Approximation of birth-death processes

The index set $\mathbb{N}$ of the transition matrix is typically referred to as the minimal state space. The “real” state space of a $Q$-process is its one-point compactification (see, e.g., [14, §2.1])

where $\overline{\mathbb{N}}$ can be metrized with the metric

This establishes a topological homeomorphism between $\overline{\mathbb{N}}$ and the set

equipped with the relative topology of ${\mathbb{R}}$. Since we do not always consider honest $Q$-processes, it is necessary to introduce a cemetery point $\partial$, which lies outside the state space $\overline{\mathbb{N}}$. It should be emphasized that, unless explicitly stated otherwise, $\partial$ is always treated as an isolated point distinct from $\overline{\mathbb{N}}$. For instance, we can define the distance between a state $n\in\overline{\mathbb{N}}$ and the cemetery point as $r(n,\partial)=\left|\frac{1}{n+1}+1\right|$ ($\frac{1}{\infty}:=0$), where the inclusion of $\partial$ into $\overline{\mathbb{N}}$ is equivalent to adding the point $-1$ to (2.2).

Given the metric $r$, the set

forms a compact, separable metric space, and its subspace

is a locally compact, separable metric space. For every bounded function $f$ defined on either $\overline{\mathbb{N}}_{\partial}$ or $\mathbb{N}_{\partial}$, we define

Let $C(\overline{\mathbb{N}}_{\partial})$ denote the family of all continuous functions on $\overline{\mathbb{N}}_{\partial}$, where we emphasize that for $f\in C(\overline{\mathbb{N}}_{\partial})$, the value of $f(\partial)$ may no be equal to $0$. Define

The families $C_{b}(\mathbb{N}_{\partial})$, $C_{0}(\mathbb{N}_{\partial})$, and $C_{c}(\mathbb{N}_{\partial})$ consist of all continuous functions on $\mathbb{N}_{\partial}$ that are bounded, that vanish at infinity, and that have compact support, respectively. In particular, a function $f$ defined on $\mathbb{N}_{\partial}$ belongs to $C_{0}(\mathbb{N}_{\partial})$ (resp. $C_{c}(\mathbb{N}_{\partial})$) if $\lim_{n\rightarrow\infty}f(n)=0$ (resp. if there exists an integer $N$ such that $f(n)=0$ for all $n\geq N$). Further, we define $C_{0}(\mathbb{N}):=\{f\in C_{0}(\mathbb{N}_{\partial}):f(\partial)=0\}$ and $C_{c}(\mathbb{N}):=\{f\in C_{c}(\mathbb{N}_{\partial}):f(\partial)=0\}$.

Given the density matrix (2.1), there exists a particular birth-death process known as the minimal $Q$-process, which is denoted by $X^{\text{min}}=\left(X^{\text{min}}_{t}\right)_{t\geq 0}$ and whose transition matrix is denoted by $\left(p^{\text{min}}_{ij}(t)\right)_{i,j\in\mathbb{N}}$. Analytically, this process corresponds to the minimal solution to the Kolmogorov backward equation (as discussed in [7, §10]). From a probabilistic perspective, it represents a $Q$-process with minimal information, where the trajectories are terminated as they approach $\infty$ at the first time.

We introduce further notations for the minimal $Q$-process. Let $\zeta^{\text{min}}:=\inf\{t>0:X^{\text{min}}_{t}=\partial\}$ denote the lifetime of $X^{\text{min}}$, and define

The resolvent matrix of this process is given by

Then, it is straightforward to verify the following relation:

Similar to regular diffusions on an interval, the minimal $Q$-process can be fully characterized by two parameters on $\mathbb{N}$ derived from the matrix (2.1): a scale function and a speed measure. The scale function $(c_{k})_{k\in\mathbb{N}}$ is given by

the speed measure $\mu$ is

The process $X^{\text{min}}$ is symmetric with respect to $\mu$ in the sense that $\mu_{i}p_{ij}^{\text{min}}(t)=\mu_{j}p_{ji}^{\text{min}}(t)$ for any $i,j\in\mathbb{N}$ and $t\geq 0$. Thus, the speed measure $\mu$ is also known as the symmetric measure of $X^{\text{min}}$. Further details and related results are referred to [14, §3.1], which provides a characterization involving time change transformation of Brownian motion.

Using another two parameters derived from the scale function and the speed measure, we can classify the boundary point $\infty$ in accordance with Feller’s approach. Specifically, we define the following two quantities:

The following classification for the boundary point $\infty$ is very well known.

It is crucial to highlight that $Q$-processes are unique if and only if $\infty$ is classified as an entrance or natural boundary; for more details, refer to, e.g., [14, Theorem 3.5]. In this paper, however, we focus on the non-uniqueness case, where $\infty$ is either regular or an exit. This non-uniqueness implies the condition $c_{\infty}<\infty$. Furthermore, it holds that

see, e.g., [14, Lemma 4.1].

From now on, we will assume that $\infty$ is either regular or an exit. In addition to the minimal one, consideration of other $Q$-processes, such as Doob processes and the $(Q,1)$-process (which is only applicable in the regular case), is warranted.

It was first examined by Feller in [7] and then firmly established by Yang in 1965 (see [20, Chapter 7]) that each (non-minimal) $Q$-process can be uniquely determined, up to a multiplicative constant, by a triple of parameters $(\gamma,\beta,\nu)$. Here, $\gamma,\beta\geq 0$ are two non-negative constants and $\nu=(\nu_{k})_{k\in\mathbb{N}}$ is a positive measure on $\mathbb{N}$ satisfying the conditions:

where $|\nu|:=\sum_{k\in\mathbb{N}}\nu_{k}$, and

Let $\mathscr{Q}$ denote the set of all triples $(\gamma,\beta,\nu)$ satisfying (2.8) and (2.9). More precisely, the resolvent matrix

of the $Q$-process $(p_{ij}(t))_{i,j\in\mathbb{N}}$ corresponding to $(\gamma,\beta,\nu)\in\mathscr{Q}$ can be expressed as (see, e.g., [14, Theorem B.1])

where $(\Phi^{\text{min}}_{ij})$ is the minimal resolvent matrix (2.4) and $u^{\text{min}}_{\alpha}$ is defined as (2.3). The matrix (2.11) is referred to as the $(Q,\gamma,\beta,\nu)$-resolvent matrix. It should be noted that for any constant $M>0$, the $(Q,M\gamma,M\beta,M\nu)$-resolvent matrix is identical to the $(Q,\gamma,\beta,\nu)$-resolvent matrix.

We need to provide some observations regarding the expression of the resolvent matrix presented in (2.11). First, the first inequality in (2.8) is equivalent to

see, e.g., [20, §7.10]. Second, $\mu(u^{\text{min}}_{\alpha}):=\sum_{k\in\mathbb{N}}\mu_{k}u^{\text{min}}_{\alpha}(k)$ is finite, if and only if $\infty$ is regular or an entrance; see, e.g., [7, Theorem 7.1]. Thus, the condition (2.9) guarantees that the resolvent matrix (2.11) is well-defined in the exit case. Third, when $\gamma>0$ while $\beta=|\nu|=0$, the resolvent matrix (2.11) reduces to the minimal one (although the second inequality in (2.8) is not satisfied).

In a recent article [14], it was demonstrated, using Ray-Knight compactification, that every $Q$-process $X$ possesses a càdlàg modification $\bar{X}$ on $\overline{\mathbb{N}}_{\partial}$. This modified version is a Ray process on $\overline{\mathbb{N}}_{\partial}$. In this paper, we do not differentiate between $X$ and its Ray-Knight compactification $\bar{X}$. Additionally, [14, Corollary 5.2] classifies all non-minimal $Q$-processes into two categories: Doob processes and Feller $Q$-processes.

According to, e.g., [14, Theorem B.1], the $Q$-process $X$ is a Doob process, if and only if its determining triple $(\gamma,\beta,\nu)$ belongs to

In this case, whenever it approaches $\infty$, the process $X$ refreshes at a randomly determined location from a distribution $\pi$ given by (1.1) on $\mathbb{N}_{\partial}$. In other words, $X$ can be obtained by the piecing out (see [11]) of the minimal $Q$-process $X^{\text{min}}$ with respect to the instantaneous distribution $\pi$, as established in [14, §5]. The readers are also referred to [14, Appendix A] for a detailed description of the piecing out transformation.

When $(\gamma,\beta,\nu)\in\mathscr{Q}_{F}:=\mathscr{Q}\setminus\mathscr{Q}_{D}$, the corresponding $Q$-process $X$ is a Feller process on $\overline{\mathbb{N}}$ (with $\partial$ being the cemetery point). Following [14], this $Q$-process will be termed a Feller $Q$-process. The infinitesimal generator of the Feller $Q$-process $X$ is derived in [14, Theorem 6.3], and the crucial point is the boundary condition at $\infty$ satisfied by the functions $F$ in the generator domain:

where $F^{+}$ represents the discrete gradient of $F$ with respect to the scale function $(c_{k})_{k\in\mathbb{N}}$; see [14, (6.2)]. Based on this boundary condition, the parameters $(\gamma,\beta,\nu)$ can be utilized to explain three possible types of boundary behaviours of $X$ at $\infty$: killing, reflecting, and jumping. Please refer to [14, §2.3] and [15] for more details.

Let us introduce some notations for a $Q$-process in the context of general Markov processes.

Given the transition matrix $(p_{ij}(t))_{t\geq 0}$ and its resolvent matrix $(\Phi_{ij}(\alpha))_{\alpha>0}$ as provided in (2.10), we define

for every $f\in C(\overline{\mathbb{N}})$ (with $f(\partial)=0$) and $\alpha>0$. According to (2.11), $R_{\alpha}f(i)$ for $f\in C(\overline{\mathbb{N}})$ and $i\in\mathbb{N}$ can be expressed in terms of the parameters $(\gamma,\beta,\nu)$ as

where $R^{\text{min}}_{\alpha}f(i):=\sum_{j\in\mathbb{N}}\Phi^{\text{min}}_{ij}(\alpha)f(i)$. By utilizing (2.5) and (2.7), we can obtain that

Define $R_{\alpha}f(\infty)$ as this limit. For $f\in C(\overline{\mathbb{N}}_{\partial})$ where $f(\partial)$ may not be equal to $0$, it holds that $f_{0}:=f-f(\partial)\in C(\overline{\mathbb{N}})$. We define

As established in [14, §4],

is a Ray resolvent in the sense of, e.g., [4, Definition 8.1]. According to, e.g., [4, Theorem 8.2], there exists a Borel measurable Markov transition semigroup $(P_{t})_{t\geq 0}$ on $\overline{\mathbb{N}}_{\partial}$, having $(R_{\alpha})_{\alpha>0}$ as the resolvent, such that $t\mapsto P_{t}f(x)$ is right-continuous for all $f\in C(\overline{\mathbb{N}}_{\partial})$ and $x\in\overline{\mathbb{N}}_{\partial}$. This transition semigroup $(P_{t})_{t\geq 0}$ is known as a Ray semigroup. Note that for $f\in C(\overline{\mathbb{N}})$ and $t\geq 0$, it holds that $P_{t}f(i)=\sum_{j\in\mathbb{N}}p_{ij}(t)f(j)$ for $i\in\mathbb{N}$ and $P_{t}f(\partial)=0$.

In the case where $(p_{ij}(t))_{t\geq 0}$ is a Feller $Q$-process, the semigroup $(P_{t})_{t\geq 0}$ is a Feller semigroup in the sense that $P_{t}C(\overline{\mathbb{N}}_{\partial})\subset C(\overline{\mathbb{N}}_{\partial})$ and

Particularly, a Feller $Q$-process satisfies the normal property on $\overline{\mathbb{N}}_{\partial}$, i.e., $P_{0}(x,\cdot)=\delta_{x}$ for all $x\in\overline{\mathbb{N}}_{\partial}$. However, in the case where $(p_{ij}(t))_{t\geq 0}$ is a Doob process, neither (2.16) nor the normal property is satisfied. More precisely, $P_{0}(x,\cdot)=\delta_{x}$ only holds for $x\in\mathbb{N}_{\partial}$, and $\infty$ is a branching point of the $Q$-process in the sense of, e.g., [4, Definition 8.3]. Specifically, $P_{0}(\infty,\cdot)=\pi$, which is given by (1.1); see [14, Theorem 5.1].

As a Ray process on $\overline{\mathbb{N}}_{\partial}$, the $Q$-process has a.s. càdlàg trajectories on $\overline{\mathbb{N}}_{\partial}$ according to, e.g., [4, Theorem 8.6] (or [19, Theorem 9.13]). Therefore, we can define the trajectory space $\Omega$ as the set of all càdlàg functions $\omega$ from $[0,\infty)$ to $\overline{\mathbb{N}}_{\partial}$ such that $\omega(t)=\partial$ for all $t\geq\zeta(\omega):=\inf\{t\geq 0:\omega(t)=\partial\}$. We can then define the projection maps

for all $t\geq 0$. The translation operators $(\theta_{t})_{t\geq 0}$ on $\Omega$ are defined by $\theta_{t}\omega(s):=\omega(t+s)$ for all $t,s\geq 0$. Let ${\mathscr{F}}^{0}:=\sigma\left\{X_{s}:s\geq 0\right\}$ and ${\mathscr{F}}^{0}_{t}:=\sigma\left\{X_{s}:0\leq s\leq t\right\}$, the $\sigma$-algebras on $\Omega$ generated by $\{X_{s}:s\geq 0\}$ and $\{X_{s}:0\leq s\leq t\}$, respectively. These $\sigma$-algebras are known as the natural filtration on $\Omega$. According to [4, Theorem 8.6] (see also [19, Theorem 9.13]), for any probability measure $\lambda$ on $\overline{\mathbb{N}}_{\partial}$, there exists a probability measure $\mathbb{P}_{\lambda}$ on $(\Omega,{\mathscr{F}}^{0})$ such that

forms a Markov process on $\overline{\mathbb{N}}_{\partial}$ with initial distribution $\lambda P_{0}$ (not $\lambda$!) and transition semigroup $(P_{t})_{t\geq 0}$. Here, $\lambda P_{0}(A):=\int_{\overline{\mathbb{N}}_{\partial}}P_{0}(x,A)\lambda(dx)$ for $A\subset\overline{\mathbb{N}}_{\partial}$. If $\lambda=\delta_{x}$ for $x\in\overline{\mathbb{N}}_{\partial}$, we write $\mathbb{P}_{\lambda}$ as $\mathbb{P}_{x}$. Additionally, note that $\mathbb{P}_{\lambda}(\cdot)=\int_{\overline{\mathbb{N}}_{\partial}}\mathbb{P}_{x}(\cdot)\lambda(dx)$. The natural filtration $({\mathscr{F}}^{0},{\mathscr{F}}^{0}_{t})$ can be augmented using the standard approach described in [19, I§6], resulting in the augmented natural filtration $({\mathscr{F}},{\mathscr{F}}_{t})$ on $\Omega$. Finally, we obtain a collection

which forms a realization of the $Q$-process $(p_{ij}(t))_{i,j\in\mathbb{N}}$.

The lifetime of $X$ is $\zeta=\inf\{t>0:X_{t}=\partial\}$. Note that $\mathbb{P}_{\partial}(X_{t}=\partial,\forall t\geq 0)=1$ and $\mathbb{P}_{i}(X_{0}=i)=1$ for all $i\in\mathbb{N}$. In the case where $X$ is a Feller $Q$-process, it also holds that $\mathbb{P}_{\infty}(X_{0}=\infty)=1$. In contrast, for a Doob process, $\mathbb{P}_{\infty}(X_{0}\in\cdot)=\pi$ since $P_{0}(\infty,\cdot)=\pi$. However, when the Doob process is restricted to $\mathbb{N}_{\partial}$, it transforms into a Borel right process that satisfies the normal property (see [19, Theorem 9.13]). It is worth noting that in the context of Borel right process, the cemetery point $\partial$ is commonly considered as the compactification point of the state space $\mathbb{N}$, which slightly differs from the setup in §2.1.

Let $D_{\overline{\mathbb{N}}_{\partial}}[0,\infty)$ denote the set of all càdlàg functions from $[0,\infty)$ to $\overline{\mathbb{N}}_{\partial}$. According to [6, §3, Theorem 5.6], the space $D_{\overline{\mathbb{N}}_{\partial}}[0,\infty)$ equipped with the metric $d$ inducing the Skorohod topology (defined in [6, §3(5.2)]) is a complete, separable metric space. In addition, utilizing [6, §3, Proposition 7.1], we can identify the Borel $\sigma$-algebra $\mathscr{B}\left(D_{\overline{\mathbb{N}}_{\partial}}[0,\infty);d\right)$ on $D_{\overline{\mathbb{N}}_{\partial}}[0,\infty)$ with respect to the Skorohod topology as $\sigma\{\pi_{t}:t\geq 0\}$, which is generated by all the projection maps $\pi_{t}:D_{\overline{\mathbb{N}}_{\partial}}[0,\infty)\rightarrow\overline{\mathbb{N}}_{\partial},w\mapsto w(t)$.

It is evident that $\Omega\subset D_{\overline{\mathbb{N}}_{\partial}}[0,\infty)$, where $\Omega$ is the trajectory space in the realization (2.17) of the $Q$-process. Since $\mathscr{B}\left(D_{\overline{\mathbb{N}}_{\partial}}[0,\infty);d\right)=\sigma\{\pi_{t}:t\geq 0\}$, it is straightforward to verify that the embedding map

is measurable. Thus, for any probability measure $\lambda$ on $\overline{\mathbb{N}}_{\partial}$, $\mathbb{P}_{\lambda}$ induces an image probability measure $\mathbb{P}_{\lambda}\circ\mathcal{X}^{-1}$ on $\left(D_{\overline{\mathbb{N}}_{\partial}}[0,\infty),\mathscr{B}\left(D_{\overline{\mathbb{N}}_{\partial}}[0,\infty)\right);d\right)$. Since $\mathbb{P}_{\lambda}\circ\mathcal{X}^{-1}$ can be regarded as the extension of $\mathbb{P}_{\lambda}$ to $D_{\overline{\mathbb{N}}_{\partial}}[0,\infty)$ by defining $\mathbb{P}_{\lambda}(D_{\overline{\mathbb{N}}_{\partial}}[0,\infty)\setminus\Omega):=0$, we will still denote this image measure by $\mathbb{P}_{\lambda}$ if no ambiguity arises.