Feller and ergodic properties of jump–move processes with applications to interacting particle systems

2.1. Iterative construction

Let E be a Polish space equipped with the Borel $\sigma$ -algebra $\mathcal{E}$ and a distance d. Let $(\Omega,\mathcal{F})$ be a measurable space and $(\mathbb{P}_x)_{x \in E}$ a family of probability measures on $(\Omega,\mathcal{F}).$ In order to define a jump–move process $(X_t)_{t\geq 0}$ on E, we need three ingredients:

1. (i) An intensity function $\alpha \;:\; E \rightarrow \mathbb{R}_+$ that governs the inter-jump waiting times.

2. (ii) A transition kernel K for the jumps, defined on $E \times \mathcal{E}.$

3. (iii) A continuous homogeneous Markov process $((Y_t)_{t \geq 0}, (\mathbb{P}_x)_{x \in E})$ on E, the distribution of which will drive the inter-jump motion of $(X_t)_{t\geq 0}$ .

Throughout this paper, we will work under the assumption that $\alpha \;:\; E \rightarrow \mathbb{R}_+ $ is continuous and bounded by $\alpha^*>0$ , i.e., for all $x\in E$ ,

(2.1) \begin{equation}0 \leq \alpha(x) \leq \alpha^*.\end{equation}

We denote by $(Q_t^Y)_{t \geq 0}$ the transition kernel of $(Y_t)_{t\geq 0}$ , given by

$$Q_t^Y(x,A)=\mathbb{P}_x(Y_t \in A), \quad x \in E,\ A \in \mathcal{E}.$$

The following iterative construction provides clear intuition for the dynamics of the process $(X_t)_{t\geq 0}$ . It follows closely the presentation in the supplementary material of [Reference Lavancier and Le Guével14], where an algorithm of simulation on a finite time interval is also derived. However, since $\alpha$ is not lower-bounded from zero, unlike in the previous reference, it is possible that eventually there are no more jumps, a situation taken into account by Equation (2.2) below. The algorithm of simulation adapts straightforwardly to this case.

Let $\big(Y_t^{(j)}\big)_{t \geq 0}$ , $j\geq 0$ , be a sequence of processes on E identically distributed as $(Y_t)_{t \geq 0}$ . Set $T_0=0$ and let $x_0\in E$ . Then $(X_t)_{t\geq 0}$ can be constructed as follows. For $j\geq 0$ , iteratively do the following:

1. (i) Given $ X_{T_j}=x_j$ , generate $\big(Y_t^{(j)}\big)_{t \geq 0}$ conditional on $ Y_0^{(j)}=x_j$ according to the kernel $(Q_t^{Y}(x_j,.))_{t \geq 0}$ .

2. (ii) Given $ X_{T_j}=x_j$ and $\big(Y_t^{(j)}\big)_{t \geq 0}$ , generate $ \tau_{j+1}$ according to the following distribution on $\mathbb{R}_+\cup\{+\infty\}$ :

   (2.2) \begin{equation} \begin{cases} \mathbb{P}(\tau_{j+1}\leq t)=1-\exp\!\left ( - \int_0^t \alpha \left ( Y_u^{(j)} \right ) \, \text{d} u \right ) \text{ for all } t\in\mathbb{R}_+, \\ \mathbb{P}(\tau_{j+1}=+\infty)=\exp\!\left ( - \int_0^{\infty} \alpha \left ( Y_u^{(j)} \right ) \, \text{d} u \right ).\end{cases} \end{equation}

3. (iii) Given $ X_{T_j}=x_j$ , $\big(Y_t^{(j)}\big)_{t \geq 0}$ and $ \tau_{j+1}$ ,

   if $\tau_{j+1}=\infty$ then set $X_t=Y_{t-T_j}^{(j)}$ for all $t\geq T_j$ (and stop the iterative construction),

   else generate $x_{j+1}$ according to the transition kernel $ K (Y_{ \tau_{j+1}}^{(j)}, .)$ .

1. (iv) Set $ T_{j+1}=T_j+\tau_{j+1}$ , $ X_t=Y_{t-T_j}^{(j)}$ for $t \in [T_j,T_{j+1})$ , and $ X_{T_{j+1}}= x_{j+1}$ .

We denote by $(\mathcal{F}_t^Y)_{t \geq 0}$ the natural filtration of $(Y_t)_{t\geq 0}$ , i.e. $\mathcal{F}_t^Y= \sigma ( Y_u, u \leq t)$ , and by $(\mathcal{F}_t)_{t >0}$ the natural filtration of $(X_t)_{t \geq 0}$ . We make these filtrations complete (see [Reference Bass3, Section 20.1]) and abusively use the same notation. The jump–move process $((X_t)_{t \geq 0}, (\mathbb{P}_x)_{x \in E})$ constructed above is a homogeneous Markov process with respect to $(\mathcal{F}_t)_{t>0}$ . The trajectories of $(X_t)_{t \geq 0}$ are continuous except at the jump times $(T_j)_{j \geq 1}$ , where they are right-continuous. The specific form (2.2) implies that the intensity of jumps is $\alpha(X_{t})$ . Denote by $N_t=\sum_{j\geq 0} {\mathbf 1}_{T_j\leq t}$ the number of jumps before $t\geq 0$ . Under the assumption (2.1), for any $n \geq 0$ and $t \geq 0$ we have

(2.3) \begin{equation} \mathbb{P}(N_t > n) \leq \mathbb{P}(N^*_t >n),\end{equation}

where $N^*_t$ follows a Poisson distribution with rate $\alpha^*t$ . This in particular implies that $(N_t)_{t \geq 0}$ is a non-explosive counting process. All of the aforementioned properties of $(X_t)_{t \geq 0}$ are either immediate or verified in [Reference Lavancier and Le Guével14].

Note that the above construction only implies the weak Markov property of $(X_t)_{t\geq 0}$ in general, at least because the process $(Y_t)_{t\geq 0}$ is only assumed to be a (weak) Markov process. A more abstract construction obtained by ‘piecing out’ strong Markov processes is introduced in [Reference Ikeda, Nagasawa and Watanabe11], leading to a strong Markov jump–move process. The strong Markov property can also be obtained in our case by strengthening the assumptions; see Section 3.1.

The transition kernel of $(X_t)_{t \geq 0}$ will be denoted, for any $t \geq 0$ , $x \in E$ , and $A \in \mathcal{E}$ , by

$$Q_t(x,A) = \mathbb{P}(X_t \in A | X_0=x)=\mathbb{P}_x(X_t \in A).$$

Also, for $f \in M_b(E)$ , where $M_b(E)$ is the set of real-valued bounded and measurable functions on E, we will write $Q_tf(x)=\mathbb{E}_x [ f(X_t)]=\int_E Q_t(x, \text{d} y) f(y)$ . Similarly we will write $Q_t^Yf(x)=\mathbb{E}_x^Y(f(Y_t))$ .

2.2. Special case of the birth–death–move process

A birth–death–move process is the particular case of a jump–move process in which E takes the form $E= \bigcup_{n=0}^{\infty} E_n$ , with $(E_n)_{n \geq 0}$ a sequence of disjoint Polish spaces, and in which the jumps are only births and deaths. We assume that each $E_n$ is equipped with the Borel $\sigma$ -algebra $\mathcal{E}_n$ , so that E is associated with the $ \displaystyle \sigma$ -field $\mathcal{E}=\sigma \left ( \bigcup_{n=0}^{\infty} \mathcal{E}_n \right ) $ . We further assume that $E_0$ consists of a single element, denoted by $\text{\O}$ . In this setting, the Markov process $(Y_t)_{t \geq 0}$ driving the motions of $(X_t)_{t \geq 0}$ is supposed to satisfy

\[ \mathbb{P}_x( (Y_t)_{t \geq 0} \subset E_n)= {\mathbf 1}_{x\in E_n}, \quad \forall x \in E, \, \forall n \geq 0. \]

We introduce a birth intensity function $\beta \;:\; E \rightarrow \mathbb{R}_+$ and a death intensity function $\delta \;:\; E \rightarrow \mathbb{R}_+$ , both assumed to be continuous on E and satisfying $\alpha = \beta + \delta.$ We prevent a death in $E_0$ by assuming that $\delta(\text{\O})=0.$ The probability transition kernel K for the jumps then reads, for any $x \in E$ and $A \in \mathcal{E}$ ,

(2.4) \begin{equation} K(x,A)=\frac{\beta(x)}{\alpha(x)}K_{\beta}(x,A)+ \frac{\delta(x)}{\alpha(x)}K_{\delta}(x,A), \end{equation}

where $K_{\beta} \;:\; E \times \mathcal{E} \rightarrow [0,1]$ is a probability transition kernel for a birth and $K_{\delta} \;:\; E \times \mathcal{E} \rightarrow [0,1]$ is a probability transition kernel for a death. They satisfy, for $x \in E$ and $n \geq 0$ ,

\[ K_{\beta}(x,E_{n+1})={\mathbf 1}_{x\in E_n} \quad \text{and} \quad K_{\delta}(x,E_{n})={\mathbf 1}_{x\in E_{n+1}}. \]

Notice that a simple birth–death process is the particular case in which $E=\mathbb{N}$ , $E_n= \{ n \}$ and the intensity functions $\beta$ and $\delta$ are sequences. More general examples of the inter-jump process Y, of the intensity functions $\beta$ and $\delta$ , and of the kernels $K_{\beta}$ and $K_{\delta}$ are presented in Sections 2.4 and 5; see also [Reference Lavancier and Le Guével14].

For later purposes, when $E= \bigcup_{n=0}^{\infty} E_n$ as in the present section, we define the function $n(.) \;:\; E \rightarrow \mathbb{N}$ by $n(x)=k$ when $x \in E_k$ , so that $x \in E_{n(x)}$ is always satisfied.

2.3. Kolmogorov backward equation

The goal of this section is to present the Kolmogorov backward equation for the transition kernel of the general jump–move process $(X_t)_{t \geq 0}$ of Section 2.1, providing a more probabilistic viewpoint on its dynamics, and to show that the solution exists and is unique. To obtain these results we use methods similar to those used in [Reference Feller9] for pure jump processes; see also [Reference Preston22]. The key assumption is the boundedness (2.1) of the intensity $\alpha$ , which prevents the explosion of the process. The proofs are postponed to Section S-1 of the supplementary material.

Theorem 1. For all $x\in E$ and all $A \in \mathcal{E}$ , the function $t \mapsto Q_t(x,A)$ , for $t>0$ , satisfies the following Kolmogorov backward equation:

(2.5) \begin{multline} Q_t(x,A) = \mathbb{E}_x^Y \left [ {\mathbf 1}_{Y_t \in A} \,\text{e}^{ - \int_0^t \alpha(Y_u) \, \text{d} u } \right ] \\+ \int_0^t \int_E Q_{t-s}(y,A) \mathbb{E}_x^Y \left [ K \left (Y_s,dy \right ) \, \alpha(Y_s) \text{e}^{- \int_0^s \alpha(Y_u) \, \text{d} u} \right ] ds.\end{multline}

In the case of the birth–death–move process of Section 2.2, the above equation reads, for $x \in E_n$ ,

(2.6) \begin{multline} Q_t(x,A) = \mathbb{E}_x^Y \left [ {\mathbf 1}_{Y_t \in A} \, \text{e}^{ - \int_0^t \alpha(Y_u) \, \text{d} u } \right ] \\ + \int_0^t \int_{E_{n+1}} Q_{t-s}(y,A) \, \mathbb{E}_x^Y \left [ \beta \left (Y_s \right ) K_\beta \left ( Y_s ,dy \right ) \, \text{e}^{ - \int_0^s \alpha(Y_u) \, \text{d} u } \right ] \text{d} s \\ + \int_0^t \int_{E_{n-1}} Q_{t-s}(y,A) \, \mathbb{E}_x^Y \left [ \delta \left (Y_s \right ) K_\delta \left ( Y_s ,dy \right ) \, \text{e}^{ - \int_0^s \alpha(Y_u) \, \text{d} u } \right ] \text{d} s.\end{multline}

To show the existence of a unique solution to (2.5), let $Q_{t,p}(x,A)\;:\!=\;\mathbb{P}_x(X_t \in A , T_p >t)$ be the transition probability from state x to A in time t with less than p jumps. Notice that we can define $Q_{t,\infty}= \lim \limits_{p \to \infty} Q_{t,p}$ , because $Q_{t,p} \leq Q_{t,p+1} \leq 1.$ In the following proposition we use a minimality argument as in [Reference Feller9] to prove that $Q_{t , \infty}$ is the unique solution to (2.5).

To conclude this section, we present an interpretation of $Q_{t,\infty}$ for the birth–death–move process of Section 2.2, which is much in the spirit of [Reference Preston22]. We write $Q_{t,(p)}(x,A)$ for the transition probability from x to A in time t without having entered $ \bigcup_{k=p+1}^{\infty} E_k$ ; that is,

\[ Q_{t,(p)}(x,A)=\mathbb{P}_x\left(X_t \in A,\, \forall s\in[0,t], \ n(X_s)\leq p \right). \]

We can also define $Q_{t,(\infty)}(x,A) = \lim \limits_{p \rightarrow \infty} \ Q_{t,(p)}(x,A) \leq 1$ by monotonicity.

2.4. Systems of interacting particles in $\mathbb{R}^d$

In this section, we focus on the dynamics of a system of interacting particles in $\mathbb{R}^d$ . We provide general examples of birth kernels, death kernels, and inter-jump motions in this setting, which in our opinion constitute realistic models for applications and are actually already used in some domains. Some of them, moreover, lead to an explicit Gibbs stationary measure of the dynamics, as we will show in Section 5. These running examples will serve in the rest of the paper to illustrate the theoretical results and make explicit our assumptions.

Let $W \subset \mathbb{R}^d$ be a closed set where the particles live, equipped with a $\sigma$ -field $\mathcal{B}$ . A collection of n particles in W is a point configuration for which the ordering does not matter. For this reason, for $n \geq 1,$ we will identify two elements $(x_1, \dots,x_n)$ and $(y_1,\dots,y_n)$ of $W^n$ if there exists a permutation $\sigma$ of $\{1,\dots,n\}$ such that $x_i=y_{\sigma(i)}$ for any $1 \leq i \leq n$ . Following [Reference Löcherbach15], [Reference Preston22], and others, we thus define $E_n$ as the space obtained by this identification. Specifically, denoting by $\pi_n \;:\; (x_1,\dots,x_n) \in W^n \mapsto \{ x_1,\dots,x_n \}$ the associated projection, for $n\geq 1$ the space $E_n$ corresponds to $E_n = \pi_n( W^n) $ equipped with the $\sigma$ -field $\mathcal{E}_n=\pi_n (\mathcal{B}^{\otimes n})$ , while $E_0=\{\text{\O}\}$ consists of just the empty configuration. The general state space of a system of particles is then $E = \cup_{n \geq 0} E_n$ equipped with the $\sigma$ -field $ \displaystyle \mathcal{E} = \sigma \left ( \cup_{n \geq 0} \mathcal{E}_n \right ).$ This formalism allows us to go back and forth quite straightforwardly between the space $E_n$ and the space $W^n$ , the latter being in particular more usual for defining the inter-jump motion of n particles, as detailed below. Note that an alternative formalism consists in viewing a configuration of particles as a finite point measure in W, in which case E becomes the set of finite point measures in W; see for instance [Reference Kallenberg12]. We choose in this paper to adopt the former point of view. We denote by $\|.\|$ the Euclidean norm on $\mathbb{R}^d$ . If $x=\{x_1,\dots,x_n\} \in E_n$ and $\xi \in W$ , then $x \cup \xi$ stands for $\{x_1,\dots,x_n,\xi \} \in E_{n+1}$ , and if $1 \leq i \leq n$ , we write $x \, \backslash \, x_i$ for $\{ x_1, \dots, x_{i-1},x_{i+1},\dots, x_n \} \in E_{n-1}.$

As long as we are concerned with continuous inter-jump motions, we need to equip E with a distance. Following [Reference Schuhmacher and Xia26], we consider the distance $d_1$ defined for $x=\{x_1,\dots,x_{n(x)} \}$ and $y=\{y_1,\dots,y_{n(y)} \}$ in E such that $n(x) \leq n(y)$ by

(2.7) \begin{equation} d_1(x,y) = \frac{1}{n(y)} \left ( \min_{\sigma \in \mathcal{S}_{n(y)}} \sum_{i=1}^{n(x)} (\|x_i-y_{\sigma(i)}\| \wedge 1) + (n(y)-n(x)) \right ), \end{equation}

with $d_1(x,\text{\O})=1$ and where $\mathcal{S}_n$ denotes the set of permutations of $\{1,\dots,n\}$ . The paper [Reference Schuhmacher and Xia26] and Section S-4 in the supplementary material detail some topological properties of $(E,d_1)$ . For the purposes of this section, let us quote in particular that $n(.) \;:\; (E,d_1) \rightarrow (\mathbb{N},|.|)$ is continuous and that $\pi_n$ is continuous. Note that distances other than $d_1$ could also be chosen, provided these two last properties (at least) are preserved. Incidentally, the Hausdorff distance, which is a common choice of distance between random sets, does not satisfy these properties (see the supplementary material) and is not appropriate in our setting.

We now show how we can easily construct a continuous Markov process $(Y_t)_{t \geq 0}$ on E from continuous Markov processes on $W^n$ for any $n \geq 1$ . We focus on the case where, for any $x \in E$ and $n\geq 0$ , $\mathbb{P}_x( (Y_t)_{t \geq 0} \subset E_n)= {\mathbf 1}_{x\in E_n}$ , as we required for birth–death–move processes in Section 2.2. It is then enough to define a process $Y^{|n}$ on each $E_n$ . To do so, consider a continuous Markov process $(Z_t^{|n})_{t \geq 0}$ on $W^n$ whose distribution is permutation-equivariant with respect to its initial value $Z^{ \, |n}_0$ . This means that for any permutation $\sigma \in \mathcal{S}_n$ , the law of $Z^{ \, |n}_t = ( Z^{ \, |n}_{t,1}, \dots , Z^{ \, |n}_{t,n} )$ given $Z^{ \, |n}_0=(z_{\sigma(1)},\dots,z_{\sigma(n)})$ is the same as the law of $(Z^{ \, |n}_{t,\sigma(1)}, \dots ,Z^{ \, |n}_{t,\sigma(n)} )$ given $Z^{ \, |n}_0=(z_1,\dots,z_n)$ . Let $x= \{x_1,\dots,x_n \} \in E_n$ and take the process $Z^{ \, |n}_{t}$ with initial state $Z^{ \, |n}_0=(x_1,\dots,x_n)$ . Note that, from the previous permutation-equivariance property, the choice of ordering for the coordinates of this initial state does not matter, as will become clear below. We finally define the process $Y^{|n}_t$ on $E_n$ starting from x as

(2.8) \begin{equation} Y^{|n}_t = \pi_n \left ( Z^{ \, |n}_t \right ) = \left \{ Z^{ \, |n}_{t,1}, \dots, Z^{ \, |n}_{t,n} \right \}.\end{equation}

Note that the continuity of $t\to Y^{|n}_t$ (with respect to $d_1$ ) follows from the continuity of $t\to Z^{ \, |n}_t$ and the continuity of $\pi_n$ . The continuity of $t\to Y_t$ is then implied by the continuity of $n(.)$ .

With this construction, the transition kernel of Y reads, for any $f \in M_b(E)$ ,

\begin{align*}Q_t^Yf(x) & = \sum_{n \geq 0}\mathbb{E} \left [ f(Y^{|n}_t) \, |Y^{|n}_0=x\right]{\mathbf 1}_{x \in E_n} \\& = \sum_{n \geq 0}\mathbb{E} \left ( f(\pi_n(Z^{ \, |n}_t)) \left | \right. Z^{ \, |n}_0=(x_{1},\dots,x_{n}) \right ){\mathbf 1}_{x \in E_n},\end{align*}

so that, denoting by $Q_t^{Z^{ \, |n}}$ the transition kernel of $Z^{ \, |n}$ in $W^n$ , we have

(2.9) \begin{equation} Q_t^Y f(x) = \sum_{n \geq 0} Q_t^{Z^{ \, |n}}(f \circ \pi_n)((x_1,\dots, x_n)){\mathbf 1}_{x \in E_n}.\end{equation}

Note that if we had chosen another ordering for the initial state, i.e. $Z^{ \, |n}_0=(x_{\sigma(1)},\dots,x_{\sigma(n)})$ for some $\sigma \in \mathcal{S}_n$ , then the transition kernel of Y would have remained the same, since by permutation-equivariance

(2.10) \begin{equation}\mathbb{E} \!\left ( f(\pi_n(Z^{ \, |n}_t)) \left | \right. \!Z^{ \, |n}_0=(x_{\sigma(1)},\dots,x_{\sigma(n)}) \right ) =\mathbb{E} \!\left ( f(\pi_n(Z^{ \, |n}_{t,\sigma(1)}, \dots ,Z^{ \, |n}_{t,\sigma(n)} )) \left | \right. Z^{ \, |n}_0=(x_{1},\dots,x_{n}) \right )\!,\end{equation}

which is $\mathbb{E} \left ( f(\pi_n(Z^{ \, |n}_t)) \left | \right. Z^{ \, |n}_0=(x_{1},\dots,x_{n}) \right )$ .

We are now in a position to present general examples of jump transition kernels and inter-jump motions for a system of particles in W. The first example introduces a death transition kernel where an existing particle dies with a probability that may depend on the distance to the other particles. The next two examples focus on birth transition kernels, driven either by a mixture of densities around each particle or by a Gibbs potential. The last two examples apply the above construction of $(Y_t)_{t \geq 0}$ on E to introduce inter-jump Langevin diffusions and growth interaction processes.

Example 1 (death kernel): Let $g \;:\; \mathbb{R}_+ \rightarrow \mathbb{R}_+^*$ be a continuous function. For $x=\{x_1,\dots,x_n\}\in E_n$ , set $w(x_1,x)=1$ if $n=1$ , and if $n\geq 2$ , for any $i\in\{1,\dots,n\}$ set

\[ w(x_i,x)=\frac{1}{z(x)} \displaystyle \sum_{k \neq i} g \left ( \|x_k-x_i\| \right ),\]

with $z(x)=\sum_{i=1}^n \sum_{k \neq i} g ( \|x_k-x_i\| )$ . A general example of a death transition kernel is

\[\displaystyle K_\delta(x,A)= \sum_{i=1}^{n(x)} w(x_i,x) {\mathbf 1}_{\{x \, \backslash \, x_i\in A\}},\quad x\in E,\ A \in \mathcal{E}.\]

The probability $w(x_i,x)$ that $x_i$ disappears then depends on the distance between $x_i$ and the other particles in x through g. Uniform deaths correspond to the particular case $w(x_i,x)=1/n(x)$ .

Example 2 (birth kernel as a mixture): Let $\varphi$ be a density function on W, and let $\phi_1 \;:\; W\to \mathbb{R}$ and $\phi_2\;:\;\mathbb{R}_+\to\mathbb{R}$ be two continuous functions. For $x=\{x_1,\dots,x_{n(x)}\}\in E\setminus E_0$ we set $ v(x_i,x) = \exp\left(\phi_1(x_i)+\sum_{k\neq i} \phi_2 \left ( \|x_k-x_i\| \right )\right)$ , and we consider the birth kernel defined for $\Lambda \subset W$ and $x \in E\setminus E_0$ by $K_\beta(\text{\O},\Lambda)=\int_\Lambda \varphi(\xi) \text{d} \xi$ and

\[ K_\beta (x,\Lambda \cup x) = \dfrac{1}{n(x)} \sum_{i=1}^{n(x)} \dfrac{1}{z(x_i,x)} \int_\Lambda \varphi \left ( \frac{\xi-x_i}{v(x_i,x)} \right ) \, \text{d} \xi,\]

where $\Lambda \cup x =\{\{u\} \cup x, u\in \Lambda\}$ and $z(x_i,x)=\int_W \varphi \left ( (\xi-x_i)/v(x_i,x)\right )\text{d} \xi$ . Note that $z(x_i,x)=v(x_i,x)^d$ if $W=\mathbb{R}^d$ . It is easily checked that $K_\beta (x,E_{n+1})=K_\beta ( x, W \cup x)= 1$ for $x \in E_n$ , and in particular this kernel is a genuine birth kernel in the sense that the transition from $E_n$ to $E_{n+1}$ is due only to the addition of a new particle, the existing ones remaining unchanged. Moreover, the new particle is generated as a mixture of distributions driven by $\varphi$ , each of them centred at the existing particles. The term $v(x_i,x)$ quantifies the dispersion of births around the particle $x_i$ , and it depends on the distance between $x_i$ and the other particles through $\phi_2$ . A natural example is a mixture of isotropic Gaussian distributions on $\mathbb{R}^d$ (restricted to W), respectively centred at $x_i$ with standard deviation $v(x_i,x)$ .

Example 3 (birth kernel based on a Gibbs potential): We introduce a measurable function $V\;:\; E \rightarrow \mathbb{R}$ , called a potential, satisfying $ z(x)\;:\!=\;\int_W \exp\!(\!-\!(V(x \cup \xi)-V(x))) \, \text{d} \xi < \infty$ for all $x\in E$ , and we consider the birth kernel defined for $\Lambda \subset W$ and $x \in E$ by

\[K_\beta(x,\Lambda \cup x) = \frac{1}{z(x)} \int_\Lambda \text{e}^{-(V(x \cup \xi)-V(x))} \, \text{d} \xi.\]

Note that $K_\beta ( x, W \cup x)= 1$ for $x \in E$ . With this kernel, given a configuration x, a new particle is more likely to appear in the vicinity of points $\xi\in W$ that make $V(x \cup \xi)-V(x)$ minimal. This kind of kernel $K_\beta$ was introduced in [Reference Preston22] for spatial birth–death processes, the case of a birth–death–move process with no move. Their importance is due to the fact that the invariant measure of a spatial birth–death process associated to $K_\beta$ , with uniform deaths and specific birth and death intensities, has been explicitly obtained in [Reference Preston22] and corresponds to the Gibbs measure with potential V. This result is the basis of perfect simulation of spatial Gibbs point process models; see [Reference Møller and Waagepetersen19]. We will similarly show in Section 5 that the same Gibbs measure is also invariant for a birth–death–move process associated to the same characteristics for the jumps and a well chosen inter-jump move process $(Y_t)_{t \geq 0}$ constructed as in the next example.

Example 4 (Langevin diffusions as inter-jump motions): Let $g \;:\; \mathbb{R}^d \rightarrow \mathbb{R}^d$ be a globally Lipschitz continuous function, $\beta >0$ , and $\{ B_{t,i} \} _{1 \leq i \leq n}$ , $n\geq 1$ , a collection of n independent Brownian motions on $\mathbb{R}^d$ . We start from the following system of SDEs, usually called overdamped Langevin equations:

\[ \text{d} Z_{t,i}^{\, |n} = - \sum_{j \neq i} g ( Z_{t,i}^{\, |n}- Z_{t,j}^{\, |n}) \, \text{d} t + \sqrt{2 \beta^{-1}} \, \text{d} B_{t,i},\quad 1 \leq i \leq n.\]

For $z=(z_1,\dots,z_n) \in (\mathbb{R}^d)^n$ , denoting by $\Phi_n \;:\; (\mathbb{R}^d)^n \rightarrow (\mathbb{R}^d)^n$ the function defined by $\Phi_n(z) = (\Phi_{n,1}(z),\dots,\Phi_{n,n}(z))$ with $\Phi_{n,i}(z)= \sum_{j \neq i} g ( z_{i}- z_{j})$ , this system of SDEs reads

(2.11) \begin{equation} \text{d} Z_t^{\, |n} = - \Phi_n(Z_t^{\, |n}) \, \text{d} t + \sqrt{2 \beta^{-1}} \, \text{d} B_t^{\, |n},\end{equation}

where $B_t^{\, |n}=(B_{t,1}, \dots, B_{t,n})$ . Since $\Phi_n$ is a permutation-equivariant function, that is, for any $\sigma \in \mathcal{S}_n$ ,

\[\Phi_n(z_{\sigma(1)},\dots,z_{\sigma(n)})=(\Phi_{n,\sigma(1)}(z),\dots,\Phi_{n,\sigma(n)}(z)),\]

and since $B_t^{\, |n}$ is exchangeable, we can verify by writing (2.11) in integral form that the law of $Z_t^{|n}$ is permutation-equivariant with respect to its initial state. So when $W=\mathbb{R}^d$ , we can define each inter-jump process $Y^{\, |n}$ in $E_n$ from $Z^{\, |n}$ as in (2.8), yielding $(Y_t)_{t \geq 0}$ on E. The same construction can be generalized if $W\subsetneq \mathbb{R}^d$ by considering a Langevin equation with reflecting boundary conditions [Reference Fattler and Grothaus8]. This inter-jump dynamics, associated with the birth kernel of Example 3 and a drift function g related to the potential V, converges to a Gibbs measure on W with potential V (see Section 5).

Example 5 (growth interaction processes): This example is motivated by models used in ecology [Reference Comas5, Reference Häbel, Myllymäki and Pommerening10, Reference Renshaw, Comas and Mateu23, Reference Renshaw and Särkkä24]. Each particle consists of a plant located in $S\subset\mathbb{R}^d$ and associated with a positive mark, which typically represents the size of the plant, so that $W=S\times \mathbb{R}^+$ here. Births and deaths of plants occur according to a spatial birth-and-death process, while a deterministic growth applies to their mark. Specifically, when a plant appears, its mark is set to zero or generated according to a uniform distribution on $[0,\varepsilon]$ for some $\varepsilon >0$ [Reference Renshaw and Särkkä24]. Then the mark increases over time, in interaction with the other marks. In order to formally define this inter-jump dynamics, let us denote by $(U_i(t),m_i(t))_{t \geq 0}$ , for $i = 1, \dots , n$ , the components of the process $(Z_t^{|n})_{t \geq 0}$ , where $U_i(t) \in S$ and $m_i(t)>0$ , so that $Z_t^{|n} \in W^n$ . We introduce the system

(2.12) \begin{align} & \dfrac{\text{d} Z^{\, |n}_t}{\text{d} t} = \left ((0,F_{1,n}(Z^{\, |n}_t)),\dots,(0,F_{n,n}( Z^{\, |n}_t)) \right ),\end{align}

where, for all $1\leq i\leq n$ , $F_{i,n}$ is a function from $W^n$ into $\mathbb{R}_+$ . We thus have $U_i(t) = U_i(0)$ for all i, and the evolution of the marks $(m_1(t), \dots ,m_n(t))$ is driven by a deterministic differential equation depending on $(U_1(0),\dots,U_n(0))$ as expected. To define $Y^{\, |n}$ by (2.8), we finally assume permutation-equivariance, namely that $F_{\sigma(i),n}(z_1,\dots,z_n)=F_{i,n}(z_{\sigma(1)},\dots,z_{\sigma(n)})$ for all i and all $\sigma \in \mathcal{S}_n$ , which is satisfied in all of the examples in the aforementioned references.

3.1. Feller properties

We assume henceforth that E is a locally compact Polish space. Let $C_b(E)$ be the set of continuous and bounded functions on E, and let $C_0(E)$ be the set of continuous functions that vanish at infinity in the sense that for all $ \epsilon>0$ , there exists a compact set $B\in E$ such that $x\notin B \Rightarrow |f(x)|<\epsilon$ .

Following [Reference Dynkin7] and [Reference Øksendal20], we say that the jump–move process $(X_t)_{t \geq 0}$ on E with transition kernel $Q_t$ is Feller continuous (or Feller on $C_b(E)$ ) if $Q_t C_b(E) \subset C_b(E)$ , and we say that it is Feller (or Feller on $C_0(E)$ ) if both $ \lim_{t \rightarrow 0} \| Q_tf - f \|_{\infty}= 0$ for any $f \in C_0(E)$ (strong continuity) and $Q_t C_0(E) \subset C_0(E)$ .

The following proposition, proved in Section S-2 of the supplementary material, provides information on the continuity property of $Q_t$ when t goes to 0.

By the second item above, the strong continuity of $Q_t$ is implied by the strong continuity of $Q_t^Y$ , which in turn holds automatically if $Q_t^Y \, C_0(E) \subset C_0(E)$ by continuity of $Y_t$ . We thus obtain the following natural conditions for the jump–move process on E to be Feller continuous or Feller. The proof is given in Section S-2 of the supplementary material.

We deduce in particular from this theorem that if $Q_t^Y \, C_b(E) \subset C_b(E)$ and $K \, C_b(E) \subset C_b(E)$ (or alternatively with $C_0(E)$ instead of $C_b(E)$ ), then $(X_t)_{t \geq 0}$ is a strong Markov process for the filtration $(\mathcal{F}_t)_{t \geq 0}$ , a property implied by the Feller continuous and Feller properties; see [Reference Bass3]. The Feller property will also be useful to us in Section 4 to construct a coupling between a birth–death–move process and a simple birth–death process on $\mathbb{N}$ , with a view to establishing ergodic properties.

In Section 3.3 we investigate the conditions of Theorem 2 for the examples of dynamics of systems of interacting particles in $\mathbb{R}^d$ introduced in Section 2.4. For these examples, the conditions turn out to be generally satisfied under mild assumptions.

3.2. Infinitesimal generator

In this section we compute the infinitesimal generator associated to the jump–move process $(X_t)_{t \geq 0}$ . We first introduce some notation and recall the definition of the generator; see for instance [Reference Dynkin7]. In connection with this, recall that the family $(Q_t)_{t \geq 0}$ of transition operators is a semigroup on $(M_b(E),\|.\|_{\infty}).$ If moreover the process $(X_t)_{t \geq 0}$ is Feller continuous (resp. Feller), then $(Q_t)_{t \geq 0}$ is a semigroup on $(C_b(E),\|.\|_{\infty})$ (resp. $(C_0(E),\|.\|_{\infty})$ ).

Let $L \subset M_b(E)$ and $(U_t)_{t \geq 0}$ be a semigroup on $(L,\|.\|_{\infty})$ . We set

\begin{equation*} L_0 = \left\{f \in L \;:\; \lim_{t \rightarrow 0}\limits \|U_t f-f\|_{\infty}=0\right\}\text{ and }\mathcal{D}_{\mathcal{A}} = \left\{f \in L \;:\; \lim_{t \rightarrow 0}\limits \frac{U_t f-f}{t} \; \text{exists in } (L,\|.\|_{\infty})\right\}.\end{equation*}

For $f \in \mathcal{D}_\mathcal{A}$ , define $\mathcal{A} f = \lim_{t \searrow 0} (U_t f -f)/t.$ The operator $\mathcal{A} \;:\; \mathcal{D}_{\mathcal{A}} \rightarrow L $ is called the infinitesimal generator associated to the semigroup $(U_t)_{t \geq 0}$ and $\mathcal{D}_{\mathcal{A}}$ is called the domain of the generator $\mathcal{A}.$

In the following we denote by $L_0$ (resp. $L_0^Y$ ) and $\mathcal{A}$ (resp. $\mathcal{A}^Y$ ) the set and the infinitesimal operator associated to $(Q_t)_{t \geq 0}$ (resp. $(Q_t^Y)_{t \geq 0}$ ). Note that $L_0=L_0^Y$ by Proposition 3.

Theorem 3. Let $(X_t)_{t \geq 0}$ be a general jump–move process on a Polish space E. Suppose that if $ f \in L_0^Y$ , then $\alpha \times f \in L_0^Y$ and $Kf \in L_0^Y$ . Then $\mathcal{D}_\mathcal{A}=\mathcal{D}_{\mathcal{A}^Y}$ , and for any $f \in \mathcal{D}_{\mathcal{A}^Y}$ ,

\[ \mathcal{A} f = \mathcal{A}^Y f + \alpha \times Kf - \alpha \times f . \]

This result, proved in Section S-3 of the supplementary material, shows that the generator $\mathcal{A}$ of the jump–move process $(X_t)_{t \geq 0}$ is simply the sum of the generator of the move $\mathcal{A}^Y$ and the generator of the jump, specifically of a pure jump Markov process with intensity $\alpha$ and transition kernel K, i.e. $\alpha \times (K- Id)$ (see [Reference Feller9]).

Note that for a pure jump process, $Q_t^Y=Id$ for any $t \geq 0$ , $L_0^Y=\mathcal{D}_{\mathcal{A}^Y}=M_b(E)$ , and $\mathcal{A}^Y \equiv 0$ , so that all assumptions of Theorem 3 are trivially true in this setting. More generally, consider a jump–move process with a Feller inter-jump process, i.e. $Q_t^Y \, C_0(E) \subset C_0(E)$ , and a Feller jump transition, i.e. $K \, C_0(E) \subset C_0(E)$ , so that $(X_t)_{t \geq 0}$ is Feller by Theorem 2. Then we can take $L_0^Y=C_0(E)$ , and again the assumptions of Theorem 3 are satisfied since $\alpha$ is bounded.

3.3. Application to systems of interacting particles in $\mathbb{R}^d$

We go back to the setting of Section 2.4, namely, systems of interacting particles in $W\subset \mathbb{R}^d$ , in order to investigate whether the examples of dynamics presented therein are (continuous) Feller or not. To do so and be able to check the conditions of Theorem 2, we need to first clarify what the sets $ C_b(E)$ and $ C_0(E) $ are in this framework. Remember that in this setting $E = \cup_{n \geq 0} E_n$ , where $E_n = \pi_n( W^n)$ corresponds to the set of unordered n-tuples of W, and we have equipped E with the distance $d_1$ defined by (2.7). As a first result, it can be verified that $(E,d_1)$ is a locally compact Polish space; see [Reference Schuhmacher and Xia26, Proposition 2.2] and Section S-4 in the supplementary material. To characterize the elements of $ C_b(E)$ , we shall use the following proposition, proved in the supplementary material.

Proposition 4. Let $x \in E$ and let $(x^{(p)})_{p \geq 1}$ be a sequence converging to x, i.e. $d_1(x^{(p)},x)\to 0$ as $p\to\infty$ . Then there exists $p_0 \geq 1$ such that for all $p \geq p_0$ , $n(x^{(p)})=n(x)$ and, when $n(x)\geq 1$ , there exists a sequence $(\sigma_{p})_{p \geq p_0}$ of $\mathcal{S}_{n(x)}$ such that for any $i \in \{ 1 , \dots, n(x) \},$

(3.1) \begin{equation} \|x_{\sigma_{p}(i)}^{(p)}-x_{i}\| \underset{p \rightarrow \infty}{\longrightarrow} 0. \end{equation}

To deal with $C_0(E)$ , we provide a characterization of the compact sets of each $E_n$ , for $n \geq 1$ , and an important property of the compact sets of E.

The previous two propositions are the main tools we need to investigate the continuous Feller and Feller properties of the jump kernel K of a jump–move process. Concerning the inter-jump move process $(Y_t)_{t\geq 0}$ , recall that we can easily define it on each $E_n$ from a continuous process $(Z^{|n}_t)_{t\geq 0}$ on $W^n$ through the projection (2.8). In general, properties on E are not simply derived from properties on $E_n$ ; for instance, we deduce from Proposition 5 that $\sum_n f_n {\mathbf 1}_{E_n}$ for $f_n\in C_0(E_n)$ is not necessarily in $C_0(E)$ (take W compact and $f_n(x)=n(x)$ ). Nonetheless, the formula (2.9) implies that the Feller properties of $(Y_t)_{t\geq 0}$ on $(E,d_1)$ are inherited from those of $(Z^{|n}_t)_{t\geq 0}$ on $W^n$ .

By this result, standard inter-jump motions are Feller continuous and Feller, as is the case under mild assumptions for our examples 4 and 5, detailed below. Concerning the jump kernels, the global picture is as follows. They are generally Feller continuous, but not necessarily Feller even if the underlying space W is compact, as shown in the following example. However, if we restrict ourselves to birth kernels, then they are generally Feller (see Examples 2 and 3 below). On the other hand, if we restrict ourselves to death kernels, then they are Feller if W is compact, but not otherwise; see Example 1 below. Notice that a birth-and-death jump kernel as in (2.4) is (continuous) Feller when the birth kernel $K_\beta$ and the death kernel $K_\delta$ are. So it is generally continuous Feller, and if W is compact, it is generally Feller.

Let us make these informal claims more precise through some examples. The first example presents a jump kernel on a set W, possibly compact, that is continuous Feller but not Feller. The other ones correspond to the examples introduced in Section 2.4.

To prove the first property, take $x\in E$ , a sequence $(x^{(p)})_{p \geq 0}$ converging to x, and $p_0$ and $(\sigma_p)_{p\geq p_0}$ from Proposition 4. Then it is not difficult to verify that $\lim_{p \rightarrow \infty} w(x_{\sigma_p(i)}^{(p)},x^{(p)}) = w(x_i,x)$ by continuity of g. Moreover, $ d_1 \left ( x^{(p)} \, \backslash \, x_{ \sigma_p(i)}^{(p)}, x \, \backslash \, x_{i} \right ) \leq \sum_{j \neq i} \|x_{\sigma_p(j)}^{(p)} - x_{j} \|/(n-1)$ , which shows that $x^{(p)} \, \backslash \, x_{\sigma_p(i)}^{(p)} \underset{p \rightarrow \infty}{\longrightarrow} x \, \backslash \, x_{i}$ . Therefore, for any $f\in C_b(E)$ ,

\begin{align*} \lim \limits_{p \rightarrow \infty} K_\delta f(x^{(p)}) & = \lim \limits_{p \rightarrow \infty} \sum_{i=1}^{n(x)} w\!\left(x_i^{(p)},x^{(p)}\right) f\!\left(x^{(p)} \, \backslash \, x^{(p)}_i\right) \\ & = \lim \limits_{p \rightarrow \infty} \sum_{i=1}^{n(x)} w\!\left(x_{\sigma_p(i)}^{(p)},x^{(p)}\right) f\!\left(x^{(p)} \, \backslash \, x^{(p)}_{\sigma_p(i)}\right) \\ & = \sum_{i=1}^{n(x)} w(x_{i},x) f(x \backslash \, x_{i}) \\ &= K_\delta f(x).\end{align*}

Let us now consider the second claim, (ii). Take $f \in C_0(E)$ and $\varepsilon >0$ . We fix a compact set A of $(E,d_{1})$ such that $|f(x)|< \varepsilon$ for $x \notin A$ . By Proposition 5, $ A \subset \bigcup_{n=0}^{n_0} E_n$ for some $n_0$ . As a straightforward consequence of Proposition 5 (see the supplementary material), the set $ B \;:\!=\; \bigcup_{n=0}^{n_0+1} E_n $ is a compact set when W is compact and it satisfies $K_\delta(x,A)=0$ for $x \notin B$ . This implies that for $x\notin B$ ,

(3.2) \begin{align} |K_\delta f(x)| & \leq \left | \int_A f(y) K_\delta (x,\text{d} y) \right | + \left | \int_{A^c} f(y) K_\delta (x, \text{d} y) \right | \nonumber \\ & \leq ||f||_{\infty} K_\delta (x,A) + \varepsilon \, K_\delta (x,A^c) \leq \varepsilon,\end{align}

and so $K_\delta f\in C_0(E)$ . Let us finally show that this result is no longer valid if W is not compact. Assume without loss of generality that $0\in W$ , and consider as in the previous example the function $f\in C_0(E)$ defined by $f(x)=\max(1-\|x\|,0){\mathbf 1}_{n(x)=1}$ . Let B be any compact subset of E. Then $B_2=B\cap E_2$ is compact because $E_2$ is closed and, by Proposition 5, for any $x=\{x_1,x_2\} \in B_2$ , there exists $R > 0$ such that $\max\{\|x_1\|,\|x_2\|\} \leq R$ . Take $y=\{0,y_2\}$ in $E_2$ such that $\|y_2\|>R+1$ , which is possible since W is not compact. Then $y\notin B$ but $K_\delta f(y)=w(y_2,y)$ , proving that $Kf\notin C_0(E)$ .

Example 2 (continued) (birth kernel as a mixture): For this example, we shall prove that if $\mathring W \neq \varnothing$ and if the dispersion function v is continuous, then $K_\beta C_b (E) \subset C_b(E)$ and $K_\beta C_0(E)\subset C_0(E)$ . Take $f\in C_b (E)$ , $x\in E$ , and a sequence $(x^{(p)})_{p \geq 0}$ converging to x, from which we define $p_0$ and $(\sigma_p)_{p\geq p_0}$ from Proposition 4. We have, for $p\geq p_0$ ,

\begin{align*} &K_\beta f(x^{(p)}) = \dfrac{1}{n(x)} \sum_{i=1}^{n(x)} \dfrac{1}{z(x_{\sigma_p(i)}^{(p)},x^{(p)})} \int_{W} f(x^{(p)} \cup \{ \xi \} )\varphi \left ( \frac{\xi-x_{\sigma_p(i)}^{(p)}}{v(x_{\sigma_p(i)}^{(p)},x^{(p)})} \right ) d\xi\\[5pt] &=\dfrac{1}{n(x)} \sum_{i=1}^{n(x)} \frac{\int_{\mathbb{R}^d} {\mathbf 1}_{\{x_{\sigma_p(i)}^{(p)}+ v(x_{\sigma_p(i)}^{(p)} ,x^{(p)})\xi \in W\}} f(x^{(p)} \cup \{ x_{\sigma_p(i)}^{(p)}+ v(x_{\sigma_p(i)}^{(p)} ,x^{(p)})\xi \} ) \varphi(\xi) d\xi}{\int_{\mathbb{R}^d} {\mathbf 1}_{\{x_{\sigma_p(i)}^{(p)}+ v(x_{\sigma_p(i)}^{(p)} ,x^{(p)})\xi \in W\}} \varphi(\xi) d\xi}. \end{align*}

By continuity of v, the indicator functions involved tend to ${\mathbf 1}_{\{x_i+ v(x_i ,x)\xi \in W\}}$ for any $x_i+ v(x_i ,x)\xi\in \mathring W$ . On the other hand, for any $i\in \{1,...,n(x)\}$ and any $\xi$ ,

\begin{align*} & d_1 \left ( x^{(p)} \cup \{ x_{\sigma_p(i)}^{(p)}+ v(x_{\sigma_p(i)}^{(p)} ,x^{(p)})\xi \}, x \cup \{ x_{i} + v(x_{i},x) \, \xi \} \right ) \\ & \leq \frac{1}{n(x)+1} \left ( \sum_{j =1}^{n(x)} \|x_{\sigma_p(j)}^{(p)} - x_{j} \| + \|x_{\sigma_p(i)}^{(p)} + v(x_{\sigma_p(i)}^{(p)},x^{(p)}) \, \xi - x_{i} - v(x_{i},x) \, \xi \| \right ) \\ & \leq \frac{1}{n(x)+1} \left ( \sum_{j =1}^{n(x)} \|x_{\sigma_p(j)}^{(p)} - x_{j} \| + \|x_{\sigma_p(i)}^{(p)} - x_{i} \| + \|\xi\| \, |v(x_{\sigma_p(i)}^{(p)},x^{(p)})-v(x_{i},x)| \right ), \end{align*}

which tends to 0 as $p\to\infty$ . So by continuity of f, $ f(x^{(p)} \cup \{ x_{\sigma_p(i)}^{(p)}+ v(x_{\sigma_p(i)}^{(p)} ,x^{(p)})\xi \} )$ tends to $f(x \cup \{x_i + v(x_i,x)\xi)$ . We conclude by the dominated convergence theorem, since f is bounded and $\varphi$ is a density, that $K_\beta f(x^{(p)})$ converges to $ K_\beta f(x)$ as $p\to\infty$ , which proves that $K_\beta C_b (E) \subset C_b(E)$ .

Let us now prove that $K_\beta C_0(E) \subset C_0(E).$ Let $f \in C_0(E)$ and $\varepsilon >0$ . We fix a compact set $A \subset E$ such that $x \notin A \Rightarrow | f(x) | < \varepsilon$ . By Proposition 5, $A \subset \bigcup_{n=0}^{n_0} E_n$ for some $n_0$ . Letting $A_n=A \cap E_n$ , for $n=0,\dots,n_0,$ we observe that $A_n$ is a compact set because $E_n$ is closed. By Proposition 5, there exists $R_n \geq 0$ such that for every $a=\{a_1,\dots,a_n\} \in A_n,$ $\max_{1 \leq k \leq n} \|a_k\| \leq R_n$ . Now let $B_{n}= \{ x \in E_{n}, \sum_{k=1}^{n} \|x_k\|/n \leq R_n \}$ and $B = \bigcup_{n=0}^{n_0-1} B_{n}$ . We can verify (see the proof of Proposition 5) that $B_n$ is compact and so is B. We claim that if $x\notin B$ , then $K_\beta(x,A)=0$ . Indeed, if $K_\beta (x,A)>0$ , then $K_\beta (x,A_n) >0$ for some $n\in\{0,\dots,n_0\}$ , but since $ K_\beta(x,A_0) \leq K_\beta(x, \{ \text{\O} \})=0$ , it cannot be $n=0$ . Now, for $n=1,\dots,n_0$ , $K_\beta (x,A_n) >0$ implies that $x \in E_{n-1}$ and $A_n \subset \{ z \cup x, \, z \in W \}$ since $K_\beta ( x, W \cup x)= 1$ . So $\max_{1 \leq k \leq n-1} \|x_k\| \leq R_n$ , whereby $x\in B_{n-1}$ . This shows that if $K_\beta (x,A)>0$ then $x\in B$ , as we claimed. We deduce that for any $x\notin B$ , $|K_\beta f(x)|\leq \epsilon$ as in (3.2).

4.1. Coupling of birth–death–move processes

Let $(X_t)_{t \geq 0}$ be a birth–death–move process, as defined in Section 2.2, and let $(\eta_t)_{t \geq 0}$ be a simple birth–death process on $\mathbb{N}$ with birth rate $\beta_n$ and death rate $\delta_n$ given by (4.1). Note that $(\eta_t)_{t \geq 0}$ can be viewed as a birth–death–move process on $\mathbb{N}$ having a constant move process $y_t =y_0$ , for all $t \geq 0$ . We denote by $(t_j)_{j \geq 1}$ the jump times of $(\eta_t)_{t \geq 0}$ and by $n_t \;:\!=\; \sum_{j \geq 1} {\mathbf 1}_{t_j \leq t}$ the number of jumps before $t \geq 0$ . We also denote by $q_t$ the transition kernel of $(\eta_t)_{t \geq 0}$ , i.e. $q_t(n,S)=\mathbb{P}(\eta_t\in S|\eta_0=n)$ for any $n\in \mathbb{N}$ and $S\in \mathcal{P}(\mathbb{N})$ .

We define the coupled process $\check{C}=(X',\eta')$ as a jump–move process on the state space $\check{E} = E \times \mathbb{N}$ equipped with the $\sigma$ -algebra $\check{\mathcal{E}} = \mathcal{E} \otimes \mathcal{P} (\mathbb{N})$ . Denoting by d the distance on E, we also equip $\check{E}$ with the distance $\check{d}((x,k);\;(y,n))\;:\!=\;d(x,y)+ |n-k|/(n\wedge k){\mathbf 1}_{nk \neq 0}$ . To fully characterize $\check{C}$ , we now specify its jump intensity function $ \check{\alpha}$ , its jump kernel $\check{K}$ , and its inter-jump move process $\check{Y}$ .

The intensity function $\check{\alpha} \;:\; E \times \mathbb{N} \rightarrow \mathbb{R}_+$ is given by

\begin{equation*} \check{\alpha} (x,n)= \left \{ \begin{array}{l@{\quad}l} \beta(x) + \delta(x) + \beta_n + \delta_n & \text{if } x \in E_m, \, m \neq n , \\ \beta_n + \delta(x) & \text{if } x \in E_n.\end{array} \right.\end{equation*}

One can easily check that $\check{\alpha} $ is a continuous function on $\check{E}$ , bounded by $2\alpha^*$ .

The transition kernel $\check{K} \;:\; \check{E} \times \check{\mathcal{E}} \rightarrow [0,1]$ takes the same specific form as in [Reference Preston22]:

1. (i) If $x \in E_m, \, m \neq n$ :

   \[ \check{K}((x,n);\; A \times\{n\}) = \dfrac{\alpha (x)}{ \check{\alpha} (x,n)} K(x,A); \]
   \[ \check{K}((x,n);\; \{x\}\times\{n+1\}) = \dfrac{\beta_n}{ \check{\alpha} (x,n)} ;\]
   \[ \check{K}((x,n);\; \{x\}\times\{n-1\}) = \dfrac{\delta_n}{ \check{\alpha} (x,n)}. \]

2. (ii) If $x \in E_n$ :

   \[ \check{K}((x,n);\; A\times\{n+1\}) = \dfrac{\beta (x)}{\check{\alpha} (x,n)} K_\beta(x,A) ;\]
   \[ \check{K}((x,n);\; \{x\}\times\{n+1\}) = \dfrac{\beta_n-\beta(x)}{\check{\alpha} (x,n)}; \]
   \[ \check{K}((x,n);\; A\times\{n-1\}) = \dfrac{\delta_n}{\check{\alpha} (x,n)}K_\delta(x,A); \]
   \[ \check{K}((x,n);\; A\times\{n\}) = \dfrac{\delta(x)-\delta_n}{\check{\alpha} (x,n)}K_\delta(x,A) . \]

The inter-jump move process $\check{Y}$ is finally obtained by an independent coupling of $(Y_t)_{ t \geq 0}$ and $(y_t)_{t \geq 0}$ ; specifically, its transition kernel $Q_t^{\check{Y}}$ is given, for any $(x,p)\in\check{E}$ and $A\times S\in \check{\mathcal{E}}$ , by

(4.2) \begin{multline} Q_t^{\check{Y}}((x,p);\; A \times S)= \mathbb{P}( \check{Y}_t \in A \times S | \check{Y}_0=(x,p))\\ =\mathbb{P}(Y_t \in A | Y_0=x) {\mathbf 1}_{p\in S} = Q_t^{Y}(x,A) {\mathbf 1}_{p\in S}.\end{multline}

This means that $\check{Y}_t = (Y'_{\!\!t},y'_{\!\!t})=(Y'_{\!\!t},y'_{\!\!0})$ for any $t\geq 0$ , where $(Y'_{\!\!t})_{t\geq 0}$ and $(y'_{\!\!t})_{t\geq 0}$ are independent and follow the same distribution as $(Y_t)_{t\geq 0}$ and $(y_t)_{t\geq 0}$ , respectively. Since Y is a continuous Markov process for the distance d, we can choose a version of Y ^′ such that $\check{Y}$ is also continuous for $\check{d}$ . Observe moreover that $(\check{Y}_t)_{t \geq 0}$ satisfies

$$ \mathbb{P}( ( \check{Y}_t)_{t \geq 0} \subset E_n \times \{ k \} \, | \, \check{Y}_0=(x,p))= {\mathbf 1}_{x\in E_n} \, {\mathbf 1}_{k=p}, \; \; \; \forall x \in E, \, \forall n \geq 0. $$

Given $ \check{\alpha}$ , $\check{K}$ , and $\check{Y}$ as above, the jump–move process $\check{C}$ is well defined and can be constructed as in Section 2.1. We denote by $\check{Q}_t$ its transition kernel, by $(\check{T}_j)_{j \geq 1}$ its jump times, and by $\check{N}_t \;:\!=\; \sum_{j \geq 1} {\mathbf 1}_{ \check{T}_j \leq t }$ the number of jumps before $t \geq 0$ . We also set $ \check{\tau}_{j}=\check{T}_j - \check{T}_{j-1}$ . The fact that $\check{C}$ defines a genuine coupling of X with $\eta$ is the object of the following theorem.

If the move $(Y_t)_{t \geq 0}$ is constant, which is the setting in [Reference Preston22], then the proof is easy under (2.1) by use of the derivative form of the Kolmogorov backward equation. In the general case of a birth–death–move process, this strategy no longer works, and the statement becomes more challenging to prove. We manage to prove it by exploiting the generator of $(X_t)_{t \geq 0}$ ; see Theorem 3, which explains the Feller conditions in Theorem 4.

Proof of Theorem 4. To prove the first part of the theorem, we use the following lemmas and corollary, proved in the appendix. Fix $(x,n) \in E \times \mathbb{N}$ and $p \geq 0$ , and let

$$\psi_p \;:\; t \in \mathbb{R}_+ \mapsto \check{Q}_t((x,n),E \times \{ p \}).$$

Lemma 2. For any $(x,n) \in E \times \mathbb{N}$ and $p \geq 0$ , $\psi_p$ is right-differentiable and satisfies

\begin{align*} \dfrac{\partial_+}{\partial t} \psi_p(t) = - \alpha_p \, \psi_p(t) + \beta_{p-1} \, \psi_{p-1}(t) + \delta_{p+1} \, \psi_{p+1}(t). \end{align*}

Corollary 1. For any $(x,n) \in E \times \mathbb{N}$ and $p \geq 0$ ,

\begin{align*} \psi_p(t) = {\mathbf 1}_{p=n} + \int_0^t \left ( - \alpha_p \, \psi_p(s) + \beta_{p-1} \, \psi_{p-1}(s) + \delta_{p+1} \, \psi_{p+1}(s) \right ) \, \text{d} s; \end{align*}

in particular, $\psi_p$ is differentiable.

Now let $ w_s(x,n) = \check{Q}_{t-s} ( {\mathbf 1}_E \times q_s({\mathbf 1}_{\{p\}}))(x,n)$ for $s \in [0,t].$ Then, using Corollary 1, we have the following lemma.

Since $ w_0(x,n) = \check{Q}_t((x,n);\; E \times \{ p \})$ and $w_t(x,n) = q_t(n, \{ p \})$ , Lemma 3 implies that these two quantities are equal. The first part of Theorem 4 then follows from the decomposition

$$ q_t(n,S) = \sum_{p \in S} q_t(n, \{ p \} ) = \sum_{p \in S} \check{Q}_t((x,n);\; E \times \{ p \}) = \check{Q}_t((x,n);\; E \times S). $$

We turn to the proof of the second part of Theorem 4. Like the first part, it is based on three results that are proved in the appendix. For $(x,n) \in E \times \mathbb{N}$ and $f \in C_0(E)$ , we set

$$\psi_f \;:\; t \in \mathbb{R}_+ \mapsto \check{Q}_t(f \times {\mathbf 1}_\mathbb{N})(x,n).$$

Lemma 5. Suppose that $(Y_t)_{t \geq 0}$ is a Feller process and that $K C_0(E) \subset C_0(E)$ . Then for any $(x,n) \in E \times \mathbb{N}$ and any $f \in \mathcal{D}_{\mathcal{A}^Y}$ , the function $\psi_f$ is right-differentiable and satisfies

(4.3) \begin{equation} \dfrac{\partial_+}{\partial t} \psi_f(t) = \psi_{\mathcal{A} f}(t), \end{equation}

where $\mathcal{A}$ is the infinitesimal generator of X given by Theorem 3.

Corollary 2. Suppose that $(Y_t)_{t \geq 0}$ is a Feller process and that $K C_0(E) \subset C_0(E)$ . Then for any $(x,n) \in E \times \mathbb{N}$ and any $f \in \mathcal{D}_{\mathcal{A}^Y}$ ,

(4.4) \begin{align} \psi_f(t) = f(x) + \int_0^t \psi_{\mathcal{A} f}(s) \, \text{d} s; \end{align}

in particular, the function $\psi_f$ is differentiable with derivative corresponding to (4.3).

By the Dynkin theorem, the second part of Theorem 4 is implied by the equality $\check{Q}_t((x,n);\; U \times \mathbb{N}) = Q_t(x,U)$ for any open set $U\subset E$ , or equivalently

(4.5) \begin{equation}\check{Q}_t(g \times {\mathbf 1}_\mathbb{N})(x,n) = Q_t (g)(x)\end{equation}

for $g={\mathbf 1}_U$ . We prove (4.5) first for $g\in \mathcal{D}_{\mathcal{A}^Y}$ and then for $g \in C_0(E)$ , before getting the result for $g={\mathbf 1}_U$ .

Let $g\in \mathcal{D}_{\mathcal{A}^Y}$ , and for $s\in[0,t]$ define $v_s(x,n)=\psi_{Q_s g}(t-s) = \check{Q}_{t-s} \left ( Q_s g \times {\mathbf 1}_\mathbb{N} \right )(x,n)$ . We shall prove that $s\mapsto v_s$ is differentiable with $v'_{\!\!s}=0$ . For any $h\in \mathbb{R}$ , write $(v_{s+h}(x,n)-v_s(x,n))/h=A_1+A_2+A_3$ with

\begin{align*}A_1 &= \frac1 h\left(\psi_{Q_{s+h} g}(t-s-h) - \psi_{Q_s g}(t-s-h)\right) - \psi_{\mathcal{A} Q_s g}(t-s-h), \\A_2 &= \psi_{\mathcal{A} Q_s g}(t-s-h) - \psi_{\mathcal{A} Q_s g}(t-s), \\A_3 &= \frac 1 h \left(\psi_{Q_s g}(t-s-h) - \psi_{Q_s g}(t-s)\right)+ \psi_{\mathcal{A} Q_s g}(t-s).\end{align*}

We know by Theorem 3, with $L_0^Y=C_0(E)$ , that $\mathcal{D}_{\mathcal{A}} = \mathcal{D}_{\mathcal{A}^Y}$ , and since $Q_s \mathcal{D}_{\mathcal{A}}\subset \mathcal{D}_{\mathcal{A}}$ (see [Reference Dynkin7, Chapter 1, Section 2]), we deduce from Corollary 2 that $A_3$ tends to $-\partial \psi_{Q_s g}(t-s)/\partial t + \psi_{\mathcal{A} Q_s g}(t-s)=0$ as $h\to 0$ . Regarding $A_2$ , note that $Q_s g\in \mathcal{D}_{\mathcal{A}}$ implies that $\mathcal{A} Q_s g\in C_0(E)$ (again see [Reference Dynkin7]), so that Lemma 4 applies and $A_2\to 0$ as $h\to 0$ . Regarding $A_1$ , using the linearity of $\psi_f(t)$ in f, we can write

$$|A_1|=|\psi_{(Q_{s+h} - Q_s) g/h - \mathcal{A} Q_s g}(t-s-h)|\leq \| (Q_{s+h} - Q_s) g/h - \mathcal{A} Q_s g\|_\infty,$$

which also tends to 0 as $h\to 0$ . We therefore obtain that $v'_{\!\!s}=0$ and so $v_t(x,n) = (Q_t g \times {\mathbf 1}_\mathbb{N}) (x,n) = \check{Q}_t(g \times {\mathbf 1}_\mathbb{N})(x,n) = v_0(x,n)$ , proving (4.5) when $g\in \mathcal{D}_{\mathcal{A}^Y}$ .

Now let $g \in C_0(E).$ By our assumptions and Theorem 2, $ (X_t)_{t \geq 0}$ is Feller, which implies that $C_0(E) = \overline{\mathcal{D}_\mathcal{A}}$ (see [Reference Dynkin7]). So there exists a sequence of functions $(g_m)_{m \geq 0}$ in $\mathcal{D}_{\mathcal{A}^Y}$ such that $\|g_m-g\|_{\infty} \to 0$ as $m\to\infty$ . The two linear operators