\documentclass[amsart,notheorems,compress,14pt]{beamer}
\usepackage{CJK}
%\usepackage{graphicx}
\usepackage{xy}
\usepackage{amsfonts, bbm}
%\usepackage{mathrsfs}
\DeclareGraphicsRule{*}{mps}{*}{}
%\usepackage{subfigure}
\usepackage{xmpmulti}
\usepackage{colortbl,dcolumn}
\usepackage[english]{babel}
\usepackage{pgf,pgfarrows,pgfnodes,pgfautomata,pgfheaps}
%\usepackage{amsmath,amssymb}
%\usepackage[latin1]{inputenc}
\usepackage{times}
%\usepackage[T1]{fontenc}
\usefonttheme{serif}
\usepackage{graphicx}
\usepackage{latexsym}
%\usepackage{amsfonts}
\usepackage{mathrsfs}
 \usepackage{times,amsmath,amssymb,mathrsfs}


\setbeamerfont{frametitle}{size=\large,series=\bfseries}

\mode<presentation> {
  %\usetheme{Warsaw}
  \usetheme{Madrid}
  %\usetheme{Darmstadt}
\usefonttheme{structurebold}
  \setbeamercovered{transparent}
}

\colorlet{redshaded}{red!3!bg}
\colorlet{shaded}{black!5!bg}
\colorlet{shadedshaded}{black!5!bg}
\colorlet{blackshaded}{black!5!bg}
\colorlet{darkred}{red!5!black}
\colorlet{darkblue}{blue!20!black}
\colorlet{darkgreen}{green!20!black}

\def\red{\color{red}}
\def\green{\color{green}}
\def\blue{\color{blue}}
\def\brown{\color{brown}}
\def\pink{\color{pink}}
\def\cyan{\color{cyan}}
\def\orange{\color{orange}}
\def\gray{\color{gray}}

\def\black{\color{black}}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{hypothesis}{Hypothesis}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{example}{Example}[section]
\newtheorem{question}{Question}[section]

\setbeamercolor{title}{fg=white, bg=blue!70!black}
\setbeamercolor{block title}{fg=white, bg=green!50!black}
\setbeamercolor{frametitle}{fg=white, bg=blue!70!black}

 \def\bhypothesis{\begin{hypothesis}\sl{}\def\ehypothesis{\end{hypothesis}}}
 \def\bdefinition{\begin{definition}\sl{}\def\edefinition{\end{definition}}}
 \def\blemma{\begin{lemma}\sl{}\def\elemma{\end{lemma}}}
 \def\bproposition{\begin{proposition}\sl{}\def\eproposition{\end{proposition}}}
 \def\btheorem{\begin{theorem}\sl{}\def\etheorem{\end{theorem}}}
 \def\bcorollary{\begin{corollary}\sl{}\def\ecorollary{\end{corollary}}}
 \def\bremark{\begin{Remark}\rm{}\def\eremark{\end{Remark}}}
 \def\bexample{\begin{example}\rm{}\def\eexample{\end{example}}}

 \def\beqlb{\begin{eqnarray}}\def\eeqlb{\end{eqnarray}}
 \def\beqnn{\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}}

%\newtheorem{dingli}[theorem]{\CJKfamily{hei}定理}
%\newtheorem{mingti}[theorem]{\CJKfamily{hei}命题}
%\newtheorem{yinli}[theorem]{\CJKfamily{hei}引理}
%\newtheorem{tuilun}[theorem]{\CJKfamily{hei}推论}
%\newtheorem{wenti}[theorem]{\CJKfamily{hei}问题}
 \def\wi{\widetilde}
 %\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
%\renewcommand{\labelenumi}{[\arabic{enumi}]}
\def\ar{\!\!\!&}

 \def\mcr{\mathscr}
 \def\mbb{\mathbb}
 \def\term{\it}
\def\rf{\ref}
 \def\az{\alpha}
\def\bz{\beta}
\def\gz{\gamma}
\def\sz{\sigma}
\def\dz{\delta}
\def\ez{\epsilon}
\def\lz{\lambda}
\def\oz{\omega}
\def\vz{\varphi}
\def\wz{\wedge}
\def\A{{\cal A}}
\def\B{{\cal B}}
\def\g{G}
\def\H{\mathcal {H}}
\def\K{{\cal K}}
\def\U{{\cal U}}
\def\V{{\cal V}}
\def\W{{\cal W}}
\def\M{{\cal M}}
\def\I{{\cal I}}
\def\J{{\cal J}}
\def\L{{\cal L}}
\def\R{{\cal R}}
\def\P{{\cal P}}
\def\Q{{\cal Q}}
\def\N{{\cal N}}
\def\S{{\cal S}}
\def\T{{\overline{T}}}
\def\l{\underline{l}}
\def\n{{\underline{n}}}
\def\m{{\underline{m}}}
\def\e{\underline{e}}
\def\nn{{\n\times\n}}
\def\mm{{\m\times\m}}
\def\tt{{t\times t}}
\def\KM{(\K,\M)}
\def\rad{\mbox{rad}}
\def\ud{\underline}
\def\wt{\widetilde}
\def\lra{\longrightarrow}
\def\lla{\longleftarrow}
\def\ra{\rangle}
\def\la{\langle}
\def\rar{\rightarrow}
\def\lar{\leftarrow}
\def\Hom{\mbox{Hom}}
\def\diag{\mbox{diag}}
\def\op{\mbox{op}}
\def\Ext{\mbox{Ext}}
\def\End{\mbox{End}}
\def\dim{\mbox{dim}}
\def\mod{\mbox{mod}}
\def\ind{\mbox{ind}}
\def\Im{\mbox{Im}}
\def\Ker{\mbox{Ker}}
\def\Coker{\mbox{Coker}}
\def\add{\mbox{add}}
\def\Supp{\mbox{Supp}}
\def\top{\mbox{top}}
\def\soc{\mbox{soc}}
\def\rank{\mbox{rank}}
\def\mf{\mathfrak}

 \def\beqlb{\begin{eqnarray}}\def\eeqlb{\end{eqnarray}}
 \def\beqnn{\begin{eqnarray*}}\def\eeqnn{\end{eqnarray*}}

 \newcommand{\bcen}{\begin{center}}
 \newcommand{\ecen}{\end{center}}
 \newcommand{\bgeqn}{\begin{equation}}
 \newcommand{\edeqn}{\end{equation}}


 \def\mcr{\mathscr}\def\mbb{\mathbb}
 \def\mbf{\mathbf}\def\mrm{\mathrm}

\renewcommand{\raggedright}{\leftskip=0pt \rightskip=0pt plus 0cm}
\raggedright

\def\hilite<#1>{%
\temporal<#1>{\color{blue!25}}{\color{magenta}}%
{\color{blue!55}}}

%\newcolumntype{H}{>{\columncolor{blue!20}}c!{\vrule}}
%\newcolumntype{H}{>{\columncolor{blue!20}}c}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%重定义字体、字号命令 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\songti}{\CJKfamily{song}}        % 宋体
\newcommand{\fangsong}{\CJKfamily{fs}}        % 仿宋体
\newcommand{\kaishu}{\CJKfamily{kai}}         % 楷体
\newcommand{\heiti}{\CJKfamily{hei}}          % 黑体
\newcommand{\lishu}{\CJKfamily{li}}           % 隶书
\newcommand{\youyuan}{\CJKfamily{you}}       % 幼圆
\newcommand{\sihao}{\fontsize{14pt}{\baselineskip}\selectfont}      % 字号设置
\newcommand{\xiaosihao}{\fontsize{12pt}{\baselineskip}\selectfont}  % 字号设置
\newcommand{\wuhao}{\fontsize{10.5pt}{\baselineskip}\selectfont}    % 字号设置
\newcommand{\xiaowuhao}{\fontsize{9pt}{\baselineskip}\selectfont}   % 字号设置
\newcommand{\liuhao}{\fontsize{7.875pt}{\baselineskip}\selectfont}  % 字号设置
\newcommand{\qihao}{\fontsize{5.25pt}{\baselineskip}\selectfont}    % 字号设置
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\begin{CJK*}{GBK}{kai}


%\textcolor[rgb]{0.00,0.55,0.50}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%            题页            %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title[SBP]{Single Birth Processes}

\author[Yu-Hui Zhang]{Yu-Hui Zhang}

\date[April 28, 2014]{\wuhao (April 28, 2014, Institute of Mathematics, Academia Sinica) }

\institute[BNU]{\sihao Beijing Normal Univesity}

\frame{\titlepage}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%            目录            %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{目录}
  %  \frame{\frametitle{目录}\tableofcontents}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%         背景及问题        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\large
%----------------------------------------------------------------------------------------2

\frame{
 \frametitle{Outline: I}\small

\begin{enumerate}
\item Background
\item Minimal nonnegative solution
\item Classical problems
\item Criterion on uniqueness of SBP
\item Criterion on recurrence
\item Criteria on ergodicity and strong ergodicity
\item Stationary distribution
%\item Criterion on $\ell$-ergodicity
\end{enumerate}
}
%----------------------------------------------------------------------------------------3

\frame{
 \frametitle{Outline: II}\small

\begin{enumerate}
\item Poisson equation
\item Uniqueness
\item Recurrence, extinction/return probability
\item Ergodicity, strong ergodicity, and the first moment of return time
\item Polynomial moments of hitting time and life time
\item Exponential ergodicity and Laplace transform of return time
\item Examples
\item Related topics and applications
\end{enumerate}
}

%----------------------------------------------------------------------------------------4
\frame{ \frametitle{1. Background}\small
\begin{itemize}
\item $Q$-process: on $\mathbb Z_+:=\{0, 1, 2, \dots\}$, suppose that sub-Markov transition probability matrix $P(t)=(p_{ij}(t), i, j\in\mathbb Z_+)$ satisfies

(1) Normal condition: $p_{ij}(t)\ge 0,\ \sum_jp_{ij}(t)\le 1,\  i, j\in\mathbb Z_+, t\ge 0$.

(2) Chapman-Kolmogorov equation: $p_{ij}(t+s)=\sum_kp_{ik}(t)p_{kj}(s)$.

(3) Jump condition: $\lim_{t\to 0}p_{ij}(t)=\delta_{ij}, \  i, j\in\mathbb Z_+.$

(4) $Q$-condition: $\lim_{t\to 0}(p_{ij}(t)-\delta_{ij})/t=q_{ij}, \  i, j\in\mathbb Z_+,$ i.e. the $Q$-matrix $Q=(q_{ij})$ is the derivative matrix at time $0$ of $P(t)$,
 $$0\leqslant q_{ij}<\infty,\ i\ne j;\ \sum_{j\ne i}q_{ij}\leqslant -q_{ii}=:q_i\leqslant \infty,\ i\in \mathbb Z_+.$$
\item totally stable: $q_i<\infty, \ i\in\mathbb Z_+$.
\item conservative: $q_i=\sum_{j\ne i}q_{ij},\ i\in\mathbb Z_+$, i.e. $\sum_{j\in\mathbb Z_+}q_{ij}=0$.
\end{itemize}}
%----------------------------------------------------------------------------------------5
\frame{\frametitle{1. Background }\small
Single birth process: $Q=(q_{ij}:i,j\ge 0)$: $q_{i,i+1}>0,\ q_{i,i+j}=0,\ i\ge 0,j\ge 2$.
%\vspace{0.3cm}
\begin{center}
\setlength{\unitlength}{1cm}
\begin{picture}(11,2)
\put(4.9,-0.5){$i$}
\put(3.55,-0.5){$i-1$}
\put(5.55,-0.5){$i+1$}
\put(6.55,0.5){$i+2$}
\put(-0.15,-0.5){$0$}
\put(0.85,-0.5){$1$}
\put(0,0){\line(1,0){11}}
\qbezier(1,0)(3,2)(5,0)
\qbezier(0,0)(2.5,2)(5,0)
\qbezier(4,0)(4.5,2)(5,0)
\qbezier(6,0)(5.5,2)(5,0)
\put(0.15,0.12){\vector(-1,-1){0.1}}
\put(1.12,0.12){\vector(-1,-1){0.1}}
\put(4.1,0.25){\vector(-1,-2){0.1}}
\put(5.9,0.25){\vector(1,-2){0.1}}
\put(0,0){\circle*{0.2}}
\put(1,0){\circle*{0.2}}
\put(4,0){\circle*{0.2}}
\put(5,0){\circle*{0.2}}
\put(6,0){\circle*{0.2}}
\put(7,0){\circle*{0.2}}
\put(9,0){\circle*{0.2}}
\put(10,0){\circle*{0.2}}
\end{picture}
\end{center}

\vskip0.3cm
$$Q=\left(%
\begin{array}{ccccc}
  - & + & 0 & 0 & \cdots \\
  * & - & + & 0 & \cdots \\
  * & * & - & + & \cdots\\
  \vdots & \vdots & \vdots & \vdots & \ddots \\
  \end{array}%
\right)$$
}
%----------------------------------------------------------------------------------------6
\frame{\frametitle{1. Background}\small

\begin{enumerate}
\item The exit boundary of the process consists at most one single extremal point
\vskip0.1cm
\item Fundamental comparison tool in studying more complex processes, such as
infinite-dimensional reaction-diffusion processes
\vskip0.1cm
\item Usually the single birth process is non-symmetric (irreversible)
\item Applications in other fields
\end{enumerate}
}
%----------------------------------------------------------------------------------------7
\frame{\frametitle{1. Background}\small
\begin{enumerate}
\item Upwardly skip-free processes (W.J. Anderson, 1991);
\item generalized birth and death processes (J.K. Zhang, 1984);
\item Population processes: birth and death processes with catastrophes; birth, death and catastrophe processes
$$\aligned i&\to i+1\\
&\to i-1\\
&\to i-2\\
&\to\cdots\\
&\to 0\endaligned
\quad
\aligned \hbox{at rate}\quad & {\blue b_i}\\
& {\blue a_i}+{ c_i}{\red f_{i,i-1}}\\
& { c_i}{\red f_{i,i-2}}\\
&\cdots\\
& { c_i}{\red f_{i0}}\endaligned$$
where $\sum_{j=0}^{i-1}{\red f_{ij}}=1$.
\end{enumerate}
}
%----------------------------------------------------------------------------------------8
\frame{\frametitle{1. Background}\small

Brockwell, Gani, Resnick and Pakes et al (1982-1986) for some special
catastrophes $f_{ij}$ ($b_i=b+\lambda i$, $a_i=0$, $c_i=ci$, $i\ge 0$).

\vskip0.2cm
geometric: $f_{ij}=p(1-p)^{i-1-j}\,(1\le j<i); f_{i0}=(1-p)^{i-1}$;
\vskip0.2cm
uniform: $f_{ij}=1/i\,(0\le j<i)$;
\vskip0.2cm
binomial: $f_{ij}=\binom{i-1}{j}p^j(1-p)^{i-1-j}\,(0\le j<i)$.

\vskip0.3cm
Aim: extinction times and probability of extinction.
\vskip0.3cm
Keys: generating function of $Q$-resolvent.
}
%----------------------------------------------------------------------------------------9
\frame{\frametitle{1. Background}\small

\vskip1cm
B. Cairns \& P. Pollett (2004).
$$\aligned i&\to i+1\\
&\to i-1\\
&\to\cdots\\
&\to 1\\
&\to 0\endaligned
\quad
\aligned \hbox{at rate}\quad & {\blue g_i} b\\
& {\blue g_i}{\red d_{1}}\\
&\cdots\\
& {\blue g_i}{\red d_{i-1}}\\
& {\blue g_i}\hbox{\red $\sum_{k\ge i}d_k$}\endaligned$$
where $b+\sum_{k\ge 1}d_k=1$.
}
%----------------------------------------------------------------------------------------10
\frame{\frametitle{1. Background}\small
single birth process:

Yan, S.-J. and Chen, M.-F.(1986), Multidimensional $Q$-processes, Chinese Ann. Math.,
\textbf{7(B)}(1), 90-110.

\vskip0.2cm
Chen, M.-F.(2004), From Markov Chains to Non-Equilibrium Particle Systems, Second Edition,
World Scientific, Singapore.

\vskip0.2cm
陈木法, 毛永华(2007), 随机过程导论, 高等教育出版社, 北京.

\vskip0.4cm
Population processes:

Anderson, W. J.(1991), Continuous-Time Markov Chains, Springer-Verlag, New York.
}
%----------------------------------------------------------------------------------------11
\frame{\frametitle{2. Minimal nonnegative solution}\small
For $a_{ij}\ge 0, b_i\ge 0$, consider the equation
$$x_i=\sum_{k}a_{ik}x_k+b_i,\ i\in\mathbb Z_+.$$

\begin{block}{Theorem[the first successive approximation scheme]} The minimal solution $(x_i^*)$ always exists uniquely. Furthermore, it can be obtained by the following procedure
$$x^{(0)}_i=0,\ x^{(n+1)}_i=\sum_{k}a_{ik}x^{(n)}_k+b_i,\ i\in\mathbb Z_+,n\ge 0,$$
then $x_i^{(n)}\uparrow x_i^*$ as $n\to\infty$.
\end{block}
}
%----------------------------------------------------------------------------------------12
\frame{%\frametitle{2. Minimal nonnegative solution}
\small

\begin{block}{Comparison Theorem} Assume that $\tilde a_{ij}\ge a_{ij}, \tilde b_i\ge b_i$. Then for any solution $(\tilde x_i)$ to
$$\tilde x_i=\sum_{k}\tilde a_{ik}\tilde x_k+\tilde b_i,\ i\in\mathbb Z_+,$$
we have $\tilde x_i\ge x_i^*$.
\end{block}

\begin{block}{Theorem[the second successive approximation scheme]} Assume that $b^{(n)}_i\ge 0$. Define
$$\tilde x^{(1)}_i=b^{(1)}_i,\ \tilde x^{(n+1)}_i=\sum_ka_{ik}\tilde x^{(n)}_k+b^{(n+1)}_i,\ i\in\mathbb Z_+,n\ge 1.$$
If $b_i^{(n)}\uparrow b_i$ (or, $\sum_{n=1}^\infty b_i^{(n)}=b_i$), then $\tilde x_i^{(n)}\uparrow x_i^*$ (or $x_i^*=\sum_{n=1}^\infty \tilde x_i^{(n)}$). In particular,
if we set  $\tilde x_i^{(1)}=b_i,\ \tilde x^{(n+1)}_i=\sum_ka_{ik}\tilde x^{(n)}_k$,
then $x_i^*=\sum_{n=1}^\infty \tilde x_i^{(n)}$.
\end{block}
}
%----------------------------------------------------------------------------------------13
\frame{%\frametitle{2. Minimal nonnegative solution}
\small

\begin{block}{Localization Theorem} Let $G\subset\mathbb Z_+$ and $(\tilde x^*_i: i\in G (\text{resp.} i\in\mathbb Z_+))$ be the minimal solution to
$$x_i=\sum_{k\in G}a_{ik}x_k+\sum_{k\in G^c}a_{ik}x_k^*+b_i,\ i\in G (\text{resp.} i\in\mathbb Z_+).$$
Then we have $\tilde x_i^*= x_i^*, i\in G (\text{resp.} i\in\mathbb Z_+)$.
\end{block}

\begin{block}{Theorem} Assume that $\sum_ja_{ij}+b_i\le 1$. Then the maximal solution to
$$ x_i=\sum_{k} a_{ik} x_k+ b_i,\ 0\le x_i\le 1, i\in\mathbb Z_+,$$
can be obtained by the following procedure
$$x^{(0)}_i=1,\ x^{(n+1)}_i=\sum_{k}a_{ik}x^{(n)}_k+b_i,\ i\in\mathbb Z_+,n\ge 0,$$
then $x_i^{(n)}\downarrow x_i^*$ as $n\to\infty$.
\end{block}
}
%----------------------------------------------------------------------------------------14
\frame{\frametitle{3. Classical problems}\small

$Q$-process: $P'(t)|_{t=0}=Q$.

Backward Kolmogorov equation: $P'(t)=QP(t)$.

Forward Kolmogorov equation: $P'(t)=P(t)Q$.

$$p_{ij}(t)=\sum_{k\ne i}\int_0^tq_{ik}e^{-q_i(t-s)}p_{kj}(s)ds+\delta_{ij}e^{-q_it},$$
$$p_{ij}(t)=\sum_{k\ne j}\int_0^tp_{ik}(s)q_{kj}e^{-q_j(t-s)}ds+\delta_{ij}e^{-q_jt},$$

\begin{itemize}
\item $Q$-process exist? unique?
\end{itemize}

\begin{block}{Theorem} Let $Q=(q_{ij})$ be totally stable and conservative. Every $Q$-process satisfies the backward Kolmogorov equation.
\end{block}
}
%----------------------------------------------------------------------------------------15-1
\frame{\frametitle{3. Classical problems}\small

Existence, uniqueness, recurrence, positive recurrence.

\begin{block}{Existence Theorem} Given a totally stable and conservative $Q$-matrix, there always exist a $Q$-process. In details, the minimal solution is a $Q$-process. Moreover, BE and FE have the same minimal solution.
\end{block}

Laplace transform: $p_{ij}(\lambda)=\int_0^\infty e^{-\lambda t}p_{ij}(t)dt$.
$$p_{ij}(\lambda)=\sum_{k\ne i}\frac{q_{ik}}{\lambda+q_i}p_{kj}(\lambda)+\frac{\delta_{ij}}{\lambda+q_i},$$
$$p_{ij}(\lambda)=\sum_{k\ne j}p_{ik}(\lambda)\frac{q_{kj}}{\lambda+q_j}+\frac{\delta_{ij}}{\lambda+q_i}.$$
}
%----------------------------------------------------------------------------------------15-2
\frame{\small
$$\bar p_{ij}^{(1)}(\lambda)=\frac{\delta_{ij}}{\lambda+q_i},\quad
\bar p_{ij}^{(n+1)}(\lambda)=\sum_{k\ne i}\frac{q_{ik}}{\lambda+q_i}\bar p_{kj}^{(n)}(\lambda),\quad n\ge 1.$$
$$p_{ij}^{\text{min}}(\lambda)=\sum_{n=1}^\infty\bar p_{ij}^{(n)}(\lambda).$$


$$\tilde p_{ij}^{(1)}(\lambda)=\frac{\delta_{ij}}{\lambda+q_i},\quad
\tilde p_{ij}^{(n+1)}(\lambda)=\sum_{k\ne j}\tilde p_{ik}^{(n)}(\lambda)\frac{q_{kj}}{\lambda+q_j},\quad n\ge 1.$$

$$\bar p_{ij}^{(n)}(\lambda)=\tilde p_{ij}(\lambda).$$

Consider $P(\lambda)-P^{\text{min}}(\lambda)$:
}
%----------------------------------------------------------------------------------------16

\frame{%\frametitle{3. Classical problems}
\small
\begin{block}{}
$$u_i=\sum_{k\ne i}\frac{q_{ik}}{\lambda+q_i}u_{k}, \ 0\le u_i\le 1, i\in\mathbb Z_+.$$
\end{block}

\begin{block}{Uniqueness Theorem} For a given totally stable and conservative $Q$-matrix $Q=(q_{ij})$, the $Q$-process is unique iff the equation has only the trivial solution $u_i\equiv0$ for some (equivalently, for all) $\lambda>0$.
\end{block}

Feller(1957) and Reuter(1957)

}
%----------------------------------------------------------------------------------------17
\frame{%\frametitle{2. Poisson equation}
\small
\begin{itemize}
\item $P(t)$ is recurrent if for each $h>0$, $P(h)$ is recurrent. Equivalently, $\int_0^\infty p_{ii}(t)dt=\infty$.
\end{itemize}

embedding chain $(\Pi_{ij})$:
$$\Pi_{ij}=I_{\{q_i\ne0\}}(1-\delta_{ij})\frac{q_{ij}}{q_i}+I_{\{q_i=0\}}\delta_{ij}.$$

\begin{block}{Recurrence Theorem(Feller,1957)} For a given totally stable and conservative $Q$-matrix $Q=(q_ij)$,
$$\int_0^\infty p^{\text{min}}_{ij}(t)dt=\sum_{n=0}^\infty \frac{\Pi_{ij}^{(n)}}{q_j}.$$
In particular, if $Q$ is irreducible and regular, then $P(t)$ is recurrent iff so is its embedding chain.
\end{block}

regular: totally stable, conservative and $Q$-process is unique.
}
%----------------------------------------------------------------------------------------18-1
\frame{%\frametitle{2. Poisson equation}
\small

For irreducible chain $P=(p_{ij})$,
$(\mathbb P_i[\sigma_{j_0}<\infty]:i\in\mathbb Z_+)$ is the minimal solution to the equation
$$x_i=\sum_{k\ne j_0}p_{ik}x_k+p_{ij_0},\ i\in\mathbb Z_+.$$

$$f_{ij_0}^{(1)}=p_{ij_0}=\mathbb P_i[\sigma_{j_0}=1],$$
$$f_{ij_0}^{(n+1)}=\sum_{k\ne j_0}p_{ik}f_{kj}^{(n)}=\mathbb P_i[\sigma_{j_0}=n+1],\ n\ge 1.$$
$$f_{ij_0}^{*}=\sum_{n=1}^\infty f_{ij_0}^{(n)}=\sum_{n=1}^\infty\mathbb P_i[\sigma_{j_0}=n]=\mathbb P_i[\sigma_{j_0}<\infty].$$

$P$ is recurrent iff $f_{ij_0}^*\equiv 1$.
}
%----------------------------------------------------------------------------------------18-2
\frame{%\frametitle{2. Poisson equation}
\small

For embedding chain $(\Pi_{ij})$,
$(f_{ij_0}^*=\mathbb P_i(\sigma_{j_0}<\infty):i\in\mathbb Z_+)$ is the minimal solution to the equation
$$x_i=\sum_{k\ne j_0}\Pi_{ik}x_k+\Pi_{i,j_0},\ i\in\mathbb Z_+.$$

$(\Pi_{ij})$ is recurrent iff $f_{ij_0}^*\equiv 1$.

\begin{block}{Theorem} For a regular and irreducible $Q=(q_{ij})$,  $P(t)$ is recurrent iff for some
(equivalently, all) $j_0$,
$$
x_i=\sum_{j\ne j_0,i}\frac{q_{ij}}{q_i}x_j,\qquad 0\le x_i\le 1,\,i\ge 0
$$
has only a trivial solution.\end{block}
}

%----------------------------------------------------------------------------------------19
\frame{
\small

Let $Q=(q_{ij})$ be a regular irreducible $Q$-matrix. Then the limit
$$\lim_{t\to\infty}p_{ij}(t)=:\pi_j$$
exists and it is independent of $i$. Moreover, we have either
$\sum_j\pi_j=1$ or $\sum_j\pi_j=0$.

\begin{itemize}
\item $P(t)$ is positive recurrent or ergodic if so is $P(h)$ for every $h>0$. Equivalently, $\lim_{t\to\infty}p_{ii}(t)=\pi_i>0$.
\item $P(t)$ is strongly ergodic or uniformly ergodic if
$$\lim_{t\to\infty}\sup_i|p_{ij}(t)-\pi_j|=0.$$
\end{itemize}
}
%----------------------------------------------------------------------------------------20
\frame{\small
Let $Q=(q_{ij})$ be a regular irreducible $Q$-matrix and $H$ be a non-empty finite subset of $\mathbb Z_+$. Define
$\sigma_H=\inf\{t\ge\text{\,the first jump}: X(t)\in H\}$.

\begin{block}{Theorem(Issacson and Arnold, 1978)}

(1) The $Q$-process is ergodic iff $\mathbb E_i\sigma_H<\infty$ for all $i\in H$.

(2) The $Q$-process is strongly ergodic iff $\sup_i\mathbb E_i\sigma_H<\infty$.
\end{block}

\begin{block}{Theorem(Tweedie, 1981)}
 The $Q$-process is ergodic (resp. strongly erergodic) iff the equation
 $$\begin{cases}\sum_jq_{ij}y_j\le -1,\ i\notin H\\\sum_{i\in H}\sum_{j\ne i}q_{ij}y_j<\infty
 \end{cases}$$
 has a finite (resp. bounded) nonnegative solution.
\end{block}

\begin{block}{Theorem($y_i=1/c(i\ne k), y_k=0$)} Given a regular and irreducible $Q$-matrix $Q=(q_{ij})$. If there exists some $k$ so that $c:=\inf_{i\ne k}q_{ik}>0$, then
the $Q$-process is str. erg.
\end{block}

}
%----------------------------------------------------------------------------------------21
\frame{\small
Assume that $Q$ is irreducible. $\pi=(\pi_i)$ is stationary distribution.
\begin{itemize}
\item Ordinary ergodicity: $\displaystyle\lim_{t\to\infty}|p_{ij}(t)-\pi_j|=0$.

\item Exponential ergodicity: $\displaystyle\lim_{t\to\infty}{\red e^{\alpha t}}|p_{ij}(t)-\pi_j|=0$.

\item Strong ergodicity: $\displaystyle\lim_{t\to\infty}\sup_i|p_{ij}(t)-\pi_j|=0$

\hskip3cm
$\Longleftrightarrow\displaystyle\lim_{t\to\infty}{\red e^{\beta t}}\sup_i|p_{ij}(t)-\pi_j|=0$.

\item Strong ergodicity $\Rightarrow$ Exponential ergodicity $\Rightarrow$
Ordinary ergodicity.
\end{itemize}
}

%----------------------------------------------------------------------------------------22
\frame{\frametitle{4. Criterion on uniqueness of SBP}\small
For $0\leq k<n$, define $q_n^{(k)}=\sum_{j=0}^k q_{nj}$ and
$${ m}_0=\frac{1}{q_{01}},\quad
{ m}_n=\frac 1{q_{n,n+1}}
\bigg(1+\sum_{k=0}^{n-1}{ q}_n^{(k)}{ m}_k\bigg),\quad n\ge 1.$$

\begin{block}{Uniqueness(J.K. Zhang, 1984; S.J. Yan \& M.F. Chen, 1986)}
Given a totally stable and conservative single birth $Q$-matrix $Q=(q_{ij})$, the process is unique $($non-explosive$)$ iff
$\sum_{n=0}^\infty m_n=\infty$.
\end{block}

\begin{block}{}
$$(\lambda+q_i)u_i=\sum_{k\ne i}q_{ik}u_{k}, \ 0\le u_i\le 1, i\in\mathbb Z_+.$$
\end{block}
}
%----------------------------------------------------------------------------------------23
\frame{\small
For birth-death $(a_i,b_i)$: $a_i=q_{i,i-1}>0, i\ge 1; b_i=q_{i,i+1}>0, i\ge 0$.
$$m_n=\frac 1{\mu_nb_n}\mu[0,n],\qquad n\ge 0,$$
where
$$\mu_0=1, \qquad \mu_i=\frac{b_0b_1\cdots b_{i-1}}{a_1a_2\cdots a_i},\quad i\ge 1,$$
and $\mu[i,k]=\sum_{j=i}^k\mu_j$.

\begin{block}{Uniqueness Criterion for BDP}
Given a totally stable and conservative birth-death $Q$~matrix $(a_i, b_i)$. The the birth-death is unique iff
$R:=\sum_{n=0}^\infty \frac 1{\mu_nb_n}\mu[0,n]=\infty$.\end{block}

}
%----------------------------------------------------------------------------------------24
\frame{\frametitle{5. Criterion on recurrence of SBP}\small

Define
 $$F_i^{(i)}=1, \ F_n^{(i)}=\frac 1{q_{n,n+1}} \sum_{k=i}^{n-1}q_n^{(k)}F_k^{(i)},\ 0\le i<n.$$

\begin{block}{Recurrence(S.J. Yan \& M.F. Chen,1986)} Assume the single birth $Q$-matrix $Q=(q_{ij})$ is regular and irreducible.
Then the process is recurrent iff $\sum_{n=0}^\infty F_n^{(0)}=\infty$.
\end{block}

\begin{block}{}
$$
x_i=\sum_{j\ne 0,i}\frac{q_{ij}}{q_i}x_j,\qquad 0\le x_i\le 1,\,i\ge 0
$$
\end{block}
}
%----------------------------------------------------------------------------------------25
\frame{\small

For birth-death $Q$~matrix $(a_i,b_i)$:

$$F_n^{(0)}=\frac{b_0}{\mu_nb_n},\ n\ge 0.$$

\begin{block}{Recurrence Criterion for BDP}
Given a regular birth-death $Q$-matrix $(a_i, b_i)$. The the birth-death is recurrent iff
$\sum_{n=0}^\infty \frac{1}{\mu_nb_n}=\infty$.
\end{block}
}
%----------------------------------------------------------------------------------------23
\frame{\frametitle{6. Criteria on ergodicity and strong ergodicity}\small
Define
$$d_0=0,\ d_n=\frac 1{q_{n,n+1}}\Big(1+\sum_{k=0}^{n-1}q_n^{(k)}d_k\Big),\ n\ge 1.$$

\begin{block}{Ergodicity(S.J. Yan \& M.F. Chen,1986; Zhang, 2001)}
Assume the single birth $Q$-matrix $Q=(q_{ij})$ is regular and irreducible. Then
the process is ergodic iff
$d:=\sup_{k\ge 0}\frac{\sum_{n=0}^k d_n}{\sum_{n=0}^k F_n^{(0)}}<\infty$;
the process is strongly ergodic iff
$S:=\sup_{k\ge 0}\sum_{n=0}^k(F^{(0)}_nd-d_n)<\infty$.
\end{block}

\begin{block}{$H=\{0\}$}
$$\begin{cases}\sum_j q_{ij}y_j\le -1,\quad i\notin H&\\
\sum_{i\in H}\sum_{j\ne i}q_{ij}y_j<\infty&
\end{cases}$$
\end{block}
}
%----------------------------------------------------------------------------------------26
\frame{\small
\begin{block}{}$$\sum_{j\ne i} q_{ij}(y_j-y_i)+1\le 0,\ i\ge 1.$$\end{block}
Sketch of the proof for ergodicity:
\begin{itemize}
\item Assume that $(u_i)$ be a finite nonnegative solution and $u_0=0$. By induction,
$$u_{n+1}-u_n\leq F_n^{(0)}u_1-d_n,\ n\ge 0.$$
\item
$$0\le u_{n+1}\le u_1\sum_{k=0}^nF_k^{(0)}-\sum_{k=0}^nd_k,\ n\ge 0.$$
\item $d\le u_1<\infty$.
\item Assume $d<\infty$. Define
$$u_0=0,\quad u_1=d,\quad u_{n+1}=u_n+F_n^{(0)}u_1-d_n,\qquad n\ge 0.$$
\item $(u_n)$ is a finite nonnegative.
\end{itemize}
}
%----------------------------------------------------------------------------------------27
\frame{\small
Sketch of the proof for strong ergodicity:

\begin{itemize}
\item  the equation
$$
y_i=\sum_{j\ne i}\frac{q_{ij}}{q_i}y_j+\frac{1}{q_i}, \quad i\ge1;\quad y_0=0 \quad (*)
$$
has a bounded nonnegative solution iff $S<\infty$.
$$d=\lim_{n\to\infty}\sum_{k=0}^nd_k\bigg/\sum_{k=0}^nF_k^{(0)},$$
and the unique solution $(u_i^*)$ (the minimal solution) of ($*$).
\item Assume that the process is strongly ergodic. Then there exists a bounded nonnegative solution $(u_i)$, i.e.,
$$u_i\ge\sum_{j\ne i}\frac{q_{ij}}{q_i}u_j+\frac{1}{q_i},\ i\ge1;\ u_0\ge 0.$$
By Comparison Theorem, $u_i\ge u_i^*\,(i\ge 0)$.
Hence, $(u_i^*)$ is bounded $\Rightarrow S<\infty$.\end{itemize}
}

%----------------------------------------------------------------------------------------28
\frame{\small
For birth-death $Q$~matrix $(a_i,b_i)$:
$$d_n=\frac 1{\mu_nb_n}\mu[1,n],\qquad n\ge 0.$$
\begin{block}{Ergodicity Criteria for BDP}
Given a regular birth-death $Q$-matrix $(a_i, b_i)$. The the birth-death chain is ergodic iff
$\mu[0,\infty)<\infty$; it is strongly ergodic iff $S:=\sum_{n=0}^\infty\frac 1{\mu_n b_n}\sum_{k=n+1}^\infty\mu_k
<\infty$.
\end{block}

H.J. Zhang, X. Lin \& Z.T. Hou, 2000; Tweedie, 1981.
}
%----------------------------------------------------------------------------------------29
\frame{\frametitle{7. Stationary distribution}\small
\begin{block}{Stationary distribution(Y.H. Zhang, 2004)} Assume that the process is ergodic. Define
$$c_k:=\sup_{i\ge k}\frac{\sum_{j=k}^i m_j}{\sum_{j=k}^iF_j^{(k)}}.\quad \text{Then}\
\pi_k=\frac 1{q_{k,k+1}c_k},\quad k\ge 0.$$
\end{block}
\begin{block}{Example 1} $b_i=q_{i,i+1}>0\,(i\ge 0)$, $a_i=q_{i,i-1}>0\,(i\ge 1)$.
Define $\mu_0=1$, $\mu_i=b_0\cdots b_{i-1}/a_1\cdots a_i\,(i\ge 1)$ and $\mu=\sum_{i=0}^\infty\mu_i$.
Then $c_i=\mu/(\mu_ib_i)$, $\pi_i=\mu_i/\mu\,(i\ge 0)$.
\end{block}
\begin{block}{Example 2} $q_{i,i+1}=1\,(i\ge 0)$ and $q_{i0}=1\,(i\ge 1)$.
Then $c_k=2^{k+1}$, $\pi_k=2^{-k-1}\,(k\ge 0)$.\end{block}
}
%----------------------------------------------------------------------------------------30
\frame{\small

\begin{block}{Example 3} $q_{i,i+1}=1\,(i\ge 0)$, $q_{10}=1$ and $q_{i,i-2}=1\,(i\ge 2)$. Then $\{F_j^{(k)}\}_{j=k}^\infty$ is a Fibonacci sequence:
$F_{k+n}^{(k)}=\frac 1{\sqrt 5}[A^{n+1}-(-B)^{n+1}],\ n\ge 0, k\ge 0$,
where $A=(\sqrt 5 +1)/2, B=(\sqrt 5 -1)/2$.
Then $c_k=A^{k+2}$, $\pi_k=B^{k+2}\,(k\ge 0)$.\end{block}

\begin{block}{Example 4(uniform catastrophes)} $q_{i,i+1}=\lambda_i:=a+\lambda i\,(i\ge 0)$,
$q_{ij}=\beta\,(0\le j<i)$, where $a>0$, $\lambda>0$, $\beta>0$. Then
$$\pi_0=\frac{\beta}{a+\beta},\ \pi_k= \frac{(k+1)\beta}{a+\beta}\prod_{j=0}^{k-1}
\frac{\lambda_j}{\lambda_{j+1}+(j+2)\beta},\ k\ge1.$$
In particular, when $a=\lambda$, then $\pi_k=\beta a^k/(a+\beta)^{k+1}\,(k\ge0)$.
Furthermore, when $a=\lambda=\beta=1$, then $\pi_k=2^{-k-1}\,(k\ge0)$.
\end{block}
}
%----------------------------------------------------------------------------------------31
\frame{\small

Define $\sigma_j=\inf\{t\ge$ the first jumping time: $X(t)=j\}$ and
$$\tau_j=\inf\{t\ge 0: X(t)=j\}.$$

\begin{block}{Theorem} Fix $i_0\ge 0$. For recurrent single process, we have
$$
\mathbb E_i\tau_{i_0}=\sum_{j=i}^{i_0-1}m_j,\ i<i_0;$$
$$
\mathbb E_i\tau_{i_0}=\sum_{j=i_0}^{i-1}(F_j^{(i_0)}c_{i_0}-m_j),\  i\ge i_0+1.
$$
\end{block}
}
%----------------------------------------------------------------------------------------32
\frame{\small
\begin{block}{} For recurrent $Q$-process, $(\mathbb E_i\tau_{i_0}:i\in\mathbb Z_+)$ is the minimal solution to the equation
$$x_{i_0}=0, \ x_i=\sum_{j\ne i,i_0}\frac{q_{ij}}{q_i}x_j+\frac 1{q_i},\ i\ne i_0.$$
\end{block}


$$\mathbb E_i\sigma_i=\frac{q_{i,i+1}c_i}{q_i},\ i\ge 0.$$

$$\pi_i q_i\mathbb E_i\sigma_i=1\Rightarrow\pi_i=\frac 1{q_{i,i+1}c_i}.$$
\begin{block}{}
$$
\mathbb E_1\tau_0=d,\ \mathbb E_i\tau_0=\sum_{j=0}^{i-1}(F_j^{(0)}d-d_j),
\ i\ge 1.$$
$$
\mathbb E_0\sigma_0=q_{01}^{-1}+\mathbb E_1\tau_0=q_{01}^{-1}+d.
$$
\end{block}
}

%----------------------------------------------------------------------------------------33
\frame{\small
Probabilistic meaning:

\begin{itemize}

\item Uniqueness. $R=\sum_{n=0}^\infty m_n=\sum_{n=0}^\infty \mathbb E_n\sigma_{n+1}=\mathbb E_0\sigma_\infty=\infty$.

\item Recurrence. $\mathbb P_0(\sigma_0<\infty)=1-(\sum_{n=0}^\infty F_n^{(0)})^{-1}=1$.

\item Ergodicity. $d=\mathbb E_1\tau_0<\infty\Leftrightarrow\mathbb E_0\sigma_0<\infty$.

\item Strong ergodicity. $S=\sup_{i\ge 1}\mathbb E_i\tau_0<\infty$.\end{itemize}
}
%----------------------------------------------------------------------------------------33-1
\frame{\small
\begin{block}{Uniqueness}
$$(\lambda+q_i)u_i=\sum_{k\ne i}q_{ik}u_{k}, \ 0\le u_i\le 1, i\in E.$$
\end{block}
\begin{block}{Recurrence}
$$x_i=\sum_{k\ne 0,i}\frac{q_{ik}}{q_i}x_k,\qquad 0\le x_i\le 1,\,i\ge 0$$
\end{block}
\begin{block}{(Strong) ergodicity}
$$x_i=\sum_{k\ne  0, i} \frac{q_{ik}}{q_i}x_k+\frac{1}{q_i}, \qquad i\in E. $$
\end{block}
\begin{block}{Extinction probability}
$$x_i=\sum_{k\ne 0, i}\frac{q_{ik}}{q_i}x_k + \frac{q_{i0}}{q_i}(1-\delta_{i0}),\qquad i\in E.$$
\end{block}
}

%%----------------------------------------------------------------------------------------34
%
%\frame{\frametitle{8. Criterion on $\ell$-ergodicity}\small
%\begin{block}{}
%For a positive integer $\ell$, the recurrent chain $P(t)$ is said to be
%$\ell$-ergodic if {\red
%$\mathbb E_j\sigma_j^\ell<\infty$} for some (and hence for all) $j\in E$.
%\end{block}
%\begin{block}{}
%{\blue $1$}-ergodic $=$ positive recurrent (ergodic), {\blue $0$}-ergodic $=$ null recurrent.
%\end{block}
%
%Discrete time:
%\begin{itemize}
%\item J.G. Kemeny, J.L. Snell and A.W. Knapp, 1976
%\item Y.H. Mao, 2003
%\item Z.T. Hou and Y.Y. Liu, 2003. Queue Theory
%\end{itemize}
%
%Continuous time:
%\begin{itemize}
%\item P. Coolen-Schrijner and E.A. van Doorn, 2002
%\item Y.H. Mao, 2004
%\end{itemize}
%}
%%----------------------------------------------------------------------------------------35
%\frame{\small
%
%\begin{itemize}
%\item $\ell$-ergodicity provides an {\red algebraic} convergence rate:
%$p_{ij}(t)-\pi_j=o({\blue t^{-(\ell-1)}})$ as $t\to\infty$.
%\item Given a regular birth-death $(a_i,b_i)$. Assume that $P(t)$ is recurrent.
%Then $P(t)$ is $\ell$-ergodic iff $\sum_{i=1}^\infty\mu_i\mathbb E_{i}\sigma_0^{\ell-1}<\infty$,
%
%$$
%\mathbb E_{i}\sigma_0^{n}=n\sum_{j=0}^{i-1}\frac1{\mu_jb_j}\sum_{k=j+1}^\infty\mu_k\mathbb E_{k}\sigma_0^{n-1},\ i\ge 1,\,n\ge 1.
%$$
%\end{itemize}
%
%\begin{block}{}
%$Q$-process is $\ell$-ergodic iff $\sum_{i=1}^\infty\pi_i \mathbb E_{i}\sigma_0^{\ell-1}<\infty$.
%\end{block}
%
%\begin{block}{}
%$Q$-process is $\ell$-regodic if and only if for some (and then for all) $j\in E$,
%the following system has a nonnegative finite solution
%$$\begin{cases}
%\sum_k q_{ik}y_k\le-\ell \mathbb E_{i}\sigma_j^{\ell-1},&i\ne j,\\
%\sum_{k\ne j}q_{jk}y_k<\infty.
%\end{cases}$$
%
%\end{block}
%}
%%----------------------------------------------------------------------------------------36
%\frame{\small
%Define
%$$d^{(\ell)}_0=0,\ d^{(\ell)}_i=\frac 1{q_{i,i+1}}\Big( \mathbb E_{i}\sigma_0^{\ell-1}+\sum_{k=0}^{i-1}q_i^{(k)}d^{(\ell)}_k\Big),
%\ i\ge 1.$$
%$$d^{(\ell)}:=\sup_{i\ge 0}\frac{\sum_{j=0}^id^{(\ell)}_j}{\sum_{j=0}^iF_j^{(0)}}.$$
%
%\begin{block}{Theorem} Given a regular irreducible single birth $Q$-matrix $Q=(q_{ij})$. Assume that the $Q$-process is $(\ell-1)$-ergodic for some $\ell\ge 1$. Then
%the single birth process is $\ell$-ergodic if and only if $d^{(\ell)}<\infty$.
%\end{block}
%}
%%----------------------------------------------------------------------------------------37
%\frame{\small
%\begin{block}{} Given a regular irreducible $Q$-matrix $Q=(q_{ij})$. Assume that the $Q$-process is $(n-1)$-ergodic
%Then $(\mathbb E_{i}\tau_0^{n})$ is the minimal solution to the equation:
%$$
%x_0=0, \ x_i=\sum_{j\ne i}\frac{q_{ij}}{q_i}x_j+\frac n{q_i} \mathbb E_{i}\sigma_0^{n-1},\ i\ge 1.
%$$
%\end{block}
%
%$$
%\textrm{Key}: \mathbb E_{1}\sigma_0^{\ell}=\ell d^{(\ell)},\ \mathbb E_{i}\sigma_0^{\ell}=\ell\sum_{j=0}^{i-1}(F_j^{(0)}d^{(\ell)}- d^{(\ell)}_j).
%$$
%$$\mathbb E_{0}\sigma_0^{\ell}=\ell!\Big(q_{01}^{-\ell}+\sum_{k=1}^\ell(k!q_{01}^{\ell-k})^{-1}\mathbb E_{1}\sigma_0^{k}\Big),\ n\ge 1.$$
%}
%----------------------------------------------------------------------------------------38
\frame{\frametitle{Outline: II}\small

\begin{enumerate}
\item Poisson equation
\item Uniqueness
\item Recurrence, extinction/return probability
\item Ergodicity, strong ergodicity, and the first moment of return time
\item Polynomial moments of hitting time and life time
\item Exponential ergodicity and Laplace transform of return time
\item Examples
\item Related topics and applications
\end{enumerate}
}
%----------------------------------------------------------------------------------------39
\frame{\frametitle{1. Poisson equation}\small
Given a function $c$, define
$$\Omega g=Qg+cg, \quad \text{where}\quad (Qg)_i=\sum_{j}q_{ij}(g_j-g_i).$$
Clearly, if $c\le 0$, then $\Omega$ is an operator corresponding to a single birth
process with killing rates $(-c_i)$. Define
$$
\widetilde F_i^{(i)}=1,\qquad  \widetilde F_n^{(i)}=\frac 1{q_{n,n+1}}
\sum_{k=i}^{n-1}\tilde q_n^{(k)}\widetilde F_k^{(i)},\qquad n>i\ge 0,$$
$$\tilde q_n^{(k)}=q_n^{(k)}-c_n:=\sum_{j=0}^kq_{nj}-c_n,\qquad 0\le k<n.$$
}

%----------------------------------------------------------------------------------------40
\frame{\small

\begin{block}{Theorem 1} Given a single-birth $Q$-matrix $Q=(q_{ij})$ and functions $c$ and $f$,
the solution $g$ to the Poisson equation
$$
\Omega g=f
$$
has the following representation:
$$
g_n=g_0+\sum_{0\le k\le n-1}\sum_{0\le j\leq k}\frac{{\widetilde F}_k^{(j)}(f_j-c_jg_0)}{q_{j,j+1}},\quad n\ge 0.
$$
In particular, the harmonic function of $\Omega$ can be represented as
$$g_n=g_0\Bigg(1-\sum_{0\le k\le n-1}\sum_{0\le j\le k}\frac{\widetilde F_k^{(j)}c_j}{q_{j,j+1}}\Bigg),\quad n\ge 0.$$\end{block}
}
%---------------------------------------------------------41
\frame{\small
\noindent{\it Proof}\quad For each $i\ge 0$, we have
\begin{align}
(\Omega g)_i&=q_{i,i+1}(g_{i+1}-g_i)-\sum_{0\le j\le i-1} q_{ij}\sum_{k=j}^{i-1}(g_{k+1}-g_k)+c_ig_i\nonumber\\
&=q_{i,i+1}(g_{i+1}-g_i)-\sum_{0\le j\le i-1}\sum_{j=0}^{k}q_{ij}(g_{k+1}-g_k)+c_ig_i\nonumber\\
&=q_{i,i+1}(g_{i+1}-g_i)-\sum_{0\le j\le i-1}\Bigg(\sum_{j=0}^{k}q_{ij}-c_i\Bigg)(g_{k+1}-g_k)+c_ig_0\nonumber\\
&=q_{i,i+1}(g_{i+1}-g_i)-\sum_{0\le j\le i-1}\tilde q_i^{(k)}(g_{k+1}-g_k)+c_ig_0.
\nonumber\end{align}
$$g_{i+1}-g_i=\frac 1{q_{i,i+1}}\Bigg(\sum_{0\le k \le i-1}\tilde q_{i}^{(k)}(g_{k+1}-g_k)+f_i-c_ig_0\Bigg).$$
}

%--------------------------------------------------------------------42-1
\frame{\small
$$g_{i+1}-g_i=\sum_{j=0}^i\frac{\widetilde F_i^{(j)}(f_j-c_jg_0)}{q_{j,j+1}},\qquad i\ge 0.$$

\begin{block}{Proposition1} For given $f$, the sequence $(h_n)$ defined successively by
$$h_n=\frac{1}{q_{n, n+1}}\bigg(f_n+\sum_{i\le k \le n-1}{\tilde q}_n^{(k)} h_k\bigg),\qquad n\ge i$$
has an unified expression as follows
$$h_n=\sum_{k=i}^n \frac{{\widetilde F}_n^{(k)}f_k}{q_{k, k+1}} ,\qquad n\ge i. $$
\end{block}
}
%----------------------------------------------------------------------------42-2
\frame{\small

\begin{block}{Lemma} For given real numbers $(\alpha_{nk})_{n-1\ge k\ge 0}$ and $(f_n)_{n\ge 0}$, the solution $(g_n)_{n\ge 0}$ to the recursive
inhomogeneous equations
$$
g_n=\sum_{0\le k\le n-1}\alpha_{nk}g_k+f_n,\qquad n\ge 0
$$
can be represented as
$$
g_n=\sum_{0\le k\le n}\gamma_{nk}f_k,\qquad n\ge 0,
$$
where for fixed $k\ge 0$, $(\gamma_{nk})_{n\ge k}$ with $\gamma_{kk}=1$ is the solution to the recursive equations
$$\gamma_{nk}=\sum_{k\le j\le n-1}\alpha_{nj}\gamma_{jk},\qquad n>k.
$$
\end{block}
}
%----------------------------------------------------------------------------43
\frame{\small
$${\widetilde F}_i^{(i)}=1,\quad  {\widetilde F}_n^{(i)}=\sum_{k=i+1}^n
\frac{{\widetilde F}_n^{(k)}{\tilde q}_k^{(i)}}{q_{k, k+1}},\quad n\ge i+1. $$

Roughly speaking, the unified treatment consists of the following three steps.
\begin{itemize}
\item[(a)] Find out the Poisson equation corresponding to the problem .
\item[(b)] Apply Theorem 1 to get the solution to Poisson equation.
\item[(c)] Work out a criterion for the problem using the solution obtained in (b).
\end{itemize}
}
%---------------------------------------------44
\frame{\small\vskip0.5cm
\begin{center}{\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|c|c|c|}
  \hline
  Problem                  & $c_i\in {\mathbb R}$                   & $f_i\in {\mathbb R}$
  \\ \hline
  Harmonic function               & $c_i\in {\mathbb R}$   &
  $f_i\equiv 0$ \\ \hline
  Uniqueness                & $c_i\equiv -\lambda<0$   & $f_i\equiv 0$ \\ \hline
  Recurrence                & $c_i\equiv 0$            & $f_i=q_{i0}(1-\delta_{i0})$\\ \hline
  Ext./return probability    &$c_i\equiv 0$             & $f_i=q_{i0}(1-\delta_{i0})(g_0-1)$ \\ \hline
  Ergodicity                & $c_i\equiv 0$            & $f_i=q_{i0}(1-\delta_{i0})g_0-1$\\ \hline
  Strong ergodicity         & $c_i\equiv 0$            & $f_i=q_{i0}(1-\delta_{i0})g_0-1$\\ \hline
  Polynomial moment         & $c_i\equiv 0$            & $f_i^{(\ell)}$\\ \hline
  Exp. moment/ergodicity        & $c_i\equiv\lambda>0$     & $f_i=q_{i0}(1-\delta_{i0})(g_0-1)$ \\ \hline
  Laplace transform  & $c_i\equiv -\lambda<0$ & $f_i=q_{i0}(1-\delta_{i0})(g_0-1)$ \\ \hline
\end{tabular}}\end{center}
where $f_i^{(\ell)}=q_{ii_0}(1-\delta_{ii_0})g_{i_0}-\ell\mathbb E_i\sigma_{i_0}^{\ell-1}$.
}
%----------------------------------------------------------------------------------45
\frame{\frametitle{2. Uniqueness}\small
Define
$${\widetilde m}_0=\frac{1}{q_{01}},\quad
{\widetilde m}_n=\frac 1{q_{n,n+1}}
\bigg(1+\sum_{k=0}^{n-1}{\tilde q}_n^{(k)}{\widetilde m}_k\bigg),\quad n\ge 1.$$

$${\widetilde m}_n
=\sum_{k=0}^n\frac{{\widetilde F}_n^{(k)}}{q_{k,k+1}},\qquad n\ge 0.$$


\begin{block}{Proposition}
Given a totally stable and single birth conservative $Q$-matrix $Q=(q_{ij})$, the process is unique $($non-explosive$)$ iff
$\sum_{n=0}^\infty m_n=\infty$.\end{block}
}

%----------------------------------------------------------46
\frame{\small
Proof.  unique iff the solution $(u_i)$ to the equation
\begin{block}{}$$(\lambda +q_i)u_i=\sum_{j\ne i}q_{ij}u_j, \qquad i\ge 0;\qquad u_0=1
$$\end{block}
is unbounded for some (equivalently for all) $\lambda>0$. Rewrite it as
\begin{block}{}
$$\Omega u=Qu- \lambda u=0;\qquad u_0=1.$$\end{block}
Applying Theorem 1 to $c_i\equiv -\lambda$ and $f_i\equiv 0$, we obtain:
$$u_n=1+\lambda\sum_{0\le k \le n-1}\sum_{j=0}^k\frac{\widetilde F_k^{(j)}}{q_{j,j+1}}=1+\lambda\sum_{0\le k \le n-1}\widetilde m_k,\qquad n\ge 0.$$
Clearly, $u_n$ is increasing in $n$ and then is unbounded iff $\sum_n {\widetilde  m}_n=\infty$. Thus,
it remains to show that $\sum_n {\widetilde  m}_n=\infty$
iff $\sum_n {m}_n=\infty$.
}
%----------------------------------------------------------47
\frame{\frametitle{3. Recurrence, extinction/return probability}\small


\begin{block}{Proposition} Assume the single birth $Q$-matrix $Q=(q_{ij})$ is non-explosive and irreducible. Then the process is recurrent iff $\sum_{n=0}^\infty F_n^{(0)}=\infty$.\end{block}

Proof. recurrent iff the equation
\begin{block}{}$$
x_i=\sum_{k\ne 0}\Pi_{ik}x_k,\qquad 0\le x_i\le 1,\qquad i\ge 0
$$\end{block}
has only zero solution, where $\Pi_{ik}=(1-\delta_{ik})q_{ik}/q_i$. It is easily seen that the equation has a non-trivial solution iff the equation
\begin{block}{}$$x_i=\sum_{k\ne 0}\Pi_{ik}x_k,\qquad i\ge 0; \qquad x_0=1$$\end{block}
has a nonnegative bounded solution.
}

%----------------------------------------------------------48
\frame{\small

Rewrite the previous equation as
\begin{block}{}
$$(Qx)_0=0,\qquad (Qx)_i=q_{i0},\qquad i\ge 1;\qquad x_0=1.$$\end{block}
Applying Theorem 1 to $c_i\equiv 0$ and $f_i=q_{i0}(1-\delta_{i0})$, we obtain
the unique solution as follows
$$x_0=1,\quad x_n=1+\sum_{k=1}^{n-1}F_k^{(0)}=\sum_{k=0}^{n-1}F_k^{(0)},\quad n\ge 1.$$
Clearly, $(x_n)$ is bounded iff $\sum_{k=0}^\infty F_k^{(0)}<\infty$. In other words,
the equation has only a trivial solution iff $\sum_{k=0}^\infty F_k^{(0)}=\infty$.
}
%----------------------------------------------------------49
\frame{\small

P. J. Brockwell(1986), W. J. Anderson(1991). Hou \& Guo(1978, 1988)

\begin{block}{Proposition}Let the single birth $Q$-matrix $Q=(q_{ij})$ be non-explosive and irreducible. Then the return/extinction probability is as follows:
$$\mathbb P_0(\sigma_0<\infty)=\frac{\sum_{k=1}^{\infty}F_k^{(0)}}{\sum_{k=0}^\infty F_k^{(0)}},\quad \mathbb P_n(\sigma_0<\infty)=\frac{\sum_{k=n}^{\infty}F_k^{(0)}}{\sum_{k=0}^\infty F_k^{(0)}},\quad n\ge 1.$$
Furthermore, $\mathbb P_n(\sigma_0<\infty)=1$ for all $n\ge 0$ iff $\mathbb P_0(\sigma_0<\infty)=1$, equivalently iff $\sum_{n=0}^\infty F_n^{(0)}=\infty$.
\end{block}
}
%----------------------------------------------------------50
\frame{\small
Proof. $(\mathbb P_i(\sigma_0<\infty): i\in E)$ is the minimal nonnegative solution to the equation
\begin{block}{}$$x_i=\sum_{k\ne 0, i}\frac{q_{ik}}{q_i}x_k + \frac{q_{i0}}{q_i}(1-\delta_{i0}),\qquad i\in E.$$\end{block}
The last equation is equivalent to
\begin{block}{}$$(Qx)_i=q_{i0}(1-\delta_{i0})(x_0-1),\qquad i\ge 0.$$\end{block}
Applying Theorem 1 to $c_i\equiv 0$ and $f_i= q_{i0}(1-\delta_{i0})(x_0-1)$,
we obtain the solution to the last equation:
$$ x_n=x_0\sum_{0\le k\le n-1} F_k^{(0)}-\sum_{1\le k\le n-1}F_k^{(0)},\qquad n\ge 0.$$
$$x_0^*=1-\frac 1{\sum_{k=0}^\infty F_k^{(0)}},\qquad x_n^*=1-\frac{\sum_{k=0}^{n-1}F_k^{(0)}}{\sum_{k=0}^\infty F_k^{(0)}},\qquad n\ge 1.$$
}
%----------------------------------------------------------51
\frame{\frametitle{\small 4. Ergodicity, strong ergodicity, and the first moment of return time}\small
Define
$$\tilde d_0=0,\quad \tilde d_n=\frac 1{q_{n,n+1}}\Bigg(1+\sum_{k=0}^{n-1}\tilde q_n^{(k)}\tilde d_k\Bigg),
\quad n\ge 1,$$
$$\tilde d_n=\sum_{1\le j\le n}\frac{\widetilde F_n^{(j)}}{q_{j,j+1}},\quad n\ge 0.$$

\begin{block}{Proposition} Assume that the single birth $Q$-matrix $Q=(q_{ij})$ is irreducible and corresponding process is recurrent. Then
$$\mathbb E_0\sigma_0=\frac 1{q_{01}}+d,\qquad \mathbb E_n\sigma_0=\sum_{k=0}^{n-1}\big(F_k^{(0)}d-d_k\big),\quad n\ge 1,$$\end{block}
}

%----------------------------------------------------------52
\frame{\small
\begin{block}{Proposition(contin.)}
where
$$d=\varlimsup_{k\to\infty} \frac{\sum_{n=0}^k d_n}{\sum_{n=0}^k F_n^{(0)}}
=\lim_{n\to\infty} \frac{d_n}{F_n^{(0)}}\;\;\text{\rm if the limit exists}.$$
Furthermore, the process is ergodic $($i.e. positive recurrent$)$ iff $d<\infty$;
and it is strongly ergodic iff $\sup_{k\in E} \sum_{n=0}^k\big(F^{(0)}_nd-d_n\big)<\infty$.
Actually, for the last conclusion, the recurrence assumption can be replaced by the
uniqueness one.\end{block}

Yan \& Chen(1986), Zhang(2001, 2003)
}

%----------------------------------------------------------53
\frame{\small

Proof. $({\mathbb E}_i \sigma_0: i\in E)$ is
the minimal nonnegative solution $(x_i^*)$ to the equation
\begin{block}{}$$
x_i=\frac{1}{q_i}\sum_{k\notin  \{0, i\}}
  q_{ik}x_k+\frac{1}{q_i}, \qquad i\in E. $$\end{block}
Then, we have
\begin{block}{}$$(Qx)_i=q_{i0}x_0-1,\qquad i\ge 1;\qquad (Qx)_0=-1.$$\end{block}
Applying Theorem 1 to $c= 0$ and $f_i=q_{i0}(1-\delta_{i0})x_0-1$ ($i\ge 0$),
we obtain the solution to the last equation:
$$x_n=\sum_{k=0}^{n-1}\bigg[F_k^{(0)}\bigg(x_0-\frac{1}{q_{01}}\bigg)-d_k\bigg],\quad n\ge 1.$$
}

%---------------------------------------------------------54
\frame{\small
$$x_0^*
=\frac 1{q_{01}}+\sup_{n\ge 1}\frac{\sum_{k=0}^{n-1} d_k}{\sum_{k=0}^{n-1}F_k^{(0)}}$$
and
$$x_n^*=\sum_{k=0}^{n-1}\left(F_k^{(0)}\sup_{n\ge 1}\frac{\sum_{j=0}^{n-1} d_j}{\sum_{j=0}^{n-1}F_j^{(0)}}
-d_k\right),\quad n\ge 1.$$
}
%---------------------------------------------------------------------------------55
\frame{\frametitle{\small 5. Polynomial moments of hitting time and life time}\small
\begin{block}{Proposition}  Assume that the single birth $Q$-matrix $Q=(q_{ij})$ is irreducible and the
corresponding process is $(\ell-1)$-ergodic ($\ell\ge 1$), i.e. ${\mathbb E}_i\sigma_{i_0}^{\ell-1}<\infty$ for every $i\ge 0$. When $\ell=1$, assume additionally that
the process is unique. Then we have
$$\mathbb E_n\sigma_{i_0}^\ell=\begin{cases}\ell \sum_{n\le k\le i_0-1} v_k^{(\ell)}+\big[1-\sum_{n\le k\le i_0-1} u_k\big]{\mathbb E}_{i_0}\sigma_{i_0}^\ell\quad & 0\le n\le i_0;\\
-\ell\sum_{i_0\le k\le n-1} v_k^{(\ell)}+\big[1+\sum_{i_0\le k\le n-1}u_k\big]{\mathbb E}_{i_0}\sigma_{i_0}^\ell,\quad & n> i_0;\end{cases}$$
\end{block}
}

%---------------------------------------------------------------------------------56
\frame{\small
\begin{block}{Proposition(contin.)}
\begin{gather}
u_k={\begin{cases}
\sum_{j=i_0-1}^k {q_{j, j+1}}^{-1} {F_k^{(j)}} q_{ji_0}(1-\delta_{ji_0}),\qquad &k\ge i_0,\nonumber\\
1, &k=i_0-1,\\
0,  &0\le k\le i_0-2\end{cases}}\\
v_k^{(\ell)}=\sum_{j=0}^k \frac{F_k^{(j)}}{q_{j, j+1}} {\mathbb E}_j\sigma_{i_0}^{\ell-1},\qquad k\ge 0,\nonumber
\end{gather}
$$
{\mathbb E}_{i_0}\sigma_{i_0}^\ell=\ell\varlimsup_{n\to\infty} \bigg({\sum_{i_0\le k\le n} v_k^{(\ell)}}\bigg)\bigg[{1+\sum_{i_0\le k\le n}u_k}\bigg]^{-1}
=\ell\lim_{n\to\infty} \frac{v_n^{(\ell)}}{u_n}\; \text{\rm if}\; \exists.$$
\end{block}

Zhang(2004, 2013)
}
%---------------------------------------------------------------------------------57
\frame{\small

Proof.
$(y_i^*:={\mathbb E}_i\sigma_{i_0}^\ell: i\in E)$ is the the minimal nonnegative solution to the following equation:
\begin{block}{}$$y_i=\sum_{k\ne i, i_0}\frac{1}{q_i} q_{ik}y_k+\frac{\ell}{q_i} {\mathbb E}_i \sigma_{i_0}^{\ell-1},\qquad i\in E.$$\end{block}
Then, we obtain the Poisson equation:
\begin{block}{}$$(Qy)_i= q_{ii_0}(1-\delta_{ii_0})y_{i_0}-\ell\mathbb E_i\sigma_{i_0}^{\ell-1},\qquad i\in E. $$\end{block}
Applying Theorem 1 to $c= 0$ and
$f_i= q_{ii_0}(1-\delta_{ii_0})y_{i_0}-\ell\mathbb E_i\sigma_{i_0}^{\ell-1}$, it follows that
the solution to the last equation is as follows:
$$y_n=y_0+\sum_{0\le k\le n-1}\sum_{j=0}^k\frac{F_k^{(j)}f_j}{q_{j,j+1}}
=y_0+ y_{i_0}\!\!\sum_{0\le k\le n-1}u_k-\ell\!\! \sum_{0\le k\le n-1} v_k^{(\ell)},\quad n\ge 0.$$
}

%-------------------------------------------------------------------------58
\frame{\small

$$y_0=\ell \sum_{0\le k\le i_0-1} v_k^{(\ell)}+ y_{i_0} \bigg(1-\sum_{0\le k\le i_0-1} u_k\bigg).$$
$$
y_n={\begin{cases}
-\ell\sum_{i_0\le k\le n-1} v_k^{(\ell)}+y_{i_0} \big[1+\sum_{i_0\le k\le n-1}u_k\big],\ & n\ge i_0+1\\
\ell \sum_{n\le k\le i_0-1} v_k^{(\ell)}+y_{i_0} \big[1-\sum_{n\le k\le i_0-1} u_k\big], \ & n\le i_0.
\end{cases}} $$
$$y_{i_0}^*= \ell\varlimsup_{n\to\infty} \frac{\sum_{i_0\le k\le n} v_k^{(\ell)}}{1+\sum_{i_0\le k\le n}u_k}$$
}
%-------------------------------------------------------------------------59
\frame{\small

Zhang(2013)

\begin{block}{Proposition}
Let the single birth $Q$-matrix $Q=(q_{ij})$ be irreducible and explosive
(i.e. $\sum_n m_n<\infty$). Assume that
the minimal process has finite $(\ell-1)$-th moments of $\tau_\infty$ for some integer $\ell\ge 1$
(i.e. $E_i\tau_\infty^{\ell-1}<\infty$ for all $i\ge 0$). Then
$$\mathbb E_n\tau_\infty^\ell=\ell\sum_{k\ge n}\overline m_k^{(\ell)},\qquad n\ge 0,$$
where
$$\overline m_n^{(\ell)}=\frac{1}{q_{n,n+1}}\bigg[{\mathbb E}_n\tau_{\infty}^{\ell-1}+\sum_{0\le k\le n-1}q_n^{(k)}{\overline m}_k^{(\ell)}\bigg]
=\sum_{j=0}^n\frac{F_n^{(j)}\mathbb E_j\tau_\infty^{\ell-1}}{q_{j,j+1}},\quad n\ge 0.$$
\end{block}
}
%--------------------------------------------------------------------60

\frame{\small
Proof. $(\mathbb E_i\tau_\infty^\ell: i\in E)$
is the the minimal nonnegative solution $(y_i^*: i\in E)$ to the following equation:
\begin{block}{}$$y_i=\sum_{k\ne i}\frac{1}{q_i} q_{ik}y_k+\frac{\ell}{q_i} {\mathbb E}_i \tau_\infty^{\ell-1},\qquad i\in E.$$\end{block}
That is,
\begin{block}{}$$(Qy)_i= -\ell\mathbb E_i\tau_\infty^{\ell-1},\qquad i\in E. $$\end{block}
Applying Theorem 1 to $c= 0$ and $f_i=-\ell\mathbb E_i\tau_\infty^{\ell-1}$ ($i\ge 0$),
$$y_n=y_0-\ell\sum_{k=0}^{n-1}\overline m_k^{(\ell)},\qquad n\ge 1.$$
By the nonnegative and minimal properties, it follows that
$$y_0^*=\sup_{n\ge 1}\Bigg(\ell\sum_{k=0}^{n-1}\overline m_k^{(\ell)}\Bigg)=\ell\sum_{k=0}^\infty\overline m_k^{(\ell)}, \quad y_n^*=\ell\sum_{k=n}^\infty\overline m_k^{(\ell)},\quad n\ge 1.$$
}
%----------------------------------------------------------------61
\frame{\frametitle{\small 6. Exponential ergodicity and Laplace transform of return time}\small

\begin{block}{Proposition} Let the single birth $Q$-matrix $(q_{ij})$ be irreducible. Assume that its process is ergodic. Define $\big(\widetilde F_k^{(i)}\big)$ and $\big(\tilde d_k\big)$ by setting $c_i\equiv\lambda>0$. Then for small $\lambda$,
$$\mathbb E_0\text{e}^{\lambda\sigma_0}\!=\!\frac {q_{01} (1+\lambda\tilde d)}{q_{01}-\lambda}\!<\!\infty,\
\mathbb E_n\text{e}^{\lambda\sigma_0}\!=\!1+ \lambda \sum_{k=0}^{n-1}\Big(\widetilde F_k^{(0)}\tilde d-\tilde d_k\Big)\!<\!\infty,\ n\ge 1$$
iff
$${\tilde d}:=\varlimsup_{n\to \infty}
\mathbbm{1}_{\big\{{\sum_{k=0}^{n}\widetilde F_k^{(0)}}>0\big\}}
\frac{\sum_{k=0}^{n}\tilde d_k}{\sum_{k=0}^{n}\widetilde F_k^{(0)}}<\infty$$
and
\begin{equation}\label{e4.4-0}
{\tilde d}\, \sum_{k=0}^{n-1}\widetilde F_k^{(0)}>\sum_{k=0}^{n-1} \tilde d_k\quad \text{  whenever } \sum_{k=0}^{n-1}\widetilde F_k^{(0)}\le 0 \quad\text{  for }\ n\ge 2.
\end{equation}
\end{block}
}
%--------------------------------------------------------62
\frame{\small

\begin{block}{Proposition(contin.)}
Furthermore, once ${\widetilde F}_n^{(0)}>0$ for large enough $n$ and $\sum_n {\widetilde F}_n^{(0)}=\infty$, we have $${\tilde d}=\lim_{n\to\infty} \frac{{\tilde d}_n}{\widetilde F_n^{(0)}}\;\;\text{\rm if the limit exists}.$$
Finally, the process is exponentially ergodic iff both $\tilde d<\infty$ and (\ref{e4.4-0}) holds.
\end{block}

Zhang(2003)
}
%----------------------------------63
\frame{\small
Proof. $(\mathbb E_i\text{e}^{\lambda\sigma_0}: i\in E)$ is the minimal solution $(x_i^*)$ of the following equation
\begin{block}{}$$x_i=\frac{1}{q_i-\lambda}\sum_{k\notin \{0,i\}}
  q_{ik}x_k+\frac{q_{i0}(1-\delta_{i0})}{q_i-\lambda}, \qquad x_i\ge1,\qquad i\in E. $$\end{block}
Then, we have
\begin{block}{}$$
(Qx)_i+\lambda x_i=q_{i0}(x_0-1),\quad i\ge 1;\quad (Qx)_0+\lambda x_0=0.
$$\end{block}
Applying Theorem 1 to $c_i\equiv\lambda$ and $f_i=q_{i0}(1-\delta_{i0})(x_0-1)$ for all $i\ge 0$,
$$x_n=x_0\bigg(1-\frac \lambda{q_{01}}\bigg)\sum_{k=0}^{n-1}\widetilde F_k^{(0)}-\sum_{k=0}^{n-1}\big(\widetilde F_k^{(0)}+\lambda\tilde d_k\big)+1,\quad n\ge 1.$$
}
%----------------------------------------------------64
\frame{\small
\begin{block}{Proposition}
Define $\big(\widetilde F_k^{(i)}\big)$ and $\big(\tilde d_k\big)$ with $c_i\equiv -\lambda<0$.
Let the single birth process be recurrent. Then
the Laplace transform of $\sigma_0$ is given by
$$\mathbb E_0\text{e}^{-\lambda\sigma_0}=\frac {q_{01} (1-\lambda\tilde d)}{q_{01}+\lambda},\ \mathbb E_n\text{e}^{-\lambda\sigma_0}=1- \lambda \sum_{k=0}^{n-1}\Big(\widetilde F_k^{(0)}\tilde d-\tilde d_k\Big),\ n\ge 1,$$
where
$$\tilde d=\lim_{n\to\infty}\frac{\sum_{k=0}^{n-1}\tilde d_k}{\sum_{k=0}^{n-1}\widetilde F_k^{(0)}}
=\lim_{n\to\infty}\frac{{\tilde d}_n}{ {\widetilde F}_n^{(0)}} \quad \text{\rm if the limit exists}.$$
\end{block}

P. J. Brockwell(1986), W. J. Anderson(1991)
}
%%----------------------------------------------------------------------------------------65
%\frame{\small
%\begin{block}{Proposition} Assume that the single birth $Q$-matrix $Q=(q_{ij})$ is explosive and irreducible. Define $(\widetilde m_k)$ with $c_i\equiv \lambda$.
%For the corresponding minimal process,
%
%(i) if $\exists\lambda>0$ such that $\lambda\sum_{k=0}^{n-1}\widetilde m_k<1$ for every $n>1$,
%then
%$$\mathbb E_n\text{e}^{\lambda\tau_\infty}=1+ \lambda \Bigg[\bar c\Bigg(1-\lambda\sum_{k=0}^{n-1}\widetilde m_k\Bigg)-\sum_{k=0}^{n-1}\widetilde  m_k\Bigg],\quad n\ge 0,$$
%$$\bar c=\varlimsup_{n\to\infty }\frac{\sum_{k=0}^{n}\widetilde m_k}{1-\lambda\sum_{k=0}^{n}\widetilde m_k}.$$
%Furthermore, the process decays exponentially fast provided $\bar c<\infty$.
%\end{block}
%}
%%----------------------------------------------------------------------------------------66
%\frame{\small
%\begin{block}{Proposition(contin.)} (ii) For $\lambda>0$, the Laplace transform of $\tau_\infty$ is given by
%$$\mathbb E_n\text{e}^{-\lambda\tau_\infty}=\frac{1+\lambda\sum_{0\le k\le n-1}\widetilde m_k}{1+\lambda\sum_{k\ge 0}\widetilde  m_k},\quad n\ge 0.$$
%\end{block}
%
%Proof. Define
%$$e_{i\infty}(\lambda)=\int_0^\infty\text{e}^{\lambda t}\mathbb P_i(\tau_\infty>t)\,\text{d} t$$
%with $\lambda<q_i$ for all $i\ge 0$.
%$$\mathbb E_i\text{e}^{\lambda\tau_\infty}=1+\lambda e_{i\infty}(\lambda).$$
%$(e_{i\infty}(\lambda))$ is the minimal solution to the equation
%$$x_i=\frac{q_i}{q_i-\lambda}\sum_{k}\Pi_{ik}x_k+\frac 1{q_i-\lambda}, \quad i\ge 0.$$
%}
%%---------------------------------------------67
%\frame{\small
%$$(Qx)_i+\lambda x_i=-1,\qquad i\ge 0.$$
%Applying Theorem 1 to $c_i\equiv \lambda$ and $f_i\equiv -1$, the solution of the equation has
%the form:
%$$x_n=x_0\Bigg(1-\lambda\sum_{k=0}^{n-1}\widetilde m_k\Bigg)-\sum_{k=0}^{n-1}\widetilde  m_k,\quad n\ge 1.$$
%Note that $\lambda<q_0=q_{01}$ and $\lambda\widetilde m_0<1$. If there exists a positive
%$\lambda$ small enough so that $\lambda\sum_{k=0}^{n-1}\widetilde m_k<1$ for every $n>1$,
%then one gets
%$$
%e_{0\infty}(\lambda)=\sup_{n\ge 1}\frac{\sum_{k=0}^{n-1}\widetilde m_k}{1-\lambda\sum_{k=0}^{n-1}\widetilde m_k}=
%\varlimsup_{n\to\infty}\frac{\sum_{k=0}^{n}\widetilde m_k}{1-\lambda\sum_{k=0}^{n}\widetilde m_k}=:\bar c
%$$
%$$e_{n\infty}(\lambda)=\bar c\Bigg(1-\lambda\sum_{k=0}^{n-1}\widetilde m_k\Bigg)-\sum_{k=0}^{n-1}\widetilde  m_k,\quad n\ge 1.$$
%}
%---------------------------------------------68
\frame{\frametitle{\small 7. Examples}\small
\begin{block}{Example 1[uniform catastrophes]}
Let
$$q_{i,i+1}=b\, i,\qquad i\ge 0;\qquad q_{ij}=a,\qquad j=0,1,\dots, i-1;$$
and $q_{ij}=0$ for other $j>i+1$, where $a$ and $b$ are positive constants. Then the extinction of the process
has an exponential distribution
$${\mathbb E}_n\text{e}^{-\lambda\tau_0}=\frac{a}{a+\lambda},
\qquad \lambda>0,\; n\ge 1.$$
It is surprising that the distribution is independent of $b$ and the starting point $n$. Redefine $q_{01}=1$. Then the irreducible process is indeed
strongly ergodic.
\end{block}

P. J. Brockwell et al(1982)
}
%---------------------------------------------69
\frame{\small
\begin{block}{Example 2}
Consider the single birth $Q$-matrix $(q_{ij})$
with
$$q_{i0}>0, \;\; q_{i, i+1}>0,\;\; q_{ij}=0\;\;\text{\rm for all other }j\ne i.$$
Assume that $q_{n0}-c_n\equiv q_{10}-c_1$ for
every $n\ge 1$. Then the process is explosive if
$$\kappa':=\lim_{n\to\infty}
\frac{n(q_{n+1, n+2}-q_{n, n+1}-q_{10})}{q_{n, n+1}+q_{10}}>1$$
($q_{n, n+1}=(n+1)^{\gamma}$ for $\gamma> 1$ for example).
Otherwise, if $\kappa'<1$ ($q_{n, n+1}=(n+1)^{\gamma}$ for some $\gamma\le 1$ for instance), then the process is
unique. If so, the process is indeed strongly ergodic.
\end{block}
}
%---------------------------------------------70
\frame{\frametitle{\small 8. Related topics and applications}\small
\begin{itemize}
\item Exponential ergodicity: $\displaystyle\lim_{t\to\infty}{\red e^{\alpha t}}|p_{ij}(t)-\pi_j|=0$.
\end{itemize}

\begin{block} {Theorem(Y.H. Mao \& Y.H. Zhang, 2004)} Given a regular irreducible single
birth $Q$-matrix $Q=(q_{ij})$. If

$$\inf_i q_i>0\ \hbox{and}\ M:=\sup_{i> 0}\sum_{j=0}^{i-1}F_j^{(0)}\sum_{j=i}^\infty \frac 1{q_{j,j+1}F_j^{(0)}}
<\infty,$$
then the $Q$-process is exponential ergodicity.\end{block}

\begin{block}{Example} Let $q_{n,n+1}=1$ for all $n\ge 0$, $q_{10}=1$,
$q_{n,n-2}=1$ for all $n\ge 2$ and $q_{ij}=0$ for other $i\ne j$.
Then the single birth process is exponentially
ergodic and not strongly ergodic.
\end{block}

}
%---------------------------------------------71
\frame{\small
\begin{block} {Theorem(M.F. Chen, 2000)} Given a regular birth-death $Q$-matrix $(a_i,b_i)$. Then the process
is exponentially ergodic iff
$$\delta:=\sup_{i> 0}\sum_{j=0}^{i-1}\frac 1{\mu_jb_j}\sum_{j=i}^\infty \mu_j<\infty.$$
\end{block}

$$\frac 1{4\delta}\le\lambda_1\le\frac 1{\delta},\quad \lambda_1>0\Longleftrightarrow \delta<\infty.$$

$$\lambda_1=\ \hbox{exponential convergence rate.}$$

Key: birth-death processes are reversible!
}
%---------------------------------------------72
\frame{\small
\begin{block} {Theorem} Given a regular irreducible $Q$-matrix $Q=(q_{ij})$. Then the process
$P(t)$ is exponentially ergodicity if and only if for some $\lambda>0$ with $\lambda <q_i$ for all $i$,
$$\begin{cases}\sum_j q_{ij}y_j\le -\lambda y_i-1,\quad i\notin H&\\
\sum_{i\in H}\sum_{j\ne i}q_{ij}y_j<\infty&
\end{cases}$$
has a nonnegative finite solution $(y_i)$.
\end{block}

\vskip0.2cm
``$\inf_iq_i>0$'' is necessary essentially for exponential ergodicity.

\vskip0.2cm

Difficulty: two parameters!
}
%---------------------------------------------73
\frame{\small
Mao and Zhang (2004) gives another proof.
\vskip0.2cm
Keys:

1. Construct a test function with double summations for sufficiency.
The construction comes from the study of spectral gap essentially.

2. Necessity. Note that
$$\aligned
\mathbb E_{i}\sigma_0^{n}\ge n\bigg(&\sum_{j=0}^{i-1}\frac1{\mu_jb_j}\sum_{k=i}^\infty\mu_k\bigg)\mathbb E_{i}\sigma_0^{n-1}\\
\ge\cdots&\ge n!\bigg(\sum_{j=0}^{i-1}\frac1{\mu_jb_j}\sum_{k=i}^\infty\mu_k\bigg)^n.\\
\mathbb E_i\textrm{e}^{\lambda\sigma_0}<\infty
&\Rightarrow\sum_{n=1}^\infty\bigg(\lambda\sum_{j=0}^{i-1}\frac 1{\mu_jb_j}\sum_{k=i}^\infty\mu_k\bigg)^n<
\infty\\
&\Rightarrow\lambda\sum_{j=0}^{i-1}\frac 1{\mu_jb_j}\sum_{k=i}^\infty\mu_k< 1\Rightarrow\delta\le \lambda^{-1}<\infty.
\endaligned$$
}
%---------------------------------------------74
\frame{\small

Unfortunately, ``$M<\infty$'' is not necessary for exponential ergodicity.

\begin{block}{Example} Given $q_{01}>0$ arbitrarily. Let $q_{i,i+1}=i^\gamma$, $q_{i0}=i^{\gamma-1}$
for all $i\ge 1$ where the constant $\gamma\in (1,2)$, and $q_{ij}=0$ for other $i\ne j$.
Obviously, the $Q$-matrix is irreducible.
It is easily computed that $F_i^{(0)}=1$ for all $i\ge 0$.
So the single birth process is recurrent and furthermore
regular. Note that $\inf_{i\ge 1}q_{i0}=1>0$. It follows that the single birth
process is strongly ergodic and furthermore exponentially ergodic immediately. But
$M=\infty$.\end{block}

Now, for the single birth $Q$-matrix $(q_{ij})$,
define a conservative birth-death $Q$-matrix $(a_i, b_i)$ as follows.
$$
 a_i=\frac{q_{i,i+1}F_i^{(0)}}{F_{i-1}^{(0)}},\quad i\ge 1;\qquad b_i=q_{i,i+1},\quad i\ge 0.
$$
Then $\mathbb E_i\tau_0^n\le\mathbb E_i\overline\tau_0^n\,(i\ge 1,n\ge 0)$, furthermore
$\mathbb E_i\textrm{e}^{\lambda\tau_0}\le \mathbb E_i\textrm{e}^{\lambda\overline\tau_0}\,(i\ge 1,\lambda>0)$.
}
%---------------------------------------------75
\frame{\small
\begin{block}{Conjecture}
(1)
$$\bar\delta=\sup_{i> 0}\sum_{j=0}^{i-1}F_j^{(0)}\bigg(d-\frac {d_i}{F_i^{(0)}}\bigg)<\infty.$$

(2)
$$\delta':=\sup_{i>0}\sum_{j=0}^{i-1}F_j^{(0)}\sup_{k\ge i}\frac 1{F_k^{(0)}}\sum_{\ell=i}^k\frac{F_k^{(\ell)}}{q_{\ell,\ell+1}}<\infty.$$

(3)
$$\delta'':=\sup_{i>0}\sum_{j=0}^{i-1}F_j^{(0)}\sup_{k\ge i}\frac {\sum_{j=i}^k\sum_{\ell=i}^j\frac{F_j^{(\ell)}}{q_{\ell,\ell+1}}}{\sum_{j=i}^kF_j^{(0)}}<\infty.$$
\end{block}

For birth-death processes, $\delta=M=\bar\delta=\delta'=\delta''$.
}
%---------------------------------------------------------------------- 76
\frame{\small

\begin{itemize}
\item Single birth $Q$-matrix with absorbing boundary: $1\le N:=\max\{i+1:q_{i,i+1}=0\}<\infty$. When $N=\max\emptyset=0$, it is single birth.

M.F. Chen(1999), Single birth processes, Chin. Ann. Math., \textbf{20B}, 77-82. Uniqueness.

\item Single birth $Q$-matrix with immigration: $q_{0j}>0$ for $j\ge 2$.

Y.H. Zhang, Q.Q. Zhao, 2010.

\item Cutoff

\end{itemize}
}
%---------------------------------------------------------------------- 77
\frame{\small

\begin{itemize}
\item Applications
\end{itemize}

To study sufficient conditions for these problems of multidimensional $Q$-processes,
we use the comparison method, i.e., compare them with a single birth $Q$-processes.
The method reduces the multidimensional problems to one-dimensional ones.

\vskip0.2cm
The typical example
is Sch\"ogl's model: a model of chemical reaction with diffusion in a container.

\vskip0.2cm The method is efficient for {\red necessary condition}.

\begin{block}{} The $Q$-process corresponding to Brusselator model is not strongly ergodic.\end{block}

}
%---------------------------------------------------------------------- 78
\frame{\small
Let $E$ be a countable set. $Q=(q(x,y):x,y\in E)$ be a
conservative $Q$-matrix. Suppose that there exists a partition $\{E_k\}$ of $E$
such that $\sum_{k=0}^\infty E_k=E$ and

(i) If $q(x,y)>0$ and $x\in E_k$, then $y\in\sum_{j=0}^{k+1}E_j$ for all
$k\ge 0$.

(ii) $\sum_{y\in E_{k+1}}q(x,y)>0$ for all $x\in E_k$ and all $k\ge 0$.

(iii) $C_k:=\sup\{q(x):x\in E_k\}<\infty$ for all $k\ge 0$.
\vskip0.3cm
Define a conservative single birth $Q$-matrix $Q=(q_{ij})$:
$$q_{ij}=\begin{cases}
\sup\{\sum_{y\in E_j}q(x,y):x\in E_i\},\qquad\text{if}\ j=i+1;\\
\inf\{\sum_{y\in E_j}q(x,y):x\in E_i\},\qquad\,\text{if}\ j<i;\\
0,\qquad\qquad\qquad\qquad\,\text{other cases of}\ j\ne i.
\end{cases}$$
}
%---------------------------------------------------------------------- 79
\frame{\small
(0) {\blue If the $(q_{ij})$-process is unique (i.e. $R=\infty$), then so is the
$(q(x,y))$-process.}

\vskip0.2cm

Now suppose that $E_0=\{\theta\}$ where $\theta\in E$ is a reference point,
and that both $(q(x,y))$ and $(q_{ij})$ are irreducible and $(q_{ij})$ is regular.
\vskip0.2cm
(1) Moreover, assume that $E_k$ is finite for all $k\ge 1$.
If the $(q_{ij})$-process is recurrent (i.e. $\sum_{n=0}^\infty F_n^{(0)}=\infty$), then so is the
$(q(x,y))$-process.

(2) If $\widehat d:=\sup_{k\ge 0}d_k/F_k^{(0)}<\infty$, then both processes are ergodic.

(3) If $\inf_i q_i>0$ and $M=\sup_{i>0}\sum_{j=0}^{i-1}F_j^{(0)}\sum_{j=i}^\infty(q_{j,j+1}F_j^{(0)})^{-1}<\infty$,
then both processes are exponentially ergodic.

(4) If $\sup_{k\ge 0} \sum_{n=0}^k(F^{(0)}_n\widehat d-d_n)<\infty$,
then both processes are strongly ergodic.

\vskip0.2cm Keys: increasing solution of equations.
}
%---------------------------------------------------------------------- 80
\frame{\small

Let $S$ be a finite set and $E=\mathbb Z_+^S$. The model is defined by the
following $Q$-matrix $Q=(q(x,y):x,y\in E)$:
$$q(x,y)=\begin{cases} \lambda_1\binom{x(u)}2+\lambda_4,\qquad&\text{if}
\ y=x+e_u,\\
\lambda_2 \binom{x(u)}3+\lambda_3x(u),&\text{if}
\ y=x-e_u,\\
x(u)p(u,v), &\text{if}\ y=x-e_u+e_v,\\
0,&\text{other}\ y\ne x,
\end{cases}$$
$q(x)=-q(x,x)=\sum_{y\ne x}q(x,y)$,
where $x=(x(u):u\in S)$, $\binom nk$ is the usual combination,
$\lambda_1,\cdots,\lambda_4$ are positive constants. $(p(u,v):u,v\in S)$ is a
transition probability matrix on $S$ and $e_u$ is the element in $E$ having
value $1$ at $u$ and $0$ elsewhere.

\begin{block}{Conclusion}
The $Q$-process corresponding to finite dimensional Schl\"ogl model is strongly ergodic.
\end{block}

}
%%---------------------------------------------------------------------- 81
%\frame{\small Main references
%\begin{enumerate}
%
%\item  Anderson W J. Continuous-Time Markov Chains: An Applications-Oriented Approach. New York: Springer-Verlag, 1991\\[-6.5mm]
%
%\item Brockwell P J. The extinction time of a general birth and death processes with catastrophes. J Appl Prob, 1986, 23: 851--858\\[-6.5mm]
%
%\item Brockwell P J, Gani J, Resnick S I. Birth, immigration and catastrophe processes. Adv Appl Prob, 1982, 14: 709--731\\[-6.5mm]
%
%\item Chen M F. From Markov Chains to Non-Equilibrium Particle Systems (2nd Edition). Singapore: World Scientific, 2004\\[-6.5mm]
%
%\item Chen M F. Single birth processes. Chinese Ann Math, 1999, 20B: 77--82\\[-6.5mm]
%
%\item Chen M F. Explicit criteria for several types of ergodicity. Chinese J Appl Prob Stat, 2001, 17(2): 1--8\\[-6.5mm]
%
%\item Chen M F, Zhang X. Isospectral operators. 2014, preprint\\[-6.5mm]
%
%\item Chen M F, Zhang Y H. Unified representation of formulas for single birth processes. Front Math China, 2014, preprint\\[-6.5mm]
%\end{enumerate}
%}
%%---------------------------------------------------------------------- 82
%\frame{\small Main references(contin.)
%
%\begin{enumerate}
%
%\item Mao Y H, Zhang Y H. Exponential ergodicity for single-birth processes. J Applied Probab, 2004, 41: 1022--1032\\[-6.5mm]
%
%\item Yan S J, Chen M F. Multidimensional $Q$-processes. Chinese Ann Math, 1986, 7B: 90--110\\[-6.5mm]
%
%\item Zhang J K. On the generalized birth and death processes \text{\rm (I)}. Acta Math Sci, 1984, 4: 241--259\\[-6.5mm]
%
%\item Zhang Y H. Strong ergodicity for single-birth processes. J Appl Prob, 2001, 38(1): 270--277\\[-6.5mm]
%
%\item Zhang Y H. Moments of the first hitting time for single birth processes. J Beijing Normal Univ, 2003, 39(4): 430--434 (in Chinese)\\[-6.5mm]
%
%\item Zhang Y H. The hitting time and stationary distribution for single birth processes. J Beijing Normal Univ, 2004, 40(2): 157--161 (in Chinese)\\[-6.5mm]
%\end{enumerate}
%}
%
%%----------------------------------------------------83
\frame{

\centerline{ \emph{\textbf{\color{blue} Thank you for your attention!}}}
\vskip0.5cm
\centerline{Homepage: http://math.bnu.edu.cn/${}^\sim$zhangyh/}

}

\end{CJK*}

\end{document}
