Birth and death processes in interactive random environments

Abstract

This paper studies birth and death processes in interactive random environments where the birth and death rates and the dynamics of the state of the environment are dependent on each other. Two models of a random environment are considered: a continuous-time Markov chain (finite or countably infinite) and a reflected (jump) diffusion process. The background is determined by a joint Markov process carrying a specific interactive mechanism, with an explicit invariant measure whose structure is similar to a product form. We discuss a number of queueing and population-growth models and establish conditions under which the above-mentioned invariant measure can be derived. Next, an analysis of the rate of convergence to stationarity is performed for the models under consideration. We consider two settings leading to either an exponential or a polynomial convergence rate. In both cases we assume that the underlying environmental Markov process has an exponential rate of convergence, but the convergence rate of the joint Markov process is determined by certain conditions on the birth and death rates. To prove these results, a coupling method turns out to be useful.

Appendix

1.1 Appendix A: Proofs of Theorems 2.1 and 3.1

Proof of Theorem 2.1

The irreducibility and aperiodicity properties are straightforward. For the measure $\eta (n,z)$ in (2.9) to be finite, by Assumption 2.2,

$$\begin{aligned} \sum _{(n,z)} \eta (n,z) = \sum _{(n,z)} r_n(z) v(z) =\Xi < \infty . \end{aligned}$$

To verify that $\eta (n,z)$ in (2.9) is an invariant measure, we prove that $\eta ' {\mathbf {R}}=0$:

$$\begin{aligned} \begin{aligned} -\eta (n,z)&R[(n,z), (n,z)] \\&= \eta (n-1,z) R[(n-1,z),(n,z)] + \eta (n+1,z) R[(n+1,z),(n,z)] \\&\qquad + \sum _{z'\ne z} \eta (n,z') R[(n,z'), (n,z)], \quad n=1,2,\ldots ,\quad z \in {\mathcal {Z}};\\ -\eta (0,z)&R[(0,z),(0,z)] \\&= \eta (1,z) R[(1,z),(0,z)] + \sum _{z'\ne z} \eta (0,z') R[(0,z'), (0,z), n = 0, \quad z \in {\mathcal {Z}}. \end{aligned} \end{aligned}$$

(6.1)

For (6.1) with $n \ge 1$, the left-hand side is

$$\begin{aligned}&\eta (n,z) \sum _{(n',z') \ne (n,z)} R[(n,z),(n',z')] \\&= \eta (n,z) \big ( R[(n,z),(n+1,z)] + R[(n,z), (n-1,z)] + \sum _{z' \ne z} R[(n,z), (n,z')] \big ) \\&= r_n(z) v(z) \big ( \lambda _n(z) + \mu _n(z) + \sum _{z'\ne z} r_n(z)^{-1} \tau _n(z,z') \big ) \\&= r_n(z) v(z) \big ( \lambda _n(z) + \mu _n(z) \big ) + v(z) \sum _{z'\ne z} \tau _n(z,z'), \end{aligned}$$

and the right-hand side is equal to

$$\begin{aligned}&r_{n-1}(z) v(z)\lambda _{n-1}(z) + r_{n+1}v(z) \mu _{n+1}(z) + \sum _{z'\ne z} r_n(z') v(z') r_n(z')^{-1} \tau _n(z',z) \\&= v(z)\big ( r_{n-1}(z) \lambda _{n-1}(z) + r_{n+1} \mu _{n+1}(z) \big ) + \sum _{z'\ne z} v(z') \tau _n(z',z). \end{aligned}$$

We get equality thanks to the assumption in (2.6) and the detailed balance equation in (2.3). For (6.1) with $n = 0$, the left-hand side is

$$\begin{aligned}&\eta (0,z) \sum _{z'\ne z} R[(0,z),(0,z')] = r_0(z) v(z) \big (R[(0,z),(1,z)] + \sum _{z' \ne z} R[(0,z), (0,z')] \big ) \\&= r_0(z) v(z) \big (\lambda _0(z) + \sum _{z' \ne z} r_0(z)^{-1} \tau _0(z,z') \big ) = r_0(z) v(z) \lambda _0(z) + v(z) \sum _{z' \ne z} \tau _0(z,z') \big ), \end{aligned}$$

and the right-hand side is

$$\begin{aligned}&r_1(z) v(z) \mu _1(z) + \sum _{z'\ne z} r_0(z') v(z') r_0(z')^{-1} \tau _0(z',z)\\&\quad = r_1(z) v(z) \mu _1(z) + \sum _{z'\ne z} v(z') \tau _0(z',z). \end{aligned}$$

This again leads to the equality thanks to (2.6) and (2.4) for $n=0$. Thus we have shown that $\pi (n,z)$ in (2.8) is an invariant probability measure. The positive recurrence property follows from [29, Theorem 3.5.3] (see also [38, Theorem 2.7.18]). The ergodicity property of convergence in total variation follows from [26]. $\square$

Proof of Theorem 3.1

The proof follows from an analogous argument as that of Theorem 3.1 in [31], so we only highlight the differences. We apply [20], and use their notation as follows: let $E = {{\mathbb {N}}}\times {\mathcal {Z}}$ and $U=\{0,1,\ldots ,m\}$, where “0" indicates ${\mathcal {Z}}$ and $i=1,\ldots ,m$ for the faces $F_1,\ldots , F_m$ of the boundary, and for $n\in {{\mathbb {N}}}$, $z\in {\mathcal {Z}}$ and $u \in U$,

$$\begin{aligned} \mu _0(\{u\}\times \{n\}\times \mathrm{d}z)&= \mathbf{1}_{u=0} r_n(z) \nu (\mathrm{d}z),\quad \mu _1(\{u\}\times \{n\}\times \mathrm{d}z) = \mathbf{1}_{u\ne 0} r_n(z) \nu _i({\textbf {d}}z),\\ \mu _0^E(\{n\}\times \mathrm{d}z)&= r_n(z) \nu (\mathrm{d}z),\quad \nu _1^E(\{n\}\times \mathrm{d}z) = r_n(z) \big (\nu _{F_1}(\mathrm{d}z) + \cdots + \nu _{F_m}(\mathrm{d}z) \big ),\\ \eta _0((n,z), \{u\})&= \mathbf{1}_{u=0}, \quad \eta _1((n,z), \{u\}) = \mathbf{1}_{u\ne 0}, \\ Af((n,z),u)&:= \beta _n^{-1} r_n(z) {\mathcal {L}} f(n,z),\\ Bf((n,z),u)&:= \mathbf{1}_{u\ne 0, z \in \partial D_i, i=1,\ldots ,m} \gamma _u(z) \cdot \nabla f(z). \end{aligned}$$

To check [20, Condition 1.2] on the absolutely continuous generator A and the singular generator B, we can verify the conditions (i)–(v) in the same way as in the proof of [31, Theorem 3.1]. For the main condition in [20, Theorem 1.7, (1.17)], we need to show that the generators A and B satisfy

$$\begin{aligned} \int _{E\times U} Af(x,u) \mu _0(\mathrm{d}x\times \mathrm{d}u) + \int _{E\times U} Bf(x,u) \mu _1(\mathrm{d}x\times \mathrm{d}u) =0. \end{aligned}$$

(6.2)

By the definitions of A and B, we can write the left-hand side as

$$\begin{aligned}&\sum _{n=0}^\infty \int _{{\mathcal {Z}}}\beta _n^{-1} r_n(z) {\mathcal {M}}_zf(n,z) \nu (\mathrm{d}z)\\&+\sum _{n=0}^\infty \bigg ( \int _{{\mathcal {Z}}} {\mathcal {A}}f(n,z) \nu (\mathrm{d}z) + \sum _{i=1}^m \int _{F_i} \gamma _i(z) \cdot \nabla f(z) \nu _{F_i} (\mathrm{d}z)\bigg ). \end{aligned}$$

The sum of the last two terms is equal to zero, because the basic adjoint relationship holds for the reflected jump diffusion process ${\widetilde{Z}}$ (see, e.g., [49]), that is, for $f\in C^2_b({\mathcal {Z}})$,

$$\begin{aligned} \int _{{\mathcal {Z}}} {\mathcal {A}}f(n,z) \nu (\mathrm{d}z) + \sum _{i=1}^m \int _{F_i} \gamma _i(z) \cdot \nabla f(z) \nu _{F_i} (\mathrm{d}z) =0. \end{aligned}$$

For each $z\in {\mathcal {Z}}$, the birth–death process N(t) has the stationary distribution given in (2.2) and (2.5), which satisfy $r_n(z) {\mathcal {M}}_zf(n,\cdot ) =0$ for each $z \in {\mathcal {Z}}$ and $n \in {{\mathbb {Z}}}$. Multiplying this by $\beta _n^{-1}$ and integrating over $z \in {\mathcal {Z}}$, we get:

$$\begin{aligned} \sum _{n=0}^\infty \int _{{\mathcal {Z}}}\beta _n^{-1} r_n(z) {\mathcal {M}}_zf(n,z) \nu (\mathrm{d}z) = 0. \end{aligned}$$

Thus we have verified that (6.2) holds. The rest of the proof follows the same argument as the proof of [31, Theorem 3.1]. $\square$

1.2 Appendix B: A comparison lemma

Lemma 6.1

(Lemma 5.1 in [31]) Fix constants $\alpha > 1$, $\beta , \gamma > 0$. Take two independent random variables $\xi \sim Exp(\beta )$ and $\eta > 0$ with ${\mathbb {P}}(\eta > u) \le \alpha e^{-\gamma u}$ for $u \ge 0$. Then

$$\begin{aligned} {\mathbb {P}}(\eta < \xi ) \ge \alpha ^{-\beta /\gamma }\frac{\gamma }{\beta + \gamma }. \end{aligned}$$

(6.3)

For $a \in [0, \beta + \gamma )$, the moment-generating function for $\xi \wedge \eta$ satisfies

$$\begin{aligned} {\mathbb {E}}\big [e^{a(\xi \wedge \eta )}\big ] \le \theta (\alpha , \beta , \gamma , a), \end{aligned}$$

(6.4)

where the function $\theta$ is defined in (4.5).