%  First, use latex to compile the file. Second, use dvi2ps to become .ps file. You can use Gsview to watch it. But
%  I suggest you use ps2pdf to become .pdf file. Because .ps file is for making a final film.
%  There are nine types of backgrounds. You can revise the second parameter. Now I use the darkblue background. All
%  backgrounds are the following:
%  alienglow, autumn, azure, contemporain, darkblue, frames, lignesbleues, nuancegris, troispoints,
% prettybox,rico, winter, corners, whitecross, wj, blends, alcatel,capsules, fyma, gyom, mancini, serpaggi, thomasd.
% Of course, you can make your background if you have the ability and you wish.

\documentclass[pdf,wj,slideColor,colorBG]{prosper}

\usepackage[latin1]{inputenc}
\usepackage{pstricks,pst-node,pst-text,pst-3d}
\usepackage{amsmath}
\usepackage{CJK}
% Definition of new colors
\newrgbcolor{LemonChiffon}{1. 0.98 0.8}
\newrgbcolor{LightBlue}{0.68 0.85 0.9}

% > BEGIN OF OVERLAPPED COLORS
% Code below devised by Pingji Deng (BNU/ - China, dengpj@eyou.com)
\newrgbcolor{LemonChiffon}{1. 0.98 0.8}
\newrgbcolor{LightBlue}{0.68 0.85 0.9}
\makeatletter
\newdimen\pst@dimz

% Draw two overlapped surfaces, with computation of the mixed color for
% the intersection of the surfaces
% #1=first  surface, #2=color of first  surface,
% #3=second surface, #4=color of second surface
\def\ColoredOverlappedSurfaces#1#2#3#4{%
\psset{fillstyle=solid}
% Decode the three components of the first RGB color
\DecodeRGBFirstColor{\csname color@#2\endcsname}%
\psset{fillcolor=#2}
% Draw first surface
#1
% Decode the three components of the second RGB color
\DecodeRGBSecondColor{\csname color@#4\endcsname}%
% Compute the mixed color
\BuildMixedColor
% Draw second surface
\psclip{\psset{fillcolor=#4}#3}
\psset{fillcolor=MixedColor}
% Redraw overlapped surface in the mixed color
#1
\endpsclip}

% Get the three components of the first color
\def\DecodeRGBFirstColor#1{%
\pst@expandafter\pst@getnumiii{#1} {} {} {} {}\@nil
\edef\pst@FirstColorR{\pst@tempg}%
\edef\pst@FirstColorG{\pst@temph}%
\edef\pst@FirstColorB{\pst@tempi}%
%\typeout{Color 1=\pst@tempg,\pst@temph,\pst@tempi}% Debug
}

% Get the three components of the second color
\def\DecodeRGBSecondColor#1{%
\pst@expandafter\pst@getnumiii{#1} {} {} {} {}\@nil
\edef\pst@SecondColorR{\pst@tempg}%
\edef\pst@SecondColorG{\pst@temph}%
\edef\pst@SecondColorB{\pst@tempi}%
%\typeout{Color 2=\pst@tempg,\pst@temph,\pst@tempi}% Debug
}

% Build the mixed RBG color (by means of each three components)
\def\BuildMixedColor{%
% Resulting R component
\pst@dimz=\pst@FirstColorR pt
\advance\pst@dimz\pst@SecondColorR pt
\divide\pst@dimz\tw@
\pst@dimtonum{\pst@dimz}{\pst@MixedColorR}%
% Resulting G component
\pst@dimz=\pst@FirstColorG pt
\advance\pst@dimz\pst@SecondColorG pt
\divide\pst@dimz\tw@
\pst@dimtonum{\pst@dimz}{\pst@MixedColorG}%
% Resulting B component
\pst@dimz=\pst@FirstColorB pt
\advance\pst@dimz\pst@SecondColorB pt
\divide\pst@dimz\tw@
\pst@dimtonum{\pst@dimz}{\pst@MixedColorB}%
% Definition of the mixed color MixedColor
\newrgbcolor{MixedColor}{%
\pst@MixedColorR\space \pst@MixedColorG\space \pst@MixedColorB}
%\typeout{Mixed color=\csname color@MixedColor\endcsname}% Debug
}
\makeatother
% < END OF OVERLAPPED COLORS

\begin{document}
\begin{CJK*}{GBK}{song}

%---------------------------------------------------------------------- SLIDE 0-

\begin{slide}[Box]{\small\blue È«¹úÊýÑ§µÚÊ®¶þ½ìÑÐ¾¿ÉúÊîÆÚÑ§Ð£, 2007}
\vspace{1.5cm}

\centerline{\Large\bf µ¥Éú¹ý³ÌµÄ±éÀúÀíÂÛ}

\vskip1cm \centerline{\large ÕÅÓà»Ô }

\vskip0.5cm \centerline{ ±±¾©Ê¦·¶´óÑ§ÊýÑ§¿ÆÑ§Ñ§Ôº }
\vskip0.5cm \centerline{\tiny (Îäºº´óÑ§, 2007Äê8ÔÂ3ÈÕÖÁ4ÈÕ)}
\end{slide}

%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 1-
\begin{slide}[Split]{Contents}
\vskip0.2cm
\begin{itemize}
\item {\blue Background}
\item {\blue Criteria for uniqueness}
\item {\blue Criteria for recurrence}
\item {\blue Criteria for ergodicity}
\item {\blue Criteria for strong ergodicity}
\item {\blue Stationary distribution}
\item Criteria for $\ell$-ergodicity
\item Exponential ergodicity
\item A class of single birth processes
\item Applications
\item Related topics
\end{itemize}
\end{slide}
%------------------------------------------------------------------------------


%---------------------------------------------------------------------- SLIDE 31-
\begin{slide}[Blinds]{Criteria for $\ell$-ergodicity}
\vskip1cm Set {\blue $m_{ij}^{(\ell)}=\mathbb E_i\sigma_j^\ell$}.

\vskip0.3cm For a positive integer $\ell$, the recurrent chain $P(t)$ is said to be
$\ell$-ergodic if {\red
$m_{jj}^{(\ell)}<\infty$} for some (and hence for all) $j\in E$.

\vskip0.3cm
{\blue $1$}-ergodic $=$ positive recurrent (ergodic),

{\blue $0$}-ergodic $=$ null recurrent.

\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 32-
\begin{slide}[Box]{Criteria for $\ell$-ergodicity}
\vskip1cm
Discrete time:
\begin{itemize}
\item Kemeny, J.G., Snell, J.L. and Knapp, A.W.(1976)
\item Mao, Y.-H.(2003)
\item Hou, Z.-T. and Liu, Y.-Y.(2003). Queue Theory
\end{itemize}
\vskip0.5cm
Continuous time:
\begin{itemize}
\item Coolen-Schrijner, P. and van Doorn, E.A.(2002)
\item Mao, Y.-H.(2004)
\end{itemize}
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 33-
\overlays{2}{
\begin{slide}[Wipe]{Criteria for $\ell$-ergodicity}
\vskip1cm
\begin{itemstep}
\item $\ell$-ergodicity provides an {\red algebraic} convergence rate:

$p_{ij}(t)-\pi_j=o({\blue t^{-(\ell-1)}})$ as $t\to\infty$.
\vskip0.3cm
\item Given a regular birth-death $(a_i,b_i)$. Assume that $P(t)$ is recurrent.
\vskip0.2cm
\centerline{$P(t)$ is $\ell$-ergodic if and only if $\sum_{i=1}^\infty\mu_i m_{i0}^{(\ell-1)}<\infty$,}

\vskip0.2cm
$$
m_{i0}^{(n)}=n\sum_{j=0}^{i-1}\frac1{\mu_jb_j}\sum_{k=j+1}^\infty\mu_km_{k0}^{(n-1)},\quad i\ge 1,\,n\ge 1.
$$
\end{itemstep}
\end{slide}
}
%------------------------------------------------------------------------------


%---------------------------------------------------------------------- SLIDE 34-
\overlays{3}{%
\begin{slide}[Box]{Criteria for $\ell$-ergodicity}
\vskip1cm
$Q$-process is $\ell$-ergodic if and only if $\sum_{i=1}^\infty\pi_i m_{i0}^{(\ell-1)}<\infty$.

$$d_0=0,\quad d_i=\frac 1{q_{i,i+1}}\Big(
\fromSlide*{2}{\rnode{NA}{\pscirclebox[linecolor=red]{1}}}
\onlySlide*{1}{\rnode{NA}{\pscirclebox[linecolor=red,linestyle=none]{1}}}
+\sum_{k=0}^{i-1}q_i^{(k)}d_k\Big),\quad i\ge 1.$$

\FromSlide{2}
$$d^{(\ell)}_0=0,\quad d^{(\ell)}_i=\frac 1{q_{i,i+1}}\Big(
\fromSlide*{3}{\rnode{NB}{\pscirclebox[linecolor=red]{m_{i0}^{(\ell-1)}}}}
\onlySlide*{2}{\rnode{NB}{\pscirclebox[linecolor=red,linestyle=none]{m_{i0}^{(\ell-1)}}}}
+\sum_{k=0}^{i-1}q_i^{(k)}d^{(\ell)}_k\Big),
\quad i\ge 1.$$
\fromSlide*{3}{\nccurve[linecolor=red,angleA=270,angleB=90]{->}{NA}{NB}}
\OnlySlide{3}

\end{slide}
}
%------------------------------------------------------------------------------
%---------------------------------------------------------------------- SLIDE 35-
\overlays{4}{%
\begin{slide}[Dissolve]{Criteria for $\ell$-ergodicity}

\vskip0.5cm
$$d^{(\ell)}:=\sup_{i\ge 0}\frac{\sum_{j=0}^i
\fromSlide*{2}{\rnode{NA}{\pscirclebox[linecolor=red]{d^{(\ell)}_j}}}
\onlySlide*{1}{\rnode{NA}{\pscirclebox[linecolor=red,linestyle=none]{d^{(\ell)}_j}}}
}{\sum_{j=0}^iF_j^{(0)}}.$$

\FromSlide{2}
$$d:=\sup_{i\ge 0}\frac{\sum_{j=0}^i
\fromSlide*{3}{\rnode{NB}{\pscirclebox[linecolor=red]{d_j}}}
\onlySlide*{2}{\rnode{NB}{\pscirclebox[linecolor=red,linestyle=none]{d_j}}}
}{\sum_{j=0}^i F_j^{(0)}}.$$
\fromSlide*{3}{\nccurve[linecolor=red,angleA=90,angleB=270]{->}{NB}{NA}}

\vskip0.5cm
\OnlySlide{4}
For $\ell\ge 1$, the single birth process is $\ell$-ergodic if and only if $d^{(\ell)}<\infty$.

\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 36-
\begin{slide}[Split]{Criteria for $\ell$-ergodicity}
\vskip1cm
$Q$-process is $\ell$-regodic if and only if for some (and then for all) $j\in E$,
the system of inequalities
$$\begin{cases}
\sum_k q_{ik}y_k\le-\ell m_{ij}^{(\ell-1)},&i\ne j,\\
\sum_{k\ne j}q_{jk}y_k<\infty
\end{cases}$$
has a nonnegative finite solution.

\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 37-
\begin{slide}[Split]{Criteria for $\ell$-ergodicity}
\vskip0.5cm
¸ø¶¨ÕýÔò²»¿ÉÔ¼~$Q$~¹ý³Ì. Ôò~$(m_{i0}^{(n)})$~ÊÇÏÂÁÐ·½³ÌµÄ×îÐ¡·Ç¸º½â:
$$
x_0=0, \qquad x_i=\sum_{j\ne i}\frac{q_{ij}}{q_i}x_j+\frac n{q_i} m_{i0}^{(n-1)},\quad i\ge 1.
$$
\vskip0.2cm

$$
\textrm{Key}: m_{10}^{(\ell)}=\ell d^{(\ell)},\quad m_{i0}^{(\ell)}=\ell\sum_{j=0}^{i-1}(F_j^{(0)}d^{(\ell)}- d^{(\ell)}_j).
$$
$$m_{00}^{(\ell)}=\ell!\Big(q_{01}^{-\ell}+\sum_{k=1}^\ell(k!q_{01}^{\ell-k})^{-1}m_{10}^{(k)}\Big),\quad n\ge 1.$$

\end{slide}
%------------------------------------------------------------------------------


%---------------------------------------------------------------------- SLIDE 38-
\overlays{1}{%
\begin{slide}[Replace]{Exponential ergodicity}
\vskip0.5cm
Exponential ergodicity:
$\displaystyle\lim_{t\to\infty}{\red e^{\alpha t}}|p_{ij}(t)-\pi_j|=0$.

\vskip0.2cm
Mao, Y.-H. and Zhang, Y.-H.(2004).
$$\inf_i q_i>0\ \hbox{and}\ M:=\sup_{i> 0}\sum_{j=0}^{i-1}F_j^{(0)}\sum_{j=i}^\infty \frac 1{q_{j,j+1}F_j^{(0)}}
<\infty.$$
$\Longrightarrow$ exponential ergodicity.
\vskip0.2cm
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{slide}
}
%------------------------------------------------------------------------------


%---------------------------------------------------------------------- SLIDE 39-
\overlays{1}{%
\begin{slide}[Replace]{Exponential ergodicity}
\vskip1cm
Chen, M.-F.(2000). For birth-death processes,

$$\hbox{exponentially ergodic } \Longleftrightarrow
\delta:=\sup_{i> 0}\sum_{j=0}^{i-1}\frac 1{\mu_jb_j}\sum_{j=i}^\infty \mu_j<\infty.$$

$$\lambda_1>0\Longleftrightarrow \delta<\infty.\quad (\delta=M)$$

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

Key: birth-death processes are reversible!
\end{slide}
}
%------------------------------------------------------------------------------
%---------------------------------------------------------------------- SLIDE 40-
\overlays{2}{%
\begin{slide}[Replace]{Exponential ergodicity}
\vskip0.5cm

$P(t)$ 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 $(y_i)$.
\vskip0.2cm
``$\inf_iq_i>0$'' is necessary essentially for exponential ergodicity.

\vskip0.2cm
\OnlySlide{2}
Difficulty: two parameters!
\end{slide}
}
%------------------------------------------------------------------------------


%---------------------------------------------------------------------- SLIDE 41-
\overlays{1}{%
\begin{slide}[Replace]{Exponential ergodicity}
\vskip0.5cm
Mao and Zhang (2004) gives another proof.
\vskip0.5cm
Keys:
\vskip0.2cm
1. Construct a test function with double summations for sufficiency.

\vskip0.2cm
The construction comes from the study of spectral gap essentially.
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 42-
\overlays{1}{%
\begin{slide}[Replace]{Exponential ergodicity}
2. Necessity. Note that
$$\aligned
m_{i0}^{(n)}\ge n\bigg(&\sum_{j=0}^{i-1}\frac1{\mu_jb_j}\sum_{k=i}^\infty\mu_k\bigg)m_{i0}^{(n-1)}\\
\ge\cdots&\ge n!\bigg(\sum_{j=0}^{i-1}\frac1{\mu_jb_j}\sum_{k=i}^\infty\mu_k\bigg)^n.\\
\mathbb E_i\textrm{e}^{\lambda\sigma_0}<\infty
&\Rightarrow\sum_{n=1}^\infty\bigg(\lambda\sum_{j=0}^{i-1}\frac 1{\mu_jb_j}\sum_{k=i}^\infty\mu_k\bigg)^n<
\infty\\
&\Rightarrow\lambda\sum_{j=0}^{i-1}\frac 1{\mu_jb_j}\sum_{k=i}^\infty\mu_k< 1\\
&\Rightarrow\delta\le \lambda^{-1}<\infty.
\endaligned$$

\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 43-
\overlays{2}{%
\begin{slide}[Replace]{Exponential ergodicity}
\vskip0.5cm

Unfortunately, ``$M<\infty$'' is not necessary for exponential ergodicity.
\vskip0.5cm
\OnlySlide{2}
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{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 44-
\overlays{2}{%
\begin{slide}[Replace]{Exponential ergodicity}
%\vskip0.2cm
Conjecture that the criterion should be

$$\delta':=\sup_{i> 0}\sum_{j=0}^{i-1}F_j^{(0)}\bigg(d-\frac {d_i}{F_i^{(0)}}\bigg)<\infty.$$

For birth-death processes, $\delta=M=\delta'$.

\vskip0.2cm
\OnlySlide{2}
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
$\mathbb E_i\textrm{e}^{\lambda\tau_0}\le \mathbb E_i\textrm{e}^{\lambda\overline\tau_0}\,(i\ge 1,\lambda>0)$.

\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 45-
\begin{slide}[Replace]{Exponential ergodicity}

\vskip0.5cm
$\ell$-ergodicty and exponential ergodicity:

\vskip0.5cm

Theorem. Given a regular and irreducible single birth $Q$-matrix $(q_{ij})$ with
$\inf_i q_i>0$. Suppose that the corresponding single birth process $X(t)$ is recurrent.
Then $X(t)$ is exponentially
ergodic if and only if $d^{(\ell)}/(\ell-1)!\le \gamma^\ell$ for some positive constant $\gamma$ and
all sufficiently large $\ell$;
equivalently, $\varlimsup_{\ell\to\infty}\root{\ell}\of{d^{(\ell)}}/\ell<\infty$.
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 46-
\begin{slide}[Replace]{Exponential ergodicity}

ÏÖÔÚ»Øµ½~$0$~µÄÊ×ÖÐÊ±µÄÖ¸Êý½×¾Ø: $\mathbb E_i\textrm{e}^{\lambda\tau_0}$, ÆäÖÐ~$\lambda$ÊÇÒ»Õý³£Êý. ¶ÔËùÓÐµÄ~$0\le j<i$,
¶¨Òå~$q_i^{(j)}(\lambda):=q_i^{(j)}-\lambda$,
$$\aligned
&F_n^{(n)}(\lambda)=1,\  F_n^{(i)}(\lambda)=\frac 1{q_{n,n+1}}
\sum_{k=i}^{n-1}q_n^{(k)}(\lambda)F_k^{(i)}(\lambda),\quad 0\le i<n,\\
&d_0(\lambda)=0,\ d_n(\lambda)=\frac
1{q_{n,n+1}}\bigg(1+\sum_{k=0}^{n-1}q_n^{(k)}(\lambda)d_k(\lambda)\bigg),
\quad n\ge 1.
\endaligned$$
Áî
$$
d(\lambda)=\sup_{i\ge 1}\frac{\sum_{j=0}^{i-1}d_j(\lambda)}{\sum_{j=0}^{i-1} F_j^{(0)}(\lambda)}
=\sup_{i\ge 0}\frac{\sum_{j=0}^i d_j(\lambda)}{\sum_{j=0}^i
F_j^{(0)}(\lambda)}.
$$
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 47-
\overlays{2}{%
\begin{slide}[Replace]{Exponential ergodicity}
\vskip0.5cm
ÀàËÆµØ:
$$
F_i^{(j)}(\lambda)=\sum_{k=j+1}^i\frac{F_i^{(k)}(\lambda)q_k^{(j)}(\lambda)}{q_{k,k+1}},\ i>j\ge 0;$$
$$
d_i(\lambda)=\sum_{k=1}^i\frac {F_i^{(k)}(\lambda)}{q_{k,k+1}},\
i\ge 1.
$$
\vskip0.2cm\OnlySlide{2}
¶¨Àí. ¼Ù¶¨´æÔÚ³ä·ÖÐ¡µÄ~$\lambda>0$~Ê¹µÃ~$\sum_{j=0}^{i-1}F_j^{(0)}(\lambda)>0$~¶ÔÒ»ÇÐ~$i\ge 1$~³ÉÁ¢.
Ôòµ¥Éú¹ý³ÌÖ¸Êý±éÀúµ±ÇÒ½öµ±~$d(\lambda)<\infty$,
ÆäÖÐ~$0<\lambda<q_i\,(i\ge 0)$.
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 48-
\overlays{2}{%
\begin{slide}[Replace]{Exponential ergodicity}
\vskip0.5cm
Example. Let $q_{i,i+1}=:\beta_i>0\,(i\ge 0)$, $q_{i0}=\alpha_i>0\,(i\ge 1)$ and $q_{ij}=0$ for other $i\ne j$.
Obviously, the $Q$-matrix is irreducible. Suppose that the $Q$-matrix is regular.
Now $\sum_{j=0}^{i-1}F_j^{(0)}(\lambda)>0$ for all $\lambda>0$ and all $i\ge 1$.
Then the single birth
process is exponentially ergodic if and only if
$$d(\lambda)=\sum_{i=1}^\infty\frac 1{\beta_i}\prod_{j=1}^i\frac{\beta_j}{q_k-\lambda}<\infty$$
for some $\lambda>0$ with $\lambda<q_i\,(i\ge 0)$.
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 49-
\begin{slide}[Dissolve]{Criteria for ergodicity}
\vskip1cm
Àý: ¸ø¶¨ÕýÔò²»¿ÉÔ¼µ¥Éú~$Q$~¾ØÕó~$Q=(q_{ij})$. Èô~$c:=\inf_{k\ge 1}q_{k0}>0$,
È¡~$\lambda\in (0, c)$. Ôò¶ÔÒ»ÇÐ~$0\le j<i$~ÓÐ~$q_i^{(j)}(\lambda)>0$;
¼Ì¶ø, ¶ÔËùÓÐ~$i\ge 0$~ÓÐ~$F_i^{(0)}(\lambda)>0$. Òò´Ë, È¡¶¨~$0<\lambda<q_i\,(i\ge 0)$.
ÓÉºÏ·Ö±ÈÐÔÖÊ, µÃµ½
$$d(\lambda)\le\sup_{i\ge 1}\frac{d_i(\lambda)}{F_i^{(0)}(\lambda)}\le
\sup_{k\ge 1}\frac 1{q_k^{(0)}(\lambda)}=\frac 1{c-\lambda}<\infty.$$
´Ëµ¥Éú¹ý³ÌÊÇÖ¸Êý±éÀúµÄ. ÒÑÖªÊÇÇ¿±éÀúµÄ!
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 50-
\overlays{1}{%
\begin{slide}[Replace]{A class of single birth processes}
\vskip0.5cm
Given a regular and irreducible single birth $Q$-matrix $(q_{ij})$:
$$
q_{i+1,j}=p_iq_{ij},\qquad 0\le j\le i-1,
$$
where $0\le p_i\le 1\,(i\ge 1)$. In convention $p_0=0$. Assume that the
single birth is recurrent. Then
\vskip0.5cm
(1) The process is $\ell$-ergodic if and only if $\sum_{k=1}^\infty(m_{k0}^{(\ell-1)}-p_{k-1}
m_{k-1,0}^{(\ell-1)})/(q_{k,k+1}F_k^{(0)})<\infty$. In particular,
the process is ergodic if and only if $d=\sum_{i=1}^\infty (1-p_{i-1})/(q_{i,i+1}F_i^{(0)})<\infty$,
where
$F_0^{(0)}=1$, $F_i^{(0)}=\prod_{j=1}^{i}(q_{j,j-1}+p_{j-1}q_{j-1})/q_{j,j+1}\,(i\ge 1)$.
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 51-
\overlays{1}{%
\begin{slide}[Replace]{A class of single birth processes}
\vskip1cm
(2) The process is strongly ergodic if and only if
$$\sum_{i=0}^\infty F_i^{(0)}\sum_{j=i+1}^\infty\frac{1-p_{j-1}}{q_{j,j+1}F_j^{(0)}}<\infty.$$
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 52-
\overlays{1}{%
\begin{slide}[Replace]{A class of single birth processes}
\vskip0.5cm
(3) If the process is exponentially ergodic, then $q:=\inf_{i\ge 0}q_i>0$ and
$$
\delta':=\sup_{i\ge 1}\sum_{j=0}^{i-1}F_j^{(0)}\sum_{j=i}^\infty\frac
{1-p_{j-1}}{q_{j,j+1}F_j^{(0)}}<\infty.
$$
\vskip0.4cm
(4) Moreover, assume that $\sup_{i\ge 1}p_i<1$. If $q>0$ and $\delta'<\infty$,
then the process is exponentially ergodic.

\end{slide}
}
%------------------------------------------------------------------------------
%---------------------------------------------------------------------- SLIDE 53-
\overlays{2}{%
\begin{slide}[Replace]{A class of single birth processes}
\vskip0.5cm
\begin{itemstep}
\item When $p_i=0\,(i\ge1)$, it is the birth-death case: $\delta'=\delta$.
\vskip0.5cm
\item When $c:=\sup_{i\ge 1}p_i<1$, exp.erg.$\Longleftrightarrow\delta'<\infty$ and $q>0$.

\vskip0.5cm
But $(1-c)M\le\delta'\le M$.

So exp.erg.$\Longleftrightarrow M<\infty$ and $q>0$.

\vskip0.5cm
Question: how to improve for some $p_i=1$?
\end{itemstep}
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 54-
\begin{slide}[Replace]{Applications}
\vskip0.5cm
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.

\vskip0.5cm
The typical example
is Sch\"ogl's model: a model of chemical reaction with diffusion in a container.
\end{slide}
%------------------------------------------------------------------------------
%---------------------------------------------------------------------- SLIDE 55-
\begin{slide}[Replace]{Applications}
\vskip0.2cm

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.
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 56-
\begin{slide}[Replace]{Applications}
\vskip0.5cm
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 $Q=(q_{ij})$:
$$q_{ij}=\begin{cases}
\sup\{\sum_{y\in E_j}q(x,y):x\in E_i\},\qquad\text{if}\ j=i+1;\\
\inf\{\sum_{y\in E_j}q(x,y):x\in E_i\},\qquad\,\text{if}\ j<i;\\
0,\qquad\qquad\qquad\qquad\,\text{other cases of}\ j\ne i.
\end{cases}$$
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 57-
\begin{slide}[Replace]{Applications}
\vskip1cm
(0) {\blue If the $(q_{ij})$-process is unique (i.e. $R=\infty$), then so is the
$(q(x,y))$-process.}

\vskip0.5cm

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.
\vskip0.5cm
(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.

\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 58-
\begin{slide}[Replace]{Applications}
\vskip0.4cm
(2) If $\widehat d:=\sup_{k\ge 0}d_k/F_k^{(0)}<\infty$, then both processes are ergodic.
\vskip0.5cm
(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.
\vskip0.5cm
(4) If $\sup_{k\ge 0} \sum_{n=0}^k(F^{(0)}_n\widehat d-d_n)<\infty$,
then both processes are strongly ergodic.
\vskip0.5cm
Keys: increasing solution of equations.
\end{slide}
%------------------------------------------------------------------------------
%---------------------------------------------------------------------- SLIDE 59-
\begin{slide}[Replace]{Applications}
\vskip0.5cm
The $Q$-process corresponding to finite dimensional Schl\"ogl model is strongly ergodic.
\vskip0.7cm
The method is efficient for {\red necessary condition}.
\vskip0.5cm
The $Q$-process corresponding to Brusselator model is not strongly ergodic.
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 60-
\overlays{2}{
\begin{slide}[Glitter]{Related topics}
\vskip0.5cm
\begin{itemstep}
\item ÔÊÐí~$N:=\max\{i+1:q_{i,i+1}=0\}<\infty$. µ±~$N=\max\emptyset=0$~Ê±, »Øµ½µ¥Éú~$Q$~¾ØÕó.
µ±~$N\ge 1$~Ê±, ³ÆÖ®Îªµ¥ÉúÐÍ~$Q$~¾ØÕó. ÓÐÎüÊÕÌ¬.
\vskip0.2cm
Chen, M.-F.(1999), Single birth processes, Chin. Ann. Math., \textbf{20B}, 77-82.

Í³Ò»´¦ÀíÎ¨Ò»ÐÔ, $\mathbb P_i(\tau_N<\infty)=1$, $\mathbb E_i\tau_N<\infty\,(i\ge N)$,
ÕâÀï~$\tau_N=\inf\{t\ge0:X(t)\le N-1\}$.
\vskip0.3cm

\item ÔÊÐí~$q_{0j}\ge 0\,(j\ge 1)$, ³ÆÖ®Îª´øÒÆÃñµÄµ¥Éú~$Q$~¾ØÕó.
\vskip0.2cm
ÕÅÓà»Ô, ÕÔÙ»Ù», ´øÒÆÃñµÄµ¥Éú¹ý³Ì, Ô¤Ó¡±¾, 2007.
\vskip0.2cm
´¦ÀíÕâÐ©ÎÊÌâ: Î¨Ò»ÐÔ, ³£·µÐÔ, ¸÷ÖÖ±éÀúÐÔ
\end{itemstep}
\end{slide}
}
%------------------------------------------------------------------------------
%---------------------------------------------------------------------- SLIDE 61-
\overlays{5}{
\begin{slide}[Glitter]{Related topics}
\vskip0.5cm
\begin{itemstep}
\item ¹ãÒåÉúÃð¹ý³Ì
\vskip0.2cm
ÎâÈºÓ¢, ÕÅºº¾ý(2003), ¹ãÒåÉúÃð¹ý³Ì, ÏµÍ³¿ÆÑ§ÓëÊýÑ§, \textbf{23}(4), 517-528.
\vskip0.2cm
ÎâÈºÓ¢(2004), ¹ãÒåÉúÃð¹ý³Ì, ¿ÆÑ§³ö°æÉç, ±±¾©.
\vskip0.2cm
\item ±éÀúËÙ¶È: Ö¸Êý, Ç¿±éÀú
\vskip0.2cm
\item Æ×¿ÕÏ¶
\vskip0.2cm

\item Lipschitz norm
\vskip0.2cm
Ma, Y.-T., Lipschitz norm for $Q$-matrix of single birth processes, preprint, 2007

\item Ó¦ÓÃ: Á£×ÓÏµÍ³
\end{itemstep}
\end{slide}
}
%------------------------------------------------------------------------------


%---------------------------------------------------------------------- SLIDE 62-
\overlays{2}{
\begin{slide}[Glitter]{}

\vskip3cm
\centerline{ \Huge\blue Thanks}
\OnlySlide{2}
\vskip0.5cm
\centerline{http://math.bnu.edu.cn/${}^\sim$zhangyh/}
\end{slide}
}
%------------------------------------------------------------------------------

\clearpage
\end{CJK*}
\end{document}
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