220244

# Non-autonomous interacting particle systems in continuum

Martin Friesen  Department of Mathematics, University of Bielefeld, Bielefeld,
Germany<EMAIL_ADDRESS>
(Date: 24/03/2016; Revised 15/06/2016)

###### Abstract.

A conservative Feller evolution on continuous bounded functions is constructed
from a weakly continuous, time-inhomogeneous transition function describing a
pure jump process on a locally compact Polish space. The transition function
is assumed to satisfy a Foster-Lyapunov type condition. The results are
applied to interacting particle systems in continuum, in particular to general
birth-and-death processes (including jumps). Particular examples such as the
BDLP and Dieckmann-Law model are considered in the end.

###### Key words and phrases:

Interacting particle systems, Feller evolution, pure jump process,
configuration space, Foster-Lyapunov criterion, Kolmogorov equation.

###### 2010 Mathematics Subject Classification:

Primary 60J75; Secondary 60G55

##  1\. Introduction

Classical birth-and-death dynamics are described by a system of ordinary
differential equations, also known as Kolmogorov’s differential equations, and
are usually studied by semigroup methods on (weighted) spaces of summable
real-valued sequences, cf. [25, 18, 19, 24]. More recent attempts study such
equations on the spaces $\ell^{p}$ for $p\in[1,\infty)$, see [2, 3, 32]. In
contrast to many real world models, see e.g. the Bolker, Dieckmann, Law,
Pacala model [7, 8, 13, 14] (short BDLP), such equations do not include the
positions of the described particles. Other models coming from ecology and the
modeling of mutations can be found in [30, 6, 26, 31, 21] and references
therein.

The simplest possibility to include spatial structure is to assign to each
particle a fixed site of a graph (e.g. from the lattice $\mathbb{Z}^{d}$).
This are the so-called lattice models. For such models a rigorous study by
semigroup methods is adequate and a detailed presentation can be found in the
classical book [28] and references therein. Several models, such as the BDLP
model, require that the positions of the particles be not a priori fixed. This
means that $\mathbb{Z}^{d}$ should be replaced by a continuous location space,
e.g. $\mathbb{R}^{d}$.

For the modeling of interacting particle systems in continuum the theory of
pure point processes is commonly used. Such processes share several properties
with the processes associated to lattice models, but also include numerous
unexpected features and require essentially different techniques for their
mathematical treatment. Taking into account that they describe real-world
particles it leads to the natural assumption that all particles are
indistinguishable and any two particles cannot occupy the same position in the
location space, say for simplicity $\mathbb{R}^{d}$. A microscopic state
$\eta$ is then, by definition, a finite linear combination of point-masses
$\delta_{x}$, where $x\in\mathbb{R}^{d}$ is the position of a particle in the
system. Such Markov dynamics can be analyzed by a measure-valued
generalization of the Kolmogorov’s differential equations. This equations have
been first analyzed in [20] and have been afterwards further investigated in
the next 60 years, cf. [17] and many others. A summary with applications to
interacting particle systems is provided in the book [10]. In this work we
identify $\eta$ with a subset of $\mathbb{R}^{d}$, i.e. we consider the
microscopic state as a finite collection of positions $x\in\mathbb{R}^{d}$.
The state space (= configuration space) is therefore the space of all finite
configurations, which is given by

$\Gamma_{0}=\\{\eta\subset\mathbb{R}^{d}\ |\ |\eta|<\infty\\}.$

Here and in the following we write $|A|$ for the number of elements in
$A\subset\mathbb{R}^{d}$.

Interacting particle systems in continuum with state space $\Gamma_{0}$ are
heuristically described by a Markov (pre-)generator on a proper set of
functions $F$. The general form of such operator is given by the heuristic
expression

$\displaystyle(LF)(\eta)=\sum_{\xi\subset\eta}\int_{\Gamma_{0}}(F(\eta\backslash\xi\cup\zeta)-F(\eta))K(\xi,\eta,d\zeta),\quad\eta\in\Gamma_{0},$
(1.1)

where $K(\xi,\eta,d\zeta)\geq 0$ is a transition kernel which will be
specified in the third section. The associated (backward) Kolmogorov equation

$\frac{\partial F_{t}}{\partial t}=LF_{t},\quad F_{t}|_{t=0}=F_{0}$

can be solved on the space of continuous bounded functions, see [15, 16, 27].
The solutions determine therefore a Markov process on $\Gamma_{0}$. This
process describes the evolution of the microscopic state $\eta$, where each
group $\xi\subset\eta$ of particles may disappear and simultaneously a new
group of particles $\zeta\in\Gamma_{0}$ may appear somewhere in
$\mathbb{R}^{d}$. The distribution of the new particles and the intensity of
this event are both described by the transition kernel $K(\xi,\eta,d\zeta)$.
To this end we define for any $\eta\in\Gamma_{0}$ and Borel measurable set
$A\subset\Gamma_{0}$ a new transition kernel by

$Q(\eta,A):=\sum_{\xi\subset\eta}\int_{\Gamma_{0}}1\kern-2.84526pt{\text{\rm
I}}_{A}(\eta\backslash\xi\cup\zeta)K(\xi,\eta,d\zeta),$

where $1\kern-2.84526pt{\text{\rm I}}_{A}(\eta):=\begin{cases}1,&\eta\in A\\\
0,&\eta\not\in A\end{cases}$. Then, under some additional conditions, it is
possible to rewrite above operator to

$\displaystyle(LF)(\eta)=\int_{\Gamma_{0}}(F(\xi)-F(\eta))Q(\eta,d\xi),\quad\eta\in\Gamma_{0}.$
(1.2)

In such a case we have to assume that

$\displaystyle
q(\eta):=Q(\eta,\Gamma_{0})=\sum_{\xi\subset\eta}K(\xi,\eta,\Gamma_{0})$

is finite for all $\eta\in\Gamma_{0}$. Hence the process described by the
operator $L$ is a pure jump Markov process and techniques coming from the
theory of Markov chains are applicable. Such approach has been investigated in
the last 20 years, a comprehensive summary of the obtained results can be
found in [10]. In the case of birth-and-death dynamics, i.e. a particular
choice of $K(\xi,\eta,d\zeta)$, an alternative approach is provided by solving
certain stochastic equations, cf. [4, 5].

This work is organized as follows. The second section is devoted to the study
of an abstract pure jump process described by the time-dependent operator

$(L(t)F)(x)=\int_{E}(F(y)-F(x))Q(t,x,dy),\quad x\in E,\quad t\geq 0.$

Here $E$ is an abstract locally compact Polish space and it is assumed that
the operator $L(t)$ satisfies a Foster-Lyapunov type condition, see [29].
Moreover, we suppose that the transition function $Q(t,x,dy)$ is weakly
continuous and satisfies some additional technical conditions. Based on the
classical works [20, 23, 17], we construct an associated conservative Feller
evolution system $U(s,t)$ on the space of continuous bounded functions. Hence
by [33] this Feller evolution is associated to a Hunt process with state space
$E$. For a countable state space $E$ such result was obtained by martingale
techniques in [35]. We show that $U(s,t)$ provides existence and uniqueness of
solutions to the Kolmogorov equations and establish the relation to the jump
process by the associated Martingale problem. Additional works dealing with
properties of (time-homogeneous) Markov processes can be found, e.g., in [12,
11, 9, 34].

The third section is devoted to particular examples of interacting particle
systems on $\Gamma_{0}$. We apply above results for time-dependent
(pre-)generators given by

$\displaystyle(L(t)F)(\eta)=\sum_{\xi\subset\eta}\int_{\Gamma_{0}}(F(\eta\backslash\xi\cup\zeta)-F(\eta))K_{t}(\xi,\eta,d\zeta).$
(1.3)

To this end we use a similar representation to (1.2) for the time-dependent
transition function

$Q(t,\eta,A)=\sum_{\xi\subset\eta}\int_{\Gamma_{0}}1\kern-2.84526pt{\text{\rm
I}}_{A}(\eta\backslash\xi\cup\zeta)K_{t}(\xi,\eta,d\zeta),\quad t\geq
0,\quad\eta\in\Gamma_{0}.$

As a consequence we are able to construct, under some reasonable conditions,
an associated conservative Feller evolution system $U(s,t)$ on continuous
bounded functions $F:\Gamma_{0}\longrightarrow\mathbb{R}$. Hence for any
initial probability measure $\mu$ on $\Gamma_{0}$ the action of the adjoint
evolution system $U^{*}(t,s)\mu$ yields a weak solution to the Fokker-Planck
equation

$\frac{\partial}{\partial
t}\int_{\Gamma_{0}}F(\eta)\mu_{t}(d\eta)=\int_{\Gamma_{0}}L(t)F(\eta)\mu_{t}(d\eta),\quad
t\geq s,$

where $F:\Gamma_{0}\longrightarrow\mathbb{R}$ is continuous and bounded.
Sufficient conditions for which $L^{1}(\Gamma_{0},d\lambda)$ is invariant for
$U^{*}(t,s)$ are given. By construction, the restriction
$U^{*}(t,s)|_{L^{1}(\Gamma_{0},d\lambda)}$ becomes strongly continuous.

The rest of the third section is devoted to the study of particular examples.
First we consider the BDLP model with time-dependent and non-translation
invariant kernels. The considerations are afterwards extended to the
Dieckmann-Law model and a generalization of this model.

##  2\. General jump processes

Let $E$ be a locally compact Polish space and denote by $\mathcal{B}(E)$ the
Borel-$\sigma$-algebra on $E$. Denote by $BM(E)$ the Banach space of all
bounded measurable functions and by $C_{b}(E)$ the subspace of all continuous
bounded functions. A pure jump process is determined by its (infinitesimal)
transition function, i.e. a function $Q:\mathbb{R}_{+}\times
E\times\mathcal{B}(E)\longrightarrow\mathbb{R}_{+}$ with the following
properties:

  1. 1\. For all $t\geq 0$, $x\in E$, $A\longmapsto Q(t,x,A)$ is a finite Borel measure with $Q(t,x,\\{x\\})=0$.

  2. 2\. For all $A\in\mathcal{B}(E)$, $(t,x)\longmapsto Q(t,x,A)$ is measurable.

  3. 3\. For all $T>0$ and all compacts $B\subset E$

$\displaystyle\sup_{(t,x)\in[0,T]\times B}Q(t,x,E)<\infty.$ (2.1)

Let $BM_{loc}(E)$ be the space of locally bounded measurable functions and
$C(E)$ be the space of continuous functions. Define for any
$F:E\longrightarrow\mathbb{R}$

$(Q(t)F)(x):=\int_{E}F(y)Q(t,x,dy),\quad t\geq 0,\quad x\in E,$

whenever it makes sense, i.e. $\int_{E}|F(y)|Q(t,x,dy)<\infty$ for all $t\geq
0$, $x\in E$. Then $Q(t):BM(E)\longrightarrow BM_{loc}(E)$ is a well-defined
positive linear operator and $q(t,x):=(Q(t)1)(x)=Q(t,x,E)$ is locally bounded.
If $Q(\cdot)C_{b}(E)\subset C(\mathbb{R}_{+}\times E)$, i.e. for any $F\in
C_{b}(E)$ the function $Q(\cdot)F$ is jointly continuous in $(t,x)$, then we
say that $Q$ is jointly continuous. This simply means that, by definition,
$Q(t,x,dy)$ is weakly continuous in $(t,x)$. In such a case (2.1) is
automatically satisfied.

We briefly recall the results obtained in [20, 17]. Let $Q$ be a transition
function. For $0\leq s\leq t,x\in E$ and $A\in\mathcal{B}(E)$ let
$P^{(0)}(s,x;t,A):=\delta(x,A)e^{-\int_{s}^{t}q(r,x)dr}$, and for $n\geq 1$

$\displaystyle
P^{(n+1)}(s,x;t,A):=\int_{s}^{t}e^{-\int_{s}^{r}q(\tau,x)d\tau}\Big{(}\int_{E}P^{(n)}(r,y;t,A)Q(r,x,dy)\Big{)}dr.$
(2.2)

Here $\delta(x,A):=1\kern-2.84526pt{\text{\rm I}}_{A}(x)=\delta_{x}(A)$. Then
$P(s,x;t,A)=\sum_{n=0}^{\infty}P^{(n)}(s,x;t,A)$ is a sub-Markov transition
function. Moreover, for fixed $A\in\mathcal{B}(E)$ and $x\in E$ it is
absolutely continuous in $s$ and $t$, respectively such that
$P(s,x;t,A)\to\delta(x,A)$ holds uniformly in $A\in\mathcal{B}(E)$ whenever
$s\to t^{-}$ or $t\to s^{+}$. For any $A\in\mathcal{B}(E)$ it is a.e.
differentiable in $s\in[0,t]$ and satisfies

$\displaystyle\frac{\partial P(s,x;t,A)}{\partial
s}=q(s,x)P(s,x;t,A)-\int_{E}P(s,y;t,A)Q(s,x,dy).$ (2.3)

Likewise, for any compact $A\subset E$ it is differentiable for a.a.
$t\in[s,\infty)$ and satisfies

$\displaystyle\frac{\partial P(s,x;t,A)}{\partial
t}=-\int_{A}q(t,y)P(s,x;t,dy)+\int_{E}Q(t,y,A)P(s,x,t,dy).$ (2.4)

It follows from [17] that $P$ is the minimal solution to (2.3) and (2.4).
Moreover, if $P(s,x;t,E)=1$, then this solution is also unique.

The main point of our interest is to study the (sub-)Markovian evolution
system

$\displaystyle U(s,t)F(x):=\int_{E}F(y)P(s,x;t,dy),\quad 0\leq s\leq t$ (2.5)

on the space of bounded measurable functions and extensions of it. Such an
evolution system is a family of positive bounded linear operators such that
$U(s,s)F=F$ and $U(s,r)U(r,t)F=U(s,t)F$ for $0\leq s\leq r\leq t$. For $F\in
BM(E)$ let

$\displaystyle L(t)F(x)=\int_{E}(F(y)-F(x))Q(t,x,dy),\quad t\geq 0$ (2.6)

be the (formal) generator of $U(s,t)$. Since in general $U(s,t)F$ is not
continuous w.r.t. the norm on $BM(E)$ or $C_{b}(E)$, we cannot expect that
some extension of $L(t)$ is a generator. However, for our needs it is
sufficient to consider only the weaker concept of pointwise generator The
precise statement is given in the proposition below.

Denote by $\mathcal{C}$ the collection of compact sets on $E$ and by
$\mathcal{C}_{1}$ the collection of compacts in $\mathbb{R}_{+}\times E$. For
a given non-negative function $V\in C(E)$ let $\|F\|_{V}:=\sup_{x\in
E}\frac{|F(x)|}{1+V(x)}$ and denote by $BM_{V}(E)$ the space of all measurable
functions for which $\|F\|_{V}$ is finite. Denote by $C_{V}(E):=BM_{V}(E)\cap
C(E)$ its closed subspace of continuous functions. Below we state the main
result for this section.

###### Proposition 2.1

Assume that there exists a continuous function
$V:E\longrightarrow\mathbb{R}_{+}$ such that $(t,x)\longmapsto Q(t)F(x)$ is
continuous for any $F\in C_{V}(E)$. Moreover, suppose that there exists a
continuous function $c:\mathbb{R}_{+}\longrightarrow\mathbb{R}_{+}$ such that
the properties below are satisfied.

  1. 1.For all $T>0$ there exists $a(T)>0$ such that $q(t,x)\leq a(T)V(x)$ holds for all $t\in[0,T]$ and $x\in E$.

  2. 2\. The Foster-Lyapunov estimate

$\displaystyle\int_{E}V(y)Q(t,x,dy)\leq c(t)V(x)+q(t,x)V(x),\quad t\geq
0,\quad x\in E$ (2.7)

is satisfied.

  3. 3\. For all $\varepsilon>0$, $B\in\mathcal{C}$ and $T>0$ there exists $A\in\mathcal{C}$ such that

$\displaystyle\int_{0}^{T}Q(r,x,A^{c})\,dr<\varepsilon,\quad x\in B$ (2.8)

is fulfilled.

Then $U(s,t)$ is a conservative Feller evolution system, i.e. $U(s,t)1=1$ and

$(s,t,x)\longmapsto U(s,t)F(x)$

is continuous for any $F\in C_{b}(E)$. Moreover, $U(s,t)$ can be extended to
$BM_{V}(E)$ so that

$\displaystyle\|U(s,t)F\|_{V}\leq e^{\int_{s}^{t}c(r)dr}\|F\|_{V},\quad 0\leq
s\leq t.$ (2.9)

The relation to the Kolmogorov equations is given by the statements below:

  1. (a) For any $F\in BM(E)$, $t>0$ and $x\in E$, $[0,t]\ni s\longmapsto U(s,t)F(x)$ is continuously differentiable and a solution to

$\displaystyle\frac{\partial}{\partial s}U(s,t)F(x)=-L(s)U(s,t)F(x).$ (2.10)

If in addition $F\in C_{V}(E)$, then $s\longmapsto U(s,t)F(x)$ is absolutely
continuous and satisfies (2.10) a.e.

  2. (b) Let $F\in BM(E)$. Then for any $x\in E$, $s\geq 0$, $[s,\infty)\ni t\longmapsto U(s,t)F(x)$ is absolutely continuous and satisfies for a.a. $t\geq s$

$\displaystyle\frac{\partial}{\partial t}U(s,t)F(x)=U(s,t)L(t)F(x).$ (2.11)

  3. (c) Let $V(s,t)$ be a Feller evolution system on $C_{b}(E)$. If for any $F\in C_{b}(E)$, $V(s,t)F$ is a solution to (2.10) or (2.11), then $V(s,t)=U(s,t)$ holds.

The time-homogeneous case was, e.g., treated in [10, 27]. Condition (2.7) can
be reformulated to

$\int_{E}(V(y)-V(x))Q(t,x,dy)\leq c(t)V(x),\quad t\geq 0,\quad x\in E.$

A transition function $Q$ with property (2.8) is said to have the localization
property. Property (c) means that $U(s,t)$ is the unique Feller evolution
system associated with the operator $L(t)$. The rest of this section is
devoted to the proof of above statement.

Suppose from now on the conditions given in Proposition 2.1 to be satisfied
and let $\alpha\in(0,1)$. Applying the iteration (2.2) to $(q(t,x),\alpha
Q(t,x,dy))$ yields the sub-probability function given by

$\displaystyle
P_{\alpha}(s,x;t,dy)=\sum_{n=0}^{\infty}\alpha^{n}P^{(n)}(s,x;t,dy).$ (2.12)

Let $U_{n}(s,t)F(x):=\int_{E}F(y)P^{(n)}(s,x;t,dy)$, then
$U_{\alpha}(s,t)F(x):=\sum_{n=0}^{\infty}\alpha^{n}U^{(n)}(s,t)F(x)$ defines
an evolution system. We will call $U_{\alpha}(s,t)$ the regularized evolution
system associated to $Q$. Clearly, above series converges uniformly in
$(s,t,x)$. The next lemma establishes the Feller property for
$U_{\alpha}(s,t)$, whereas the limit $\alpha\to 1$ will be considered at the
end of this section.

###### Lemma 2.2

$(U_{\alpha}(s,t))_{0\leq s\leq t}$ is a Feller evolution on $C_{b}(E)$.

###### Proof.

It suffices to show that for any $n\geq 0$ and each $F\in C_{b}(E)$ the
function $U^{(n)}(s,t)F(x)$ is continuous in all variables. Since
$U^{(0)}(s,t)F(x)=F(x)e^{-\int_{s}^{t}q(r,x)dr}$, by Lemma 3.11 this clearly
holds for $n=0$. Assume the assertion holds for some $n\geq 0$. By (2.2) we
get

$\displaystyle
U^{(n+1)}(s,t)F(x)=\int_{s}^{t}\int_{E}e^{-\int_{s}^{r}q(\tau,x)d\tau}(U^{(n)}(r,t)F)(y)Q(r,x,dy)\,dr.$
(2.13)

By induction hypothesis $e^{-\int_{s}^{r}q(\tau,x)d\tau}(U^{(n)}(r,t)F)(y)$ is
continuous in all variables. Moreover, due to
$|U^{(n)}(r,t)F(y)|\leq\|F\|_{\infty}$ this function is bounded and hence by
Lemma 3.12 we see that also

$(s,t,x)\longmapsto\int_{E}e^{-\int_{s}^{r}q(\tau,x)d\tau}(U^{(n)}(r,t)F)(y)Q(r,x,dy)$

is continuous. Thus Lemma 3.11 yields the continuity of $U^{(n+1)}(s,t)F(x)$
in the variables $(s,t,x)$. ∎

The next result studies stability of the Feller evolution $U_{\alpha}(s,t)$
with respect to $Q$. That is given a sequence of transition functions
$(Q_{j})_{j\in\mathbb{N}}$, we are interested in conditions such that
$U_{\alpha,j}(s,t)F\longrightarrow U_{\alpha}(s,t)F$ as $j\to\infty$, where
$U_{\alpha,j}(s,t)$ are the regularized evolution systems defined as in
(2.12). For functions $f\in C_{b}(E\times E)$ let
$(Q(t)f(x,\cdot))(y):=\int_{E}f(x,w)Q(t,y,dw)$.

###### Lemma 2.3

Let $(Q_{j})_{j\in\mathbb{N}}$ be a family of transition functions and assume
that $Q_{j}$ is weakly continuous for any $j\in\mathbb{N}$. Moreover, suppose
that the following conditions below are satisfied.

  1. 1\. Let $q_{j}(t,x):=Q_{j}(t,x,E)$, then $\sup_{\genfrac{}{}{0.0pt}{}{j\geq 1}{(t,x)\in B}}q_{j}(t,x)<\infty$ holds for all $B\in\mathcal{C}_{1}$.

  2. 2\. For any $f\in C_{b}(E\times E)$ the convergence

$\displaystyle(Q_{j}(t)f(x,\cdot))(x)\longrightarrow(Q(t)f(x,\cdot))(x),\quad
j\to\infty$ (2.14)

is uniform in $(t,x)\in B$ for any $B\in\mathcal{C}_{1}$.

Then for any $0\leq s\leq t$ and $F\in C_{b}(E)$

$\displaystyle U_{\alpha,j}(s,t)F\longrightarrow U_{\alpha}(s,t)F,\quad
j\to\infty$ (2.15)

holds uniformly on compacts. If instead of (2.14) the stronger convergence in
the total variation norm holds, i.e.

$\sup_{(t,x)\in[0,T]\times B}\|Q_{j}(t,x,\cdot)-Q(t,x,\cdot)\|\to 0,\quad
j\to\infty$

for any $T>0$, then the convergence (2.15) is uniform on any
$A\in\mathcal{C}_{1}$ and on $\|F\|_{\infty}\leq 1$.

###### Proof.

Since $Q,Q_{j}$ are transition functions, it follows that $U_{\alpha,j}(s,t)$
and $U_{\alpha}(s,t)$ are Feller evolution systems on $C_{b}(E)$ obtained by

$U_{\alpha,j}(s,t)F(x)=\sum_{n=0}^{\infty}\alpha^{n}U_{j}^{(n)}(s,t)F(x)$

and

$U_{\alpha}(s,t)F(x)=\sum_{n=0}^{\infty}\alpha^{n}U^{(n)}(s,t)F(x).$

Since $|U^{(n)}(s,t)F(x)|,|U_{j}^{(n)}(s,t)F(x)|\leq\|F\|_{\infty}$ the
convergence of the series is also uniform in $j\geq 1$. As a consequence it is
enough to show for any $0\leq s\leq t$, any compact $B\subset E$, $n\geq 0$
and $F\in C_{b}(E)$

$\displaystyle\lim_{j\to\infty}\sup_{x\in B}\
|U_{j}^{(n)}(s,t)F(x)-U^{(n)}(s,t)F(x)|=0.$ (2.16)

For $n=0$ this follows from (2.14) and

$|U_{j}^{(0)}(s,t)F(x)-U^{(0)}(s,t)F(x)|\leq\|F\|_{\infty}\int_{s}^{t}|q_{j}(r,x)-q(r,x)|dr.$

Assume that (2.16) holds for one $n\geq 0$, proceeding by induction we obtain
for $x\in B,0\leq s\leq t$ and $F\in C_{b}(E)$

$|U_{j}^{(n+1)}(s,t)F(x)-U^{(n+1)}(s,t)F(x)|\leq I_{1}+I_{2}+I_{3}\ ,$

where we have used (2.13) and

$\displaystyle I_{1}$
$\displaystyle=\int_{s}^{t}\int_{E}\Big{|}e^{-\int_{s}^{r}q_{j}(\tau,x)d\tau}-e^{-\int_{s}^{r}q(\tau,x)d\tau}\Big{|}|U_{j}^{(n)}(r,t)F(y)|Q_{j}(r,x,dy)\,dr$
$\displaystyle I_{2}$
$\displaystyle=\Big{|}\int_{s}^{t}\int_{E}(U_{j}^{(n)}(r,t)F(y)-U^{(n)}(r,t)F(y))e^{-\int_{s}^{r}q(\tau,x)d\tau}Q_{j}(r,x,dy)\,dr\Big{|}$
$\displaystyle I_{3}$
$\displaystyle=\Big{|}\int_{s}^{t}\int_{E}e^{-\int_{s}^{r}q(\tau,x)d\tau}U^{(n)}(r,t)F(y)(Q_{j}(r,x,dy)-Q(r,x,dy))\,dr\Big{|}.$

The first integral can be estimated by using
$|U_{j}^{(n)}(r,t)F(y)|\leq\|F\|_{\infty}$ and $q_{j}(r,x)\leq
q^{*}:=\sup_{j\geq 1}\sup_{(\tau,x)\in[s,t]\times B}\ Q_{j}(\tau,x,E)$ for
each $r\in[s,t]$, which yields

$\displaystyle I_{1}$
$\displaystyle\leq\|F\|_{\infty}\int_{s}^{t}q_{j}(r,x)\Big{|}\int_{s}^{r}(q_{j}(\tau,x)-q(\tau,x))\,d\tau\Big{|}dr$
$\displaystyle\leq\|F\|_{\infty}(t-s)^{2}q^{*}\sup_{(\tau,x)\in[s,t]\times
B}|q_{j}(\tau,x)-q(\tau,x)|.$

To estimate $I_{2}$ we need the following lemma.

###### Lemma 2.4

For any $\varepsilon>0$, $T>0$ there exists a compact $A\subset E$ and
$j_{0}\geq 1$ such that

$\int_{0}^{T}Q_{j}(t,x,A^{c})\,dt\leq\varepsilon,\quad x\in B,\quad j\geq
j_{0}.$

###### Proof.

Since $Q$ has the localization property we can find a compact $A_{1}\subset E$
such that

$\int_{0}^{T}Q(t,x,A_{1}^{c})\,dt\leq\frac{\varepsilon}{2},\quad x\in B.$

Choose compacts $A,A_{2}\subset E$ such that
$A_{1}\subset\overset{\circ}{A_{2}}\subset
A_{2}\subset\overset{\circ}{A}\subset A$, since $(\overset{\circ}{A_{2}})^{c}$
and $(\overset{\circ}{A})^{c}$ are closed there exists a continuous function
$\varphi$ with $1\kern-2.84526pt{\text{\rm
I}}_{(\overset{\circ}{A})^{c}}\leq\varphi\leq 1\kern-2.84526pt{\text{\rm
I}}_{(\overset{\circ}{A_{2}})^{c}}$. We obtain

$\int_{0}^{T}Q_{j}(t,x,A^{c})\,dt\leq\int_{0}^{T}Q_{j}(t,x,(\overset{\circ}{A})^{c})\,dt\leq\int_{0}^{T}\int_{E}\varphi(y)Q_{j}(t,x,dy)\,dt$

and by (2.14) there exists $j_{0}\geq 1$ such that for $j\geq j_{0}$, $x\in B$
and $t\in[0,T]$

$\int_{E}\varphi(y)Q_{j}(t,x,dy)\leq\frac{\varepsilon}{2T}+\int_{E}\varphi(y)Q(t,x,dy).$

Therefore the assertion follows from

$\displaystyle\int_{0}^{T}\int_{E}\varphi(y)Q_{j}(t,x,dy)\,dt$
$\displaystyle\leq\frac{\varepsilon}{2}+\int_{0}^{T}\int_{E}\varphi(y)Q(t,x,dy)\,dt$
$\displaystyle\leq\frac{\varepsilon}{2}+\int_{0}^{T}Q(t,x,(\overset{\circ}{A_{2}})^{c})\,dt\leq\frac{\varepsilon}{2}+\int_{0}^{T}Q(t,x,A_{1}^{c})\,dt\leq\varepsilon.$

∎

Take $A\subset E$ and $j_{0}\geq 1$ as in above lemma, then for any $j\geq
j_{0}$ and $x\in B$

$\displaystyle I_{2}$
$\displaystyle\leq\int_{s}^{t}\int_{A}|U_{j}^{(n)}(r,t)F(y)-U^{(n)}(r,t)F(y)|Q_{j}(r,x,dy)\,dr+2\|F\|_{\infty}\int_{s}^{t}Q_{j}(r,x,A^{c})\,dr$
$\displaystyle\leq q^{*}\int_{s}^{t}\underset{y\in A}{\sup}\
|U_{j}^{(n)}(r,t)F(y)-U^{(n)}(r,t)F(y)|\,dr+2\|F\|_{\infty}\int_{s}^{t}Q_{j}(r,x,A^{c})\,dr$
$\displaystyle\leq q^{*}\int_{s}^{t}\underset{y\in A}{\sup}\
|U_{j}^{(n)}(r,t)F(y)-U^{(n)}(r,t)F(y)|\,dr+2\|F\|_{\infty}\varepsilon.$

The integrand tends for each fixed $r\in[s,t]$ to zero as $j\to\infty$ and
since

$\underset{y\in A}{\sup}\ |U_{j}^{(n)}(r,t)F(y)-U^{(n)}(r,t)F(y)|\leq
2\|F\|_{\infty}$

also the integral tends to zero. Altogether this shows the assertion for
$I_{2}$. For the last integral observe that $(r,x,y)\longmapsto
e^{-\int_{s}^{r}q(\tau,x)d\tau}U^{(n)}(r,t)F(y)$ is continuous and moreover
bounded by $\|F\|_{\infty}$. Therefore by (2.14) for any $r\in[s,t]$

$F_{j}(r,s,t):=\sup_{x\in
B}\Big{|}\int_{E}e^{-\int_{s}^{r}q(\tau,x)d\tau}U^{(n)}(r,t)F(y)(Q_{j}(r,x,dy)-Q(r,x,dy))\Big{|}\to
0,\quad j\to\infty$

and since $F_{j}(r,s,t)\leq 2\|F\|_{\infty}q^{*}$ we obtain the assertion by
dominated convergence. The second assertion can be proved very similarly, here
only $I_{3}$ should be estimated again. ∎

As a consequence we can show that $U_{\alpha}(s,t)$ satisfies a Chernoff
product formula. That is $U_{\alpha}(s,t)$ can be approximated by evolution
systems $U_{\alpha,n}(s,t)$ with piecewise constant (in time) transition
functions $Q_{n}$. More precisely, take for any $n\in\mathbb{N}$ a sequence
$0=t_{0}^{(n)}\leq t_{k}^{(n)}<t_{k+1}^{(n)}$ with $\sup_{k\geq 0}\
(t_{k+1}^{(n)}-t_{k}^{(n)})\to 0$ as $n\to\infty$ and
$t_{k}^{(n)}\longrightarrow\infty$, $k\to\infty$ for all $n\in\mathbb{N}$.
Define piecewise constant transition functions by

$Q_{n}(t,x,dy)=Q(t_{k}^{(n)},x,dy),\quad t_{k}^{(n)}\leq t<t_{k+1}^{(n)},\quad
k\geq 0,$

then $Q_{n}$ is weakly continuous in $x$ for any fixed $t\geq 0$ and $n\geq
1$. Denote by $U_{\alpha,n}(s,t)$ the regularized Feller evolutions on
$C_{b}(E)$ constructed above, cf. Theorem 2.2. For fixed $r\geq 0$ set
$Q^{r}(x,dy):=Q(r,x,dy)$, then $Q^{r}$ is a weakly continuous transition
function and its associated regularized Feller evolution on $C_{b}(E)$ can be
represented by a Feller semigroup $T_{\alpha,r}(t)$.

###### Lemma 2.5

Let $F\in C_{b}(E)$ and for $0\leq s\leq t$ choose $m_{0},m_{1}\geq 1$ such
that

$\displaystyle t_{m_{0}}^{(n)}\leq s<t_{m_{0}+1}^{(n)}<\cdots
t_{m_{1}}^{(n)}\leq t<t_{m_{1}+1}^{(n)}.$ (2.17)

Then

$\displaystyle
U_{\alpha,n}(s,t)F(x)=T_{\alpha,t_{m_{0}}^{(n)}}(t_{m_{0}+1}^{(n)}-s)\cdots
T_{\alpha,t_{m_{1}}^{(n)}}(t-t_{m_{1}}^{(n)})F\to U_{\alpha}(s,t)F,\quad
n\to\infty$ (2.18)

holds uniformly on compacts.

###### Proof.

First observe that for any compact $B\subset E$, $T>0$ and $f\in C_{b}(E\times
E)$ Lemma 3.12 implies that $F(r,x):=\int_{E}f(x,y)Q(r,x,dy)$ is continuous.
Therefore

$\int_{E}f(x,y)Q_{n}(r,x,dy)\longrightarrow\int_{E}f(x,y)Q(r,x,dy),\quad
n\to\infty$

holds uniformly in $(x,r)\in B\times[0,T]$ and hence (2.14) follows. Applying
Lemma 2.3 we obtain for all $F\in C_{b}(E)$ and $0\leq s\leq t$:
$U_{\alpha,n}(s,t)F\to U_{\alpha}(s,t)F$ as $n\to\infty$ uniformly on
compacts. By the evolution system property it follows that

$U_{\alpha,n}(s,t)=U_{\alpha}(s,t_{m_{0}+1}^{(n)})\cdots
U_{\alpha}(t_{m_{1}}^{(n)},t)$

holds. For each pair $r_{0}<r_{1}$ with $t_{m}^{(n)}\leq r_{0}<r_{1}\leq
t_{m+1}^{(n)}$ for some $m\geq 1$, by (2.2) and (2.13) it follows that
$U_{\alpha,n}(r_{0},r_{1})=T_{\alpha,t_{m}^{(n)}}(r_{1}-r_{0})$ and hence

$U_{\alpha,n}(s,t)=T_{\alpha,t_{m_{0}}^{(n)}}(t_{m_{0}+1}^{(n)}-s)\cdots
T_{\alpha,t_{m_{1}}^{(n)}}(t-t_{m_{1}}^{(n)})$

implies the assertion. ∎

In the following we consider the limit $\alpha\to 1$ and deduce from that
$U(s,t)1=1$.

###### Theorem 2.6

The evolution system $U(s,t)$ is conservative and can be extended to
$BM_{V}(E)$ so that

$\displaystyle\|U(s,t)F\|_{V}\leq e^{\int_{s}^{t}c(r)dr}\|F\|_{V},\quad 0\leq
s\leq t.$

###### Proof.

Denote by $T_{\alpha,r}(t)$ the regularized semigroups with piecewise constant
(in the time variable) transition functions, see Theorem 2.5 and by $T_{r}(t)$
their counterparts with $\alpha=1$. Then $T_{\alpha,r}(t)V(x)\leq
T_{r}(t)V(x)$. The moment condition (2.7) and the results obtained in [10, 27]
imply that for any $r\geq 0$, $x\in E$ and $t\geq 0$

$T_{r}(t)V(x)\leq e^{c(r)t}V(x).$

Now given $0\leq s<t$ and $n\in\mathbb{N}$ we can find $m_{0},m_{1}\geq 0$
with (2.17). For $m\geq 0$ let $V_{m}(x):=V(x)\wedge m$, then $V_{m}\in
C_{b}(E)$ and hence

$\displaystyle U_{\alpha,n}(s,t)V_{m}(x)$
$\displaystyle=T_{\alpha,t_{m_{0}}^{(n)}}(t_{m_{0}+1}^{(n)}-s)\cdots
T_{\alpha,t_{m_{1}}^{(n)}}(t-t_{m_{1}}^{(n)})V_{m}(x)$ $\displaystyle\leq
T_{t_{m_{0}}^{(n)}}(t_{m_{0}+1}^{(n)}-s)\cdots
T_{t_{m_{1}}^{(n)}}(t-t_{m_{1}}^{(n)})V(x)$ $\displaystyle\leq
V(x)\exp\left(c(t_{m_{1}}^{(n)})(t-t_{m_{1}}^{(n)})+\dots+c(t_{m_{0}}^{(n)})(t_{m_{0}+1}^{(n)}-s)\right).$

Letting $n\to\infty$ yields

$U_{\alpha}(s,t)V_{m}(x)\leq V(x)\exp\Big{(}\int_{s}^{t}c(r)\,dr\Big{)}.$

The sequence $(U_{\alpha}(s,t)V_{m}(x))_{m\in\mathbb{N}}$ is increasing and
bounded, so by monotone convergence it follows that

$\displaystyle\int_{E}V(x)P_{\alpha}(s,x;t,dy)\leq
V(x)\exp\Big{(}\int_{s}^{t}c(r)\,dr\Big{)}$ (2.19)

is satisfied. The right-hand side is increasing in $\alpha$, hence taking the
limit $\alpha\to 1$ yields that $U(s,t)$ can be extended to $BM_{V}(E)$. By
[20] the evolution system $U(s,t)$ is conservative if and only if for any
$s<t$, $x\in E$

$\int_{s}^{t}\int_{E}q(r,y)P^{(n)}(s,x;r,dy)\,dr\longrightarrow 0,\quad
n\to\infty.$

Let $T>0$ such that $[s,t]\subset[0,T]$, then by $q(r,y)\leq a(T)V(y)$ for
$r\in[s,t]$ and (2.19) with $\alpha=1$

$\int_{s}^{t}\int_{E}q(r,y)P(s,x;r,dy)\,dr\leq
a(T)V(x)\int_{s}^{t}\exp\Big{(}\int_{s}^{r}c(\tau)\,d\tau\Big{)}dr<\infty$

follows. The assertion follows from the representation
$P(s,x;r,dy)=\sum_{n=0}^{\infty}P^{(n)}(s,x;r,dy)$. ∎

The next result shows that $U(s,t)$ is differentiable in $s$.

###### Theorem 2.7

For any $F\in BM(E)$, $t>0$ and $x\in E$, $[0,t]\ni s\longmapsto U(s,t)F(x)$
is continuously differentiable and a solution to

$\displaystyle\frac{\partial}{\partial s}U(s,t)F(x)=-L(s)U(s,t)F(x).$

Moreover, for any $F\in C_{V}(E)$ the function $U(s,t)F(x)$ is absolutely
continuous in $s$ and solves above equation a.e.. Let $V(s,t)$ be a Feller
evolution system on $C_{b}(E)$ and assume that $V(s,t)F$ is a solution to
(2.10) for any $F\in C_{b}(E)$, then $V(s,t)=U(s,t)$ is fulfilled.

###### Proof.

By (2.3) we obtain for any $A\in\mathcal{B}(E)$ and $0\leq s\leq t$

$P(s,x;t,A)=\delta(x,A)+\int_{s}^{t}q(r,x)P(r,x;t,A)\,dr-\int_{s}^{t}\int_{E}P(r,y;t,A)Q(r,x,dy)\,dr$

and hence for any $F\in BM(E)$ and $x\in E$

$U(s,t)F(x)=F(x)+\int_{s}^{t}q(r,x)U(r,t)F(x)\,dr-\int_{s}^{t}\int_{E}U(r,t)F(y)Q(r,x,dy)\,dr$

follows. Clearly $q(r,x)U(r,t)F(x)$ and by Lemma 3.12 also
$\int_{E}U(r,t)F(y)Q(r,x,dy)$ are continuous in $r$, which implies that
$L(r)U(r,t)F(x)$ is continuous in $(r,t)$. Therefore

$\displaystyle U(s,t)F(x)=F(x)-\int_{s}^{t}L(r)U(r,t)F(x)\,dr$ (2.20)

implies (2.10). If $F\in C_{V}(E)$, then $U(s,t)F(x)$ is bounded and
measurable in $(s,t)$. Hence by (2.7) $L(r)U(r,t)F(x)$ is well-defined and
integrable w.r.t. $r$. In view of (2.20) it follows that $s\longmapsto
U(s,t)F(x)$ is absolutely continuous and satisfies (2.10) for any $x\in E$.

Now let $V(s,t)$ be a Feller evolution on $C_{b}(E)$ which satisfies (2.10).
By [33, Chapter 2, Theorem 2.9] $V(s,t)$ is given by

$V(s,t)F(x)=\int_{E}F(y)\widetilde{P}(s,x;t,dy),\quad x\in E,\quad 0\leq s\leq
t,$

where $\widetilde{P}$ is a transition probability function. Moreover, this
evolution system satisfies (2.20) for any $F\in C_{b}(E)$ and hence by
approximation also for any $F\in BM(E)$. Therefore for any
$F=1\kern-2.84526pt{\text{\rm I}}_{A}$, $A\in\mathcal{B}(E)$ it solves
equation (2.10) which is simply (2.3). The minimality of $P$ implies
$P\leq\widetilde{P}$ and hence $U(s,t)F\leq V(s,t)F$. Since $U(s,t)$ is
conservative it follows that $P(s,x;t,dy)$ is the unique solution to (2.3),
i.e. $P(s,x;t,dy)=\widetilde{P}(s,x;t,dy)$. ∎

###### Theorem 2.8

Let $F\in BM(E)$, then for any $x\in E$ and $s\geq 0$, $[s,\infty)\ni
t\longmapsto U(s,t)F(x)$ is absolutely continuous and satisfies for a.a.
$t\geq s$

$\displaystyle\frac{\partial}{\partial t}U(s,t)F(x)=U(s,t)L(t)F(x).$

Let $V(s,t)$ be a Feller evolution system on $C_{b}(E)$ and assume that
$V(s,t)F$ is for any $F\in C_{b}(E)$ a solution to (2.11), then
$V(s,t)=U(s,t)$ holds.

###### Proof.

For all $0\leq s\leq r\leq t<T$

$\displaystyle\int_{E}q(r,y)P(s,x;t,dy)\leq a(T)\int_{E}V(y)P(s,x;t,dy)\leq
a(T)V(x)e^{\int_{s}^{t}c(r)dr}$ (2.21)

and (2.4) implies for any $0\leq s\leq t$ and compact $A\subset E$

$P(s,x;t,A)=\delta(x,A)-\int_{s}^{t}\int_{A}q(r,y)P(s,x;r,dy)\,dr+\int_{s}^{t}\int_{E}Q(r,y,A)P(s,x;r,dy)\,dr.$

By (2.21) this implies

$U(s,t)F(x)=F(x)-\int_{s}^{t}\int_{E}q(r,y)F(y)P(s,x;r,dy)\,dr+\int_{s}^{t}\int_{E}Q(r)F(y)P(s,x;r,dy)\,dr$

and hence

$\displaystyle U(s,t)F(x)=F(x)+\int_{s}^{t}\int_{E}L(r)F(y)P(s,x;r,dy)\,dr$

holds. The first assertion is proved. Uniqueness follows by the same arguments
as for (2.10). ∎

###### Remark 2.9

It is worth noting that in the time-homogeneous case (2.10) and (2.11) are
equivalent and less restrictive conditions are sufficient to show that
$U(s,t)$ is an Feller evolution, see [27].

Since $U(s,t)$ is given by a transition probability function we see that for
each $x\in E$ and $s\geq 0$ there exists a probability space
$(\Omega,\mathcal{F}^{s},\mathbb{P}_{s,x})$ and a conservative Markov process
$(X(t))_{t\geq s}$ on this space such that

$U(s,t)F(x)=\mathbb{E}_{s,x}(F(X(t)),\quad F\in C_{b}(E),\quad t\geq s.$

This process is considered w.r.t. its natural filtration defined by
$\mathcal{F}_{\tau}^{s}=\sigma\left(X(t)\ |\ s\leq t\leq\tau\right)$ for
$s\leq\tau$. Note that this process is, by construction, a pure jump process.
The next statement completes the proof of Proposition 2.1.

###### Corollary 2.10

The following statements are true:

  1. 1\. Let $F\in BM(E)$. Then for any fixed $s\geq 0$

$M_{s,F}(t):=F(X(t))-F(X(s))-\int_{s}^{t}L(r)F(X(r))\,dr,\quad t\geq s$

is a martingale with respect to $(\mathcal{F}_{t}^{s})_{t\geq s}$ and
$\mathbb{P}_{s,x}$.

  2. 2\. For any $a>0$, $x\in E$ and $0\leq s<T$

$\mathbb{P}_{s,x}\Big{(}\sup_{t\in[s,T]}V(X(t))\geq a\Big{)}\leq
V(x)\frac{e^{\int_{s}^{T}c(r)dr}}{a}$

holds.

  3. 3\. $U(s,t)$ is a Feller evolution system.

###### Proof.

1\. This follows by the Markov property and the relation

$\mathbb{E}_{\tau,X(\tau)}(L(r)F(X(r)))=(U(\tau,r)L(r)F)(X(\tau))=\frac{\partial}{\partial
r}U(\tau,r)F(X(\tau)),$

where $0\leq s\leq\tau\leq t$.
2\. Let $E_{n}:=\\{x\in E\ |\ V(x)<n\\}$, fix $s\geq 0$ and define a family of
stopping times

$\tau_{n}:=\inf\\{t\geq s\ |\ X_{t}\not\in E_{n}\\}.$

Let $\varphi_{n}\in C(E)$ be such that $1\kern-2.84526pt{\text{\rm
I}}_{\overline{E}_{n}}\leq\varphi_{n}\leq 1\kern-2.84526pt{\text{\rm
I}}_{\overline{E}_{n+1}}$ and define a new transition function by
$Q_{n}(t,x,dy):=\varphi_{n}(x)Q(t,x,dy)$. Then

$L_{n}(t)F(x):=-\varphi_{n}(x)q(t,x)F(x)+\int_{E}F(y)\varphi_{n}(x)Q(t,x,dy)=\varphi_{n}(x)L(t)F(x)$

determines a bounded linear operator on $C_{b}(E)$ and $BM_{V}(E)$. Hence
there exists an associated conservative Feller evolution system $U_{n}(s,t)$
on $C_{b}(E)$. This evolution system can be extended to $BM_{V}(E)$. Let
$(X_{t}^{n})_{t\geq 0}$ be the corresponding Markov process, and denote by
$(\mathcal{F}_{t,n}^{s})_{t\geq s}$ its associated natural filtration. By
construction it follows for $x\in E_{n}$ and $n\geq 1$ that these processes
satisfy

$\displaystyle(X_{t})_{t<\tau_{n}}=(X_{t}^{n})_{t<\tau_{n}}$ (2.22)

in the sense of finite dimensional distributions. For $s\geq 0$ let
$g(t,x):=e^{-\int_{s}^{t}c(r)dr}V(x)$. A short computation shows that

$\frac{\partial}{\partial t}g(t,x)+L(t)g(t,x)\leq 0.$

Then

$M_{n}(s,t):=g(t,X^{n}(t))-g(s,X^{n}(s))-\int_{s}^{t}\left(\frac{\partial}{\partial
r}+L_{n}(r)\right)g(r,X^{n}(r))\,dr,\quad t\geq s$

is a $\mathcal{F}_{t,n}^{s}$-martingale w.r.t. $\mathbb{P}_{s,x}$. Fix $x\in
E_{n}$, $n\geq 1$, hence by Dynkin’s formula

$\displaystyle\mathbb{E}_{s,x}(g(t\wedge\tau_{n},X_{t\wedge\tau_{n}}^{n}))$
$\displaystyle=g(s,x)+\mathbb{E}_{s,x}\Bigg{(}\int_{s}^{t\wedge\tau_{n}}\left(\frac{\partial}{\partial
r}+L_{n}(r)\right)g(r,X_{r}^{n})\,dr\Bigg{)}\leq g(s,x)$ (2.23)

holds. Here $\frac{\partial}{\partial r}$ acts only on the first variable of
$g$. Let
$M_{t}^{n}:=e^{-\int_{s}^{t}c(\sigma)d\sigma}V(X_{t}^{n})1\kern-2.84526pt{\text{\rm
I}}_{t<\tau_{n}}$, we will show that $(M_{t}^{n})_{t\geq s}$ is a
supermartingale. Fix $s\leq r\leq t$. On
$\\{r\geq\tau_{n}\\}\in\mathcal{F}_{r,n}^{s}$ we have $M_{t}^{n}=M_{r}^{n}=0$
and hence obtain

$\mathbb{E}_{s,x}(M_{t}^{n}|\mathcal{F}_{r,n}^{s})=M_{r}^{n}=0.$

On $\\{r<\tau_{n}\\}$ we have by the Markov property and (2.23)

$\displaystyle\mathbb{E}_{s,x}(M_{t}^{n}|\mathcal{F}_{r,n}^{s})$
$\displaystyle=e^{-\int_{s}^{t}c(\sigma)d\sigma}\mathbb{E}_{r,X_{r}^{n}}(V(X_{t}^{n})1\kern-2.84526pt{\text{\rm
I}}_{t<\tau_{n}})\leq\mathbb{E}_{r,X_{r}^{n}}(g(t\wedge\tau_{n},X_{t\wedge\tau_{n}}^{n}))$
$\displaystyle\leq
g(r,X_{r}^{n})=g(r\wedge\tau_{n},X_{r\wedge\tau_{n}}^{n})=M_{r}^{n}.$

Applying Doob’s inequality yields

$\displaystyle\mathbb{P}_{s,x}\Big{(}\sup_{\genfrac{}{}{0.0pt}{}{s\leq t\leq
T}{t<\tau_{n}}}g(t,X(t))\geq a\Big{)}=\mathbb{P}_{s,x}\Big{(}\sup_{s\leq t\leq
T}M_{t}^{n}\geq
a\Big{)}\leq\frac{1}{a}\mathbb{E}_{s,x}(M_{s}^{n})=\frac{V(x)}{a}.$

As a consequence we obtain

$\displaystyle\mathbb{P}_{s,x}\Big{(}\sup_{\genfrac{}{}{0.0pt}{}{s\leq t\leq
T}{t<\tau_{n}}}V(X(t))\geq a\Big{)}$
$\displaystyle\leq\mathbb{P}_{s,x}\Big{(}\sup_{\genfrac{}{}{0.0pt}{}{s\leq
t\leq T}{t<\tau_{n}}}g(t,X(t))\geq ae^{-\int_{s}^{T}c(r)dr}\Big{)}$
$\displaystyle\leq V(x)\frac{e^{\int_{s}^{T}c(r)dr}}{a}.$

Since $(X_{t})_{t\geq s}$ is conservative it follows
$\tau_{n}\longrightarrow\infty$ when $n\to\infty$. The assertion follows by
monotone convergence and $n\to\infty$.
3\. For any $F\in C_{b}(E)$, $x\in E_{n}$ and $n\geq 1$ it follows by (2.22)

$\displaystyle|\mathbb{E}_{s,x}(F(X_{t}))-\mathbb{E}_{s,x}(F(X_{t}^{n}))|$
$\displaystyle=|\mathbb{E}_{s,x}(F(X_{t})1\kern-2.84526pt{\text{\rm
I}}_{\tau_{n}\leq t})-\mathbb{E}_{s,x}(F(X_{t}^{n})1\kern-2.84526pt{\text{\rm
I}}_{\tau_{n}\leq t})|$ $\displaystyle\leq
2\|F\|_{\infty}\mathbb{P}_{s,x}(\tau_{n}\leq t).$

By

$\displaystyle\mathbb{P}_{s,x}(\tau_{n}\leq t)$
$\displaystyle\leq\mathbb{P}_{s,x}\left(\sup_{s\leq r\leq t}V(X(r))\geq
n\right)\leq\frac{V(x)}{n}e^{\int_{s}^{t}c(r)dr}.$

and the continuity of $V$ we see that $U_{n}(s,t)F(x)\longrightarrow
U(s,t)F(x)$ uniformly on compacts which implies the assertion. ∎

We close this section with the relation to the evolution of measures. Let
$\mathcal{M}(E)$ be the space of all finite, signed Borel measures on $E$
equipped with the total variation norm. Define bounded linear operators
$(U^{*}(t,s))_{0\leq s\leq t}$ on $\mathcal{M}(E)$ by

$U^{*}(t,s)\mu(dx)=\int_{E}P(s,y;t,dx)\mu(dy).$

Then $U^{*}(t,t)=\mathrm{id}_{\mathcal{M}(E)}$,
$U^{*}(t,r)U^{*}(r,s)=U^{*}(t,s)$ holds for $0\leq s\leq r\leq t$ and

$\int_{E}F(y)U^{*}(t,s)\mu(dy)=\int_{E}U(s,t)F(y)\mu(dy),\quad F\in
C_{b}(E),\quad\mu\in\mathcal{M}(E).$

Previous considerations show that $U^{*}(t,s)\mu$ is the unique weak solution
to the Fokker-Planck equation

$\frac{\partial}{\partial
t}\int_{E}F(y)U^{*}(t,s)\mu(dy)=\int_{E}L(t)F(y)U^{*}(t,s)\mu(dy),$

where $\mu\in\mathcal{M}(E)$ is such that $\int_{E}V(x)|\mu|(dx)<\infty$.

##  3\. Interacting particle systems in continuum

### Preliminaries

The configuration space $\Gamma_{0}$ is the space of all finite subsets of
$\mathbb{R}^{d}$, i.e.

$\Gamma_{0}=\\{\eta\subset\mathbb{R}^{d}\ |\ |\eta|<\infty\\},$

where $|\eta|$ denotes the number of elements in the set $\eta$. This space
has a natural decomposition into $n$-particle spaces,
$\Gamma_{0}=\bigsqcup_{n=0}^{\infty}\Gamma_{0}^{(n)}$, where
$\Gamma_{0}^{(n)}=\\{\eta\subset\mathbb{R}^{d}\ |\ |\eta|=n\\},\ \ n\geq 1$
and in the case $n=0$ we set $\Gamma_{0}^{(0)}=\\{\emptyset\\}$. For a compact
$\Lambda\subset\mathbb{R}^{d}$ let

$\Gamma_{\Lambda}=\\{\eta\in\Gamma_{0}\ |\ \eta\subset\Lambda\\}$

and $\Gamma_{\Lambda}^{(n)}=\\{\eta\in\Gamma_{0}^{(n)}\ |\
\eta\subset\Lambda\\}$. Denote by $\widetilde{(\mathbb{R}^{d})^{n}}$ the space
of all sequences $(x_{1},\dots,x_{n})\in(\mathbb{R}^{d})^{n}$ with $x_{i}\neq
x_{j}$ for $i\neq j$. $\Gamma_{0}^{(n)}$ can be identified with
$\widetilde{(\mathbb{R}^{d})^{n}}$ via the symmetrization map

$\mathrm{sym}_{n}:\widetilde{(\mathbb{R}^{d})^{n}}\longrightarrow\Gamma_{0}^{(n)},\
(x_{1},\dots,x_{n})\longmapsto\\{x_{1},\dots,x_{n}\\},$

which defines a topology on $\Gamma_{0}^{(n)}$. Namely, a set
$A\subset\Gamma_{0}^{(n)}$ is open if and only if
$\mathrm{sym}_{n}^{-1}(A)\subset\widetilde{(\mathbb{R}^{d})^{n}}$ is open. On
$\Gamma_{0}$ we define the topology of disjoint unions, i.e. a set
$A\subset\Gamma_{0}$ is open iff $A\cap\Gamma_{0}^{(n)}$ is open in
$\Gamma_{0}^{(n)}$ for all $n\in\mathbb{N}$. Then $\Gamma_{0}$ is a locally
compact Polish space. Let $\mathcal{B}(\Gamma_{0})$ stand for the
Borel-$\sigma$-algebra on $\Gamma_{0}$. With respect to this topology the
function

$\eta\longmapsto\langle f,\eta\rangle:=\sum_{x\in\eta}f(x)$

is continuous whenever $f\in C_{b}(\mathbb{R}^{d})$. Therefore convergence of
a sequence $(\eta_{n})_{n\in\mathbb{N}}\subset\Gamma_{0}$ to
$\eta\in\Gamma_{0}$ can be rewritten to: there exists $N\in\mathbb{N}$ such
that for all $n\geq N$: $\eta_{n}=\\{x_{1}^{(n)},\dots,x_{l}^{(n)}\\},\
\eta=\\{x_{1},\dots,x_{l}\\}$ and

$x_{j}^{(n)}\longrightarrow x_{j},\ \ n\to\infty,\ \forall
j\in\\{1,\dots,l\\}$

is fulfilled. For given $\delta>0$, $N\in\mathbb{N}_{0}$ and a compact
$\Lambda\subset\mathbb{R}^{d}$ the set

$\displaystyle B=\\{\eta\in\Gamma_{\Lambda}\ |\ \forall x\neq y,\ x,y\in\eta:\
|x-y|\geq\delta,\ \ |\eta|\leq N\\}$ (3.1)

is compact. Conversely, for any compact set $A\subset\Gamma_{0}$ there exist
$\delta,N,\Lambda$ such that $A$ is contained in a compact $B$ defined above.
Denote by $dx$ the Lebesgue measure on $\mathbb{R}^{d}$ and by $d^{\otimes
n}x$ the product measure on $(\mathbb{R}^{d})^{n}$. The image measure of
$d^{\otimes n}x$ on $\Gamma_{0}^{(n)}$ via $\mathrm{sym}_{n}$ is then denoted
by $d^{(n)}x$. The Lebesgue-Poisson measure is defined by

$\lambda=\delta_{\emptyset}+\sum_{n=1}^{\infty}\frac{1}{n!}\,d^{(n)}x.$

Given a measurable function
$G:\Gamma_{0}\times\Gamma_{0}\longrightarrow\mathbb{R}$, then

$\displaystyle\int_{\Gamma_{0}}\sum_{\xi\subset\eta}G(\xi,\eta\backslash\xi)\,d\lambda(\eta)=\int_{\Gamma_{0}}\int_{\Gamma_{0}}G(\xi,\eta)d\lambda(\xi)\,d\lambda(\eta)$
(3.2)

holds, provided one side of the equality is finite for $|G|$. Here and in the
following we write $\eta\backslash x$, $\eta\cup x$, instead of
$\eta\backslash\\{x\\}$ and $\eta\cup\\{x\\}$. The decomposition
$\Gamma_{0}=\bigsqcup_{n=0}^{\infty}\Gamma_{0}^{(n)}$ implies that any
measurable function $G:\Gamma_{0}\longrightarrow\mathbb{R}$ can be represented
as a sequence of symmetric measurable functions $(G^{(n)})_{n=0}^{\infty}$,
where $G^{(n)}:(\mathbb{R}^{d})^{n}\longrightarrow\mathbb{R}$. Such functions
are uniquely determined on the off-diagonal part
$\widetilde{(\mathbb{R}^{d})^{n}}$ and integration w.r.t. to the Lebesgue-
Poisson measure is simply determined by the identity

$\int_{\Gamma_{0}}G(\eta)\,d\lambda(\eta)=G^{(0)}+\sum_{n=1}^{\infty}\frac{1}{n!}\int_{(\mathbb{R}^{d})^{n}}G^{(n)}(x_{1},\cdots,x_{n})\,dx_{1}\cdots
dx_{n}.$

For a given measurable function $f:\mathbb{R}^{d}\longrightarrow\mathbb{R}$
the Lebesgue-Poisson exponential is defined by

$e_{\lambda}(f;\eta):=\prod_{x\in\eta}f(x)$

and satisfies the combinatorial formula

$\sum_{\xi\subset\eta}e_{\lambda}(f;\xi)=e_{\lambda}(1+f;\eta).$

For computations we will use the identity

$\int_{\Gamma_{0}}e_{\lambda}(f;\eta)\,d\lambda(\eta)=\exp\Big{(}\int_{\mathbb{R}^{d}}f(x)\,dx\Big{)},$

whenever $f\in L^{1}(\mathbb{R}^{d})$.

### General statement

The class of Markov jump processes we are interested in are given by a Markov
pre-generator of the form

$\displaystyle(L(t)F)(\eta)=\sum_{\xi\subset\eta}\int_{\Gamma_{0}}(F(\eta\backslash\xi\cup\zeta)-F(\eta))K_{t}(\xi,\eta,d\zeta),\quad\eta\in\Gamma_{0},\quad
t\geq 0.$ (3.3)

Such Kolmogorov operator includes death, birth and jumps of groups of
particles. We will say $K_{t}$ satisfies the usual conditions if the
conditions given below are satisfied.

  1. 1\. For all $\eta,\xi\in\Gamma_{0}$ and $t\geq 0$: $K_{t}(\xi,\eta,\cdot)\geq 0$ is a finite, non-atomic Borel measure.

  2. 2\. For all $A\in\mathcal{B}(\Gamma_{0})$, the map $(t,\xi,\eta)\longmapsto K_{t}(\xi,\eta,A)$ is measurable.

For $t\geq 0$, $\eta\in\Gamma_{0}$ and $A\in\mathcal{B}(\Gamma_{0})$ define
$Q(t,\eta,d\omega)$ by

$\displaystyle
Q(t,\eta,A)=\sum_{\xi\subset\eta}\int_{\Gamma_{0}}1\kern-2.84526pt{\text{\rm
I}}_{A}(\eta\backslash\xi\cup\zeta)K_{t}(\xi,\eta,d\zeta).$ (3.4)

The cumulative intensity is defined by
$q(t,\eta):=Q(t,\eta,\Gamma_{0})=\sum_{\xi\subset\eta}K_{t}(\xi,\eta,\Gamma_{0})$.
We will work with the following conditions:

  1. (A)

For any $\varepsilon>0,\ T>0$ and any compact $B\subset\Gamma_{0}$ there
exists another compact $A\subset\Gamma_{0}$ such that

$\int_{0}^{T}Q(r,\eta,A^{c})\,dr<\varepsilon,\quad\eta\in B$

is satisfied.

  2. (B)

There exist continuous functions $V:\Gamma_{0}\longrightarrow\mathbb{R}_{+}$
and $c:\mathbb{R}_{+}\longrightarrow\mathbb{R}_{+}$ such that

$\displaystyle\sum_{\xi\subset\eta}\int_{\Gamma_{0}}V(\eta\backslash\xi\cup\zeta)K_{t}(\xi,\eta,d\zeta)\leq
c(t)V(\eta)+q(t,\eta)V(\eta),\quad t\geq 0,\quad\eta\in\Gamma_{0}$ (3.5)

holds.

  3. (C)

For any $F\in C(\Gamma_{0})$ with
$\sup_{\eta\in\Gamma_{0}}\frac{|F(\eta)|}{1+V(\eta)}<\infty$

$(t,\eta)\longmapsto\sum_{\xi\subset\eta}\int_{\Gamma_{0}}F(\eta\backslash\xi\cup\zeta)K_{t}(\xi,\eta,d\zeta)$

is continuous.

  4. (D)

For any $T>0$ there exists $a(T)>0$ such that $q(t,\eta)\leq a(T)V(\eta)$
holds for all $\eta\in\Gamma_{0}$ and $t\in[0,T]$.

  5. (E)

For any $T>0$ there exists $b(T)>0$ such that $q(t,\eta)\geq b(T)q(T,\eta)$
holds for all $\eta\in\Gamma_{0}$ and $t\in[0,T]$.

As in the previous section let $BM_{V}(\Gamma_{0})$ stand for the Banach space
of all measurable functions $F$ equipped with the norm
$\|F\|_{V}=\sup_{\eta\in\Gamma_{0}}\frac{|F(\eta)|}{1+V(\eta)}$. Denote by
$C_{V}(\Gamma_{0})$ the closed subspace of all continuous functions for which
$\|\cdot\|_{V}$ is finite. Then condition (D) simply states that for any $F\in
C_{V}(\Gamma_{0})$ the action $L(t)F$, cf. (3.3), is continuous in $(t,\eta)$.

###### Proposition 3.1

Let $K_{t}$ be a transition function with the usual conditions and assume that
conditions (A)–(D) hold. Then there exists a unique associated conservative
Feller evolution $U(s,t)$ on $C_{b}(\Gamma_{0})$. This evolution system can be
extended to $BM_{V}(\Gamma_{0})$ so that

$\displaystyle|U(s,t)F(\eta)|\leq\|F\|_{V}V(\eta)e^{\int_{s}^{t}c(r)dr}.$
(3.6)

Moreover the following assertions are true:

  1. 1\. For any $F\in BM(\Gamma_{0})$, $t>0$ and $\eta\in\Gamma_{0}$, $U(s,t)F(\eta)$ is a solution to

$\frac{\partial}{\partial s}U(s,t)F(\eta)=-L(s)U(s,t)F(\eta),\quad s\in[0,t).$

  2. 2\. Let $F\in BM(\Gamma_{0})$. Then for any $s\geq 0$ and $\eta\in\Gamma_{0}$, $U(s,t)F(\eta)$ is a solution to

$\frac{\partial}{\partial t}U(s,t)F(\eta)=U(s,t)L(t)F(\eta),\quad\text{ a.a.
}t\geq s.$

###### Proof.

The assertion follows by Proposition 2.1. ∎

The considerations of the first section imply that $U(s,t)F$ is given by a
transition probability function $P(s,\eta;t,d\omega)$, that is

$\displaystyle U(s,t)F(\eta)=\int_{\Gamma_{0}}F(\omega)P(s,\eta;t,d\omega)$
(3.7)

holds. The adjoint evolution system on $\mathcal{M}(\Gamma_{0})$ is given by

$U^{*}(t,s)\mu(A)=\int_{\Gamma_{0}}P(s,\eta;t,A)\,d\mu(d\eta).$

The action of the adjoint evolution $U^{*}(t,s)\mu$ provides a weak solution
to the Fokker-Planck equation

$\frac{\partial}{\partial
t}\int_{\Gamma_{0}}F(\eta)\mu_{t}(d\eta)=\int_{\Gamma_{0}}L(t)F(\eta)\,d\mu_{t}(\eta),\quad
F\in C_{b}(\Gamma_{0}).$

In particular, if conditions (A)–(D) are satisfied, then $U(t,s)^{*}$ is
unique with such property.

Here and in the following we identify the space of densities
$L^{1}(\Gamma_{0},d\lambda)$ with its image in $\mathcal{M}(\Gamma_{0})$ given
by the (isometric) embedding

$L^{1}(\Gamma_{0},d\lambda)\ni R\longmapsto
Rd\lambda\in\mathcal{M}(\Gamma_{0}).$

The next theorem states conditions for which $U^{*}(t,s)$ leaves the space of
densities invariant and its restriction to $L^{1}(\Gamma_{0},d\lambda)$ is
strongly continuous.

###### Theorem 3.2

Assume that $K_{t}(\xi,\eta,d\zeta)$ satisfies the usual conditions, is
absolutely continuous with respect to the Lebesgue-Poisson measure and the
conditions (A)–(E) hold. Then $U^{*}(t,s)$ leaves $L^{1}(\Gamma_{0},d\lambda)$
invariant and is strongly continuous on $L^{1}(\Gamma_{0},d\lambda)$.

###### Proof.

Denote by
$K_{t}(\xi,\eta,\zeta)=\frac{dK_{t}(\xi,\eta,d\zeta)}{d\lambda(\zeta)}$ and
let $L^{*}(t)$ be the adjoint operator with respect to the duality of
$BM(\Gamma_{0})$ and $\mathcal{M}(\Gamma_{0})$. Then $L^{*}(t)$ is given by
$L^{*}(t)=-q(t,\cdot)+Q(t)$ with $(-q(t,\cdot)R)(\eta)=-q(t,\eta)R(\eta)$ and

$\displaystyle
Q(t)R(\eta)=\sum_{\xi\subset\eta}\int_{\Gamma_{0}}R(\eta\backslash\xi\cup\zeta)K_{t}(\zeta,\eta\backslash\xi\cup\zeta,\xi)\,d\lambda(\zeta),$
(3.8)

see (3.2). For $t\geq 0$ let

$\displaystyle D(L^{*}(t))=\\{R\in L^{1}(\Gamma_{0},d\lambda)\ |\
q(t,\cdot)R\in L^{1}(\Gamma_{0},d\lambda)\\}.$ (3.9)

First observe that $W^{*}(t,s)R(\eta)=e^{-\int_{s}^{t}q(r,\eta)dr}R(\eta)$ is
a positive contraction operator and $Q(t)$ is positive. In order to apply [1,
Theorem 2.1] it is enough to show that for a.a. $t>s$ and all $R\in
L^{1}(\Gamma_{0},d\lambda)$: $W^{*}(t,s)R\in D(L^{*}(t))$ and

$\int_{s}^{t}\|Q(r)W^{*}(r,s)R\|_{L^{1}}dr\leq\|R\|_{L^{1}}-\|W^{*}(t,s)R\|_{L^{1}}.$

The first property follows by property (E) from

$\int_{\Gamma_{0}}q(t,\eta)|W^{*}(t,s)R(\eta)|\,d\lambda(\eta)\leq\int_{\Gamma_{0}}q(t,\eta)e^{-b(t)(t-s)q(t,\eta)}|R(\eta)|\,d\lambda(\eta)\leq\frac{\|R\|_{L^{1}}}{b(t)(t-s)e}.$

For the second property let $R\in L^{1}(\Gamma_{0},d\lambda)$ and note that

$\int_{\Gamma_{0}}|Q(r)R(\eta)|\,d\lambda(\eta)\leq\int_{\Gamma_{0}}q(r,\eta)|R(\eta)|\,d\lambda(\eta)$

holds. Altogether this implies

$\displaystyle\int_{s}^{t}\|Q(r)W^{*}(r,s)R\|_{L^{1}}dr$
$\displaystyle\leq\int_{s}^{t}\int_{\Gamma_{0}}q(r,\eta)e^{-\int_{s}^{r}q(\tau,\eta)d\tau}|R(\eta)|\,d\lambda(\eta)dr$
$\displaystyle=-\int_{s}^{t}\int_{\Gamma_{0}}\frac{\partial}{\partial
r}e^{\\!-\\!\int_{s}^{r}q(\tau,\eta)d\tau}|R(\eta)|\,d\lambda(\eta)dr$
$\displaystyle=\|R\|_{L^{1}}\\!-\\!\|W^{*}(t,s)R\|_{L^{1}}.$

Hence by [1, Theorem 2.1] there exists a strongly continuous evolution family
$(V^{*}(t,s))_{0\leq s\leq t}$ on $L^{1}(\Gamma_{0},d\lambda)$. The
construction of $V^{*}(t,s)$ coincides with the construction of $U^{*}(t,s)$
restricted to $L^{1}(\Gamma_{0},d\lambda)$, i.e.
$U^{*}(t,s)R=\sum_{n=0}^{\infty}U_{n}^{*}(t,s)R$ with
$U_{0}^{*}(t,s)R=e^{-\int_{s}^{t}q(r,\eta)dr}R$ and

$U_{n+1}^{*}(t,s)R=\int_{s}^{t}U_{n}^{*}(r,t)Q(r)W^{*}(r,s)R\,dr,$

cf. [20, Section 3, Theorem 1]. ∎

###### Remark 3.3

For the application of [1, Theorem 2.1] it is necessary to show that
$t\longmapsto Q(t)R\in L^{1}(\Gamma_{0},d\lambda)$ is measurable. Since
$L^{1}(\Gamma_{0},d\lambda)$ is separable, strong measurability and weak
measurability coincide, which is the reason why we have to restrict the
evolution to the space of densities.

### Examples

In this part we study two particular models, which have applications in
ecological sciences. To simplify the proofs we consider first the case of a
Markov (pre-)generator describing only the death of particles.

###### Lemma 3.4

Consider the operator $L(t)$ given by

$(L(t)F)(\eta)=\sum_{\xi\subset\eta}(F(\eta\backslash\xi)-F(\eta))D_{t}(\xi,\eta),\quad
t\in I,$

where $(t,\xi,\eta)\longmapsto D_{t}(\xi,\eta)\geq 0$ is assumed to be
continuous. Then condition (A) and the usual conditions holds. Moreover,
$(t,\eta)\longmapsto L(t)F(\eta)$ is continuous for any $F\in C(\Gamma_{0})$.

###### Proof.

The associated function is given by
$K_{t}(\xi,\eta,d\zeta)=D_{t}(\xi,\eta)\delta_{\emptyset}(d\zeta)$ and thus
satisfies the usual conditions. The characterization of convergence in
$\Gamma_{0}$ and continuity of $D_{t}$ imply that for each $F\in
C(\Gamma_{0})$ also $L(t)F(\eta)$ is continuous in $(t,\eta)$. Concerning (A),
fix $\varepsilon>0$, $T>0$ and a compact $B\subset\Gamma_{0}$. Then there
exist $\delta_{B}>0$, $N_{B}\in\mathbb{N}$ and a compact
$\Lambda_{B}\subset\mathbb{R}^{d}$ such that for each $\eta\in B$

$\displaystyle|\eta|\leq N_{B},\quad\eta\subset\Lambda_{B},\quad\forall
x,y\in\eta,\ x\neq y:\ |x-y|\geq\delta_{B}$ (3.10)

holds. Let $A\subset\Gamma_{0}$ be a compact of the form (3.1) with
$\delta,N,\Lambda$ as in (3.10). Then for each $\eta\in B$ and
$\xi\subset\eta$ we obtain that (3.10) also holds for $\eta\backslash\xi$
instead of $\eta$. Hence $\eta\backslash\xi\in A$ and thus $Q(t,\eta,A^{c})=0$
for any $t\in[0,T]$. ∎

#### The BDLP-model

In [7, 8, 13, 14] the so called Bolker-Dieckmann-Law-Pacala model (short BDLP-
model) was introduced to study spatial patterns for certain ecological
systems. Elements $x\in\eta$ are interpreted as plants and the configuration
$\eta\in\Gamma_{0}$ describes therefore the whole ecological system. The BDLP-
model is based only on the two elementary events $\eta\longmapsto\eta\cup x$
(branching of plants) and $\eta\longmapsto\eta\backslash x$ (death of plants).
The branching is assumed to be density independent, that is any plant at
position $x\in\eta$ creates with intensity $0\leq\lambda\in
C(\mathbb{R}_{+}\times\mathbb{R}^{d})$ a new plant at position
$y\in\mathbb{R}^{d}\backslash\eta$ and the spatial probability distribution
for the new plant is given by $a^{+}(x,y)dy$, where $a^{+}\in
C(\mathbb{R}^{d}\times\mathbb{R}^{d})$. Moreover, each plant at position
$x\in\eta$ has an individual lifetime independent of the other plants. Such
lifetime is described by the intensity $0\leq m\in
C(\mathbb{R}_{+}\times\mathbb{R}^{d})$. The competition between different
plants is assumed to be of additive type and hence of the form
$\sum_{y\in\eta\backslash x}a^{-}(x,y)$, where $0\leq a^{-}\in
C(\mathbb{R}^{d}\times\mathbb{R}^{d})$ is the competition kernel. Above
description is summarized in the form of the following Markov (pre-)generator

$\displaystyle(L(t)F)(\eta)$
$\displaystyle=\sum_{x\in\eta}\Big{(}m(t,x)+\sum_{y\in\eta\backslash
x}a^{-}(x,y)\Big{)}(F(\eta\backslash x)-F(\eta))$ $\displaystyle\ \ \
+\sum_{x\in\eta}\lambda(t,x)\int_{\mathbb{R}^{d}}a^{+}(x,y)(F(\eta\cup
y)-F(\eta))\,dy.$

Such model has been analyzed in the time-homogeneous case in [22]. In
applications one is often interested in $a^{+}$ being of the form

$a^{+}(x,y)\sim\frac{1}{|x-y|^{\alpha}},\quad|x-y|\to\infty$

or

$a^{+}(x,y)\sim e^{-\nu|x-y|^{\alpha}},\quad|x-y|\to\infty.$

###### Theorem 3.5

Suppose that $m,\lambda,a^{-}$ are continuous and bounded, $a^{+}$ is
continuous with $1=\int_{\mathbb{R}^{d}}a^{+}(x,y)dy$ and for any compact
$\Lambda\subset\mathbb{R}^{d}$ there exists $a^{*}\geq 0$ with $a^{*}\in
L^{1}(\mathbb{R}^{d})$ such that

$a^{+}(x,y)\leq a^{*}(y),\quad x\in\Lambda,\quad y\in\mathbb{R}^{d}$

holds. Then conditions (A)–(D) hold for $V(\eta)=|\eta|+|\eta|^{2}$.

###### Proof.

Let $B\subset\Gamma_{0}$ be a compact and take $N_{B}\in\mathbb{N}$,
$\Lambda_{B}\subset\mathbb{R}^{d}$ and $\delta_{B}>0$ like in (3.1). Let
$A\subset\Gamma_{0}$ be another compact defined by (3.1) with
$N_{A}:=N_{B}+1$, $\Lambda_{B}\subset\Lambda_{A}$ and
$\delta_{A}\in(0,\delta_{B})$, then $B\subset A$ holds. We obtain for
$x\in\Lambda_{B}$ and $\eta\in B$

$\int_{\mathbb{R}^{d}}1\kern-2.84526pt{\text{\rm I}}_{A^{c}}(\eta\cup
y)a^{+}(x,y)\,dy\leq\int_{\Lambda_{A}^{c}}a^{+}(x,y)\,dy+\int_{B_{\delta_{A}}(\eta)}a^{+}(x,y)\,dy,$

where $B_{\delta_{A}}(\eta):=\\{w\in\mathbb{R}^{d}\ |\ \exists y\in\eta:\
|w-y|<\delta_{A}\\}$. Since $\eta\in B$ and $\delta_{B}>\delta_{A}$ we obtain
$B_{\delta_{A}}(\eta)=\bigsqcup_{y\in\eta}B_{\delta_{A}}(y)\subset\Lambda_{B}^{\delta_{B}}$
where $\Lambda_{B}^{\delta_{B}}:=\\{w\in\mathbb{R}^{d}\ |\
d(w,\Lambda_{B})\leq\delta_{B}\\}$ with $d(w,\Lambda_{B}):=\inf\ \\{|w-u|\ |\
u\in\Lambda_{B}\\}$. Let $c>0$ be such that $a^{+}(x,y)\leq c$ for all
$x\in\Lambda_{B}$ and $y\in\Lambda_{B}^{\delta_{B}}$, then

$\int_{\mathbb{R}^{d}}1\kern-2.84526pt{\text{\rm I}}_{A^{c}}(\eta\cup
y)a^{+}(x,y)\,dy\leq\int_{\Lambda_{A}^{c}}a^{*}(y)\,dy+N_{B}c|B_{\delta_{A}}|$

is satisfied, where $|B_{\delta_{A}}|$ is the Lebesgue volume of
$B_{\delta_{A}}=\\{w\in\mathbb{R}^{d}\ |\ |w|\leq\delta_{A}\\}$. Condition (A)
now follows from above estimate, Lemma 3.4 and $\lambda\in
C_{b}(\mathbb{R}_{+}\times\mathbb{R}^{d})$. Condition (B) follows from

$\displaystyle(L(t)V)(\eta)$
$\displaystyle=\sum_{x\in\eta}\lambda(t,x)+2\sum_{x\in\eta}\sum_{y\in\eta\backslash
x}a^{-}(x,y)$ $\displaystyle\ \ \
+2|\eta|\sum_{x\in\eta}(\lambda(t,x)-m(t,x))-2|\eta|\sum_{x\in\eta}\sum_{y\in\eta\backslash
x}a^{-}(x,y)$
$\displaystyle\leq\max\\{\|\lambda\|_{\infty},2\|a^{-}\|_{\infty}+2\|\lambda\|_{\infty}+2\|m\|_{\infty}\\}V(\eta).$

Condition (D) is fulfilled due to

$\displaystyle q(t,\eta)$
$\displaystyle=\sum_{x\in\eta}m(t,x)+\sum_{x\in\eta}\lambda(t,x)+\sum_{x\in\eta}\sum_{y\in\eta\backslash
x}a^{-}(x,y)$
$\displaystyle\leq\max\\{\|m\|_{\infty}+\|\lambda\|_{\infty},\|a^{-}\|_{\infty}\\}V(\eta).$

For condition (C) it is enough to show that for any continuous function $F$
such that $|F(\eta)|\leq\|F\|_{V}(1+|\eta|+|\eta|^{2})$ also
$(t,\eta)\longmapsto\sum_{x\in\eta}\lambda(t,x)\int_{\mathbb{R}^{d}}a^{+}(x,y)F(\eta\cup
y)dy$ is continuous. Since $\lambda(t,x)$ is continuous it is enough to show
that the integral is continuous. But this follows from dominated convergence
and the condition imposed on $a^{+}$. ∎

Above statement implies the following a priori estimate for the evolution of
states. Let $\mu$ be a probability measure with
$\int_{\Gamma_{0}}(1+|\eta|+|\eta|^{2})\mu(d\eta)<\infty$. Then

$\int_{\Gamma_{0}}(1+|\eta|+|\eta|^{2})U^{*}(t,s)\mu(d\eta)\leq
e^{(t-s)c}\int_{\Gamma_{0}}(1+|\eta|+|\eta|^{2})\mu(d\eta)$

holds,
$c:=\max\\{\|\lambda\|_{\infty},2\|a^{-}\|_{\infty}+2\|\lambda\|_{\infty}+2\|m\|_{\infty}\\}$.

#### Dieckmann-Law model

In contrast to the BDLP-model we discuss here one possible extension for which
the branching mechanism includes interactions of the plants. For simplicity we
suppose that all intensities are translation invariant. A plant at location
$x\in\eta$ shall now have the modified branching intensity given by

$\lambda(t)+\sum_{y\in\eta\backslash x}b^{+}(x-y),\quad t\geq 0,$

where $0\leq b^{+}\in C_{b}(\mathbb{R}^{d})$. The location of the offspring is
described by the probability density $a^{+}(x-y)$. The modified Markov
(pre-)generator is therefore given by

$\displaystyle(L(t)F)(\eta)$
$\displaystyle=\sum_{x\in\eta}\Big{(}m(t)+\sum_{y\in\eta\backslash
x}a^{-}(x-y)\Big{)}(F(\eta\backslash x)-F(\eta))$ $\displaystyle\ \
+\sum_{x\in\eta}\lambda(t)\int_{\mathbb{R}^{d}}(F(\eta\cup
w)-F(\eta))a^{+}(x-y)\,dw$ $\displaystyle\ \
+\sum_{x\in\eta}\sum_{y\in\eta\backslash
x}b^{+}(x-y)\int_{\mathbb{R}^{d}}(F(\eta\cup w)-F(\eta))a^{+}(x-w)\,dw,$

where $m,\lambda\in C(\mathbb{R}_{+})$ and $a^{-}\in C_{b}(\mathbb{R}^{d})$.
We assume that $a^{-}-b^{+}$ is a stable potential. By definition this means
that there exists a constant $b\geq 0$ such that

$\sum_{x\in\eta}\sum_{y\in\eta\backslash
x}(a^{-}(x-y)-b^{+}(x-y))\geq-b|\eta|,\quad\eta\in\Gamma_{0}.$

Let $E^{+}(\eta)=\sum_{x\in\eta}\sum_{y\in\eta\backslash x}b^{+}(x-y)$ and
$E^{-}(\eta)=\sum_{x\in\eta}\sum_{y\in\eta\backslash x}a^{-}(x-y)$, that it
above condition is equivalent to

$E^{+}(\eta)\leq b|\eta|+E^{-}(\eta),\quad\eta\in\Gamma_{0}.$

###### Theorem 3.6

Suppose that for any compact $\Lambda\subset\mathbb{R}^{d}$ there exists
$a^{*}\in L^{1}(\mathbb{R}^{d})$ which satisfies

$a^{+}(x-w)\leq a^{*}(w),\quad x\in\Lambda,\quad w\in\mathbb{R}^{d}.$

Then conditions (A)–(D) are satisfied for $V(\eta):=|\eta|+|\eta|^{2}$.
Moreover, for any $n\geq 1$ and state $\mu$ with
$\int_{\Gamma_{0}}|\eta|^{n}\mu(d\eta)<\infty$, the evolution of states
satisfies $\int_{\Gamma_{0}}|\eta|^{n}U^{*}(t,s)\mu(d\eta)<\infty$. If in
addition $m(t),\lambda(t)>0$ for all $t\geq 0$, then condition (E) holds and
$U^{*}(t,s)$ leaves the space of densities invariant.

###### Proof.

Condition (A) will be shown for a more general case later on. Concerning
condition (B) we have

$(L(t)|\cdot|)(\eta)\leq(b+\lambda(t)-m(t))|\eta|$

and by $(|\eta|+1)^{2}-|\eta|^{2}=2|\eta|-1$,
$(|\eta|-1)^{2}-|\eta|^{2}=-2|\eta|$ also

$(L(t)|\cdot|^{2})(\eta)\leq(2\lambda(t)+\|b^{+}\|_{\infty}+2b-2m(t))|\eta|^{2}+(\lambda(t)-\|b^{+}\|_{\infty})|\eta|.$

Altogether this yields

$L(t)V(\eta)\leq|\eta|(b+2\lambda(t)-m(t)-\|b^{+}\|_{\infty})+|\eta|^{2}(2\lambda(t)+\|b^{+}\|_{\infty}+2b-2m(t)),$

i.e. (3.5) is satisfied. Since

$\displaystyle q(t,\eta)$
$\displaystyle=(m(t)+\lambda(t))|\eta|+E^{+}(\eta)+E^{-}(\eta)$
$\displaystyle\leq(\|a^{-}\|_{\infty}+\|b^{+}\|_{\infty})|\eta|^{2}+|\eta|\sup_{t\in[0,T]}(m(t)+\lambda(t))$

also (D) holds. For property (C) it is enough to show that
$x\longmapsto\int_{\mathbb{R}^{d}}F(\eta\cup y)a^{+}(x-y)\,dy$ is continuous
for any continuous function $F$ with $|F(\eta)|\leq c(1+|\eta|+|\eta|^{2})$,
$\eta\in\Gamma_{0}$ and some constant $c>0$. But this follows immediately by
dominated convergence and the assumptions on $a^{+}$. Property (E) is a direct
consequence of the continuity of $m$ and $\lambda$. For the remaining
assertion it suffices to show that for any $n\geq 1$ there exist a continuous
function $c_{n}:\mathbb{R}_{+}\longmapsto\mathbb{R}_{+}$ such that

$(L(t)|\cdot|^{n})(\eta)\leq c_{n}(t)|\eta|^{n},\quad t\geq 0.$

We have $(|\eta|+1)^{n}-|\eta|^{n}=\sum_{l=0}^{n-1}\binom{n}{l}|\eta|^{l}$,
$(|\eta|-1)^{n}-|\eta|^{n}=\sum_{l=0}^{n-1}\binom{n}{l}(-1)^{n-l}|\eta|^{k}\leq
0$ and since $(L(t)|\cdot|^{n})(\emptyset)=0$ we can assume w.l.g. that
$|\eta|>0$. Hence

$\displaystyle(L(t)|\cdot|^{n})(\eta)$
$\displaystyle\leq\lambda(t)\sum_{l=0}^{n-1}\binom{n}{l}|\eta|^{l+1}+\sum_{l=0}^{n-1}\binom{n}{l}|\eta|^{l}(E^{+}(\eta)+(-1)^{n-l}E^{-}(\eta))$
$\displaystyle=\lambda(t)\sum_{l=1}^{n}\binom{n}{l-1}|\eta|^{l}+\sum_{l=1}^{n}\binom{n}{l-1}|\eta|^{l-1}(E^{+}(\eta)-(-1)^{n-l}E^{-}(\eta))$
$\displaystyle\leq|\eta|^{n}\lambda(t)\sum_{l=1}^{n}\binom{n}{l-1}+\sum_{l=1}^{n-1}\binom{n}{l-1}|\eta|^{n}(\|b^{+}\|_{\infty}+\|a^{-}\|_{\infty})$
$\displaystyle\ \ \ \ +\binom{n}{n-1}(E^{+}(\eta)-E^{-}(\eta))|\eta|^{n-1}$
$\displaystyle\leq
2^{n}(\lambda(t)+\|b^{+}\|_{\infty}+\|a^{-}\|_{\infty})n\cdot|\eta|^{n}+bn|\eta|^{n}$

implies the assertion. ∎

###### Remark 3.7

The proof shows that $U^{*}(t,s)$ maps the space of probability measures with
the constraint $\int_{\Gamma_{0}}|\eta|^{n}\mu(d\eta)<\infty$ continuously on
itself. Moreover, using Corollary 2.10 one can show that

$\int_{\Gamma_{0}}|\eta|U^{*}(t,s)\mu(d\eta)\leq
e^{b(t-s)}e^{\int_{s}^{t}(\lambda(r)-m(r))dr}\int_{\Gamma_{0}}|\eta|\mu(d\eta)$

and

$\displaystyle\int_{\Gamma_{0}}(|\eta|+|\eta|^{2})U^{*}(t,s)\mu(d\eta)$
$\displaystyle\leq
e^{(b-\|b^{+}\|_{\infty})(t-s)}e^{\int_{s}^{t}(2\lambda(r)-m(r))dr}\int_{\Gamma_{0}}|\eta|\mu(d\eta)$
$\displaystyle\ \ \
+e^{(\|b^{+}\|_{\infty}+2b)(t-s)}e^{2\int_{s}^{t}(\lambda(r)-m(r))dr}\int_{\Gamma_{0}}|\eta|^{2}\mu(d\eta)$

are valid.

#### Generalized Dieckmann-Law model

Assume that any plant at position $x\in\eta$ may create any number
$k\in\mathbb{N}$ of new plants. Their locations are, for any fixed $t\geq 0$,
distributed according to the probability measure

$a^{+}(t,x-y_{1})\cdots a^{+}(t,x-y_{k})\,dy_{1}\cdots dy_{k}.$

Therefore the (pre-)generator is assumed to be given by

$\displaystyle(L(t)F)(\eta)$
$\displaystyle=\sum_{x\in\eta}\Big{(}m(t,x)+\sum_{y\in\eta\backslash
x}a^{-}(t,x-y)\Big{)}(F(\eta\backslash x)-F(\eta))$ $\displaystyle\ \
+\frac{1}{e}\sum_{x\in\eta}\lambda(t,x)\int_{\Gamma_{0}\backslash\\{\emptyset\\}}(F(\eta\cup\zeta)-F(\eta))e_{\lambda}(a^{+}(t,x-\cdot);\zeta)\,d\lambda(\zeta)$
$\displaystyle\ \ +\frac{1}{e}\sum_{x\in\eta}\sum_{y\in\eta\backslash
x}b^{+}(t,x-y)\int_{\Gamma_{0}\backslash\\{\emptyset\\}}(F(\eta\cup\zeta)-F(\eta))e_{\lambda}(a^{+}(t,x-\cdot);\zeta)\,d\lambda(\zeta).$

The factor $\frac{1}{e}$ is a normalization factor since we have

$\int_{\Gamma_{0}}e_{\lambda}(a^{+}(t,x-\cdot);\zeta)\,d\lambda(\zeta)=e.$

###### Theorem 3.8

Let $0\leq m,\lambda,a^{-},b^{+}\in C_{b}(\mathbb{R}_{+}\times\mathbb{R}^{d})$
with $a^{+}(t,\cdot)$ being a probability density for all $t\geq 0$. Suppose
that for any compact $\Lambda\subset\mathbb{R}^{d}$ and $T>0$ there exists
$a^{*}\in L^{1}(\mathbb{R}^{d})$ which satisfies

$\displaystyle a^{+}(t,x-y)\leq a^{*}(y),\quad x\in\Lambda,\quad
t\in[0,T],\quad y\in\mathbb{R}^{d}.$ (3.11)

Moreover, assume that $b^{+}(t,x)\leq a^{-}(t,x)$ holds for all
$x\in\mathbb{R}^{d}$ and $t\geq 0$. Then conditions (A)–(D) are satisfied for
$V(\eta)=|\eta|+|\eta|^{2}$.

###### Proof.

By
$\int_{\Gamma_{0}\backslash\emptyset}|\zeta|e_{\lambda}(a^{+}(t,x-\cdot);\zeta)d\lambda(\zeta)=e$
we obtain

$\displaystyle L(t)V(\eta)$
$\displaystyle=\sum_{x\in\eta}\left((2-e^{-1})\lambda(t,x)-2m(t,x)\right)$
$\displaystyle\ \ \ +\sum_{x\in\eta}\sum_{y\in\eta\backslash
x}\left((2-e^{-1})b^{+}(t,x-y)-2a^{-}(t,x-y)\right)$ $\displaystyle\ \ \
+2|\eta|\sum_{x\in\eta}(\lambda(t,x)-m(t,x))+2|\eta|\sum_{x\in\eta}\sum_{y\in\eta\backslash
x}(b^{+}(t,x-y)-a^{-}(t,x-y))$ $\displaystyle\leq
2(\|\lambda\|_{\infty}+\|m\|_{\infty})V(\eta),$

which implies condition (B). Condition (D) follows from

$\displaystyle q(t,\eta)$
$\displaystyle=\sum_{x\in\eta}m(t,x)+\frac{e-1}{e}\sum_{x\in\eta}\lambda(t,x)$
$\displaystyle\ \ \ +\sum_{x\in\eta}\sum_{y\in\eta\backslash
x}a^{-}(t,x-y)+\frac{e-1}{e}\sum_{x\in\eta}\sum_{y\in\eta\backslash
x}b^{+}(t,x-y)$
$\displaystyle\leq(\|m\|_{\infty}+\|\lambda\|_{\infty})|\eta|+(\|a^{-}\|_{\infty}+\|b^{+}\|_{\infty})|\eta|^{2}.$

In order to see (C), observe that the assertion is clear for the contribution
from the terms of the operator $L(t)$ describing the death of plants. Because
$\lambda$ and $b^{+}$ are continuous it suffices to show for any $F\in
C_{V}(\Gamma_{0}),\eta_{n}\to\eta,t_{n}\to t$ and $x_{n}\in\eta_{n},x\in\eta$
with $x_{n}\to x$

$\displaystyle\int_{\Gamma_{0}\backslash\emptyset}F(\eta_{n}\cup\zeta)e_{\lambda}(a^{+}(t_{n},\cdot-
x_{n});\zeta)\,d\lambda(\zeta)\to\int_{\Gamma_{0}\backslash\emptyset}F(\eta\cup\zeta)e_{\lambda}(a^{+}(t,\cdot-x);\zeta)\,d\lambda(\zeta),$
$\displaystyle\quad n\to\infty.$

Since the integrand is continuous it converges for each
$\zeta\in\Gamma_{0}\backslash\emptyset$ and by (3.11) with compacts
$K=\\{t_{n}\ |\ n\geq 1\\}\cup\\{t\\}$, $B=\\{x_{n}\ |\ n\geq 1\\}\cup\\{x\\}$
we obtain by dominated convergence the assertion. Therefore it remains to show
property (A). Take $T>0$ and fix a compact $B\subset\Gamma_{0}$. Hence there
exists $\Lambda_{B}\subset\mathbb{R}^{d}$ compact, $N_{B}\in\mathbb{N}$ and
$\delta_{B}>0$ such that for any $\eta\in B$ (3.10) holds. Condition (A) was
shown for the death of plants, so let us focus on the terms contributing to
the birth. Due to the continuity of $\lambda,b^{+}$ the sum

$\sum_{x\in\eta}\Big{(}\lambda(t,x)+\sum_{y\in\eta\backslash
x}b^{+}(t,x-y)\Big{)}$

is uniformly bounded on $[0,T]\times B$. Hence it is enough to estimate the
integral. Take a compact set $A\subset\Gamma_{0}$ with the characteristics
$N_{A}>N_{B}$, $\delta_{A}<\delta_{B}$, $\Lambda_{B}\subset\Lambda_{A}$, i.e.
(3.1) and set

$B_{\delta_{A}}(\eta)=\Big{\\{}\xi\in\Gamma_{0}\ |\
\xi\subset\bigcup_{x\in\eta}B_{\delta_{A}}(x)\Big{\\}},$

where $B_{\delta_{A}}(x)=\\{y\in\mathbb{R}^{d}\ |\ |x-y|<\delta_{A}\\}$. Then
we obtain for $\eta\in B$ and $x\in\eta$, so $x\in\Lambda_{B}$

$\displaystyle\int_{\Gamma_{0}\backslash\emptyset}1\kern-2.84526pt{\text{\rm
I}}_{A^{c}}(\eta\cup\zeta)e_{\lambda}(a^{+}(t,x-\cdot);\zeta)\,d\lambda(\zeta)$
$\displaystyle\leq\Big{(}\int_{|\zeta|>N_{A}-N_{B}}+\int_{B_{\delta_{A}}(\eta)\backslash\emptyset}+\int_{\Gamma_{\Lambda_{A}^{c}}\backslash\emptyset}+\int_{C(\delta_{A})}\Big{)}e_{\lambda}(a^{+}(t,x-\cdot);\zeta)\,d\lambda(\zeta)$
$\displaystyle=I_{1}+I_{2}+I_{3}+I_{4},$

where $C(\delta_{A})=\\{\zeta\in\Gamma_{0}\ |\ \exists w\neq z,\ w,z\in\zeta:\
|w-z|<\delta_{A}\\}$. For the first integral we obtain uniformly in
$t\in[0,T],\eta\in B$ and $x\in\eta$

$\displaystyle
I_{1}\leq\int_{|\zeta|>N_{A}-N_{B}}e_{\lambda}(a^{*};\zeta)\,d\lambda(\zeta)=\sum_{n=N_{A}-N_{B}+1}^{\infty}\frac{\Big{(}\int_{\mathbb{R}^{d}}a^{*}(y)dy\Big{)}^{n}}{n!}$

and similarly for the third

$\displaystyle
I_{3}\leq\int_{\Gamma_{\Lambda_{A}^{c}}\backslash\emptyset}e_{\lambda}(a^{*};\zeta)\,d\lambda(\zeta)=\sum_{n=1}^{\infty}\frac{1}{n!}\Big{(}\int_{\Lambda_{A}^{c}}a^{*}(y)dy\Big{)}^{n}=\exp\Big{(}\int_{\Lambda_{A}^{c}}a^{*}(y)dy\Big{)}-1.$

This two terms tend uniformly in $\eta\in B$ and $t\in[0,T]$ to zero as
$N_{A}\to\infty$ and $\Lambda_{A}\to\mathbb{R}^{d}$. Denote by $c>0$ a
constant for which

$a^{+}(t,z-w)\leq c,\quad t\in[0,T],\quad z\in\Lambda_{B},\quad
w\in\Lambda_{B}^{\delta_{B}}$

with $\Lambda_{B}^{\delta_{B}}=\\{w\in\mathbb{R}^{d}\ |\
d(w,\Lambda_{B})\leq\delta_{B}\\}$ holds, where
$d(w,\Lambda_{B}):=\inf\\{d(w,u)\ |\ u\in\Lambda_{B}\\}$. For $I_{2}$ we
obtain with $|B_{\delta_{A}}|$ the Lebesgue volume of a ball with radius
$\delta_{A}$ in $\mathbb{R}^{d}$, since for any $w,z\in\eta$ with $w\neq z$:
$B_{\delta_{A}}(w)\cap B_{\delta_{A}}(z)=\emptyset$

$\displaystyle
I_{2}=\int_{B_{\delta_{A}}(\eta)\backslash\emptyset}e_{\lambda}(a^{+}(t,x-\cdot);\zeta)\,d\lambda(\zeta)\leq\Big{(}\sum_{n=1}^{\infty}\frac{c^{n}|B_{\delta_{A}}|^{n}}{n!}\Big{)}^{|\eta|}=(e^{c|B_{\delta_{A}}|}-1)^{|\eta|}.$

Finally due to $C(\delta_{A})\to\emptyset$ as $\delta_{A}\to 0$ we have shown
that for all $T>0$, all compacts $B\subset E$ and $\varepsilon>0$ there is a
compact $A\subset E$ such that

$Q(t,\eta,A^{c})<\varepsilon,\quad t\in[0,T],\quad\eta\in B,$

which is stronger then (A). ∎

###### Remark 3.9

Condition (3.11) is for instance satisfied if there exist strictly positive
continuous functions $\lambda,C>0$ and $R>0$, $\alpha>\frac{d}{2}$ such that

$a^{+}(t,x)\leq\frac{C(t)}{(\lambda(t)+|x|^{2})^{\alpha}},\quad|x|\geq R$

holds.

###### Remark 3.10

In the time-homogeneous case weaker conditions are sufficient to prove the
Feller property.

## Appendix

Set $\Delta:=\\{(s,t)\in\mathbb{R}_{+}\times\mathbb{R}_{+}\ |\ s\leq t\\}$,
the next two lemmas should be well-known and are included here only for
convenience.

###### Lemma 3.11

Let $f_{j}:\Delta\times\mathbb{R}_{+}\times E\longrightarrow\mathbb{R}$ be a
family of measurable functions indexed by $j\in M$, where $M$ is an arbitrary
non-empty index set, such that

  1. 1\. $f_{j}$ is bounded on compacts uniformly in $j\in M$.

  2. 2\. The map $(s,t,x)\longmapsto f_{j}(s,t,r,x)$ is continuous uniformly in $j\in M$ for fixed $r\in[s,t]$.

Then $(s,t,x)\longmapsto\int_{s}^{t}f_{j}(s,t,r,x)\,dr$ is continuous
uniformly in $j\in M$.

###### Proof.

Let $(s,t),(s_{n},t_{n})\in\Delta$ and $x,x_{n}\in E$ be such that $s_{n}\to
s$, $t_{n}\to t$ and $x_{n}\to x$ as $n\to\infty$. We find $T>0$ and a compact
$B\subset E$ such that $s,s_{n},t,t_{n}\in[0,T]$ and $x,x_{n}\in B$ for
$n\in\mathbb{N}$. Let $f^{*}:=\sup_{j\in
M}\sup_{(t_{1},t_{2},t_{3},x)\in\Delta\cap[0,T]^{2}\times[0,T]\times B}\
f_{j}(t_{1},t_{2},t_{3},x)<\infty$, then for any $n\in\mathbb{N}$ and $j\in M$

$\displaystyle\Big{|}\int_{s}^{t}f_{j}(s,t,r,x)\,dr-\int_{s_{n}}^{t_{n}}f_{j}(s_{n},t_{n},r,x_{n})\,dr\Big{|}$
$\displaystyle\leq|s-s_{n}|f^{*}+|t-t_{n}|f^{*}+\int_{0}^{T}|f_{j}(s,t,r,x)-f_{j}(s_{n},t_{n},r,x_{n})|\,dr.$

For each $r\in[0,T]$ the integrand on the right-hand-side tends to zero as
$n\to\infty$, and since $|f_{j}(s,t,r,x)-f_{j}(s_{n},t_{n},r,x_{n})|\leq
2f^{*}$ dominated convergence yields the assertion. ∎

The next lemma will show continuity in the case where instead of $dr$ there is
an arbitrary kernel $H(t,x,dy)$. In such a case we will need that $E$ is
locally compact.

###### Lemma 3.12

Let $E$ be a locally compact Polish space,

$f:\\{(s,r,t)\in\mathbb{R}_{+}^{3}\ |\ s\leq r\leq t\\}\times E\times
E\longrightarrow\mathbb{R}$

be continuous and bounded, and let $H:I\times
E\times\mathcal{B}(E)\longrightarrow\mathbb{R}_{+}$ be a weakly continuous
kernel, i.e. for all $F\in C_{b}(E)$, $\mathbb{R}_{+}\times
E\ni(r,x)\longmapsto\int_{E}F(y)H(r,x,dy)$ is continuous. Then

$(s,r,t,x)\longmapsto\int_{E}f(s,r,t,x,y)H(r,x,dy)$

is continuous.

###### Proof.

Let $s_{n}\leq r_{n}\leq t_{n}$ be such that $s_{n}\to s,r_{n}\to r,t_{n}\to
t$ and $x_{n}\to x$ as $n\to\infty$. Fix $\varepsilon>0$ and take $A\subset E$
compact with $H(r,x,A^{c})<\varepsilon$. Since $E$ is a locally compact space
we can find another compact $A_{1}\subset E$ with
$A\subset\overset{\circ}{A_{1}}\subset A_{1}$. Portmanteau implies then
$\underset{n\to\infty}{\limsup}\
H(r_{n},x_{n},(\overset{\circ}{A_{1}})^{c})\leq
H(r,x,(\overset{\circ}{A_{1}})^{c})\leq H(r,x,A^{c})<\varepsilon$. The
function $f$ restricted to the compact $\\{(s_{n},r_{n},t_{n})\ |\
n\in\mathbb{N}\\}\cup\\{(s,r,t)\\}\times\\{x_{n}\ |\
n\in\mathbb{N}\\}\cup\\{x\\}\times A_{1}$ is uniformly continuous and hence we
obtain for sufficiently large $n$

$\displaystyle\Big{|}\int_{E}f(s_{n},r_{n},t_{n},x_{n},y)H(r_{n},x_{n},dy)-\int_{E}f(s,r,t,x,y)H(r,x,dy)\Big{|}$
$\displaystyle\leq\int_{E}|f(s_{n},r_{n},t_{n},x_{n},y)-f(s,r,t,x,y)|H(r_{n},x_{n},dy)$
$\displaystyle\
+\Big{|}\int_{E}f(s,r,t,x,y)H(r_{n},x_{n},dy)-\int_{E}f(s,r,t,x,y)H(r,x,dy)\Big{|}$
$\displaystyle\leq
H(r_{n},x_{n},A_{1})\varepsilon+2\|f\|H(r_{n},x_{n},(\overset{\circ}{A_{1}})^{c})+\varepsilon$
$\displaystyle\leq H(r_{n},x_{n})\varepsilon+2\|f\|\varepsilon+\varepsilon.$

Due to the weak continuity of $H$ the function $H(r,x):=H(r,x,E)$ is
continuous and hence $H(r_{n},x_{n})$ is uniformly bounded in
$n\in\mathbb{N}$, which shows the assertion. ∎

Acknowledgments. The author would like to thank Viktor Bezborodov for many
fruitful discussions and critical remarks.

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