| DC 欄位 |
值 |
語言 |
| DC.contributor | 數學系 | zh_TW |
| DC.creator | 黎懷仁 | zh_TW |
| DC.creator | Le Hoai Nhan | en_US |
| dc.date.accessioned | 2023-7-17T07:39:07Z | |
| dc.date.available | 2023-7-17T07:39:07Z | |
| dc.date.issued | 2023 | |
| dc.identifier.uri | http://ir.lib.ncu.edu.tw:444/thesis/view_etd.asp?URN=106281601 | |
| dc.contributor.department | 數學系 | zh_TW |
| DC.description | 國立中央大學 | zh_TW |
| DC.description | National Central University | en_US |
| dc.description.abstract | 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。 | zh_TW |
| dc.description.abstract | 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. | en_US |
| DC.subject | 隨機遊走 | zh_TW |
| DC.subject | 條件隨機遊走 | zh_TW |
| DC.subject | Random walks | en_US |
| DC.subject | Conditional Random walks | en_US |
| DC.title | 一維和二維的标准以及條件隨機遊走的性質 | zh_TW |
| dc.language.iso | zh-TW | zh-TW |
| DC.title | Properties of One and Two Dimensional Random Walks: Simple and Conditioned | en_US |
| DC.type | 博碩士論文 | zh_TW |
| DC.type | thesis | en_US |
| DC.publisher | National Central University | en_US |