3.1. Explosivity and non-explosivity

The sequence $J=(J_n)_{n\in\mathbb{N}_0}$ of jump times of a CTMC $Y_t$ are defined by $J_0=0$ and $J_n=\inf\{t\ge J_{n-1}\,:\, Y_t\neq Y_{J_{n-1}}\}$ , $n\ge1$ , where $\inf\varnothing=\infty$ by convention. The lifetime is denoted by $\zeta=\sup_{n}J_n$ . The process $Y_t$ is said to explode (with positive probability) at $y\in\mathcal{Y}$ if $\mathbb{P}_y(\{\zeta<\infty\})>0$ . In particular, $Y_t$ explodes a.s. at $y\in\mathcal{Y}$ if $\mathbb{P}_y(\{\zeta<\infty\})=1$ , and does not explode at $y\in\mathcal{Y}$ if $\mathbb{P}_y(\{\zeta<\infty\})=0$ [Reference Meyn and Tweedie33]. Hence, $Y_t$ does not explode if $Y_0\in\partial$ (since $\partial$ is closed and finite), and $\mathbb{E}_y(\zeta)<\infty$ implies that $Y_t$ explodes at y a.s. Recall that non-explosivity and explosivity are class properties. They hold for either all or no states in $\mathcal{Y}\setminus\partial$ . Hence, we simply say $Y_t$ is explosive (respectively, explosive a.s.) if it explodes with positive probability (respectively, explodes a.s.) at some state in $\mathcal{Y}\setminus\partial$ , and $Y_t$ is non-explosive if it does not explode at some state in $\mathcal{Y}\setminus\partial$ .

We present necessary and sufficient conditions for explosivity and non-explosivity.

Explosion might occur with probability less than one for CTMCs with non-polynomial transition rates and $\partial=\varnothing$ [Reference Menshikov and Petritis32]. Reuter’s criterion and generalizations of it provide necessary and sufficient conditions for explosivity (with positive probability) for general CTMCs in terms of convergence or divergence of a series [Reference Chen, Pollet, Zhang and Cairns13, Reference Li and Li30, Reference Reuter38]. However, these conditions are not easy to check. In comparison, for CTMCs with polynomial transition rates, Theorem 1 provides an explicit and checkable necessary and sufficient condition.

3.2. Recurrence versus transience, and certain absorption

For a nonempty subset $A\subseteq\mathcal{Y}$ , let $\tau_A=\inf\{t\ge0\,:\, Y_t\in A\}$ be the hitting time of A, with the convention that $\inf\varnothing=\infty$ . If $Y_0\in A$ , then $\tau_A=0$ . Let $\tau^+_A=\inf\{t\ge J_1\,:\, Y_t\in A\}$ be the first return time to A. Obviously, $\tau_A=\tau^+_A$ if and only if $Y_0\notin A$ . The process $Y_t$ has certain absorption if the hitting time of $\partial$ is finite a.s. for all $Y_0\in\mathcal{Y}$ .

The results show that CTMCs with polynomial transition rates cannot have an infinite series of critical transitions from recurrence to transience, for varying parameter values. This stands in contrast to the case of CTMCs with non-polynomial transition rates, as discovered in [Reference Menshikov and Petritis32]. One might hope that this phenomenon carries over to CTMCs with polynomial transition rates in dimensions higher than one.

3.3. Moments of hitting times

Below we present threshold results on the existence of moments of hitting times for recurrent states only, as transient states have infinite return time. Therefore, in light of Theorem 3, we investigate the existence and nonexistence of moments of hitting times only for $\alpha<0$ and for $\alpha=0,\ \beta\le0$ . Moreover, limited by the tools we apply, we do not discuss existence and nonexistence of moments of absorption times for $\partial\neq\varnothing$ . Hence, we assume $Y_t$ is irreducible on $\mathcal{Y}$ (equivalently, $\partial=\varnothing$ ) and provide existence and nonexistence of moments of hitting times for states in $\mathcal{Y}$ .

Theorem 5. Assume $(\textbf{A1})$ – $(\textbf{A5})$ , $\partial=\varnothing$ , and that $\Omega$ is finite. Then the following hold:

(i) There exists a finite nonempty subset $B\subseteq\mathcal{Y}$ such that

(5) \begin{equation}\mathbb{E}_x\big(\tau^{\epsilon}_B\big)<+\infty,\quad \forall x\in\mathcal{Y},\quad {\forall 0<\epsilon<\delta,}\end{equation}

for $\delta>0$ , provided one of the conditions (C3), (C9), (C10) holds; for $0<\delta<1/2$ , provided (C13) holds; for $0<\delta<\tfrac\beta{\beta-\gamma}$ , provided (C14) holds; and for $0<\delta<1$ , provided (C16) holds. In particular, $\mathbb{E}_x(\tau_B)<+\infty$ , provided one of the conditions (C3), (C9), (C10), (C11) holds.

(ii) There exists a finite nonempty subset $B\subseteq\mathcal{Y}$ such that

\begin{equation*}\mathbb{E}_x\big(\tau_B^{\epsilon}\big)=+\infty,\quad \forall x\in\mathcal{Y}\setminus B,\quad {\forall \epsilon>\delta,}\end{equation*}

for $\delta>1$ , provided (C13) holds; for $\delta>\tfrac\beta{\beta-\gamma}$ , provided (C15) holds; and for $\delta>1$ , provided (C16) holds. In particular, $\mathbb{E}_x(\tau_B)=+\infty$ provided (C12) holds.

3.4. Positive recurrence and null recurrence

We provide sharp criteria for positive and null recurrence, as well as exponential ergodicity of stationary distributions and QSDs.

If $\partial=\varnothing$ , then $\tau_{\partial}=\infty$ a.s., and the conditional process $(Y_t\,:\, \tau_{\partial}>t)$ reduces to $(Y_t\,:\, t\ge 0)$ . If $\partial\neq\varnothing$ , and $\tau_{\partial}<\infty$ a.s. (that is, $Y_t$ has certain absorption), then the process conditioned to never be absorbed, $(Y_t\,:\, \tau_{\partial}>t)$ , is referred to as the Q-process [Reference Champagnat and Villemonais11, Reference Collet, Martínez and San Martín18].

The process $(Y_t\,:\, \tau_{\partial}>t)$ on $\mathcal{Y}\setminus\partial$ is said to be exponentially ergodic if there exist a probability measure $\mu_*$ and $0<\delta<1$ such that for all probability measures $\mu$ on $\mathcal{Y}\setminus\partial$ , there exists a constant $C_{\mu}>0$ such that

\begin{equation*}{|\mathbb{P}_{\mu}(Y_t\in B|\tau_{\partial}>t)-\mu_*(B)|\le C_{\mu}\delta^t,\quad \forall t>0,\ B\subseteq \mathcal{Y}\setminus\partial}\end{equation*}

(see [Reference Gibbs and Su25]). The measure $\mu^*$ is also said to be exponentially ergodic. In particular, if $C_{\mu}$ can be chosen independently of $\mu$ , then $(Y_t\,:\, \tau_{\partial}>t)$ and $\mu^*$ is said to be uniformly exponentially ergodic. Moreover, if $\partial=\varnothing$ , then $\mu_*$ is the unique ergodic stationary distribution; if $\partial\neq\varnothing$ , then $\mu_*$ is a quasi-limiting distribution (QLD) [Reference Collet, Martínez and San Martín18].

If $\partial\neq\varnothing$ , a probability measure $\nu$ on $\mathcal{Y}\setminus\partial$ is a QSD for $Y_t$ if for all $t\ge0$ and all sets $B\subseteq \mathcal{Y}\setminus\partial$ ,

\begin{equation*}\mathbb{P}_{\nu}(Y_t\in B|\tau_{\partial}>t)=\nu(B).\end{equation*}

Any QLD is a QSD [Reference Collet, Martínez and San Martín18]. The existence of a QSD implies certain absorption, and exponential ergodicity of the Q-process implies existence of a unique QSD [Reference Collet, Martínez and San Martín18]. A probability measure $\nu$ on $\mathcal{Y}\setminus\partial$ is a quasi-ergodic distribution if, for any $x\in\mathcal{Y}\setminus\partial$ and any bounded function f on $\mathcal{Y}\setminus\partial$ [Reference Breyer and Roberts10, Reference He, Zhang and Zhu27], the following limit holds:

\begin{equation*}\lim_{t\to\infty}\mathbb{E}_x\!\left(\frac{1}{t}\int_0^tf(Y_s){\text{d}}s\Big|\tau_{\partial}>t\right)=\int_{\mathcal{Y}\setminus\partial}f{\text{d}}\nu.\end{equation*}

A quasi-ergodic distribution is in general different from a QSD [Reference He, Zhang and Zhu27].

We provide some perspectives:

• The convergence (or ergodicity) in Theorem 8(ii) is uniform with respect to the initial distribution, while in contrast, the convergence in Theorem 8(i) is not uniform. Indeed, for the subcritical linear BDP, the stationary distribution is exponentially ergodic but not uniformly so [Reference Anderson5].

• Indeed, one can obtain uniform exponential ergodicity in Theorem 7(i) with (C18) or (C19) by choosing a non-reachable absorption set (potentially empty), hence imposing that the time to extinction is infinite. In this case, the QSD is in fact a stationary distribution [Reference Champagnat and Villemonais12].

• The subtle difference between the conditions for positive recurrence and for exponential ergodicity of QSDs lies in the fact that we have no a priori estimate of the decay parameter

  \begin{equation*}\psi_0=\inf\bigl\{\psi>0\,:\, \liminf_{t\to\infty}e^{\psi t}\mathbb{P}_x(X_t=x)>0\bigr\} \end{equation*}
  (which is independent of x) [Reference Champagnat and Villemonais12]. We cannot compare $\psi_0$ with $-\alpha$ when $R>1$ , or with $-\beta$ when $R>2$ and $\alpha=0$ . Refer to the constructive proofs (using Lyapunov functions) in Appendix A for details. Hence, one may believe that the condition we provide for quasi-ergodicity generically is stronger than that for ergodicity.

  The only gap cases that remain for QSDs are (C11), (C20), (C21), where neither existence of a QSD nor exponential ergodicity of the Q-process are known to occur, provided one of the three conditions hold.

3.5. Implosivity

Assume $\partial=\varnothing$ . Then $Y_t$ is irreducible on $\mathcal{Y}$ . Let $B\subsetneq\mathcal{Y}$ be a nonempty proper subset. Then $Y_t$ implodes towards B [Reference Menshikov and Petritis32] if there exists $t_*>0$ such that

\begin{equation*}\mathbb{E}_y(\tau_B)\le t_*,\quad \forall y\in \mathcal{Y} \setminus B.\end{equation*}

Implosion towards a single state $x\in\mathcal{Y}$ implies finite expected first return time to the state, and thus positive recurrence of x. Indeed,

\begin{equation*}\mathbb{E}_x\big(\tau^+_x\big)\le \mathbb{E}_x(J_1)+\sup\nolimits_{\{y\,:\, y\neq x\}}\mathbb{E}_y(\tau_x)<\infty,\end{equation*}

where $\tau_x^+=\tau_{\{x\}}^+$ , $\tau_x=\tau_{\{x\}}$ , and $J_1$ has finite expectation since x is not absorbing. Hence, $Y_t$ does not implode towards any transient or null recurrent state.

The process $Y_t$ is implosive if $Y_t$ implodes towards any state of $\mathcal{Y}$ , and otherwise, $Y_t$ is non-implosive. Hence, implosivity implies positive recurrence. If $Y_t$ implodes towards a finite nonempty subset of $\mathcal{Y}$ , then $Y_t$ is implosive (see Proposition 17).

Theorem 9. Assume $(\textbf{A1})$ – $(\textbf{A5})$ , $\partial=\varnothing$ , and that $\Omega$ is finite. Then $Y_t$ is implosive, and there exists $\epsilon>0$ such that for every nonempty finite subset $B\subseteq\mathcal{Y}$ and every $x\in \mathcal{Y}\setminus B$ ,

\begin{equation*}\mathbb{E}_x\left(\!\exp\!\left(\tau_B^{\epsilon}\right)\right)<\infty,\end{equation*}

if either (C18) or (C19) holds, while $Y_t$ is non-implosive otherwise.

3.6. Relations between absorbed CTMCs and non-absorbed CTMCs

It is worth comparing the properties of absorbed CTMCs to those of non-absorbed CTMCs, as first discussed by Karlin and McGregor [Reference Karlin and McGregor28]. Indeed, in the case of a finite set of absorbing states $\partial\neq\varnothing$ and an irreducible state space $\mathcal{Y}\setminus \partial$ , one can add positive transition rates to the transition matrix from the states in $\partial$ to a finite subset of states in $\mathcal{Y}\setminus\partial$ , such that $\mathcal{Y}$ is irreducible for the new chain. Conversely, if $\partial=\varnothing$ , then one can prescribe a finite set $\partial^{\prime}\subseteq\mathcal{Y}$ , and delete all transitions from $\partial^{\prime}$ to $\mathcal{Y}^{\prime}=\mathcal{Y}\setminus\partial$ , so that $\mathcal{Y}^{\prime}$ is irreducible and $\partial^{\prime}$ an absorbing set for the new chain. These operations can be viewed as simple extensions to those of [Reference Karlin and McGregor28], proposed in the context of BDPs. As the dynamical properties we discuss generically are determined by transitions among states of large values, the operations provide a way to link the dynamics of an absorbed CTMC with that of a corresponding non-absorbed CTMC, and vice versa.

Figure 2 shows the implications among the properties, in agreement with the parameter conditions derived in the main theorems. In Examples 2 and 3 below, counterexamples are given. It remains unknown whether exponential ergodicity of the Q-process implies implosivity, and whether ergodicity implies existence of a QSD; see Figure 2.

Figure 2.

Flow diagram of implications among the 21 conditions.

Implosivity is indeed a stronger property than positive recurrence (e.g., when $R\le1$ , $\alpha<0$ ), as shown in the following example (see also Table 1).

Example 3. Let $Y_t$ be an irreducible BDP on $\mathbb{N}_0$ with $\Omega=\{1,-1\}$ and

\begin{equation*}\lambda_{-1}(x)=x,\quad \lambda_1(x)=1,\quad x\in\mathbb{N}_0.\end{equation*}

In this case, $R=1$ and $\alpha=-1$ . By Theorems 7 and 9, $Y_t$ is positive recurrent and admits an ergodic stationary distribution, but $Y_t$ is non-implosive.

4.1. Stochastic reaction networks

A reaction network $(\mathcal{C},\mathcal{R})$ on a finite set $\mathcal{S}=\{S_1,\ldots,S_n\}$ (with elements called species) is an edge-labeled finite digraph with node set $\mathcal{C}$ (with elements called complexes) and edge set $\mathcal{R}$ (with elements called reactions), such that the elements of $\mathcal{C}$ are nonnegative linear combinations of species, $y=\sum_{i=1}^ny^i S_i$ , identified with vectors $y=\big(y^1,\ldots,y^n\big)$ in $\mathbb{N}_0^n$ . Reactions are directed edges between complexes, written as $y\to y^{\prime}$ . We assume that every species has a positive coefficient in some complex, and that every complex is in some reaction. Hence, the reaction network can be deduced from the reactions alone, and it is customary simply to list (or draw) the reactions. If $n=1$ , the reaction network is a one-species reaction network.

A stochastic reaction network (SRN) is a reaction network together with a CTMC X(t), $t\ge 0$ , on $\mathbb{N}_0^n$ , modeling the number of molecules of each species over time. A reaction $ y\to y^{\prime}$ fires with transition rate $\eta_{y\to y^{\prime}}(x)$ , in which case the chain jumps from $X(t)=x$ to $x+y^{\prime}-y$ [Reference Anderson and Kurtz4]. The Markov process with transition rates $\eta_{y\to y^{\prime}}\,:\,\mathbb{N}^n_0\to \mathbb{R}_{\geq 0}$ , ${y\to y^{\prime}}\in\mathcal{R}$ , has Q-matrix

\begin{equation*}q_{x,x+\omega}=\sum_{y \to y^{\prime}\in \mathcal{R}\,:\, y^{\prime}-y=\omega}\eta_{y\to y^{\prime}}(x).\end{equation*}

Hence, the transition rate from the state x to $x+\omega$ is

\begin{equation*}\lambda_{\omega}(x)=\sum_{y \to y^{\prime}\in \mathcal{R}\,:\, y^{\prime}-y=\omega}\eta_r(x).\end{equation*}

For (stochastic) mass-action kinetics, the transition rate for $y\to y^{\prime}$ is

\begin{equation*}\eta_{y\to y^{\prime}}(x)=\kappa_{y\to y^{\prime}}\tfrac{(x)!}{(x-y)!}1_{\{x^{\prime}\,:\, x^{\prime}\geq y\}}(x),\quad x\in\mathbb{N}^n_0\end{equation*}

(in accordance with the transition rate introduced in the introduction for one-species reaction networks), where $x!\,:\!=\,\prod_{i=1}^nx_i!$ , and $\kappa_{y\to y^{\prime}}$ is a positive reaction rate constant [Reference Anderson, Kurtz, Koeppl and Setti3, Reference Anderson and Kurtz4]. Generally, we number the reactions and write $\kappa_1,\kappa_2,\ldots$ for convenience.

In this section, we apply the results developed in Section 3 to some examples of SRNs.

Example 4. Consider the following two reaction networks:

\begin{equation*}(A)\quad \varnothing \ce{<=>[\kappa^{\textrm{A}}_1][\kappa^{\textrm{A}}_2]} \text{S},\qquad \text{and}\qquad(B)\quad \varnothing \ce{<=>[\kappa^{\textrm{B}}_1][\kappa^{\textrm{B}}_2]} \text{S},\quad 2\text{S} \ce{->[\kappa^{\textrm{B}}_3]}3\text{S},\end{equation*}

with $\Omega=\{-1,1\}$ in both cases and with transition rates

\begin{align*}\lambda^A_{-1}(x)&=\kappa^A_2 x,\qquad \lambda^A_{1}(x) =\kappa^A_1,\\\lambda^B_{-1}(x)&=\kappa^B_2 x,\qquad \lambda^B_{1}(x)=\kappa^B_1+\kappa^B_3x(x-1),\end{align*}

respectively. By Theorem 7, the first is positive recurrent and admits an exponentially ergodic stationary distribution on $\mathbb{N}_0$ , since $\alpha=-\kappa^A_2$ and $R=1$ , while by Theorem 1, the second reaction network is explosive for any initial state, since $\alpha=\kappa^B_3>0$ and $R=2$ . Indeed, these two reaction networks are structurally equivalent in the sense that there is only one irreducible component $\mathbb{N}_0$ [Reference Xu, Hansen and Wiuf48].

Example 5. Consider the following pair of SRNs from the introduction:

(6) \begin{equation}\text{S}\ce{<=>[1][2]} 2\text{S}\ce{<=>[4][4]} 3\text{S}\ce{<=>[6][1]} 4\text{S}\ce{->[1]} 5\text{S},\qquad \text{S}\ce{<=>[1][2]} 2\text{S}\ce{<=>[3][1]} 3\text{S}\ce{->[1]} 4\text{S}.\end{equation}

For the first reaction network, $R=4$ , $\alpha=0$ , and $\beta=1$ , and for the second, $R=3$ , $\alpha=0$ , and $\beta=0$ . By Theorem 1, the first is explosive for any initial state and the second does not explode for any initial state.

Example 6. (i) Consider a strongly connected reaction network:

For the underlying CTMC $Y_t$ , $\Omega=\{1,-2\}$ , and

\begin{equation*}\lambda_1(x) =\kappa_1 x+\kappa_2x(x-1),\qquad \lambda_{-2}(x)=\kappa_3 x(x-1)(x-2).\end{equation*}

Hence $Y_t$ is irreducible on $\mathbb{N}$ with 0 a neutral state. Moreover, $R=3$ and $\alpha=-2\kappa_3$ . By Theorem 7, there exists a unique exponentially ergodic stationary distribution on $\mathbb{N}$ .

(ii) Consider a similar reaction network including direct degradation of S:

The threshold parameters are the same as in (i). Let $\partial=\{0\}$ with $\mathbb{N}_0\setminus\partial=\mathbb{N}$ an irreducible component. By Theorem 7, the network has a uniformly exponentially ergodic QSD on $\mathbb{N}$ .

4.2. An extended class of branching processes

Consider an extended class of branching processes [Reference Chen16] with transition rate matrix $Q=(q_{x,y})_{x,y\in\mathbb{N}_0}$ :

(7) \begin{equation}q_{x,y}=\left\{\begin{array}{cl}r(x)\mu(y-x+1) &\quad \text{if}\quad y\ge x-1\ge0\quad \text{and}\quad y\neq x,\\[3pt] -r(x)(1-\mu(1)) &\quad \text{if}\quad y=x\ge1,\\[3pt] q_{0,y} & \quad \text{if}\quad y>x=0,\\[3pt] -q_0 &\quad \text{if}\quad y=x=0,\\[3pt] 0 &\quad \text{otherwise},\end{array}\right.\end{equation}

where $\mu$ is a probability measure on $\mathbb{N}_0$ , $q_0=\sum_{y\in\mathbb{N}}q_{0,y}$ , and r(x) is a positive finite function on $\mathbb{N}_0$ . Assume the following:

1. (H1) $\mu(0)>0$ , $\mu(0)+\mu(1)<1$ ;

2. (H2) $\sum_{y\in\mathbb{N}}q_{0,y}y<\infty$ , ${E}=\sum_{k\in\mathbb{N}_0}k\mu(k)<\infty$ ;

3. (H3) r(x) is a polynomial of degree $R\ge1$ for large x.

The next theorem follows from the results in Section 3. We would like to mention that the results below provide conditions for different dynamical regimes in terms of only R and M. In contrast, the condition for positive recurrence in [Reference Chen16] also depends on the integrability of a definite integral as well as summability of a series, which nonetheless never appear.

Proof. For all $k\in\mathbb{N}\cup\{-1\}$ , let

\begin{equation*}\lambda_k(x)=\begin{cases} r(x)\mu(k+1) \qquad\!\!\! \text{if}\ x\in\mathbb{N},\\ q_{0k} \qquad\qquad\quad\, \text{if}\ x=0.\end{cases}\end{equation*}

By ${(\textbf{H1})}$ , $\mu(k)>0$ for some $k\in\mathbb{N}$ . Note that $\mathcal{Y}\setminus\partial$ is irreducible, with $\partial=\varnothing$ if $q_0>0$ and $\partial=\{0\}$ if $q_0=0$ . Hence, regardless of $q_0$ , by positivity of r, ${(\textbf{A1})}$ – ${(\textbf{A2})}$ are satisfied with $\Omega_-=\{-1\}$ and $\Omega_+=\{y\,:\, q_{0y}>0\}\cup(\textrm{supp}\ \mu\setminus\{0,1\}-1)$ . Moreover, ${(\textbf{H2})}$ – ${(\textbf{H3})}$ imply ${(\textbf{A3})}$ – ${(\textbf{A4})}$ . Let $r(x)=ax^R+bx^{R-1}+\textrm{O}\big(x^{R-2}\big)$ with $a>0$ . Since $R\ge1$ , it coincides with $\max\{\deg\!(\lambda_{\omega})\,:\, \omega\in\Omega\}$ . It is straightforward to verify that

\begin{equation*}\alpha=a({E}-1),\quad \beta=\big(\tfrac{1}{2}a+b\big)({E}-1)-\tfrac{1}{2}a{E^{\prime}},\quad \text{and}\quad \gamma=b({E}-1),\end{equation*}

where ${E^{\prime}}=\sum_{k\in\mathbb{N}}k(k-1)\mu(k)>0$ . Hence $\alpha$ has the same sign as ${E}-1$ , and $\beta<0$ whenever ${E}=1$ (or equivalently, $\alpha=0$ ). Furthermore, $\alpha=0$ implies $\gamma=0$ . In addition, in light of the fact that $R\ge1$ , the condition ${E}\ge 1$ decomposes into three possibilities:

\begin{equation*}{E}>1,\quad \text{or}\quad {E}=1, \ R>1,\quad \text{or}\quad {E}=R=1.\end{equation*}

Then the conclusions follow directly from Theorems 1, 3, 7 and 9.

The extended branching process under more general assumptions (allowing more general forms of r) is addressed in [Reference Chen16]. In that reference, the conditions given for the dynamic behavior of the process seem more involved than here and even become void in some situations (e.g., in [Reference Chen16, Corollary 1.5(iii)], where the definite integral indeed is always infinite under ( $\textbf{H1}$ )–( $\textbf{H3}$ ).)

4.3. A general single-cell stochastic gene expression model

To model single-cell stochastic gene expression with bursty production, we propose the following one-species generalized reaction network (consisting potentially of infinitely many reactions) with mass-action kinetics:

(8) \begin{equation}m\text{S}\ce{->[c_m\mu_m(\textrm{k})]}(m+k)\text{S},\quad m=0,\ldots,J_1,\qquad m\text{S}\ce{->[r_m]}(m-1)\text{S},\quad m=1,\ldots,J_2,\end{equation}

where $c_m\ge0$ for $m=0,\ldots,J_1$ , $r_m\ge0$ for $m=1,\ldots,J_2$ , $J_1\in\mathbb{N}_0,\ J_2\in\mathbb{N}$ , and $\mu_m$ , for $m=0,\ldots,J_1$ , are probability distributions on $\mathbb{N}$ . Assume the following:

1. (H4) $J_1\le J_2$ , $c_0>0$ , $c_{J_1}>0$ , $r_1>0$ , and $r_{J_2}>0$ .

2. (H5) ${E}_m=\sum_{k=1}^{\infty}k\mu_m(k)<\infty$ , for $m=0,\ldots,J_1$ .

This network encompasses several single-cell stochastic gene expression models in the presence of bursting; see e.g. [Reference Be’er and Assaf8, Reference Chen and Jia17, Reference Mackey, Tyran-Kamińska and Yvinec31, Reference Shahrezaei and Swain41]. Ergodicity and an exact formula for the ergodic stationary distribution (when it exists) are the main concerns of these references. The first set of $J_1$ reactions accounts for bursty production of mRNA copies with transcription rate $c_m$ and burst size distribution $\mu_m$ . The second set of $J_2$ reactions accounts for degradation of mRNA with degradation rate $r_m$ [Reference Chen and Jia17, Reference Shahrezaei and Swain41].

The network (8) reduces to the specific model studied in the following references:

• [Reference Chen and Jia17, Section 4] (see also [Reference Mackey, Tyran-Kamińska and Yvinec31, Section 3.2]), when $J_1=0$ , $J_2=1$ , and $\mu_0$ is a geometric distribution.

• [Reference Schwabe, Rybakova and Bruggeman40], when $J_1=0$ , $J_2=1$ , and $\mu_0$ is a negative binomial distribution.

• [Reference Mackey, Tyran-Kamińska and Yvinec31, Example 3.6], when $J_1=J_2=1$ , and $\mu_0=\mu_1$ are geometric distributions.

• [Reference Falk, Mendler and Drossel21] when $J_1=2$ , $J_2=3$ , $\mu_0=\delta_1$ , $\mu_2=\delta_k$ for some $k\in\mathbb{N}$ , and $c_1=r_2=0$ . Here $\delta_i$ is the Dirac delta measure at i.

Proof. We have $\Omega=\{-1\}\cup\Big(\cup_{j=0}^{J_1}\textrm{supp}\ \mu_j\Big)$ , and

\begin{equation*}\lambda_{-1}(x)=\sum_{j=1}^{J_2}r_jx^{\,\underline{j}},\qquad \lambda_k(x)=\sum_{j=0}^{J_1}c_j\mu_j(k)x^{\,\underline{j}},\end{equation*}

for $k\in\mathbb{N}$ and $x\in\mathbb{N}_0$ , where $x^{\,\underline{j}}=\prod_{i=0}^{j-1}(x-i)$ is the descending factorial. By ${(\textbf{H4})}$ , ${(\textbf{A1})}$ – ${(\textbf{A2})}$ are satisfied; moreover, the irreducibility of $Y_t$ also follows from [Reference Xu, Hansen and Wiuf48]. Under ${(\textbf{H5})}$ , the mass-action kinetics yield ${(\textbf{A3})}$ – ${(\textbf{A4})}$ . Since $J_1\le J_2$ by ${(\textbf{H4})}$ , we have $R=J_2\ge1$ . Since

\begin{equation*}\sum_{\omega\in\Omega}\lambda_{\omega}(x)\omega=-\sum_{j=J_1+1}^{J_2}r_jx^{\,\underline{j}}+\sum_{j=1}^{J_1}\big(c_j{E}_j-r_j\big)x^{\,\underline{j}}+c_0{E}_0,\end{equation*}

\begin{equation*}\sum_{\omega\in\Omega}\lambda_{\omega}(x)\omega^2=\sum_{j=J_1+1}^{J_2}r_jx^{\,\underline{j}}+\sum_{j=1}^{J_1}\big(c_j{E}^{\prime}_j+r_j\big)x^{\,\underline{j}}+c_0{E}^{\prime}_0,\end{equation*}

we have $\alpha=c_{J_1}{E}_{J_1}\delta_{J_1,J_2}-r_{J_2}$ , where $\delta_{i,j}$ is the Kronecker delta. When $\alpha=0$ , we have

\begin{equation*}J_1=J_2,\quad c_{J_2}{E}_{J_2}=r_{J_2},\quad \gamma=c_{J_2-1}{E}_{J_2-1}-r_{J_2-1},\end{equation*}

\begin{equation*} \beta=c_{J_2-1}{E}_{J_2-1}-r_{J_2-1}-\frac{1}{2}c_{J_2}\big({E}_{J_2}+{E}^{\prime}_{J_2}\big).\end{equation*}

The combination of the conditions (i) and (ii) is equivalent to $\alpha<0$ ; the condition (iii) is equivalent to $R>2$ , $\alpha=0$ , $\beta\le0$ ; the condition (iv) is equivalent to $R=2$ , $\alpha=0$ , $\beta<0$ ; the condition (v) is equivalent to $R=1$ , $\alpha=0$ , $\gamma<0$ . The conclusions then follow from Theorems 7 and 9.

4.4. Stochastic populations under bursty reproduction

Two stochastic population models with bursty reproduction are investigated in [Reference Be’er and Assaf8].

The first model is a Verhulst logistic population process with bursty reproduction. The process $Y_t$ is a CTMC on $\mathbb{N}_0$ with transition rate matrix $Q=(q_{x,y})_{x,y\in\mathbb{N}_0}$ satisfying

\begin{equation*}q_{x,y}=\begin{cases} c\mu(\,j)x \qquad \text{if}\ y=x+j,\ j\in\mathbb{N},\\[3pt] \frac{c}{K}x^2+x \quad\,\, \text{if}\ y=x-1\in\mathbb{N}_0,\\[3pt] 0 \qquad\qquad\ \text{otherwise},\end{cases}\end{equation*}

where $c>0$ is the reproduction rate, $K\in\mathbb{N}$ is the typical population size in the long-lived metastable state prior to extinction [Reference Be’er and Assaf8], and $\mu$ is the burst size distribution. Assume the following:

(H6) ${E_b}=\sum_{k=1}^{\infty}k\mu(k)<\infty$ .

Approximations of the mean time to extinction and QSD are discussed in [Reference Be’er and Assaf8] against various burst size distributions of finite mean (e.g., Dirac measure, Poisson distribution, geometric distribution, negative-binomial distribution). Nevertheless, the existence of a QSD is not proved there. Here we prove the certain absorption and ergodicity of the QSD for this population model.

The second model is a runaway model of a stochastic population including bursty pair reproduction [Reference Be’er and Assaf8]. This model can be described as a generalized reaction network, where c, K, and $\mu$ are defined as in the first model. The survival probability of this population model is addressed in [Reference Be’er and Assaf8]. Nevertheless, it turns out that this model is explosive for any initial state.

5.1. Proof of Theorem 1

Hereafter, we use the notation $[m,n]_1$ ( $[m,n[_1$ , etc.) for the set of consecutive integers from m to n, with $m,n\in\mathbb{N}_0\cup\{+\infty\}$ . The notation is adopted from [Reference Xu, Hansen and Wiuf48].

Moreover, throughout the proofs, we assume without loss of generality that $\mathcal{Y}=\mathbb{N}_0$ , and $\partial\subseteq\{0\}$ for ease of exposition. Indeed, the dynamical properties of the CTMCs discussed in this paper depend only on the transition structure of the states $x\in\mathcal{Y}$ with large values of x. By Assumptions ( $\textbf{A4}$ )–( $\textbf{A5}$ ), all jumps in $\Omega$ are possible. When $\partial\neq\varnothing$ , it is standard to ‘glue’ all states in $\partial$ to be a single state 0, since $\partial$ is finite and the set of states in $\mathcal{Y}\setminus\partial$ one jump away from $\partial$ is also finite, by ( $\textbf{A2}$ ).

We prove the conclusions case by case.

(a) Assume $\partial=\varnothing$ . Then $Y_t$ is irreducible on $\mathbb{N}_0$ , and one can directly apply Propositions 2 and 3 with appropriate Lyapunov functions to be determined.

(b) Assume $\partial\neq\varnothing$ . Let $Z_t$ be the irreducible CTMC on the state space $\mathcal{Y}\setminus\partial$ with $Z_0=Y_0$ and transition operator $\widetilde{Q}$ being Q restricted to $\mathcal{Y}\setminus\partial$ :

\begin{equation*}\widetilde{q}_{x,y}=q_{x,y},\quad \text{for all}\ x,\, y\in\mathcal{Y}\setminus\partial \ \text{and}\ x\neq y.\end{equation*}

In the following, we show that $Z_t$ is explosive if and only if $Y_t$ is explosive, and hence the case (b) reduces to the case (a). This equivalence is not quite trivial. There is a positive probability that, starting from any non-absorbing state, the chain will jump to an absorbing state in a finite number of steps. So we need to show that this will not happen with probability one. Otherwise, explosivity is not possible. Assume first that $Z_t$ is explosive. Then $\widetilde{Q}v=v$ for some bounded nonnegative nonzero v. Let $u_x=v_x \unicode{x1D7D9}_{\mathcal{Y}\setminus\partial}(x),\ \forall x\in\mathcal{Y}$ . It is straightforward to verify that $Qu=u$ . By Proposition 4, $Y_t$ is also explosive. Conversely, assume that $Y_t$ is explosive; then $Qu=u$ for some bounded nonnegative nonzero u. Let $w=u|_{\partial}$ , i.e., $w_x=u_x$ for all $x\in\partial$ . Since $Q|_{\partial}$ is a lower-triangular matrix with nonpositive diagonal entries, and $w\ge0$ , it is readily deduced that $w=0$ by Gaussian elimination. This implies from $Qu=u$ that $\widetilde{Q}v=v$ with $v=u|_{\mathcal{Y}\setminus\partial}$ . Hence $Z_t$ is explosive. To sum up, $Z_t$ is explosive if and only if $Y_t$ is explosive.

Based on the above analysis, it remains to prove the conclusions for the case (a) using Propositions 2 and 3. We first prove the conclusions assuming $\Omega$ is finite.

(i) We prove explosivity by Proposition 2. Let the lattice interval $A=[0,x_0-\min\!\Omega_-[_1$ for some $x_0>1$ to be determined. Since $\#\Omega_-<\infty$ , $A\subseteq\mathbb{N}_0$ is finite. Let f be decreasing and bounded such that $f(x)=\unicode{x1D7D9}_{[0,x_0[_1}(x)+x^{-\delta}\unicode{x1D7D9}_{[x_0,\infty[_1}(x)$ for all $x\ge x_0$ , with $\delta>0$ to be determined. Obviously, Proposition 2(i) is satisfied for the set A. Next we verify the conditions in Proposition 2(ii). It is easy to verify by straightforward calculation that

\begin{equation*}Qf(x)<-\epsilon \quad \text{for all}\ x\in \mathbb{N}_0\setminus A,\end{equation*}

where $\epsilon=\delta\alpha/2$ provided (C1) holds with $\delta<R-1$ , or $\epsilon=\delta\!\left(\beta-\delta\vartheta\right)/2$ provided (C2) holds with $\delta<\min\{\beta/\vartheta,R-2\}$ , and $x_0$ is chosen large enough. Since $\delta>0$ can be arbitrarily small, in either case, there exist $\delta$ and $\epsilon$ such that the conditions in Proposition 2 are fulfilled, and thus $\mathbb{E}_x\zeta<+\infty$ for all $x\in\mathcal{Y}$ . In particular, $Y_t$ is explosive a.s.

(ii) Now we prove non-explosivity using Proposition 3. Let $f(x)=\log\log\!(x+1)$ and $g(x)=(|\alpha|+|\beta|+1)(x+M)$ for all $x\in\mathbb{N}_0$ , with some $M>0$ to be determined. One can show that all the conditions in Proposition 3 are satisfied with some large constant $M>0$ , provided neither (C1) nor (C2) holds. Hence $Y_t$ is non-explosive.

5.2. Proof of Theorem 2

We first prove explosivity under the condition (C1). Let f be as in the proof of Theorem 1(i), and let

\begin{equation*}\alpha_-=\lim_{x\to\infty}\tfrac{\sum_{\omega\in\Omega_-}\lambda_{\omega}(x)\omega}{x^R}, \qquad\alpha_+=\lim_{x\to\infty}\tfrac{\sum_{\omega\in\Omega_+}\lambda_{\omega}(x)\omega}{x^R}.\end{equation*}

Then $\alpha=\alpha_++\alpha_-$ . Since $\alpha>0$ , we have $R_+=R$ and there exists $\epsilon_0\in]0,1[$ such that $\alpha_-+(1-\epsilon_0)\alpha_+>0$ . By Proposition 1, there exist $N_0,\ u^{\prime}\in\mathbb{N}$ such that

\begin{equation*}\frac{\sum_{\omega\in\Omega_+\cap[1,N_0]_1}\lambda_{\omega}(x)\omega}{\sum_{\omega\in\Omega_+}\lambda_{\omega}(x)\omega}\ge1-\epsilon_0 \quad \text{for all}\ x\ge u^{\prime}.\end{equation*}

By ( $\textbf{A3}$ ), $\Omega\setminus]N_0,\infty[_1$ is finite. Hence, choosing $x_0\ge u^{\prime}$ large, we have for all $x\in\mathcal{Y}\setminus A$

\begin{equation*}\begin{split}Qf(x)&=\sum_{\omega\in\Omega_-}\lambda_{\omega}(x)\big((x+\omega)^{-\delta}-x^{-\delta}\big)+\sum_{\omega\in\Omega_+}\lambda_{\omega}(x)\big((x+\omega)^{-\delta}-x^{-\delta}\big)\\[3pt]&\le\sum_{\omega\in\Omega_-}\lambda_{\omega}(x)\big((x+\omega)^{-\delta}-x^{-\delta}\big)+\sum_{\omega\in\Omega_+\cap[1,N_0]_1}\lambda_{\omega}(x)\big((x+\omega)^{-\delta}-x^{-\delta}\big)\\[3pt]&\le x^{-\delta}\sum_{\omega\in\Omega_-}\lambda_{\omega}(x)\big({-}\omega\delta x^{-1}+\textrm{O}\big(x^{-2}\big)\big)+x^{-\delta}\sum_{\omega\in\Omega_+\cap[1,N_0]_1}\lambda_{\omega}(x)\big({-}\omega\delta x^{-1}+\textrm{O}\big(x^{-2}\big)\big)\\[3pt]&=-\delta\alpha_-x^{R-1-\delta}+\textrm{O}\big(x^{R-2-\delta}\big)-\delta x^{-1-\delta}\sum_{\omega\in\Omega_+\cap[1,N_0]_1}\lambda_{\omega}(x)\omega\\[3pt]&\le-\delta\alpha_-x^{R-1-\delta}+\textrm{O}\big(x^{R-2-\delta}\big)-\delta (1-\epsilon_0)x^{-1-\delta}\sum_{\omega\in\Omega_+}\lambda_{\omega}(x)\omega\\[3pt]&=-\delta\big(\alpha_-+(1-\epsilon_0)\alpha_+\big)x^{R-1-\delta}+\textrm{O}\big(x^{R-2-\delta}\big)<-\epsilon,\end{split}\end{equation*}

where

\begin{equation*}\epsilon=\frac{\delta\big(\alpha_-+(1-\epsilon_0)\alpha_+\big)}{2},\end{equation*}

and $\delta<R-1$ . The rest of the argument is the same as that of the proof of Theorem 1(i).

Next, we prove non-explosivity under (C3), (C4), or (C5). Let f and g be as in (ii) in the proof of Theorem 1. By ( $\textbf{A3}$ ), for some large $M>0$ to be determined, for all $x\in\mathbb{N}_0$ ,

\begin{align*} {Qf(x)}&=\sum_{\omega\in\Omega_-}\lambda_{\omega}(x)\big(\!\log\log\!(x+1+\omega)-\log\log\!(x+1)\big)\\[2pt]&\quad+\sum_{\omega\in\Omega_+}\lambda_{\omega}(x)\big(\!\log\log\!(x+1+\omega)-\log\log\!(x+1)\big)\\[2pt]&=\sum_{\omega\in\Omega_-}\lambda_{\omega}(x)\log\Bigg(1+\frac{\log\bigl(1+\frac{\omega}{x+1}\bigr)}{\log\!(x+1)}\Bigg)+\sum_{\omega\in\Omega_+}\lambda_{\omega}(x)\log\Bigg(1+\frac{\log\bigl(1+\frac{\omega}{x+1}\bigr)}{\log\!(x+1)}\Bigg)\\[2pt]&=\sum_{\omega\in\Omega_-}\lambda_{\omega}(x)\Bigg(\frac{\omega}{(x+1)\log\!(x+1)}+\textrm{O}\bigl((x+1)^{-2}(\!\log (x+1))^{-1}\bigr)\Bigg)\\[2pt]&\quad+\sum_{\omega\in\Omega_+}\lambda_{\omega}(x)\log\Bigg(1+\frac{\log\bigl(1+\frac{\omega}{x+1}\bigr)}{\log\!(x+1)}\Bigg)\\[2pt]& \le\sum_{\omega\in\Omega_-}\lambda_{\omega}(x)\Bigg(\frac{\omega}{(x+1)\log\!(x+1)}+\textrm{O}\Big((x+1)^{-2}(\!\log (x+1))^{-1}\Big)\Bigg)\\[2pt]&\quad+\sum_{\omega\in\Omega_+}\lambda_{\omega}(x)\frac{\omega}{(x+1)\log\!(x+1)}\\[2pt]&=\frac{1}{(x+1)\log\!(x+1)}\sum_{\omega\in\Omega}\lambda_{\omega}(x)\omega+\textrm{O}\Big((x+1)^{R-2}(\!\log (x+1))^{-1}\Big)\\[2pt]&=\alpha\frac{(x+1)^{R-1}}{\log\!(x+1)}+\textrm{O}\Big((x+1)^{R-2}(\!\log (x+1))^{-1}\Big)\le g(f(x)),\end{align*}

provided (C3), (C4), or (C5) holds. The rest of the proof is the same as that of Theorem 1(ii).

5.3. Proof of Theorem 3

Let $h_A=\mathbb{P}_{Y_0}(\tau_A<\infty)$ be the hitting probability [Reference Norris36]. In particular, $h_A$ is called the absorption probability if A is a closed communicating class. To verify conditions for certain absorption requires the following property for hitting probabilities. For any set $A\subseteq\mathcal{Y}$ , we write $h_A(i)$ for $h_A$ , to emphasize the dependence of the hitting probability on the initial state $i\in\mathcal{Y}$ . In particular, if $A=\{x\}$ is a singleton, we simply write $h_x$ for $h_A$ .

Assume without loss of generality that $\mathcal{Y}=\mathbb{N}_0$ , and $\partial\subseteq\{0\}$ .

(i) We first show recurrence and transience. The idea of applying the classical semimartingale approach originates from [Reference Lamperti29]. It suffices to show recurrence and transience for the embedded discrete-time Markov chain $\widetilde{Y}_n$ of $Y_t$ .

To show recurrence, let $Z_n=\log\log\!(\widetilde{Y}_n+1)$ . Since a one-to-one bicontinuous transformation of the state space preserves the Markov property and recurrence, it suffices to show recurrence for $Z_n$ . In light of the expression for the transition probability of $\widetilde{Y}_n$ , we have

\begin{equation*}\mathbb{E}(Z_{n+1}-Z_n|Z_n=\log\log\!(x+1))=\frac{1}{\sum_{\omega\in\Omega}\lambda_{\omega}(x)}\sum_{\omega\in\Omega}\lambda_{\omega}(x)(\!\log\log\!(x+\omega)-\log\log x).\end{equation*}

By tedious but straightforward computation, we have the following asymptotic expansion:

\begin{equation*}\mathbb{E}\big(Z_{n+1}-Z_n|Z_n=\log\log\!(x+1)\big)=\frac{\alpha x^R+\beta x^{R-1}-\vartheta x^{R-1}(\!\log x)^{-1}+\textrm{O}\big(x^{R-2}\big)}{(1+x)\log\!(1+x)\sum_{\omega\in\Omega}\lambda_{\omega}(x)}.\end{equation*}

From this asymptotic expansion, and noting that $\vartheta>0$ , we have

\begin{equation*}\mathbb{E}\big(Z_{n+1}-Z_n|Z_n=\log\log\!(x+1)\big)\le0,\quad \forall n\in\mathbb{N}_0,\ \text{for all large}\ x,\end{equation*}

provided either (C3) or (C6) holds. From Proposition 6 follows the recurrence of $Z_n$ , and thus the recurrence of $\widetilde{Y}_n$ as well.

Next, we prove the transience of $\widetilde{Y}_n$ under the reverse conditions (that is, neither (C3) nor (C6) holds). Let $Z^{\prime}_n=1-\big(1+\widetilde{Y}_n\big)^{-\delta}$ , with $\delta>0$ to be determined. Again, $Z^{\prime}_n$ is a Markov chain, and $\widetilde{Y}_n\to\infty$ if and only if $Z^{\prime}_n\to1$ , which implies that (14) is fulfilled for $Z^{\prime}_n$ with $M=1$ , since $\widetilde{Y}_n$ on a subset of $\mathbb{N}_0$ is irreducible. Similarly to the above computation, we have the asymptotic expansion

\begin{equation*}\mathbb{E}\Big(Z^{\prime}_{n+1}\!-Z^{\prime}_n|Z^{\prime}_n=1\!-(1\!+x)^{-\delta}\Big)=\frac{\delta}{(1+x)^{\delta+1}\sum_{\omega\in\Omega}\lambda_{\omega}(x)}\big(\alpha x^R+(\beta-\delta\vartheta)x^{R-1}\!+\textrm{O}\big(x^{R-2}\big)\big).\end{equation*}

Hence

\begin{equation*}\mathbb{E}\Big(Z^{\prime}_{n+1}-Z^{\prime}_n|Z^{\prime}_n=1-(1+x)^{-\delta}\Big)\ge0 \qquad \forall n\in\mathbb{N}_0\end{equation*}

for all large x (and so for all values of $z=Z^{\prime}_n$ in some interval $C\le z<1$ ), provided $\alpha>0$ or $\alpha=0,\ \beta>0$ with $\delta<\frac{\beta}{\vartheta}$ . By Proposition 7,

\begin{equation*}\mathbb{P}\Big(\lim_{n\to\infty}Z^{\prime}_n=1\Big)=1,\end{equation*}

that is,

\begin{equation*}\mathbb{P}\Big(\lim_{n\to\infty}\widetilde{Y}_n=\infty\Big)=1,\end{equation*}

meaning $\widetilde{Y}_n$ is transient.

(ii) Let $\widetilde{\omega}\in\Omega_+$ . Let $k_0=\min\{l\in\mathbb{N}\,:\, l\widetilde{\omega}\in\mathbb{N}\}$ . Define $Z_t$ to be a CTMC on $\mathbb{N}_0$ with transition matrix $\widetilde{Q}=(\widetilde{q}_{xy})$ satisfying, for all $x\neq y,\ x,y\in\mathbb{N}_0$ ,

\begin{equation*}\widetilde{q}_{xy}=\begin{cases} q_{xy} \quad \text{if}\ x\in\mathbb{N},\\ 1 \quad\ \ \, \text{if}\ x=0\ \text{and}\ y=j\widetilde{\omega},\ j=1,\ldots,k_0,\\ 0 \quad\ \,\ \text{otherwise}.\end{cases}\end{equation*}

It is easy to verify that $Z_t$ is irreducible on $\mathbb{N}_0$ . In the following, we show that the recurrence of $Z_t$ is equivalent to the absorption of $Y_t$ , which yields the conclusion.

On the one hand, applying (i) to $Z_t$ , we have that $Z_t$ is recurrent if and only if (C3) or (C6) holds. On the other hand, from Proposition 9, $Z_t$ is recurrent if and only if $h^Z_{0}(i)=1$ for all $i\in\mathcal{Y}$ , where $h^Z_{0}(i)$ is the hitting probability for $Z_t$ . By Proposition 8, $h^Z_{0}(i)=1$ for all $i\in\mathcal{Y}$ if and only if $(1,\ldots,1)$ is the minimal nonnegative solution to the linear equations

\begin{equation*}\begin{cases} x_i=1,\qquad\quad\!\qquad\qquad\quad\ i=0,\\[3pt] \sum_{j\in\partial\setminus\{i\}}\widetilde{q}_{ij}(x_i-x_j)=0,\quad i\in\mathbb{N},\end{cases}\end{equation*}

which, by the definition of $\widetilde{Q}$ , are identical to

(9) \begin{equation}\begin{cases} x_i=1,\qquad\qquad\,\quad\qquad\quad\ i=0,\\[3pt] \sum_{j\in\mathcal{Y}\setminus\{i\}}q_{ij}\big(x_i-x_j\big)=0,\quad i\in\mathbb{N}.\end{cases}\end{equation}

By Proposition 8, $Y_t$ has certain absorption if and only if $(1,\ldots,1)$ is the minimal nonnegative solution to (9). Hence the recurrence of $Z_t$ is equivalent to the certain absorption of $Y_t$ .

5.4. Proof of Theorem 4

If $\Omega$ is infinite, then by $(\textbf{A3})$ , $\Omega_+$ is also infinite, while $\Omega_-$ is finite. Hence, the asymptotic expansions of the sum over all negative jumps in $\Omega_-$ in the proof of Theorem 3 remain valid, while the asymptotic expansions of the sum over all positive jumps in $\Omega_+$ might fail, in that the sum is infinite. Nevertheless, an upper estimate of Qf(x) for certain Lyapunov functions is still possible, as demonstrated in the proof of Theorem 2. A careful examination of the arguments of Theorem 3 shows that the desired upper estimates of Qf(x) hold under the respective conditions listed in Theorem 4, by replacing the asymptotic expansions by one-sided inequalities.

5.5. Proof of Theorem 5

Assume without loss of generality that $\mathcal{Y}=\mathbb{N}_0$ .

We first prove the existence of moments of hitting times assuming $\Omega$ is finite, by applying Proposition 10(i) case by case. We now prove the existence of moments under (C13) for $0<\delta<1/2$ . Let $f(x)=\sqrt{x+1}$ for $x\in\mathbb{N}_0$ . One can directly verify that for every $0<\sigma=2\delta<2$ , there exists $0<c<+\infty$ such that $Q f^{\sigma}(x)\le -cf^{\sigma-2}(x)$ for all large x. By Proposition 10(i), there exists $a>0$ such that

\begin{equation*}\mathbb{E}_x\big(\tau^{\epsilon}_{\{f\le a\}}\big)<+\infty,\quad \forall x\in\mathcal{Y},\ \forall 0<\epsilon<\sigma/2.\end{equation*}

Moreover, $\{f\le a\}$ is finite since $\lim_{x\to\infty}f(x)=+\infty$ .

Analogous arguments apply to the cases (C16) for $0<\delta<1$ and (C10) for $\delta>0$ with $f(x)=\log (x+1)$ ; the condition (C14) for $0<\delta<\frac{\beta}{\beta-\gamma}$ with $f(x)=\sqrt{x+1}$ ; the condition (C9) with $f(x)=\log\log\!(x+1)$ , and the condition (C3) with $f(x)=x+1$ .

Next we prove the nonexistence of hitting times by Proposition 10(ii) assuming $\Omega$ is finite. For all cases, let $f(x)=g(x)$ , and specifically, let $f(x)=x+1$ in the cases (C13) for $\delta>1$ and (C15) for $\delta>\tfrac{\beta}{\beta-\gamma}$ , and $f(x)=\log\!(x+1)$ in the case (C16) for $\delta>1$ . Note that in the case (C15) for $\delta>\tfrac{\beta}{\beta-\gamma}$ , we have that $-\frac{\beta}{\vartheta}<1$ is equivalent to $\gamma>0$ . The tedious but straightforward verification of the conditions (ii-1)–(ii-4) in Proposition 10 is left to the interested reader.

5.6. Proof of Theorem 6

As alluded to in the proof of Theorem 4, the desired upper estimate of Qf(x) with the same f under (C3) still holds as in the proof of Theorem 5, by replacing the asymptotic expansions by one-sided inequalities.

5.7. Proof of Theorem 7

Assume without loss of generality that $\mathcal{Y}=\mathbb{N}_0$ and $\partial\subseteq\{0\}$ . We prove this theorem by Propositions 12–13, assuming $\Omega$ is finite. We emphasize that the nonexistence of a QSD rests only on the failure of certain absorption.

(i) Since $Y_t$ is recurrent, we have that $\widetilde{Y}_n$ is recurrent by Proposition 5. Let $\widetilde{\pi}$ be its unique (up to a scalar) invariant measure. By [Reference Norris36, Theorem 3.5.1], and since $\sum_{\omega\in\Omega}\lambda_{\omega}(x)>0$ is bounded away from zero uniformly in x,

(10) \begin{equation}\pi(x)=\frac{\widetilde{\pi}(x)}{q_x}=\frac{\widetilde{\pi}(x)}{\sum_{\omega\in\Omega}\lambda_{\omega}(x)},\quad x\in\mathbb{N}_0,\end{equation}

is a finite stationary measure for $Y_t$ . Let

\begin{equation*}b=\lim_{x\to\infty}\frac{\sum_{\omega\in\Omega}\lambda_\omega(x)}{x^R}=\sum_{d_{\omega}=R}a_{\omega}.\end{equation*}

Note that when $R=0$ , we have $\gamma=0$ , and hence $R=0$ , $\gamma\neq0$ , and $\beta\le0$ cannot occur (see Table 1). Thanks to Theorem 3, we only need to show the following:

• $Y_t$ is positive recurrent if one of (C3), (C9), (C10), (C11) holds.

• $Y_t$ is null recurrent if one of the following conditions holds:

  • ${\circ}\ $ $R=0$ , $\alpha=0$ ;

  • ${\circ}\ $ $R=1$ , $\alpha=\gamma=0$ ;

  • ${\circ}\ $ $R=2$ , $\alpha=0$ , $\beta=0$ ;

  • ${\circ}\ $ $R=1$ , $\alpha=0$ , $\beta\le 0\le\gamma$ .

First, we show positive recurrence and exponential ergodicity. Notice that (C9) $\cup$ (C10) $=$ (C18) $\cup$ (C21). Moreover, (C3) can be decomposed into (C17) and (C17)^′ $R=0$ , $\alpha<0$ .

For (C11) or (C17)^′, let $f(x)=x+1$ ; for (C21), let $f(x)=\log\!(x+1)$ . Then one can verify (using the asymptotics of Qf in Appendix A.7) that there exists $\epsilon>0$ such that $Q f(x)\le-\epsilon$ for all large x. With an appropriate finite set F, by Proposition 11, $X_t$ is positive recurrent and there exists a unique ergodic stationary distribution on $\mathbb{N}_0$ .

For (C17), let $f(x)=x+1$ ; for (C18), let $f(x)={\log\log\!(x+2)}$ . Then one can similarly verify that there exists $\epsilon>0$ such that $Q f(x)\le-\epsilon f(x)$ for all large x. By Proposition 12, $Y_t$ is positive recurrent and there exists a unique exponentially ergodic stationary distribution on $\mathbb{N}_0$ .

Next, we show null recurrence case by case, applying Proposition 16 as well as nonexistence of moments of passage times in Theorem 5.

• Assume $R=0$ , $\alpha=0$ . Then

  \begin{equation*}\lambda_{\omega}(x)\equiv a_{\omega},\quad \omega\in\Omega,\quad \sum_{\omega\in\Omega}a_{\omega}\omega=0,\end{equation*}

since $\lambda_{\omega}$ are polynomials. Let $c=-\min\!\Omega$ , $g(x)=x+c$ , $h(x)=(x+c)^{1/2}$ , and $f(x)=1$ . Hence it is easy to verify that Proposition 16(i) and Proposition 16(iii) hold. Let $\mathcal{F}_n$ be the filtration to which $\widetilde{Y}_n$ is adapted. Then

\begin{align*} \mathbb{E}\big(h\big(\widetilde{Y}_{n+1}-h\big(\widetilde{Y}_n\big)|\mathcal{F}_n\big)\big) =&\sum_{\omega\in\Omega}a_{\omega}\bigg(\sqrt{\widetilde{Y}_n+\omega+c}-\sqrt{\widetilde{Y}_n+c}\bigg)\\ =&\sum_{\omega\in\Omega}\frac{a_{\omega}\omega}{\sqrt{\vphantom{\int^{\int}}\widetilde{Y}_n+\omega+c}+\sqrt{\vphantom{\int^{\int}}\widetilde{Y}_n+c}}\\ =&\sum_{\omega\in\Omega}\left(\frac{a_{\omega}\omega}{\sqrt{\vphantom{\int^{\int}}\widetilde{Y}_n+\omega+c}+\sqrt{\vphantom{\int^{\int}}\widetilde{Y}_n+c}} -\frac{a_{\omega}\omega}{\sqrt{\vphantom{\int^{\int}}\widetilde{Y}_n+c}+\sqrt{\vphantom{\int^{\int}}\widetilde{Y}_n+c}}\right)\\ =&\sum_{\omega\in\Omega}\frac{\sqrt{\vphantom{\int^{\int}}\widetilde{Y}_n+c}-\sqrt{\vphantom{\int^{\int}}\widetilde{Y}_n+\omega+c}}{2\sqrt{\vphantom{\int^{\int}}\widetilde{Y}_n+c} \Big(\sqrt{\vphantom{\int^{\int}}\widetilde{Y}_n+\omega+c}+\sqrt{\vphantom{\int^{\int}}\widetilde{Y}_n+c}\Big)} a_{\omega}\omega\\ =&\frac{1}{2\sqrt{\vphantom{\int^{\int}}\widetilde{Y}_n+c}}\sum_{\omega\in\Omega}\frac{a_{\omega}\omega^2}{\Big(\sqrt{\vphantom{\int^{\int}}\widetilde{Y}_n+\omega+c} +\sqrt{\vphantom{\int^{\int}}\widetilde{Y}_n+c}\Big)^2}\ge0, \end{align*}

which shows that Proposition 16(ii) also holds for the finite set $A=[0,c]_1$ . Moreover,

\begin{equation*}\mathbb{E}\big(\widetilde{Y}_{n+1}-\widetilde{Y}_n|\mathcal{F}_n\big)=\sum_{\omega\in\Omega}a_{\omega}\omega\equiv0<1=f(x) \quad P_z\text{-a.s.}\quad \text{on}\ \{\tau_A>n\},\end{equation*}

which shows Proposition 16(iv) also holds. Since

\begin{equation*}\pi(\,j)=\frac{\widetilde{\pi}(\,j)}{\sum_{\omega\in\Omega}a_{\omega}},\end{equation*}

by Proposition 16, we have $\sum_{j\in\mathbb{N}_0}\pi(\,j)=\infty$ . By the uniqueness of stationary measures under the recurrence condition [Reference Miller35], we know $Y_t$ is null recurrent.

• Assume $R=1$ , $\alpha=\gamma=0$ . Then

  \begin{equation*}\mathbb{E}\big(\widetilde{Y}_{n+1}-\widetilde{Y}_n|\widetilde{Y}_n=x\big) =\frac{\sum_{\omega\in\Omega}\lambda_{\omega}(x)\omega}{\sum_{\omega\in\Omega}\lambda_{\omega}(x)}\equiv0\end{equation*}

for all large x. Applying [Reference Aspandiiarov and Iasnogorodski6, Theorem 3.5(iii)] with $\varepsilon_0=1/2$ and $a>1$ , we have

(11) \begin{equation}\sum_{j\ge2}\frac{\widetilde{\pi}(\,j)}{j\sqrt{\log j}}=\infty.\end{equation}

From (10), (11), and the fact that $\deg\big(\sum_{\omega\in\Omega}\lambda_{\omega}(x)\big)=1$ for large x, it follows that

\begin{equation*}\sum_{j\ge2}\frac{\pi(\,j)}{\sqrt{\log j}}=\infty,\end{equation*}

which implies that $\sum_{j\ge2}\pi(\,j)=\infty$ . The rest of the argument is the same as in the case $R=0$ , $\alpha=0$ .

• Assume $R=2$ , $\alpha=\beta=0$ . Let $g_{\delta}(x)=(\!\log x)^{\delta}$ for some $1<\delta\le2$ . Choose $g=g_{2}$ , $h=g_{3/2}$ , and $A=[0,a]$ for some large $a>0$ to be determined. Then it is straightforward to verify (using the asymptotics of Qf in Appendix A.7) that

\begin{equation*}\mathbb{E}\big(g_{\delta}\big(\widetilde{Y}_{n+1}\big)-g_{\delta}\big(\widetilde{Y}_n\big)|\mathcal{F}_n\big) =\delta(\delta-1)\frac{\vartheta}{b} x^{-2}(\!\log x)^{\delta-2}+\textrm{O}\big(x^{-3}(\!\log x)^{\delta-1}\big).\end{equation*}

Hence it is easy to show that h and g satisfy Proposition 16(i)–(iii).

Moreover, it is easy to show that there exist $C= C(\vartheta,b)>0$ and $f(x)=Cx^{-2}$ such that

\begin{equation*}\mathbb{E}\big(g(\widetilde{Y}_{n+1}-g\big(\widetilde{Y}_n\big)|\mathcal{F}_n)\big)\le f\big(\widetilde{Y}_n\big)\quad \mathbb{P}_x\text{-a.s.}\quad \text{on}\quad \{\tau_A>n\}.\end{equation*}

By Proposition 16,

\begin{equation*}\sum_{j\in\mathcal{Y}}f(\,j)\widetilde{\pi}(\,j)=\infty.\end{equation*}

Since $\sum_{\omega\in\Omega}\lambda_{\omega}(x)=bx^2+\textrm{O}(x)$ , substituting (10) we obtain

\begin{equation*}\sum_{j\in\mathcal{Y}}\pi(\,j)=\infty.\end{equation*}

Hence $Y_t$ is null recurrent.

• Assume $R=1$ , $\alpha=0$ , $\gamma>0$ . To prove null recurrence, it suffices to show that there exists $x\in\mathbb{N}_0$ such that $\mathbb{E}_x\big(\tau^+_{x}\big)=\infty$ . Let $B\subsetneq\mathbb{N}_0$ be as in Theorem 5(ii) and $x=\max B\in\mathbb{N}_0$ . Hence $Y_{J_1}\in\Omega+x\subsetneq\mathbb{N}_0$ a.s. and $(\Omega_++x)\cap B=\varnothing$ . By the Markov property of $Y_t$ ,

\begin{equation*}\mathbb{E}_x\big(\tau_x-J_1|Y_{J_1}=j\big)=\mathbb{E}_j(\tau_x),\quad \forall j\in\mathbb{N}_0\setminus\{x\}.\end{equation*}

Hence by the law of total probability,

\begin{align*}\mathbb{E}_x\big(\tau_x^+\big)& =\mathbb{E}_x(J_1)+\sum_{j\in\mathbb{N}_0\setminus B}\mathbb{E}_j(\tau_x)\mathbb{P}_x\big(Y_{J_1}=j\big)+\sum_{j\in B}\mathbb{E}_j(\tau_x)\mathbb{P}_x\big(Y_{J_1}=j\big)\\&\ge\sum_{j\in\mathbb{N}_0\setminus B}\mathbb{E}_j(\tau_x)\mathbb{P}_x\big(Y_{J_1}=j\big)\\&\ge\mathbb{P}_x(Y_{J_1}\in\Omega_++x)\inf_{j\in\mathbb{N}_0\setminus B}\mathbb{E}_j(\tau_x)\\&=\frac{\sum_{\omega\in\Omega_+}\lambda_{\omega}(x)}{\sum_{\omega\in\Omega}\lambda_{\omega}(x)}\cdot\infty=\infty,\end{align*}

since

\begin{equation*}\frac{\sum_{\omega\in\Omega_+}\lambda_{\omega}(x)}{\sum_{\omega\in\Omega}\lambda_{\omega}(x)}>0,\end{equation*}

and $\mathbb{E}_j(\tau_B)=\infty$ for all $j\in\mathbb{N}_0\setminus B$ , by Theorem 5(ii), under the respective conditions.

(ii) By assumption, $\partial=\{0\}$ and $\mathcal{Y}\setminus\partial=\mathbb{N}$ . We first show nonexistence of QSDs. Construct an irreducible process $Z_t$ on $\mathbb{N}_0$ with transition rate matrix $\widetilde{Q}$ as in the proof of Theorem 3. Applying the conclusion (i) to $Z_t$ , $Z_t$ is not positive recurrent when none of the conditions of (C3), (C9), (C10), (C11) holds. It thus suffices to show that the existence of a QSD for $Y_t$ implies positive recurrence of $Z_t$ . Assume that $Y_t$ has a QSD on $\mathbb{N}$ . By Proposition 14, there exists $\psi>0$ such that

\begin{equation*}\psi\mathbb{E}_i(\tau_{0})=\mathbb{E}_i(\psi\tau_{0})\le\mathbb{E}_i(\!\exp(\psi\tau_{0}))<\infty,\quad \forall i\in\mathbb{N}_0.\end{equation*}

Let $\mathbb{E}_i^Z(\tau_{0})$ be the expected hitting times for process $Z_t$ . By Proposition 15, $\big(\mathbb{E}_i^Z(\tau_{0})\big)_{i\in\mathbb{N}_0}$ is the minimal solution to the linear equations associated with $\widetilde{Q}$ . By a similar argument as in the proof of Theorem 3, $\big(\mathbb{E}_i^Z(\tau_{0})\big)_{i\in\mathbb{N}_0}$ is also the minimal solution to the linear equations associated with the transition matrix Q, and thus

\begin{equation*}\mathbb{E}_i^Z(\tau_{0})=\mathbb{E}_i(\tau_{0})<\infty,\quad \forall i\in\mathbb{N}_0.\end{equation*}

Since $Z_t$ is irreducible, we have that $Z_t$ is positive recurrent, owing to the classical fact that an irreducible CTMC that positively recurs to a finite set positively recurs everywhere (cf. [Reference Menshikov and Petritis32]).

Next, we prove the ergodicity of the Q-process. For either (C18) or (C19), $Y_t$ is non-explosive by Theorem 1; let $f(x)=(2-x^{-1})\unicode{x1D7D9}_{\mathcal{Y}\setminus\partial}(x)$ . Under the respective conditions, it is straightforward to verify that

\begin{equation*}\lim_{x\to\infty}\frac{Q f(x)}{f(x)}=-\infty,\end{equation*}

which implies that the set

\begin{equation*}D=\{x\in \mathcal{Y}\setminus\partial\,:\, \frac{Q f(x)}{f(x)}\ge-\psi_0-1\}\end{equation*}

is finite. Then, with such f, D, and $\delta$ , the conditions in Proposition 13 are satisfied and the conclusions follow. Note that $\textrm{supp}\nu=\mathbb{N}$ comes from the fact that the support of the ergodic stationary distribution of the Q-process is $\mathbb{N}$ by the irreducibility.

5.8. Proof of Theorem 8

As discussed in the proof of Theorem 4, the desired upper estimates of Qf(x) with the same f under the respective conditions still hold, by replacing the asymptotic expansions by one-sided inequalities in the proof of Theorem 7.

5.9. Proof of Theorem 9

First we prove implosivity. Assume (C18) or (C19) holds. Hence $Y_t$ is recurrent by Theorem 3. Let $f(x)=1-(x+1)^{-1}$ . One can show that the conditions in Proposition 18(i-1) are fulfilled, and implosivity is achieved.

Next we turn to non-implosivity. Assume neither (C18) nor (C19) holds. Since $Y_t$ does not implode towards any transient state, it suffices to prove non-implosivity assuming a recurrence condition, i.e., (C3) or (C6), by Theorem 3. Let $f(x)={\log\log\!(x+2)}$ . It is easy to verify that the conditions (with $\delta=2$ ) in Proposition 18(ii) are fulfilled, and $Y_t$ is non-implosive.

5.10. Proof of Theorem 10

As discussed in the proof of Theorem 4, the same functon f under the condition (C19) also serves as a Lyapunov function as in the proof of Theorem 9.