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\begin{document}

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\title[非对称随机过程：从单生到单死]{非对称随机过程：从单生到单死}

\author[张余辉]{张余辉}

\date[2017年10月]{\wuhao (中国农业大学, 2017年10月25日) }

\institute[北京师范大学]{\sihao 北京师范大学}

\frame{\titlepage}

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

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

%----------------------------------------------------------------------------------------2

\frame{
 \frametitle{Outline}\small
单生过程:
\begin{enumerate}
\item Background
\item Minimal nonnegative solution
\item Classical problems
\item Criterion on uniq., recur., erg. and str. erg.
\item Poisson equation
\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}
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\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:

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

\vskip0.2cm
M.F. Chen(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:

W.J. Anderson(1991). Continuous-Time Markov Chains, Springer-Verlag, New York.
}

%-----------------------------------------------------------6

\frame{\small
\begin{itemize}
\item Uniqueness. J.K. Zhang(1984), S.J. Yan \& M.F. Chen(1986).
\item Recurrence. Yan \& Chen(1986).
\item Extinction/return probability. P.J. Brockwell(1986), W.J. Anderson(1991), Z.T. Hou
\& Q.F. Guo(1978, 1988). Chen \& Z. (2014)
\item Ergodicity. Yan \& Chen(1986).
\item Strong ergodicity. Z.(2001).
\item Polynomial moments and stationary dist. Z.(2003, 2004, 2013)
\item Exponential moments. Z.(2003)
\item Laplace transform. Brockwell(1986), Anderson(1991)
\end{itemize}
}

%-----------------------------------------------------------6

\frame{\small
\begin{itemize}
\item Poisson equation. Chen \& Z.(2014)
\item Separationn cutoff. Mao, C. Zhang \& Z.(2016)
\item Mean occupation time. P.S. Li \& Z.(2017)
\end{itemize}
}

%----------------------------------------------------------------------------------------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}
}

%----------------------------------------------------------------------------------------13
\frame{%\frametitle{2. Minimal nonnegative solution}
\small

{\blue Z.T. Hou \& Q.F. Guo. Homogeneous Denumerable Markov Processes. Beijing: Science
Press, 1978(in Chinese); English translation, Beijing: Science Press and Springer, 1988}

\vskip0.2cm

侯振挺(2017)获得中国数学会第十三届华罗庚数学奖.
}
%----------------------------------------------------------------------------------------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
\frame{\frametitle{3. Classical problems}\small


\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: $p_{ij}(\lambda)=\int_0^\infty e^{-\lambda t}p_{ij}(t)dt$.
$$p_{ij}(\lambda)=\sum_{k\ne i}\frac{q_{ik}}{\lambda+q_i}p_{kj}(\lambda)+\frac{\delta_{ij}}{\lambda+q_i},$$
$$p_{ij}(\lambda)=\sum_{k\ne j}p_{ik}(\lambda)\frac{q_{kj}}{\lambda+q_j}+\frac{\delta_{ij}}{\lambda+q_i}.$$

Denote the minimal process by $P^{\text{min}}(\lambda)$. Consider $P(\lambda)-P^{\text{min}}(\lambda)$:
}
%----------------------------------------------------------------------------------------16

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

\begin{block}{Uniqueness Theorem[Feller(1957), Reuter(1957)]} 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}
}
%----------------------------------------------------------------------------------------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 $(\Pi_{ij})$: $\displaystyle\Pi_{ij}=I_{\{q_i\ne0\}}(1-\delta_{ij})\frac{q_{ij}}{q_i}+I_{\{q_i=0\}}\delta_{ij}.$

\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}

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

For embedding chain $(\Pi_{ij})$,
$(f_{i,j_0}=\mathbb P_i(\sigma_{j_0}<\infty):i\in\mathbb Z_+)$ 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 $f_{i,j_0}\equiv 1$.
}

%----------------------------------------------------------------------------------------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 \& 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}
}

%----------------------------------------------------------------------------------------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{4.1 Criterion on uniqueness of SBP}\small
For $0\leq k<n$, define {\blue $q_n^{(k)}=\sum_{j=0}^k q_{nj}$} and
{\blue $$F_i^{(i)}=1, \ F_n^{(i)}=\frac 1{q_{n,n+1}} \sum_{k=i}^{n-1}q_n^{(k)}F_k^{(i)},\ 0\le i<n.$$}
Define
$$m_n
=\sum_{k=0}^n\frac{F_n^{(k)}}{q_{k,k+1}},\qquad n\ge
0.$$

\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}
}

%----------------------------------------------------------------------------------------24
\frame{\frametitle{4.2 Criterion on recurrence of SBP}\small


\begin{block}{Recurrence[Yan \& 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}
}
%----------------------------------------------------------------------------------------23
\frame{\frametitle{4.3 Criteria on ergodicity and strong ergodicity}\small
Define
$$d_n=\sum_{k=1}^n\frac{F_n^{(k)}}{q_{k,k+1}},\qquad n\ge 0.$$

\begin{block}{Ergodicity[Yan \& Chen(1986); Z.(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}{$H=\{0\}$}
$$\sum_{j\ne i} q_{ij}(y_j-y_i)+1\le 0,\ i\ge 1.$$
\end{block}
}


%----------------------------------------------------------------------------------------31
\frame{\small

Define $\tau_j=\inf\{t\ge 0: X(t)=j\}.$
\begin{block}{Theorem[Z.(2004)]} 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}
\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}
}

%----------------------------------------------------------------------------------------39
\frame{\frametitle{5. Poisson equation}\small
Given a function $c$, define
$${\blue \Omega g=Qg+cg,} \quad \text{where}\quad (Qg)_i=\sum_{j}q_{ij}(g_j-g_i).$$
Clearly, if $c\le 0$, then $\Omega$ is an operator corresponding to a single birth
process with killing rates $(-c_i)$. Define
{\blue $$
\widetilde F_i^{(i)}=1,\qquad  \widetilde F_n^{(i)}=\frac 1{q_{n,n+1}}
\sum_{k=i}^{n-1}\tilde q_n^{(k)}\widetilde F_k^{(i)},\qquad n>i\ge 0,$$
$$\tilde q_n^{(k)}=q_n^{(k)}-c_n:=\sum_{j=0}^kq_{nj}-c_n,\qquad 0\le k<n.$$}
}

%----------------------------------------------------------------------------------------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}
}

%----------------------------------------------------------------------------43
\frame{\frametitle{7. Poisson equation}\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}

{\blue M.F. Chen \& Z.(2014): Unified representation of formulas for single birth processes, Frontiers of Mathematics in China,
9(4): 761-796.}
}
%---------------------------------------------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{7.1 Uniqueness}\small
Define
$${\widetilde m}_n
=\sum_{k=0}^n\frac{{\widetilde F}_n^{(k)}}{q_{k,k+1}},\qquad n\ge 0.$$


\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
$$(\lambda +q_i)u_i=\sum_{j\ne i}q_{ij}u_j, \qquad i\ge 0;\qquad u_0=1
$$
is unbounded for some (equivalently for all) $\lambda>0$. Rewrite it as
$$\Omega u=Qu- \lambda u=0;\qquad u_0=1.$$
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{7.2 Recurrence}\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
$$
x_i=\sum_{k\ne 0}\Pi_{ik}x_k,\qquad 0\le x_i\le 1,\qquad i\ge 0
$$
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
$$x_i=\sum_{k\ne 0}\Pi_{ik}x_k,\qquad i\ge 0; \qquad x_0=1$$
has a nonnegative bounded solution.
}

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

Rewrite the previous equation as
$$(Qx)_0=0,\qquad (Qx)_i=q_{i0},\qquad i\ge 1;\qquad x_0=1.$$
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 $\sum_{k=0}^\infty F_k^{(0)}=\infty$.
}

%---------------------------------------------68
\frame{\frametitle{ 7.3 One example}\small
\begin{block}{Example: uniform catastrophes}
Let $q_{i,i+1}=b\, i,\ i\ge 0;\ q_{ij}=a,\ j=0,1,\dots, i-1;$
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)
}


%---------------------------------------------------------------------- 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 $j\ge 2$.

Z. \& Q.Q. Zhao(2010).

\end{itemize}
}


%----------------------------------------------------------------------------------------38
\frame{\small
单死过程:

\begin{enumerate}
\item Motivation
\item Mean hitting time
\item Ergodicity and strong ergodicity
\end{enumerate}
}

%----------------------------------------------------------------------------------------3
\frame{\frametitle{1. Motivation}\small Single death process:
$Q=(q_{ij})$ satisfies
$$q_{i,i-1}>0,\ q_{i,i-j}=0,\ i\ge 1,j> 1.$$
%\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(9.55,0.5){$i+k$}
\put(-0.15,-0.5){$0$}
\put(0.85,-0.5){$1$}
\put(0,0){\line(1,0){11}}
\qbezier(9,0)(6.5,2)(5,0)
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\put(9,0){\circle*{0.2}}
\put(10,0){\circle*{0.2}}
\end{picture}
\end{center}

\vskip0.3cm
$$Q=\left(%
\begin{array}{ccccc}
  - & * & * & * & \cdots \\
  + & - & *& * & \cdots \\
  0 & + & - & * & \cdots\\
  \vdots & \vdots & \vdots & \vdots & \ddots \\
  \end{array}%
\right)$$
}
%----------------------------------------------------------------------------------------4

\frame{ \frametitle{1. Motivation}\small
\begin{itemize}

\item branching processes: reference model

\end{itemize}

$$\aligned
 i&\to i-1\quad\text{at rate}\quad ib_0\\
  &\to i\qquad\quad \text{at rate}\quad ib_1\\
  &\to i+1\quad\text{at rate}\quad ib_2\\
  &\to i+2\quad\text{at rate}\quad ib_3\\
  &\to \cdots
\endaligned$$
$$b_1=-\sum_{k\ne 1}b_k\ge -1.$$

}

%----------------------------------------------------------------------------------------4

\frame{ \frametitle{1. Motivation}\small
\begin{itemize}
\item single birth processes and dual approach.
\end{itemize}

{\blue Y.R. Li, A.G. Pakes, J. Li, A.H. Gu(2008). The limit behavior of dual Markov branching processes. J. Appl. Prob. 45, 176-189.}
\begin{itemize}

\item stationary distribution. zero-entrance.

\item quasi-stationary distribution: $\sup_{n\ge 1}\mathbf E_n\tau_0$.

\end{itemize}
}


%----------------------------------------------------------------------------------------24
\frame{\frametitle{2. Mean hitting time }

\small \begin{itemize}
\item single birth process
\end{itemize}


$$\mathbb E_n\tau_{n+1}=m_n:=\sum_{k=0}^n\frac{F_n^{(k)}}{q_{k,k+1}},\ n\ge
0.$$

\begin{block}{}
$$x_n=\sum_{k=0}^{n-1}\frac{q_n^{(k)}}{q_n}x_k+\frac{q_n^{(n-1)}}{q_n}x_n+\frac 1{q_{n}},\ n\ge 0.$$
\end{block}

\begin{block}{}
$$F_i^{(i)}=1, \ F_n^{(i)}=\frac 1{q_{n,n+1}} \sum_{k=i}^{n-1}q_n^{(k)}F_k^{(i)},\ 0\le i<n.$$
\end{block}
}
%----------------------------------------------------------------------------------------25
\frame{\frametitle{2. Mean hitting time }
\small \begin{itemize}
\item single death process: $\mathbb E_n\tau_{n-1}=?$
\end{itemize}
\begin{block}{}
$$x_n=\sum_{k=n+1}^{\infty}\frac{ q_n^{(k)}}{ q_n}x_k+\frac{ q_n^{(n+1)}}{ q_n}x_n+\frac 1{q_{n}},\ n\ge 1.$$
\end{block}

$$q_n^{(k)}:=\sum_{j=k}^\infty q_{nj},\  k>n\ge 0$$

\begin{block}{}
$$
G_i^{(i)}=1,\  G_n^{(i)}=\frac 1{q_{n,n-1}}
\sum_{k=n+1}^{i} q_n^{(k)} G_k^{(i)},\ 0\le n< i.
$$
\end{block}
}
%----------------------------------------------------------------------------------------25
\frame{\frametitle{2. Mean hitting time }\small

\begin{block}{}
$$\mathbb E_n\tau_{n-1}=\sum_{k=n}^\infty\frac{G_n^{(k)}}{q_{k,k-1}}=:m_n (n\ge 1)?$$
\end{block}

\begin{block}{Lemma 1}
$(m_n)$ is the minimal nonnegative solution to
$$x_n=\sum_{k=n+1}^{\infty}\frac{ q_n^{(k)}}{ q_n}x_k+\frac{ q_n^{(n+1)}}{ q_n}x_n+\frac 1{q_{n}},\ n\ge 1.$$
\end{block}

$$\mathbb E_n\tau_{n-1}\ge m_n.$$
}

%----------------------------------------------------------------------------------------25
\frame{\frametitle{2. Mean hitting time }\small

\begin{block}{Lemma 2}
$(\mathbf E_i\tau_{i_0}: i\ge i_0)$ is the minimal nonnegative solution to
$$
x_{i_0}=0,\ x_i=\sum_{j\ne i,j> i_0}\frac {q_{ij}}{q_i} x_j+\frac 1{q_i},\  i> i_0.
$$
\end{block}

$\sum_{i_0+1\le n\le i}m_n$ satisfies the equation above.

$$\mathbf E_i\tau_{i_0}\le \sum_{i_0+1\le n\le i}m_n \Rightarrow \mathbb E_n\tau_{n-1}\le m_n.$$



\begin{block}{Theorem 2}
$$\sum_{n=1}^\infty\sum_{k=n}^\infty\frac{G_n^{(k)}}{q_{k,k-1}}<\infty \Rightarrow \exists\ \text{a QSD uniquely}.$$
\end{block}

}

%----------------------------------------------------------------------------------------25
\frame{\frametitle{3. Ergodicity and strong ergodicity }\small

\begin{block}{Theorem 3} Assume that the single death $Q$ is irreducible and regular. Then
the process is ergodic if and only if
$$D:=\sum_{k\ge 1}q_{0}^{(k)}\sum_{\ell\ge k}\frac{G_k^{(\ell)}}{q_{\ell,\ell-1}}<\infty;$$
the process is strongly ergodic if and only
$$S:=\sum_{k\ge 1}\sum_{\ell\ge k}\frac{G_k^{(\ell)}}{q_{\ell,\ell-1}}<\infty.$$
\end{block}
}

\frame{\frametitle{3. Ergodicity and strong ergodicity }\small

The original branching process can be described as follows. Let $\alpha>0$ and $(p_j:j\in \mathbb Z_+)$ be a probability distribution. Then the process has death rate
$\alpha ip_0:i\to i-1\,(i\ge 1)$ and growth rate $\alpha ip_{k+1}:i\to i+k\,(k\ge 1,i\in \mathbb Z_+)$. An extended class of branching processes:
$$
q_{ij}=\begin{cases} q_{0j},\qquad &j>i=0;\\-q_0,&j=i=0;\\r_ip_0,&j=i-1,\,i\ge 1;\\
r_ip_{k+1},&j=i+k,\,i,k\ge 1;\\-r_i(1-p_1),&j=i\ge 1;\\
0,&\text{else},\quad i,j\in \mathbb Z_+.\end{cases}$$
where $r_i> 0$ for all $i\ge 1$ and $0<q_0:=\sum_{j\ge 1}q_{0j}<\infty$.

{\blue R.R. Chen(1997). An extended class of time-continuous branching processes. J. Appl. Prob., 34(1), 14-23.}
}

\frame{\frametitle{3. Ergodicity and strong ergodicity }\small
\begin{block}{Theorem 4} Assume that the extended branching $Q$ is irreducible and regular. Then
the extended branching process is ergodic if and only if
$$\sum_{\ell\ge 1}\frac 1{r_\ell }\bigg(q_0^{(\ell)}+\sum_{1\le k\le \ell-1}\frac {(\mathbf f^{*k}*\mathbf q)_{\ell-k+1}}{p_0^k}\bigg)<\infty; $$
it is strongly ergodic if and only if
$$\sum_{\ell\ge 1}\frac 1{r_\ell }\bigg(1+\sum_{1\le k\le \ell-1}\frac {(\mathbf f^{*k}*\mathbf 1)_{\ell-k+1}}{p_0^k}\bigg)<\infty.$$
where $\mathbf f=(\sum_{k\ge n}p_k,n\ge 2)$, $\mathbf 1=(1,1,\cdots)$ and $\mathbf q=(q_0^{(n-1)}, n\ge 2)$.
\end{block}

$(\mathbf a*\mathbf b)_n=\sum_{2\le m\le n}a_{n+2-m}b_m,\ n\ge 2.$
}

%----------------------------------------------------83
\frame{

\centerline{ \emph{\textbf{\color{blue} Thank you for your attention!}}}
\vskip0.5cm
\centerline{Homepage: http://math0.bnu.edu.cn/${}^\sim$zhangyh/}

}

\end{CJK*}

\end{document}
