一維和二維的标准以及條件隨機遊走的性質

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黎懷仁(Le Hoai Nhan) 查詢紙本館藏

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論文名稱

一維和二維的标准以及條件隨機遊走的性質
(Properties of One and Two Dimensional Random Walks: Simple and Conditioned)

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摘要(中)

我們考慮在 $Z^d$、$d=1,2$ 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 $x in Z^d$ 開始的簡單隨機遊走定義為
egin{equation*}
S_n = x + X_1 + X_2+ cdots + X_n,
end{equation*}
而 $hat{S}_n$ 是
egin{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
egin{equation*}
PP[S_n=y|S_{n-1}=x] = frac{1}{2d} qquad ext{if } ||y-x||=1,
end{equation*}
和
egin{equation*}
PP[hat{S}_n = y|hat{S}_{n-1}=x] = left{
egin{array}{ll}
displaystyle dfrac{1}{2d}frac{a(y)}{a(x)} & ext{if } x e 0 ext{ and } ||y-x||=1\
0& ext{otherwise.}
end{array}

ight.
end{equation*}
這裡 $a(x)$ 是 $S_n$ 的勢能核函數。
設 $ au$ 和 $hat{ au}$ 為 $Z^d$ 的連通有限子集相對於 $S$ 和 $hat{S}$ 的存活時間。 $ au$ 和 $hat{ au}$ 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。

摘要(英)

We consider random walks on $Z^d$, $d=1,2$ in case simple and conditioned on never hit the origin. The simple random walk starting at a point $x in Z^d$ is defined as
egin{equation*}
S_n = x + X_1 + X_2+ cdots + X_n
end{equation*}
whereas the conditioned one is
egin{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
egin{equation*}
PP[S_n=y|S_{n-1}=x] = frac{1}{2d} qquad ext{if } ||y-x||=1,
end{equation*}
and
egin{equation*}
PP[hat{S}_n = y|hat{S}_{n-1}=x] = left{
egin{array}{ll}
displaystyle dfrac{1}{2d}frac{a(y)}{a(x)} & ext{if } x e 0 ext{ and } ||y-x||=1\
0& ext{otherwise}
end{array}

ight.
end{equation*}
here $a(x)$ is the potential kernel of $S_n$.
Let $ au$ and $hat{ au}$ be the exiting time of a connected finite subset of $Z^d$ with respect to $S$ and $hat{S}$. $ au$ and $hat{ au}$ 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.

關鍵字(中)

★ 隨機遊走
★ 條件隨機遊走

關鍵字(英)

★ Random walks
★ Conditional Random walks

論文目次

摘要v
Abstract vii
Acknowledgement ix
Contents xi
Explanation of Symbols xiii
1 The Life Time of One Dimensional Conditional Random
Walk 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Life Time of One Dimensional Conditional Ran-
dom Walks . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Some martingales as a function of ˆ Sn . . . . . . . . . 8
2 Two Dimensional Simple Random Walks and Life Time
In A Finite Set 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Potential Kernel of Two Dimensional Simple Ran-
dom Walk . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Green Function in a Finite Set D . . . . . . . . . . . 47
2.4 Exiting Time to a Finite Set . . . . . . . . . . . . . 49
2.4.1 The Matrix Norm . . . . . . . . . . . . . . . . . . . 49
2.4.2 Statement of the problem . . . . . . . . . . . . . . . 50
2.4.3 Matrix representation . . . . . . . . . . . . . . . . . 51
2.4.4 An estimation of Ex[τD] with D is a ball . . . . . . . 77
xi
3 Two Dimensional Conditional Random Walks 79
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 79
3.2 Why ˆ Sn is called conditional simple random walk . . 79
3.3 Some martingales as a function of ˆ Sn . . . . . . . . . 83
References 89

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指導教授

方向(Fang Xiang)

審核日期

2023-7-17

推文