A system of nonlinear equations with application to large deviations for Markov chains with finite lifetime

TL;DR

This paper proves the existence of solutions to a complex nonlinear system and applies this to establish a large deviation principle for occupation times in finite lifetime Markov chains.

Contribution

It introduces a novel existence result for a class of nonlinear equations and applies it to large deviations in Markov chain occupation times.

Findings

Existence of solutions to the nonlinear system for n ≥ 3.

Application to large deviation principles for Markov chains.

Insights into occupation time distributions with finite lifetime.

Abstract

In this paper, we first show the existence of solutions to the following system of nonlinear equations \begin{eqnarray*}\left\{\begin{array}{l} a_{11}x_1+a_{12}x_2+a_{13}x_3+\cdots+a_{1n}x_{n} = b_{11}\frac{1}{x_1}+b_{12}\frac{1}{x_2}+b_{13}\frac{1}{x_3}+\cdots+b_{1n}\frac{1}{x_{n}},\\ a_{21}\frac{1}{x_1}+a_{22}\frac{x_2}{x_1}+a_{23}\frac{x_3}{x_1}+\cdots+a_{2n}\frac{x_{n}}{x _1}=b_{21}x_1+b_{22}\frac{x_1}{x_2}+b_{23}\frac{x_1}{x_3}+\cdots+b_{2n}\frac{x_1}{x_{n}},\\ a_{31}\frac{x_1}{x_2}+a_{32}\frac{1}{x_2}+a_{33}\frac{x_3}{x_2}+\cdots+a_{3n}\frac{x_{n}}{x _2}=b_{31}\frac{x_2}{x_1}+b_{32}x_2+b_{33}\frac{x_2}{x_3}+\cdots+b_{3n}\frac{x_2}{x_{n}},\\ \cdots\cdots\\ a_{n1}\frac{x_1}{x_{n-1}}+a_{n2}\frac{x_2}{x_{n-1}}+a_{n3}\frac{x_3}{x_{n-1}}+ \cdots+a_{n,n-1}\frac{1}{x_{n-1}}+a_{nn}\frac{x_{n}}{x_{n-1}}\\ =b_{n1}\frac{x_{n-1}}{x_1}+b_{n2}\frac{x_{n-1}}{x_2}+b_{n3}\frac{x_{n-1}}{x_3}+\cdots+b_{n, n-1}x_{n-1} +b_{nn}\frac{x_{n-1}}{x_{n}}, \end{array} \right. \end{eqnarray*} where $n\ge 3$ and $a_{ij},b_{ij},1\le i,j\le n$, are positive constants. Then, we make use of this result to obtain the large deviation principle for the occupation time distributions of continuous-time finite state Markov chains with finite lifetime.

Equations

$$\displaystyle\left\{\begin{array}[]{l}a_{11}x_{1}+a_{12}x_{2}+a_{13}x_{3}+\cdots+a_{1n}x_{n}=b_{11}\frac{1}{x_{1}}+b_{12}\frac{1}{x_{2}}+b_{13}\frac{1}{x_{3}}+\cdots+b_{1n}\frac{1}{x_{n}},\\ a_{21}\frac{1}{x_{1}}+a_{22}\frac{x_{2}}{x_{1}}+a_{23}\frac{x_{3}}{x_{1}}+\cdots+a_{2n}\frac{x_{n}}{x_{1}}=b_{21}x_{1}+b_{22}\frac{x_{1}}{x_{2}}+b_{23}\frac{x_{1}}{x_{3}}+\cdots+b_{2n}\frac{x_{1}}{x_{n}},\\ a_{31}\frac{x_{1}}{x_{2}}+a_{32}\frac{1}{x_{2}}+a_{33}\frac{x_{3}}{x_{2}}+\cdots+a_{3n}\frac{x_{n}}{x_{2}}=b_{31}\frac{x_{2}}{x_{1}}+b_{32}x_{2}+b_{33}\frac{x_{2}}{x_{3}}+\cdots+b_{3n}\frac{x_{2}}{x_{n}},\\ \cdots\cdots\\ a_{n1}\frac{x_{1}}{x_{n-1}}+a_{n2}\frac{x_{2}}{x_{n-1}}+a_{n3}\frac{x_{3}}{x_{n-1}}+\cdots+a_{n,n-1}\frac{1}{x_{n-1}}+a_{nn}\frac{x_{n}}{x_{n-1}}\\ =b_{n1}\frac{x_{n-1}}{x_{1}}+b_{n2}\frac{x_{n-1}}{x_{2}}+b_{n3}\frac{x_{n-1}}{x_{3}}+\cdots+b_{n,n-1}x_{n-1}+b_{nn}\frac{x_{n-1}}{x_{n}},\end{array}\right.$$

$$\sum _{j\in E}{q}_{ij}\frac{\alpha (j)}{\alpha (i)}\beta (j)=\sum _{j\in E}{q}_{ji}\frac{\alpha (i)}{\alpha (j)}\beta (j),\forall i\in E.$$

$$A^{-1}QA\beta =A{Q}^T{A}^{-1}\beta .$$

$$L_t(A)=\frac{1}{t}{\int }_0^t{1}_A(X_s)ds,A\subset E.$$

$$I(\mu ):=-\underset{u>0}{\inf }{\int }_E\frac{Qu}{u}d\mu =-\underset{u>0}{\inf }\sum _{i\in E}\frac{Qu(i)}{u(i)}\mu (\{i\}).$$

$$\underset{t\to \infty }{\mathrm{liminf}}\frac{1}{t}\log P_i(L_t\in G,t<\zeta )\ge -\underset{\mu \in G}{\inf }I(\mu ),\forall i\in E.$$

$$\underset{t\to \infty }{\mathrm{limsup}}\frac{1}{t}\log P_i(L_t\in K,t<\zeta )\le -\underset{\mu \in K}{\inf }I(\mu ),\forall i\in E.$$

$$\underset{t\to \infty }{\lim }\frac{1}{t}\log P_i(t<\zeta )=\underset{\mu \in {\mathcal{P}}_1(E)}{\sup }\left\{\underset{u>0}{\inf }\sum _{i\in E}\sum _{j\in E}\frac{q_{ij}\mu (\{i\})u(j)}{u(i)}\right\}.$$

$$f_1(x)=(a_{11}{x}_1+a_{12}{x}_2+a_{13}{x}_3)-\left(b_{11}\frac{1}{x_1}+b_{12}\frac{1}{x_2}+b_{13}\frac{1}{x_3}\right),$$

$$f_2(x)=\left(a_{21}\frac{1}{x_1}+a_{22}\frac{x_2}{x_1}+a_{23}\frac{x_3}{x_1}\right)-\left(b_{21}{x}_1+b_{22}\frac{x_1}{x_2}+b_{23}\frac{x_1}{x_3}\right),$$

$$f_3(x)=\left(a_{31}\frac{x_1}{x_2}+a_{32}\frac{1}{x_2}+a_{33}\frac{x_3}{x_2}\right)-\left(b_{31}\frac{x_2}{x_1}+b_{32}{x}_2+b_{33}\frac{x_2}{x_3}\right),$$

$$F(x)=f_1^2(x)+f_2^2(x)+f_3^2(x).$$

$$f_1(x^{\ast })=f_2(x^{\ast })=f_3(x^{\ast })=0.$$

$$\frac{\partial F}{\partial x_1}(x^{\ast })=f_1(x^{\ast })c_{11}-f_2(x^{\ast })c_{12}+f_3(x^{\ast })c_{13}=0,$$

$$\frac{\partial F}{\partial x_2}(x^{\ast })=f_1(x^{\ast })c_{21}+f_2(x^{\ast })c_{22}-f_3(x^{\ast })c_{23}=0,$$

$$\frac{\partial F}{\partial x_3}(x^{\ast })=f_1(x^{\ast })c_{31}+f_2(x^{\ast })c_{32}+f_3(x^{\ast })c_{33}=0,$$

$$f_1(x^{\ast })=f_2(x^{\ast })=f_3(x^{\ast })=0.$$

$$f_1(x^{\ast })\ast f_2(x^{\ast })\ast f_3(x^{\ast })\ne 0.$$

$$f_1(x^{\ast })=(a_{11}{x}_1^{\ast }+a_{12}{x}_2^{\ast }+a_{13}{x}_3^{\ast })-\left(b_{11}\frac{1}{x_1^{\ast }}+b_{12}\frac{1}{x_2^{\ast }}+b_{13}\frac{1}{x_3^{\ast }}\right)>0,$$

$$f_2(x^{\ast })=\left(a_{21}\frac{1}{x_1^{\ast }}+a_{22}\frac{x_2^{\ast }}{x_1^{\ast }}+a_{23}\frac{x_3^{\ast }}{x_1^{\ast }}\right)-\left(b_{21}{x}_1^{\ast }+b_{22}\frac{x_1^{\ast }}{x_2^{\ast }}+b_{23}\frac{x_1^{\ast }}{x_3^{\ast }}\right)<0,$$

$$f_3(x^{\ast })=\left(a_{31}\frac{x_1^{\ast }}{x_2^{\ast }}+a_{32}\frac{1}{x_2^{\ast }}+a_{33}\frac{x_3^{\ast }}{x_2^{\ast }}\right)-\left(b_{31}\frac{x_2^{\ast }}{x_1^{\ast }}+b_{32}{x}_2^{\ast }+b_{33}\frac{x_2^{\ast }}{x_3^{\ast }}\right)<0.$$

$$f_1(x^{\ast })>f_1(x_1^{\ast }-{\delta }_1,x_2^{\ast }-{\delta }_2,x_3^{\ast })>0,$$

$$f_2(x^{\ast })<f_2(x_1^{\ast }-{\delta }_1,x_2^{\ast }-{\delta }_2,x_3^{\ast })<0,$$

$$f_3(x^{\ast })<f_3(x_1^{\ast }-{\delta }_1,x_2^{\ast }-{\delta }_2,x_3^{\ast })<0.$$

$${\gamma }_1=f_1(x^{\ast }),{\gamma }_2=-f_2(x^{\ast }),{\gamma }_3=-f_3(x^{\ast }).$$

$${\delta }_1<x_1^{\ast },\delta {\delta }_1<x_2^{\ast },$$

$$0<f_1(x^{\ast })-f_1(x_1^{\ast }-{\delta }_1,x_2^{\ast }-{\delta }_2,x_3^{\ast })\le \frac{{\gamma }_1}{2},$$

$$0<f_2(x_1^{\ast }-{\delta }_1,x_2^{\ast }-{\delta }_2,x_3^{\ast })-f_2(x^{\ast })\le \frac{{\gamma }_2}{2},$$

$$0<f_3(x_1^{\ast }-{\delta }_1,x_2^{\ast }-{\delta }_2,x_3^{\ast })-f_3(x^{\ast })\le \frac{{\gamma }_3}{2},$$

$${\delta }_1<\min \left\{x_1^{\ast },\frac{x_2^{\ast }}{\delta }\right\},$$

$$0<(a_{11}+a_{12}\delta ){\delta }_1+\left(b_{11}+\frac{b_{12}}{\delta }\right)\left(\frac{1}{x_1^{\ast }-{\delta }_1}-\frac{1}{x_1^{\ast }}\right)\le \frac{{\gamma }_1}{2},$$

$$0<(a_{21}+a_{23}{x}_3^{\ast })\left(\frac{1}{x_1^{\ast }-{\delta }_1}-\frac{1}{x_1^{\ast }}\right)+\left(b_{21}+\frac{b_{23}}{x_3^{\ast }}\right){\delta }_1\le \frac{{\gamma }_2}{2},$$

$$0<(a_{32}+a_{33}{x}_3^{\ast })\left(\frac{1}{\delta }\right)\left(\frac{1}{x_1^{\ast }-{\delta }_1}-\frac{1}{x_1^{\ast }}\right)+\left(b_{32}+\frac{b_{33}}{x_3^{\ast }}\right)\delta {\delta }_1\le \frac{{\gamma }_3}{2}.$$

$$f_1(x^{\ast })=(a_{11}{x}_1^{\ast }+a_{12}{x}_2^{\ast }+a_{13}{x}_3^{\ast })-\left(b_{11}\frac{1}{x_1^{\ast }}+b_{12}\frac{1}{x_2^{\ast }}+b_{13}\frac{1}{x_3^{\ast }}\right)<0,$$

$$f_2(x^{\ast })=\left(a_{21}\frac{1}{x_1^{\ast }}+a_{22}\frac{x_2^{\ast }}{x_1^{\ast }}+a_{23}\frac{x_3^{\ast }}{x_1^{\ast }}\right)-\left(b_{21}{x}_1^{\ast }+b_{22}\frac{x_1^{\ast }}{x_2^{\ast }}+b_{23}\frac{x_1^{\ast }}{x_3^{\ast }}\right)>0,$$

$$f_3(x^{\ast })=\left(a_{31}\frac{x_1^{\ast }}{x_2^{\ast }}+a_{32}\frac{1}{x_2^{\ast }}+a_{33}\frac{x_3^{\ast }}{x_2^{\ast }}\right)-\left(b_{31}\frac{x_2^{\ast }}{x_1^{\ast }}+b_{32}{x}_2^{\ast }+b_{33}\frac{x_2^{\ast }}{x_3^{\ast }}\right)>0.$$

$$f_1(x^{\ast })<f_1(x_1^{\ast }+{\delta }_1,x_2^{\ast }+{\delta }_2,x_3^{\ast })<0,$$

$$f_2(x^{\ast })>f_2(x_1^{\ast }+{\delta }_1,x_2^{\ast }+{\delta }_2,x_3^{\ast })>0,$$

Full text

A system of nonlinear equations with application to large deviations for Markov chains with finite lifetime

Ze-Chun Hu

College of Mathematics, Sichuan University, Chengdu, 610064, China

E-mail: zchu@scu.edu.cn

Wei Sun

Department of Mathematics and Statistics, Concordia University,

Montreal, H3G 1M8, Canada

E-mail: wei.sun@concordia.ca

Jing Zhang

School of Mathematics and Statistics, Hainan Normal University,

Haikou, 571158, China

E-mail: zh_jing0820@hotmail.com

Abstract In this paper, we first show the existence of solutions to the following system of nonlinear equations

[TABLE]

where $n\geq 3$ and $a_{ij},b_{ij},1\leq i,j\leq n$, are positive constants. Then, we make use of this result to obtain the large deviation principle for the occupation time distributions of continuous-time finite state Markov chains with finite lifetime.

Keywords system of nonlinear equations, continuous-time Markov chain, finite lifetime, occupation time distribution, large deviation principle.

1 Introduction and main results

In a series of fundamental papers (see [1, 2, 3, 4]), Donsker and Varadhan developed the large deviation theory for the occupation time distributions of Markov processes. By virtue of Dirichlet forms, Fukushima and Takeda derived the Donsker-Varadhan type large deviation principle for a general, not necessarily conservative symmetric Markov processes (see [5], [6, Section 6.4] and the references therein). The motivation of this work is to generalize some results of Donsker-Varadhan and Fukushima-Takeda to not necessarily conservative and not necessarily symmetric Markov processes.

We denote $E=\{1,2,\dots,n\}$ for $n\geq 2$. Let $X=((X_{t})_{t\geq 0},(P_{i})_{i\in E})$ be a continuous-time Markov chain with the state space $E$. Denote by $\zeta$ the lifetime of $X$ and denote by $Q=(q_{ij})_{i,j\in E}$ the $Q$-matrix of $X$. We assume that $Q$ satisfies the following conditions:

(1) $0<-q_{ii}<\infty,\ i\in E$.

(2) $q_{ij}>0,\ i,j\in E,\ i\neq j$.

(3) $\sum_{j\in E}q_{ij}\leq 0,\ i\in E$.

In this paper, we will derive the large deviation principle for the occupation time distributions of $X$.

We discover that the large deviations for $X$ rely heavily on the existence of solutions to the following system of nonlinear equations

[TABLE]

where $n\geq 3$ and $a_{ij},b_{ij},1\leq i,j\leq n$, are constants. It is a bit surprising to us that (1.8) turns out to be undiscussed to date. In the next section, we will prove the following result.

Theorem 1.1

Suppose that $n\geq 3$ and $a_{ij},b_{ij},1\leq i,j\leq n$, are positive constants. Then, there exists a positive solution $(x_{1},x_{2},\dots,x_{n})$ to (1.8).

As a direct consequence of Theorem 1.1, we obtain the following result.

Theorem 1.2

Suppose that $n\geq 2$ and $\beta(i)>0,1\leq i\leq n$. Then, there exist $\alpha(i)>0,1\leq i\leq n$, such that

[TABLE]

The proof of Theorem 1.2 will be given in the next section.

Remark 1.3

(a) Denote by $A$ the diagonal matrix with $A_{ii}=\alpha(i)$, $1\leq i\leq n$, $A_{ij}=0$ if $i\not=j$, and denote by $\beta=(\beta(1),\dots,\beta(n))^{T}$. Hereafter T denotes transpose. Then, we can rewrite (1.9) as follows

[TABLE]

Theorem 1.2 implies that for any vector $\beta>0$ there exists a positive diagonal matrix $A$ such that (1.10) holds.

(b) If the matrix $Q$ is symmetric, then it is easy to see that $\alpha(i)\equiv 1,1\leq i\leq n$, provide a solution to (1.9). When $Q$ is non-symmetric, Theorem 1.2 seems to be a new result in the literature.

In Section 3 of this paper, we will make use of Theorem 1.2 to obtain the large deviation principle for $X$. Define the normalized occupation time distribution $L_{t}$, $t>0$, by

[TABLE]

Let $u$ be a function on $E$. We write $u=(u(1),\dots,u(n))^{T}$ and denote $u>0$ if $u(i)>0$ for each $i\in E$. For $i\in E$, we have $Qu(i)=\sum_{j\in E}q_{ij}u(j)$. Let $\mu$ be a measure on $E$. We define

[TABLE]

Denote by ${\mathcal{P}}_{1}(E)$ the set of all probability measures on $E$.

Theorem 1.4

For each open set $G$ of ${\mathcal{P}}_{1}(E)$,

[TABLE]

For each closed set $K$ of ${\mathcal{P}}_{1}(E)$,

[TABLE]

By setting $G=K={\mathcal{P}}_{1}(E)$ in Theorem 1.4, we get

Corollary 1.5

For $i\in E$,

[TABLE]

2 Proofs of Theorems 1.1 and 1.2

Proof of Theorem 1.1.

We first consider the case that $n=3$.

We define four continuous functions $f_{1},f_{2},f_{3},F$ with the domain $D_{3}:=\{x=(x_{1},x_{2},x_{3})\in\mathbf{R}^{3}|x_{i}>0,1\leq i\leq 3\}$ by

[TABLE]

It is easy to see that the function $F$ has a minimum value at some point $x^{*}=(x_{1}^{*},x_{2}^{*},x_{3}^{*})\in D_{3}$. In the following, we will prove that

[TABLE]

Since $F$ has a minimum value at $x^{*}$, we have

[TABLE]

where $c_{ij},1\leq i,j\leq 3$, are positive constants. If $f_{3}(x^{*})=0$, then we obtain by (2.1) and (2.2) that $f_{1}(x^{*})=f_{2}(x^{*})=0$. Similarly, if $f_{1}(x^{*})=0$ or $f_{2}(x^{*})=0$, we also have

[TABLE]

Thus, to prove the existence of solutions to (1.8), we need only show that there is a contradiction if

[TABLE]

Suppose that (2.4) holds. If $f_{1}(x^{*})>0$, then we obtain by (2.3) that either $f_{2}(x^{*})<0$ or $f_{3}(x^{*})<0$. Further, we obtain by (2.1) and (2.2) that both $f_{2}(x^{*})<0$ and $f_{3}(x^{*})<0$. Similarly, we can show that if $f_{1}(x^{*})<0$, then $f_{2}(x^{*})>0$ and $f_{3}(x^{*})>0$. Therefore, to prove the existence of solutions to (1.8), we need only show that neither of the following two cases can happen: Case (i). $f_{1}(x^{*})>0$, $f_{2}(x^{*})<0$ and $f_{3}(x^{*})<0$. **Case (ii). ** $f_{1}(x^{*})<0$, $f_{2}(x^{*})>0$ and $f_{3}(x^{*})>0$. Case (i) cannot happen. Suppose that

[TABLE]

In the following, we will show that there exist sufficiently small positive numbers $\delta_{1}$ and $\delta_{2}$ such that $x^{*}_{1}-\delta_{1}>0,x^{*}_{2}-\delta_{2}>0$, $\frac{x^{*}_{1}}{x^{*}_{2}}=\frac{\delta_{1}}{\delta_{2}}$, and

[TABLE]

Define $\delta=\frac{x^{*}_{2}}{x^{*}_{1}},\delta_{2}=\delta\delta_{1}$, and

[TABLE]

Then, it is sufficient to show that there exists a positive number $\delta_{1}$ such that

[TABLE]

i.e.,

[TABLE]

Obviously, there exists a positive number $\delta_{1}$ satisfying all the above conditions. For this $\delta_{1}$, we have that $F(x^{*}_{1}-\delta_{1},x^{*}_{2}-\delta_{2},x^{*}_{3})<F(x^{*})$, which contradicts that $F$ reaches its minimum at $x^{*}$. Case (ii) cannot happen. Suppose that

[TABLE]

In the following, we will show that there exist sufficiently small positive numbers $\delta_{1}$ and $\delta_{2}$ such that $\frac{x^{*}_{1}}{x^{*}_{2}}=\frac{\delta_{1}}{\delta_{2}}$, and

[TABLE]

Define $\delta=\frac{x^{*}_{2}}{x^{*}_{1}},\delta_{2}=\delta\delta_{1}$, and

[TABLE]

Then, it is sufficient to show that there exists a positive number $\delta_{1}$ such that

[TABLE]

i.e.,

[TABLE]

Obviously, there exists a positive number $\delta_{1}$ satisfying all the above conditions. For this $\delta_{1}$, we have that $F(x^{*}_{1}+\delta_{1},x^{*}_{2}+\delta_{2},x^{*}_{3})<F(x^{*})$, which contradicts that $F$ reaches its minimum at $x^{*}$.

We now consider the general case that $n\geq 4$.

We define $(n+1)$ continuous functions $f_{1},f_{2},\dots,f_{n},F$ with the domain $D_{n}:=\{x=(x_{1},x_{2},\dots,x_{n})\in\mathbf{R}^{n}|x_{i}>0,i=1,2,\dots,n\}$ by

[TABLE]

The function $F$ has a minimum value at some point $x^{*}=(x_{1}^{*},x_{2}^{*},\dots,x_{n}^{*})\in D_{n}$. In the following, we will prove that

[TABLE]

Since $F$ has a minimum value at $x^{*}$, we have

[TABLE]

It follows that

[TABLE]

where $c_{ij},1\leq i,j\leq n$, are positive constants. Note that there is exactly one minus sign in the first $(n-1)$ equations and there is no minus sign in the last equation. Case (a). Suppose that $f_{1}(x^{*})=0$. We consider the following $(n+1)$ continuous functions with the domain $\bar{D}_{n-1}=\{(x_{2},\dots,x_{n})\in\mathbf{R}^{n-1}|x_{i}>0,i=2,\dots,n\}$:

[TABLE]

Since $F$ has a minimum value at $x^{*}\in D_{n}$, $\bar{F}$ has a minimum value at $(x_{2}^{*},\dots,x_{n}^{*})\in\bar{D}_{n-1}$. Then, we have

[TABLE]

which together with $\bar{f}_{1}(x_{2}^{*},\dots,x_{n}^{*})=f_{1}(x^{*})=0$ implies that

[TABLE]

where $\bar{c}_{ij},1\leq i,j\leq n$, are positive constants. Thus, we obtain by following the same argument for the $(n-1)$ case that

[TABLE]

Therefore,

[TABLE]

Case (b). Suppose that $\prod_{i=2}^{n}f_{i}(x^{*})=0$. By symmetry, we can assume without loss of generality that $f_{n}(x^{*})=0$. Now we consider the following $(n+1)$ continuous functions with the domain $\bar{D}_{n-1}=\{(x_{1},\dots,x_{n-1})\in\mathbf{R}^{n-1}|x_{i}>0,i=1,\dots,n-1\}$:

[TABLE]

Since $F$ has a minimum value at $x^{*}\in D_{n}$, $\bar{F}(x_{1},\dots,x_{n-1})$ has a minimum value at $(x_{1}^{*},\dots,x_{n-1}^{*})\in\bar{D}_{n-1}$. Then, we have

[TABLE]

which together with $\bar{f}_{n}(x_{1}^{*},\dots,x_{n-1}^{*})=f_{n}(x^{*})=0$ implies that

[TABLE]

where $\bar{c}_{ij},1\leq i,j\leq n-1$, are positive constants. Thus, we obtain by following the same argument for the $(n-1)$ case that

[TABLE]

Therefore,

[TABLE]

Case (c). Suppose that $\prod_{i=1}^{n}f_{i}(x^{*})\neq 0$. We will show that there is a contradiction. By symmetry, we need only consider four different subcases as follows. Case (c1). Suppose that

[TABLE]

Similar to the case that $n=3$, we can find positive numbers $\delta_{1},\delta_{2},\dots,\delta_{n-1}$ such that

[TABLE]

and

[TABLE]

It follows that

[TABLE]

which contradicts that $F$ reaches its minimum at $x^{*}$. Case (c2). Suppose that for $2\leq i\leq n-1$,

[TABLE]

We fix $x_{1}^{*},\dots,x_{i-1}^{*}$ and $x_{n}^{*}$. Similar to the case that $n=3$, we can find positive numbers $\delta_{i},\cdots,\delta_{n-1}$ such that

[TABLE]

and

[TABLE]

It follows that

[TABLE]

which contradicts that $F$ reaches its minimum at $x^{*}$. Case (c3). Suppose that

[TABLE]

Similar to the case that $n=3$, we can find positive numbers $\delta_{1},\delta_{2},\dots,\delta_{n-1}$ such that

[TABLE]

and

[TABLE]

It follows that

[TABLE]

which contradicts that $F$ reaches its minimum at $x^{*}$. Case (c4). Suppose that for $2\leq i\leq n-1$,

[TABLE]

We fix $x_{1}^{*},\dots,x_{i-1}^{*}$ and $x_{n}^{*}$. Similar to the case that $n=3$, we can find positive numbers $\delta_{i},\dots,\delta_{n-1}$ such that

[TABLE]

and

[TABLE]

It follows that

[TABLE]

which contradicts that $F$ reaches its minimum at $x^{*}$.

Proof of Theorem 1.2.

Case $n=2$. Note that now equations (1.9) become

[TABLE]

Hence we can obtain a solution to (1.9) by defining $\frac{\alpha(2)}{\alpha(1)}=\sqrt{\frac{q_{21}}{q_{12}}}$. Case $n=3$. Equations (1.9) are equivalent to

[TABLE]

Multiplying the first two equations by $\beta(1)/\beta(3)$ and $\beta(2)/\beta(3)$, respectively, and then adding them up, we obtain the third equation. Define

[TABLE]

Thus, the first two equations of (2.11) become

[TABLE]

where $a_{ij},b_{ij},1\leq i,j\leq 2$, are positive constants.

We define three continuous functions $f_{1},f_{2},F$ with the domain $D_{2}:=\{x=(x_{1},x_{2})\in\mathbf{R}^{2}|x_{1}>0,x_{2}>0\}$ by

[TABLE]

It is easy to see that the function $F$ has a minimum value at some point $x^{*}=(x^{*}_{1},x^{*}_{2})\in D_{2}$. Then, we have

[TABLE]

Since all the coefficients of the above equations are positive, we must have

[TABLE]

Hence there exists a positive solution $(x_{1},x_{2})$ to (2.14) and therefore there exist $\alpha(i)>0,1\leq i\leq 3$, such that (1.9) holds. Case $n\geq 4$. Note that the last equation of (1.9) is implied by the first $(n-1)$ equations. If we define $\alpha(i)/\alpha(1)=x_{i-1}$ for $i=2,3,\dots,n$, then equations (1.9) become equations of the type (1.8). Therefore, the proof is completed by Theorem 1.1.

3 Proof of Theorem 1.4

Let $\phi>0$ be a function on $E$. We define

[TABLE]

$(L^{\phi}_{t})_{t\geq 0}$ is a supermartingale of $X$. The upper bound (1.12) can be proved by following the standard argument (see [1]). In the following, we will focus on the proof of the lower bound (1.11).

Define

[TABLE]

Let $G$ be an open subset of ${\mathcal{P}}_{1}(E)$. Denote by $m$ the measure on $E$ satisfying

[TABLE]

If $\delta>0$ is small enough, then $(1-\delta)\mu+\delta m\in G\cap{\mathcal{M}}_{0}$ for each $\mu\in G$. From the definition of $I(\mu)$, we find that

[TABLE]

Hence $\limsup_{\delta\rightarrow 0}[I((1-\delta)\mu+\delta m)]\leq I(\mu)$. Since $\mu\in G$ is arbitrary, $\inf_{\mu\in G}I(\mu)\geq\inf_{\mu\in G\cap{\mathcal{M}}_{0}}I(\mu)$ and thus $\inf_{\mu\in G}I(\mu)=\inf_{\mu\in G\cap{\mathcal{M}}_{0}}I(\mu)$. Therefore, to prove (1.11), we need only prove that

[TABLE]

Let $f$ be a function on $E$. We define

[TABLE]

The generator of the semigroup $(P^{\phi}_{t})_{t>0}$ is given by

[TABLE]

That is, for any $i\in E$, we have

[TABLE]

Then, the matrix associated with $L^{\phi}$, denoted by $Q^{\phi}=(q^{\phi}_{ij})_{i,j\in E}$, is given by

[TABLE]

Denote by $X^{\phi}$ the Markov chain associated with $L^{\phi}$. By (3.2) and the assumption that $q_{ij}>0,\ i,j\in E,i\neq j$, we find that $X^{\phi}$ is an ergodic Markov chain. Hence $X^{\phi}$ has a unique invariant distribution, which is denoted by $\nu_{\phi}$. Note that

[TABLE]

By the ergodicity of $X^{\phi}$, we obtain that

[TABLE]

We define

[TABLE]

If we can prove the following claim

[TABLE]

then we obtain by (3.3) that

[TABLE]

and thus (3.1) is proved.

In the following, we will prove claim (3.4). Let $\mu\in{\mathcal{M}}_{0}$. We write

[TABLE]

where $h$ is a function on $E$ satisfying $h(i)>0$ for each $i\in E$. To show that $\mu\in\Pi$, it is sufficient to show that there exists a function $\phi>0$ such that

[TABLE]

Note that

[TABLE]

Hence, to show that $\mu\in\Pi$, it is sufficient to show that there exist $\phi(i)>0$, $1\leq i\leq n$, such that

[TABLE]

For $i\in E$, we define

[TABLE]

Then, equations (3.5) become equations (1.9). Since the existence of solutions to equations (1.9) is guaranteed by Theorem 1.2, the proof is complete.

Acknowledgments This work was supported by National Natural Science Foundation of China (Grant No. 11371191), Natural Sciences and Engineering Research Council of Canada (Grant No. 311945-2013), Natural Science Foundation of Hainan Province (Grant No. 117096), and Scientific Research Foundation for Doctors of Hainan Normal University.