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\begin{document}
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\title[Moments of the FHTs for BDPes on trees]{Moments of the first hitting times for\\ birth-death processes on trees}

\author[Yuhui Zhang]{\sihao Yuhui Zhang}

\date[July 18, 2018]{\wuhao (The 14th Workshop on Markov processes and related topics,\\ July 16-20, 2018, Sichuan University) }

\institute[Beijing Normal Univeristy]{\wuhao Beijing Normal Univeristy}

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 \maketitle
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\begin{frame}
  \frametitle{CONTENTS}
  \tableofcontents%[sections]
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Background}


%----------------------------------------------------------------------------------------2
\frame{ \frametitle{Background}\small

The tree $T$ is a connected graph without cycles. We fix a point on $T$ as the root, denoted by $o$. For any vertex $i\in T\setminus\{o\}$, there is a unique simple path from $i$ to the root $o$.
\begin{itemize}
\item $\mathscr P(i)$: the set of all the vertices on this path (the root $o$ is excluded).

\item $|i|$: the number of segments of this path is the length of $i$.

\item $i\sim j$: two vertices $i$ and $j$ are called adjacent if they are joined by a segment.

\item When $|j|=|i|+1$ and $i\sim j$, $j$ is called one offspring of $i$ and the set of all the offsprings of $i$ is denoted by $J(i)$.

\item When $|j|=|i|-1$ and $i\sim j$, $j$ is called the father of $i$ and denoted by $i^*$.

\item Denote $T_i$ the subtree with $i$ as its root, including all the descendants of $i$.

\item Assume that any vertex has finite offsprings.
\end{itemize}
}
\frame{\frametitle{Background}
%\vskip-0.5cm
  \includegraphics[scale=0.5]{20180718-1-1.png}
}
%----------------------------------------------------------------------------------------3
\frame{\frametitle{Background}\small

We consider a birth-death process on this tree which $Q$-matrix satisfies:

 $$q_{ij}>0\quad\text{if and only if}\quad i\sim j,\ \text{i.e.},\ j=i^*\ \text{or}\ j\in J(i).$$

$$q_i:=-q_{ii}=q_{ii^*}+\sum_{j\in J(i)}q_{ij}<\infty, \quad i\in T.$$

Define a measure $(\mu_i)$ on $T$ as follows.
$$\mu_o=1,\quad \mu_i=\prod_{j\in\mathscr P(i)}\frac{q_{j^*j}}{q_{jj^*}},\quad i\in T\setminus\{0\},\quad\mu=\sum_{i\in T}\mu_i.$$
which is invariant with respect to $Q$.
In fact, $(\mu_i)$ satisfies the so-called detailed balance equation:
$$\label{e0.01} \mu_iq_{ij}=\mu_jq_{ji},\qquad i\sim j.$$
}
%----------------------------------------------------------------------------------------4
\frame{\frametitle{Background}\small
\begin{itemize}
\item For estimation of the spectral gap $\lambda_1$ on trees, Miclo (1999) proposed a more explicit quantity defined in a recursive way to bound $\lambda_1$ by a factor $16$.
\item Shao and Mao (2007) obtained a variational formula for the Dirichlet eigenvalue.
\item Miclo (2008) described the shapes of eigenfunctions associated to $\lambda_1$.
\item Ma (2010) gave some explicit upper and lower bounds for the Dirichlet eigenvalue.
\item Wang and Z. (2013) obtained three kinds of variational formulas for the Dirichlet eigenvalue.
\item For finite trees, Liu, Ma and Wu (2016) obtained a two-sided variational estimate of $\lambda_1$ by factor $2$ and identified explicitly the Lipschitzian norm of the operator
in appropriate functional space, which led to the identification of the best constant in the generalized Cheeger isoperimetric inequality on the tree, and to transportation-information inequalities.
\end{itemize}
}
%----------------------------------------------------------------------------------------5
\frame{\frametitle{Background}\small
\begin{itemize}
\item For some classic problems (uniqueness, recurrence and ergodicity etc.) on trees, Ma (2010) obtained some computable sufficient conditions or necessary ones for that by constructing two corresponding birth-death processes on $\mathbb Z_+$.
\end{itemize}

Denote the birth-death process by $\{X(t):t\geqslant 0\}$. Define the first hitting time
$$\tau_i:=\inf\{t>0:X_t=i\}$$
and the first returning time
$$\sigma_i:=\inf\{t>\text{the first jumping time }: X_t=i\}.$$
In this talk, we consider the $n$-th moments of the hitting times for birth-death processes on trees.
}

\section{Main results}
%----------------------------------------------------------------------------------------6
\frame{\frametitle{Main results}\small

\begin{block}{Theorem 1.} Assume that the process corresponding to the $Q$-matrix on the tree $T$ is recurrent. Then
$$\mathbb E_i\tau_k^n=n\sum_{j\in\mathscr P(i)\setminus\mathscr P(k)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell\mathbb E_\ell\tau_k^{n-1},
\qquad k\in \mathscr P(i),$$
$$\mathbb E_i\tau_o^n=n\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell\mathbb E_\ell\tau_o^{n-1},\qquad i\ne o.$$
\end{block}
}

%----------------------------------------------------------------------------------------7
\frame{\frametitle{Main results}\small
In particular,
$$\mathbb E_i\sigma_o=\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell,\qquad i\ne o,$$
$$\mathbb E_o\sigma_o=\frac \mu{q_o}.$$

\begin{block}{Corollary 2.} Assume that the process corresponding to the $Q$-matrix on the tree $T$ is unique. Then
the process is ergodic if and only if $\mu<\infty$;
the process is strongly ergodic if and only if
$$S:=\sup_{k\ne o}\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell<\infty.$$
\end{block}
}

%----------------------------------------------------------------------------------------8
\frame{\frametitle{Main results}\small

\begin{block}{Corollary 3.} Assume that the process corresponding to the $Q$-matrix on the tree $T$ is exponential ergodic. Then
$$\delta:=\sup_{i\in T\setminus\{o\}}\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_i}\mu_\ell<\infty.$$
\end{block}

$$\mathbb E_i\tau_o^n=n\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell \mathbb E_\ell\tau_o^{n-1},\qquad i\in T, n\geqslant 1.
$$

$$\aligned \mathbb E_i\tau_o^n&\geqslant n\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in {\blue T_i}}\mu_\ell\mathbb E_\ell\tau_o^{n-1}
\geqslant n\Bigg(\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_i}\mu_\ell\Bigg){\blue \mathbb E_i\tau_o^{n-1}}\\
&\geqslant\cdots\geqslant n!\Bigg(\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_i}\mu_\ell\Bigg)^n,
\qquad n\geqslant 1.\endaligned$$
}
%----------------------------------------------------------------------------------------9
\frame{\frametitle{Main results}\small
(continued)

From the exponential ergodicity and the Taylor expansion
$$\aligned &\infty>\mathbb E_i\text{e}^{\lambda\tau_o}=\sum_{n\geqslant 0}\frac{\lambda^n}{n!}\mathbb E_i\tau_o^n\\
&\Rightarrow\infty>\sum_{n\geqslant 0}\Bigg(\lambda\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_i}\mu_\ell\Bigg)^n\\
&\Rightarrow\lambda\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_i}\mu_\ell<1\\
&\Rightarrow
\delta=\sup_{i\in T\setminus\{o\}}\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_i}\mu_\ell\leqslant \frac 1\lambda<\infty.\endaligned$$
}
\section{Sketch of the proof}
%----------------------------------------------------------------------------------------10
\frame{\frametitle{Sketch of the proof}\small

\begin{itemize}
\item induction

\item truncation of the tree

\item approximation of minimal nonnegative solution theory


\end{itemize}
}

\subsection{$n=1$}

%----------------------------------------------------------------------------------------11
\frame{\frametitle{$n=1$}\small

\begin{block}{Theorem 1.($n=1$)} Assume that the process corresponding to the $Q$-matrix on the tree $T$ is recurrent. Then
$$\mathbb E_i\tau_k=\sum_{j\in\mathscr P(i)\setminus\mathscr P(k)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell,
\qquad k\in \mathscr P(i),$$
$$\mathbb E_i\tau_o=\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell,\qquad i\ne o.$$
\end{block}

Under the assumption of ergodicity, Martinez and Ycart (2001) obtained the same result as Theorem 1($n=1$) in a different approach but they made some mistakes in their proof.
 
\vskip0.2cm
By the single death property, we only need to consider {\blue $ \mathbb E_i\tau_{i^*}$} for any $i\ne o$.

}
%----------------------------------------------------------------------------------------12
\frame{\frametitle{$n=1$}\small
Define $q_i^{(+)}=\sum_{j\in J(i)}q_{ij}$ for all $i\in T$ and
$$h_i=\frac 1{\mu_iq_{ii^*}}\sum_{k\in T_i}\mu_kq_{kk^*},\qquad i\ne o,$$

\begin{block}{Lemma 4.} $(h_i,i\ne o)$ is the minimal nonnegative solution of the following equations
$$
y_i=\frac{q_{i}^{(+)}}{q_i}y_i+\sum_{j\in J(i)}\frac{q_{ij}}{q_i}y_j+\frac 1{q_i},\qquad i\ne o.
$$
\end{block}

$(\mathbb E_i\tau_{i^*},i\ne o)$ satisfies the equation above, so
{\blue $$\mathbb E_i\tau_{i^*}\geqslant h_i.$$}
}
%----------------------------------------------------------------------------------------13
\frame{\frametitle{$n=1$: Truncation}
%\vskip-0.5cm
  \includegraphics[scale=0.5]{20180718-2-2.png}
}

\frame{\frametitle{$n=1$}\small
{\blue Truncation}: 
Truncate the tree to be a finite tree. Denote the new $Q$-matrix by $Q^{(N)}=(\tilde q_{ij})$ in the finite tree. Now
the corresponding equation is:
$$
y_i=\frac{\tilde q_{i}^{(+)}}{\tilde q_i}y_i+\sum_{j\in J(i)\cap E^{(N)}}\frac{\tilde q_{ij}}{\tilde q_i}y_j+\frac 1{\tilde q_i},\qquad i\in E^{(N)}\setminus \{o\}.
$$

The key in the proof to construct the transition rates for the boundary of $Q^{(N)}$ so that the minimal nonnegative solution of the equation above is increasing to the one of the original equation.
}

\frame{\frametitle{$n=1$}\small
{\blue Truncation}: To ensure that, fix $N\in\mathbb Z_+$, define $Q$-matrix $Q^{(N)}=(\tilde q_{ij})$ on $E^{(N)}:=\bigcup\limits_{0\leqslant n\leqslant N}E_n$ as follows.
$$\tilde q_{ij}=\begin{cases}q_{ij}&\hbox{if}\ |i|< N, |j|\leqslant N;\\
(q_i\vee N)G_N&\hbox{if}\ |i|=N, j=i^*;\\
0&\hbox{other cases of}\ j\ne i,\end{cases}$$
where
$$G_N=1\vee \sum_{k\in E_N}\mu_k q_{kk^*},$$
which is finite by the assumption that any vertex has finite offsprings.
}

%----------------------------------------------------------------------------------------13
\frame{\frametitle{$n=1$}\small

\vskip0.5cm
{\blue Approximation}: Now, for the finite tree, rewrite the corresponding equation as
$$
y_i=\frac 1{(q_i\vee N)G_N},\quad i\in E_N;\quad y_i=\frac{q_{i}^{(+)}}{ q_i}y_i+\sum_{j\in J(i)}\frac{q_{ij}}{ q_i}y_j+\frac 1{q_i},\quad i\in \bigcup_{1\leqslant n\leqslant N-1}E_n.
$$
We get the minimal nonnegative solution $(h_i^{(N)})$ which is increasing to the minimal nonnegative solution of the original equation:
$$ h_i^{(N)}=\frac 1{\mu_iq_{ii^*}}\sum_{k\in T_i\cap E^{(N-1)}}\mu_k+\frac 1{\mu_iq_{ii^*}}\sum_{k\in T_i\cap E_N}\frac{\mu_k q_{kk^*}}{(q_k\vee N)G_N}\to \frac 1{\mu_iq_{ii^*}}\sum_{k\in T_i}\mu_k=h_i.$$
as $N\to\infty$.

{\blue $$\mathbb E_i\tau_{i^*}\geqslant h_i.$$}
}

%----------------------------------------------------------------------------------------14
\frame{\frametitle{$n=1$}\small

\begin{block}{Lemma 5.} Assume that the process is recurrent. Given $i_0\in T$ arbitrarily. Then
$$\mathbb E_i\tau_{i_0}\leqslant \sum_{\ell\in\mathscr P(i)\setminus\mathscr P(i_0)}\frac 1{\mu_\ell q_{\ell\ell^*}}\sum_{k\in T_\ell}\mu_k, \qquad i\in T_{i_0}.$$
\end{block}
By Localization Theorem and the single death property, we obtain that $(\mathbb E_i\tau_{i_0}, i\in T_{i_0})$ is the minimal nonnegative solution to the following equations
$$x_{i_0}=0,\qquad x_i=\sum_{j\in T_{i_0}\setminus\{i_0,i\}}\frac{q_{ij}}{q_i}x_j+\frac 1{q_i},\qquad i\in T_{i_0}\setminus\{i_0\}.$$

$(\sum_{\ell\in\mathscr P(i)\setminus\mathscr P(i_0)}\frac 1{\mu_\ell q_{\ell\ell^*}}\sum_{k\in T_\ell}\mu_k,i\in T_{i_0})$ satisfies the equation above, so
{\blue $$\mathbb E_i\tau_{i^*}\leqslant \frac 1{\mu_i q_{ii^*}}\sum_{k\in T_i}\mu_k=h_i,\qquad i\in T\setminus \{o\}.$$}

}

\subsection{$n>1$}
%----------------------------------------------------------------------------------------15
\frame{\frametitle{$n>1$}\small

\begin{block}{Lemma 6.} Assume that the process is recurrent. Given $i_0\in T$ arbitrarily. Then
$$\mathbb E_i\tau_{i_0}^n\leqslant n\sum_{\ell\in\mathscr P(i)\setminus\mathscr P(i_0)}\frac 1{\mu_\ell q_{\ell\ell^*}}\sum_{k\in T_\ell}\mu_k\mathbb E_k\tau_{i_0}^{n-1}, \qquad i\in T_{i_0}.$$
In particular,
{\blue $$\mathbb E_i\tau_{i^*}^n\leqslant \frac n{\mu_i q_{ii^*}}\sum_{k\in T_i}\mu_k\mathbb E_k\tau_{i^*}^{n-1},\qquad i\in T\setminus \{o\}.$$}
\end{block}
By Localization Theorem and the single death property, we obtain that $(\mathbb E_i\tau_{i_0}^n, i\in T_{i_0})$ is the minimal nonnegative solution to the following equations
$$x_{i_0}=0,\qquad x_i=\sum_{j\in T_{i_0}\setminus\{i_0,i\}}\frac{q_{ij}}{q_i}x_j+\frac n{q_i}\mathbb E_i\tau_{i_0}^{n-1},\qquad i\in T_{i_0}\setminus\{i_0\}.$$

$(n\sum_{\ell\in\mathscr P(i)\setminus\mathscr P(i_0)}\frac 1{\mu_\ell q_{\ell\ell^*}}\sum_{k\in T_\ell}\mu_k\mathbb E_k\tau_{i_0}^{n-1}, i\in T_{i_0})$ satisfies the equation above.
}

%----------------------------------------------------------------------------------------16
\frame{\frametitle{$n>1$}\small

Define
$$h_i=\frac n{\mu_iq_{ii^*}}\sum_{k\in T_i}\mu_kq_{kk^*}f_k,\qquad i\ne o,$$
where
$$f_i=\frac 1{q_{ii^*}}\Bigg(\mathbb E_i\tau_{i^*}^{n-1}+\sum_{j\in J(i)}q_{ij}\sum_{m=0}^{n-2}\binom {n-1} m\frac 1{m+1}\mathbb E_j\tau_{j^*}^{m+1}\mathbb E_{i}\tau_{i^*}^{n-1-m}\Bigg), \qquad i\ne o.$$

\begin{block}{Lemma 7.} Assume that the $Q$-matrix on the tree $T$ is regular and the process is recurrent. Then
$(h_i,i\ne o)$ is the minimal nonnegative solution of the following equation
$$
y_i=\frac{q_{i}^{(+)}}{q_i}y_i+\sum_{j\in J(i)}\frac{q_{ij}}{q_i}y_j+\frac {n q_{ii^*}}{q_i}f_i,\qquad i\ne o.
$$
and
$$h_i=\frac n{\mu_iq_{ii^*}}\sum_{k\in T_i}\mu_k\mathbb E_k\tau_{i^*}^{n-1},\qquad i\ne o.$$
\end{block}
}
%----------------------------------------------------------------------------------------17
\frame{\frametitle{$n>1$}\small
{\blue Truncation}: Fix a positive integer $N\geqslant 2$, define $Q$-matrix $Q^{(N)}=(\tilde q_{ij})$ on $E^{(N)}$
$$\tilde q_{ij}=\begin{cases}q_{ij}&\hbox{if}\ |i|< N, |j|\leqslant N;\\
\Bigg(\displaystyle\frac{q_i}{q_{ii^*}}\vee N\Bigg)G_{N}&\hbox{if}\ |i|=N, j=i^*;\\
0&\hbox{other cases of}\ j\ne i,\end{cases}$$
where
$$G_{N}=\begin{cases} 1\vee \sum_{k\in E_N}\mu_k q_{kk^*}f_k,\quad &\text{ if}\ f_i<\infty\ \text{for some}\ i\ \text{with}\ |i|>1;\\
1,&\text{otherwise}.\end{cases}$$

\vskip0.2cm
{\blue Approximation}: For finite tree, we get the representation of minimal nonnegative solution $(h_i^{(N)})$. Let $N\to\infty$, it is increasing to the minimal nonnegative solution of the original equation, which is exactly $(h_i)$.
\vskip0.5cm
{\blue Induction: $$h_i=\frac n{\mu_iq_{ii^*}}\sum_{k\in T_i}\mu_k\mathbb E_k\tau_{i^*}^{n-1},\qquad i\ne o.$$
}}
%----------------------------------------------------------------------------------------18
\frame{\frametitle{$n>1$}\small

$(\mathbb E_i\tau_{i^*}^n, i\ne o)$ satisfies the equation above, so
{\blue $$\mathbb E_i\tau_{i^*}^n\geqslant h_i=\frac n{\mu_iq_{ii^*}}\sum_{k\in T_i}\mu_k\mathbb E_k\tau_{i^*}^{n-1},\qquad i\ne o.$$}
Summing up the arguments above, we obtain that
{\blue $$\mathbb E_i\tau_{i^*}^n=\frac n{\mu_iq_{ii^*}}\sum_{k\in T_i}\mu_k\mathbb E_k\tau_{i^*}^{n-1},\qquad i\ne o.$$}
By the strong Markov property and the induction, it is derived that
{\blue
$$\mathbb E_i\tau_k^n= n\sum_{j\in\mathscr P(i)\setminus\mathscr P(k)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell\mathbb E_\ell\tau_k^{n-1},
\qquad k\in \mathscr P(i),$$
$$\mathbb E_i\tau_o^n= n\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell\mathbb E_\ell\tau_o^{n-1},\qquad i\ne o. $$}
}
%----------------------------------------------------------------------------------------18
\frame{\small
\begin{itemize}
\item It coincides with the results on the birth-death process $(a_i, b_i)$ on $\mathbb Z_+$:
$$\mathbb E_i\tau_k^n= n\sum_{k<j\leqslant i}\frac 1{\mu_ja_j}\sum_{\ell\geqslant j}\mu_\ell\mathbb E_\ell\tau_k^{n-1},
\qquad k\leqslant i.$$
\item The key property used in the arguments above is the single death property. So, for single death processes, we can prove the similar assertion:
$$\mathbb E_i\tau_k^n= n\sum_{k<j\leqslant i}\sum_{\ell\geqslant j}\frac {G_j^{(\ell)}}{q_{\ell,\ell-1}}\mathbb E_\ell\tau_k^{n-1},
\qquad k\leqslant i.$$
Z. and X.F. Zhou (2018+)
\end{itemize}
}

\section{References}
%----------------------------------------------------------------------------------------19
\frame{\frametitle{References}\small

\begin{itemize}
\item  Liu, W., Ma, Y.T., Wu, L.M. {\it Spectral gap, isoperimetry and concentration on trees}, Sci. China Math., 2016, 59(3): 539-556.
\item Ma, Y.T., {\it Birth-death processes on trees}, Sci. China Math., 2010, 53(11): 2993-3004.
\item Martinez, S., Ycart, B., {\it Decay rates and cutoff for convergence and hitting times of Markov chains with countably infinite state space}, Adv. Appl. Prob., 2001, 33(1): 188-205.
\item Miclo, L., {\it Relations entre isop\'{e}rim\'{e}trie et trou spectral pour les cha\^{i}nes de Markov finies}, Probab. Theory Related Fields, 1999, 114: 431-485.
\item Miclo, L., {\it On eigenfunctions of Markov processes on trees}, Probab. Theory Related Fields, 2008, 142: 561-594.
\item Shao, J.H., Mao, Y.H., {\it Estimation of the Dirichlet eigenvalue of birth-death process on trees}, Acta Math Sinica (Chinese Series), 2007, 50(3): 507-516.
\item Wang, L.D., Zhang, Y.H., {\it The first Dirichlet eigenvalue of birth–death process on trees}, Statistics and Probability Letters, 2013, 83(9): 1973-1982.
\end{itemize}
}

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

}

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