我們考慮在 $\ZZ^d$、$d=1,2$ 上的簡單隨機遊走 $S_n$，並考慮其限制在未到達原點，稱作 $\hat{S}_n$。 從點 $x \in \ZZ^d$ 開始的簡單隨機遊走定義為 \begin{equation*} S_n = x + X_1 + X_2+ \cdots + X_n, \end{equation*} 而 $\hat{S}_n$ 是 \begin{equation*} \hat{S}_n = x + \hat{X}_1 + \hat{X}_2+ \cdots + \hat{X}_n. \end{equation*} 它們都是具有轉移機率的馬爾可夫鏈 \begin{equation*} \PP[S_n=y|S_{n-1}=x] = \frac{1}{2d} \qquad \text{if } ||y-x||=1, \end{equation*} 和 \begin{equation*} \PP[\hat{S}_n = y|\hat{S}_{n-1}=x] = \left\{ \begin{array}{ll} \displaystyle \dfrac{1}{2d}\frac{a(y)}{a(x)} & \text{if } x \ne 0 \text{ and } ||y-x||=1\\ 0& \text{otherwise.} \end{array} \right. \end{equation*} 這裡 $a(x)$ 是 $S_n$ 的勢能核函數。 設 $\tau$ 和 $\hat{\tau}$ 為 $\ZZ^d$ 的連通有限子集相對於 $S$ 和 $\hat{S}$ 的存活時間。 $\tau$ 和 $\hat{\tau}$ 幾乎必然是有限的。 我們將根據 $D$ 上限制的轉移矩陣和 $D$ 上的格林函數給出它們的分佈和期望值的表達式。 $S_n$ 是鞅，但 $\hat{S}_n$ 是嚴格的下鞅。 我們還給出充要條件，使得 $\hat{S}_n$ 和 $n$ 的函數是鞅。;We consider random walks on $\ZZ^d$, $d=1,2$ in case simple and conditioned on never hit the origin. The simple random walk starting at a point $x \in \ZZ^d$ is defined as \begin{equation*} S_n = x + X_1 + X_2+ \cdots + X_n \end{equation*} whereas the conditioned one is \begin{equation*} \hat{S}_n = x + \hat{X}_1 + \hat{X}_2+ \cdots + \hat{X}_n. \end{equation*} They are both Markov chains with transition probabilities \begin{equation*} \PP[S_n=y|S_{n-1}=x] = \frac{1}{2d} \qquad \text{if } ||y-x||=1, \end{equation*} and \begin{equation*} \PP[\hat{S}_n = y|\hat{S}_{n-1}=x] = \left\{ \begin{array}{ll} \displaystyle \dfrac{1}{2d}\frac{a(y)}{a(x)} & \text{if } x \ne 0 \text{ and } ||y-x||=1\\ 0& \text{otherwise} \end{array} \right. \end{equation*} here $a(x)$ is the potential kernel of $S_n$. Let $\tau$ and $\hat{\tau}$ be the exiting time of a connected finite subset of $\ZZ^d$ with respect to $S$ and $\hat{S}$. $\tau$ and $\hat{\tau}$ are finite almost surely. We will give an expression of their distribution and expectation in terms of transition matrix restricted on $D$ and the Green function on $D$. The simple random walk are martingale but the conditioned is a strictly submartingale. We also give necessary and sufficiency condition such that a function of $\hat{S}_n$ and $n$ is a martingale.
