterways is not an acceptable way to get rid of them. Some good ways to give the fish away safely without harming the local nature are finding local pond hobbyists and fish collectors, or reaching out to pet stores, schools, and other places that might want the fish. When fish that do not belong in ponds are introduced, they wreak havoc on the native fish and natural ecosystem. Some fish commonly introduced through pet owners are carp and goldfish. Another way that invasive species are spread is through people moving fish from pond to pond. If parents can make sure this does not happen and make sure their kids do not accidentally do this it would greatly diminish the spread of invasive species. A very easy way to prevent species like snails from spreading is to check canoes and other water toys for snails hitching a ride. In most ponds the snails and other minor invasive species are not the largest problems, the fish that can spread fast and far cause the most harm. These fish outcompete the native fish through their feeding styles, temperature and water type resistance, and behavior. The fish push out native species through their advantages and destroy the environment through things like algae blooms that they cause. Never release your fish into ponds, and talk to family members and friends about this issue in order to educate them.
Yellowstone Fiber goes live in The Lakes: Following a ribbon cutting ceremony September 19, 2022 at the West Pavilion, The Lakes is part of the first residential district to received fiber optic internet service from Yellowstone Fiber, a nonprofit company that is installing fiber optic cable throughout Bozeman.
Update from Bozeman City Parks about activities in Valley West and The Lakes: Bozeman City Parks began managing city parks in our community in 2021. In June 2022, the City of Bozeman Parks & Recreation Department provided these helpful reports about their activities in city park space in Valley West and The Lakes. The HOA appreciates the city's work in our community!
Bozeman Library Bookmobile visits Valley West Park every other Friday from 3:30 to 4:30 PM. Visit the Bookmobile website for the current schedule. At the Bookmobile, you can get a library card, check out books and movies, and pick up "hold" items.
Let us know of events going on in our community, and we'll post them here. Book club? Block party? Walking group? Bridge club? Children's play group? Let your neighbors know!
Submit event information using the form below.
Thank you! We have received your submission.
Dogs off leash: There are too many dogs running loose! Please be mindful to always keep your dog on leash and clean up after your dog
Parking in The Lakes: The HOA receives many complaints about residents and guests who park outside the designated parking pads in woonerfs in The Lakes. Recently the HOA installed NO PARKING signs in the woonerfs as a reminder to respect our neighbors and keep these narrow streets clear for snow removal in the winter
Potholes: The HOA is assessing the asphalt conditions in the alleys and woonerfs for future repairs. Please report specific concerns (Contact)
Coming Soon in Our Area
In Bozeman City Planning: Bozeman Community Center in West Bozeman, expected to include a library branch and recreation facilities. In early 2023, the city is engaging community members about their hopes for the new facility.
Commercial Lots at Babcock & Cottonwood
Infrastructure is in place for commercial lots on the north side of Babcock, between Cottonwood and Stafford. These commercial lots were developed as part of Valley West, but they are not part of the Valley West Homeowners Assocation.
Development South of The Lakes: Norton Ranch
The Norton Ranch subdivision has begun development of the northern extension of their community, between Babcock Street and Vaughn Drive. Click for schematics of Norton Ranch Phase 5 (to the West) and Phase 6 (to the East). In these plans, Laurel Parkway, Westgate Avenue, and the east end of Vaughn Drive (as well as the multi-use trail along Aajker Creek) continue south from The Lakes into the Norton Ranch community. Open spaces will adjoin The Lakes in many sections of Norton Ranch. Phase 6 of Norton Ranch, east of Laurel Parkway, has R-4 (Residential High Density District) zoning. As of June 2022, the Bozeman Planning Department has not yet received a site plan for Norton Ranch Phase 6.
Development Southwest of Babcock and Ferguson: Six Range
On the south side of Babcock Avenue, west of Ferguson Avenue and east of Hanley Avenue, The Paine Group is building a 183-unit development, Six Range, which includes 4-story apartment buildings, 2-story triplexes (townhomes) facing Babcock, and a 3800-square-foot commercial building at the southwest corner of Babcock and Ferguson.
Development Northwest of Oak and Cottonwood: Northwest Crossing
100-Room Hotel Near Rosauers: Everhome Hotel
Planned Grocery Store at Ferguson Farm: Town & Country Grocery Store
Is EVERYTHING in our area going to be developed?
Well, not quite everything. Gallatin Valley Land Trust documents a conservation easement on land along Gooch Hill Road, to the southwest of The Lakes (shown in orange in the map below). This easement does not protect the land directly west of The Lakes, but does protect a large area of land along Gooch Hill Road, north of Huffine. The land to the west of The Lakes is currently not in the city of Bozeman. If that land is ever annexed into the city, Bozeman's Community Plan Future Land Use map designates the area west of The Lakes as "Urban Neighborhood."
For the latest, check out 2022 Member Update and Community
Valley West and The Lakes
About the HOA
© Valley West Homeowners Association. All rights reserved. Designed by Metcalfe Media. Photo credit to Krogstad Photography.

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: A problem that isn't really a problem I'm looking for a word or, more likely, a phrase--along the lines of "Hobson's choice," "busman's holiday," or "Pyrrhic victory"--to describe a problem that isn't really a problem. An example might be the "problem" of having too much evidence for something you are trying to prove.
"First-world problem" comes pretty close to what I'm looking for, but I'd appreciate a phrase that telegraphs a more focused irony and is perhaps a bit more sophisticated.

A: For the 'too much [evidence] ...' example, a fitting expression is
an embarrassment of riches.
From CDO:

an embarrassment of riches  [formal]:
so many good things or people that it is impossible to decide which
  of them you want

More generally,
'I wish I had your problems' (as bib suggests)
(meaning 'I wish I [merely] had [what you consider to be] {your} problems')
is used. A clear example is this from metrolyrics:

He said I wish I had your problems cause they're
Nothing compared to mine.

(though I wouldn't use this source very often).

A: I recently heard this described as a champagne problem.
from the Online Slang Dictionary:

champagne problem noun
a problem that, in reality, demonstrates one's good fortune.

A: Perhaps a term for the result rather than the cause is of interest  ...
If you are working on a problem that isn't really a problem you could end up doing busy work.
From wikipedia:

Busy work (also referred to as make-work and busywork) can refer to activity that is undertaken to pass time and stay busy but in and of itself has no actual value. [...]
Busy work also occurs in business, military and other settings, in situations where people may be required to be present but may lack the opportunities, skills or need to do something more productive.
People may engage in busy work to maintain an appearance of activity, in order to avoid criticism of being inactive or idle.

redpajama_set_name: RedPajamaStackExchange


This is the guide to looking after your horse in a reliable, natural and inexpensive way and in excellent health. In NATURAL HORSE CARE, Pat Coleby includes up-to-date information on the natural care and management of horses. She explains the important roles of correct feed rations, vitamins and minerals in the health of the horse, and details practical and inexpensive ways of avoiding and curing health problems. NATURAL HORSE CARE is an inexpensive book that costs less than one visit to the vet and will help you keep your horse healthy for its lifetime.


Gwynedd Cottages was posted on October 10, 2018 at 6:35 pm. This article is uploaded at the Cottage category. Gwynedd Cottages is tagged with Gwynedd Cottages, Gwynedd, Cottages..
1493 sq. mi. (3866 sq. km).
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text:
A Conversation With Mr. Cincinnati, Jim Tarbell
91.7 WVXU | By John Kiesewetter
Published March 20, 2020 at 4:00 AM EDT
From the original Ludlow Garage to Arnold's, Peanut Jim to the vice-mayor, Jim Tarbell has earned the title Mr. Cincinnati. There's never a dull moment in an interview with the man featured in the big mural on Central Parkway as you'll hear in this conversation with our Media Beat blogger, John Kiesewetter.
CommunityAround CincinnatiJim TarbellJohn KiesewetterMr. Cincinnati
John Kiesewetter joined the WVXU news team as a TV/Media blogger on July 1 2015, after nearly 30 years covering local and national broadcasting for The Cincinnati Enquirer. He'll be posting news about Greater Cincinnati TV, radio and movies; updating your favorite former local TV/radio personalities or stars who grew up here; and breaking news about national TV, radio and media trends. You'll also learn about Cincinnati's rich broadcasting history.
See stories by John Kiesewetter


You are watching the movie Confirmation 2016 produced in USA belongs in Category History, Drama , with duration 110 Min , broadcast at Cmovieshd.Net,Director by Rick Famuyiwa, Judge Clarence Thomas' nomination to the United States' Supreme Court is called into question when former colleague, Anita Hill, testifies that he had sexually harassed her.
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Supplied in raw black finish and would advise sanding and painting, comes with mounting screws to attach to vehicle - Fitting instructions are NOT included.
Polyurethane is a pliable material which means it will flex/bend - it is normal for these to slightly warp due to heat in transit. In most cases they will still be able to be installed and will correct their shape once installed. If you need to manipulate the parts use a heat gun or hot hairdryer until the polyurethane is warm and then hold in desired shape until it cools. We have found to prevent rippleing after installation - we advise a trip to bunnings and purchasing some legths of 25mm wide aluminium strip and sikaflex to stick on the underside of the skirts - Works great!


It features a frequency range of 10.7 to 12.75GHz divided into Low Band (10.70 to 11.70GHz) and High Band (11.70 to 12.75GHz). It receives waves with horizontal and vertical polarization. The LNB provides one switchable IF outputs (Single model). The output carries also power supply and control signals. Output port is F type. DiSEqC1.0 commands allow switching the output to a particular band and orbital position.
The DiSEqC ODU allocation is the following: "ODU A" corresponds to 16E, "ODU B" to 13E and "ODU C" to 19.2E. As long as no DiSEqC command has been received, the selected orbital position is 16E (ie DiSEqC A). The LNB comprises three feeds, one for each orbital position. The feeds allow mounting into the feed clamp of the satellite dish. The feeds will be marked with "13˚E", "16˚E" and "19.2˚E" respectively.
The DISEqC positions (A/B/C) and the respective orbital positions (16E/13E/19.2E) are marked on the LNB housing. The LNB is supplied with 40mm plastic ring adapter.
Low band 1.1 dB max.
High band 1.0 dB max.
50 ~ 62 dB min.
In band spurious level -55 dBm max.


Personal, thoughtful touches go a long way when you are raising the romance or seduction bar. Here's a temptingly beautiful gift set by Shunga which will bring a touch of class to any lovers armoury. Shunga Lovers Gift Set - Geisha's Secrets - Sparkling Strawberry Wine is perfect for travelling as well as for that romantic getaway you both deserve.


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