%  First, use latex to compile the file. Second, use dvi2ps to become .ps file. You can use Gsview to watch it. But
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\begin{document}
\begin{CJK*}{GBK}{song}

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

\begin{slide}[Box]{\small\blue July 29, 2006, Taiwan}
\vspace{1cm}

\centerline{\large A class of multidimensional $Q$-processes}

\vskip1cm \centerline{Yuhui Zhang }

\vskip0.5cm \centerline{ Beijing Normal University}
\vskip0.5cm \centerline{(http://math.bnu.edu.cn/${}^\sim$zhangyh/)}
\vskip0.8cm \centerline{This talk is based on a joint work with Bo Wu.}
\end{slide}

%------------------------------------------------------------------------------
%---------------------------------------------------------------------- SLIDE 1-
\begin{slide}[Split]{Contents}
\vskip1cm
\begin{itemize}
\item Background
\item Example 1: finite dimensional Schl\"ogl model
\item Example 2: Brusselator model
\item Main results
\end{itemize}
\end{slide}
%------------------------------------------------------------------------------
%---------------------------------------------------------------------- SLIDE 2-
\begin{slide}[Split]{Background}
\vskip1cm
The uniqueness, recurrence and ergodicity are three classical problems
in the theory of Markov processes with continuous time.

\vskip0.2cm
It is more difficult to study directly the properties mentioned above for
multidimensional $Q$-processes, especially for those particle system models.

\vskip0.2cm
{\blue Yan and Chen's paper: Multidimensional $Q$-processes, Chinese Ann. Math. 7B, 1986}

\vskip0.2cm
Key: reduces the multidimensional problems to one-dimensional ones. Sufficient conditions.

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

%---------------------------------------------------------------------- SLIDE 3-
\overlays{3}{
\begin{slide}[Box]{Background: Yan-Chen's method}
\vskip0.7cm
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\end{slide}
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%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 3-
\begin{slide}[Glitter]{Ex.1: finite dimen. Schl\"ogl model}
\vskip0.5cm
finite dimensional Schl\"ogl model: $S$ be a finite set and $E=\mathbf Z_+^S$.
$Q$-matrix $Q=(q(x,y):x,y\in E)$ is defined by
$$q(x,y)=\begin{cases}
\lambda_1 \binom{x(u)}2+\lambda_4,\quad&\hbox{if}
\ y=x+e_u,\\
\lambda_2 \binom{x(u)}3+\lambda_3x(u),&\hbox{if}
\ y=x-e_u,\\
x(u)p(u,v), &\hbox{if}\ y=x-e_u+e_v,\\
0,&\hbox{other cases of}\ y\ne x,
\end{cases}$$
and $q(x)=-q(x,x)=\sum_{y\ne x}q(x,y)$,
where $x=(x(u):u\in S)$, $\lambda_1,\cdots,\lambda_4$ 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 4-
\begin{slide}[Glitter]{Ex.1: finite dimen. Schl\"ogl model}
\vskip0.5cm
{\blue Theorem} (Y.H. Zhang, J. Appl. Probab. 38, 2001):

The $Q$-process corresponding to finite dimensional Schl\"ogl model is strongly ergodic.


\vskip0.4cm
Key: the following controlling birth-death process is strongly ergodic
$$\begin{cases}
a_i=\inf\{\sum_{y\in E_{i-1}}q(x,y):x\in E_i\},&\quad i\ge 1,\\
b_i=\sup\{\sum_{y\in E_{i+1}}q(x,y):x\in E_i\},&\quad i\ge 0.
\end{cases}$$
Note that
$$
a_i\ge \frac {\lambda_2}6\Big(\frac{i^3}{|S|^2}-3i^2\Big)+
\Big(\lambda_3+\frac{\lambda_2}3\Big)i,\qquad
b_i=\frac {\lambda_1}2(i^2-i)+\lambda_4|S|.
$$
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 5-
\begin{slide}[Glitter]{Ex.1: finite dimen. Schl\"ogl model}
\vskip0.5cm
Comparison method. The controlling process is a single birth (or birth-death) process. Explicit sufficient condition
for uniqueness, recurrence and ergodicity. The method is efficient for most of particle systems, especially for
uniqueness.

\vskip0.5cm
Our aims: {\blue explicit necessary conditions for uniq., recur., erg., $\ell$-erg., ecp. erg, and strong erg..}
Comparison and coupling method.
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 6-
\overlays{7}{
\begin{slide}[Wipe]{Example 2: Brusselator model }
\vskip0.5cm
Brusselator model is a typical model of reaction-diffusion process with several
species.  Activitor particles $A$ and $B$. Reaction particles $X_1$ and $X_2$. The chemical reaction is:

\vskip0.2cm
\small

\fromSlide*{2}{
\blue reaction \hskip75pt rate\ $q(x,y)$}

\fromSlide*{3}{
$A\overset{\lambda_1}\longrightarrow X_1$ \hskip71pt
$\lambda_1 a(u)$,\hskip55pt $y=x+e_{u1}$}

\fromSlide*{4}{
$B+X_1\overset{\lambda_2}\longrightarrow X_2+D$\hskip19pt
$\lambda_2 b(u)x_1(u)$, \hskip27pt $y=x-e_{u1}+e_{u2}$}

\fromSlide*{5}{
$2X_1+X_2\overset{\lambda_3}\longrightarrow 3X_1$\hskip27pt
$\lambda_3 \binom{x_1(u)}2 x_2(u)$, \hskip15pt $y=x+e_{u1}-e_{u2}$}

\fromSlide*{6}{
$X_1\overset{\lambda_4}\longrightarrow C$\hskip73pt
$\lambda_4 x_1(u)$, \hskip48pt $y=x-e_{u1}$}

\onlySlide*{7}{
{\blue diffusion\ $k=1,2,v\ne u$\hskip22pt
$x_k(u)p_k(u,v)$,\hskip24pt $y=x-e_{uk}+e_{vk}$}}

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

%---------------------------------------------------------------------- SLIDE 7-
\begin{slide}[Glitter]{Example 2: Brusselator model}
Let $S$ be a given set and $E=(\mathbf Z_+^2)^S$. The $Q$-matrix:
$$q(x,y)=\begin{cases} \lambda_1 a(u),\quad &\hbox{if}\ y=x+e_{u1};\\
\lambda_2 b(u)x_1(u), &\hbox{if}\ y=x-e_{u1}+e_{u2};\\
\lambda_3 \binom{x_1(u)}2 x_2(u), &\hbox{if}\ y=x+e_{u1}-e_{u2};\\
\lambda_4 x_1(u), &\hbox{if}\ y=x-e_{u1};\\
x_k(u)p_k(u,v),&\hbox{if}\ y=x-e_{uk}+e_{vk},\ k=1,2,\ v\ne u;\\
0, &\hbox{other cases of}\ y\ne x,\end{cases}$$
and $q(x)=\sum_{y\ne x}q(x,y)$, where
$p_k(u,v)$ is the tr. pr. on $S$ \,($k=1,2$) and
$$e_{ui}(v,j)=\begin{cases} 1,\quad&\hbox{if}\ v=u,\ j=i;\\
0,&\hbox{other cases of}\ u,v\in S,\ i,j=1,2.\end{cases}$$
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 7-
\begin{slide}[Box]{Example 2: Brusselator model}
\vskip0.5cm
Yan-Chen's method is efficient for uniqueness of Brusselator model, but unefficient for recurrence.

\vskip0.5cm
D. Han (1991) proved the recurrence for single box ($|S|=1$). No diffusion. Construct a finite nonnegative solution
for $\sum_j q_{ij}y_j\le -1 (i\notin H),\ \sum_{i\in H}\sum_{j\ne i}q_{ij}y_j<\infty$, where
$H$ is a finite nonempty set.

\vskip0.5cm
J.W. Chen (1995) proved the exponential erg. for finite boxes ($|S|<\infty$). Construct  a nonnegative compact $h$,
$C\ge 0$, $c>0$: $\Omega h\le C-ch$.

\vskip0.5cm
How about strong erg.?
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 8-
\begin{slide}[Box]{Example 2: Brusselator model}
\vskip0.5cm
{\blue Theorem} Assume that $a=\sum_{u\in S}a(u)<\infty$. Then the $Q$-process corresponding to Brusselator
model on $E_*$ is not strongly ergodic, where $E_*=\{x\in E:|x|=\sum_{u\in S}(x_1(u)+x_2(u))<\infty\}$.

\vskip0.4cm
Key: the following controlling birth-death process is not strongly ergodic
$$\begin{cases}
a_i=\sup\{\sum_{y\in E_{i-1}}q(x,y):x\in E_i\},&\quad i\ge 1,\\
b_i=\inf\{\sum_{y\in E_{i+1}}q(x,y):x\in E_i\},&\quad i\ge 0.
\end{cases}$$
Note that
$$a_i=\lambda_4 i,\quad i\ge 1; \qquad b_i=\lambda_1 a,\quad i\ge 0.$$
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 9-
\begin{slide}[Box]{Main results}
Let $E$ be a countable set and $Q=(q(x,y):x,y\in E)$ be a totally stable and conservative $Q$-matrix.
Suppose that there exists a partition $\{E_k\}$ of $E$ such that $\sum_{k=0}^\infty E_k=E$ and that

\vskip0.2cm
(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 $k\ge 0$.

\vskip0.2cm\noindent
Define a totally stable and conservative $Q$-matrix
$Q^*=(q_{ij}:i,j\in\mathbf Z_+)$ as follows:
\begin{equation*}
q_{ij}=\begin{cases}
\inf\big\{\sum_{y\in E_j}q(x,y):x\in E_i\big\},&\quad\hbox{if}\ j=i+1;\\
\sup\big\{\sum_{y\in E_j}q(x,y):x\in E_i\big\},&\quad\hbox{if}\ j<i;\\
0,&\quad\hbox{other cases of}\ j\ne i.
\end{cases}
\end{equation*}
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 10-
\begin{slide}[Box]{Main results}
\vskip0.5cm
Denote by $X(t)$ and $X^*(t)$ the $Q$-processes corresponding to $(q(x,y))$ and $(q_{ij})$ respectively.
\vskip0.5cm
{\blue Theorem 1.} (1) Uniqueness. If $X(t)$ is unique, then so is $X^*(t)$.

Suppose that $E_0=\{\theta\}$, where $\theta\in E$ is a reference point.
Moreover, suppose that both $(q(x,y))$ and $(q_{ij})$ are irreducible and that $(q(x,y))$ is regular.

(2) Recurrence. If $X(t)$ is recurrent, then so is $X^*(t)$.
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 10-
\begin{slide}[Box]{Main results}
\vskip0.5cm
{\blue Theorem 2.}(Ergodicity) Suppose that $E_0=\{\theta\}$, where $\theta\in E$ is a reference point,
and $E_k$ is finite for all $k\ge 1$. Moreover, suppose that both $(q(x,y))$ and $(q_{ij})$ are irreducible and
that $(q(x,y))$ is regular.  Define $\tau=\inf\{t\ge 0: X(t)=\theta\}$.

(1) Take an $x_i\in E_i$ such that $f_i:=\mathbf E_{x_i}\tau=\min_{x\in E_i}\mathbf E_x\tau$.
If $f$ is increasing and $X(t)$ is ergodic $($or strongly ergodic$)$, then so is $X^*(t)$.

(2) Take some $\lambda\in (0,\inf_{x\in E} q(x))$ and an $x_i\in E_i$ such that
$g_i:=\mathbf E_{x_i}\mathrm e^{\lambda\tau}=\min_{x\in E_i}\mathbf E_x\mathrm e^{\lambda\tau}$.
If $g$ is increasing and $X(t)$ is exponentially ergodic, then so is $X^*(t)$.
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 11-
\overlays{1}{
\begin{slide}[Glitter]{}

\vskip3cm
\centerline{ \Huge\blue Thanks}
\end{slide}
}

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