政大機構典藏-National Chengchi University Institutional Repository(NCCUR):Item 140.119/139556

政大機構典藏-National Chengchi University Institutional Repository(NCCUR):Item 140.119/139556

English | 正體中文 | 简体中文 | Post-Print筆數 : 27 | 全文笔数/总笔数 : 113324/144300 (79%)
造访人次 : 51150112 在线人数 : 859

RC Version 6.0 © Powered By DSPACE, MIT. Enhanced by NTU Library IR team.

搜寻范围

查询小技巧：

您可在西文检索词汇前后加上"双引号"，以获取较精准的检索结果

若欲以作者姓名搜寻，建议至进阶搜寻限定作者字段，可获得较完整数据

进阶搜寻

主页 ‧ 登入 ‧ 上传 ‧ 说明 ‧ 关于政大典藏 ‧ 管理

到手机版

政大機構典藏 > 理學院 > 應用數學系 > 學位論文 > Item 140.119/139556

请使用永久网址来引用或连结此文件: https://nccur.lib.nccu.edu.tw/handle/140.119/139556

题名:	三角點陣上的簡單隨機漫步 Simple Random Walk on Triangle Lattice
作者:	林宸旭 Lin, Chen-Hsu
贡献者:	陳隆奇 Chen, Lung-Chi 林宸旭 Lin, Chen-Hsu
关键词:	隨機漫步位勢核振盪積分鞅可選停止定理調和測度容度 Random walk Potential kernel Oscillatory integral Martingale Optional stopping theorem Harmonic measure Capacity
日期:	2022
上传时间:	2022-04-01 15:04:20 (UTC+8)
摘要:	在本篇文章中，我們將介紹在二維三角點陣上的簡單隨機漫步。我們首先介紹位勢核函數a(x)，其中x ∈ Z2，我們求得在∥x∥ 趨近於無窮下， a(x) 會近似於ln ∥x∥，並對其收斂速度進行討論。此外，假設Sn 為一在三角點陣上的簡單隨機漫步，我們觀察到a(Sn) 在不通過原點的情況下是為鞅，我們設Sn 的起始點位於大小兩圓B(R) 與B(r) 之間，利用可選停止定理，我們將a(·) 與逃脫兩圓之間機率做了連結，並且我們發現在R 趨近於無窮下先碰到大圓B(R) 的機率為O(1/ lnR)。在特別情況下，我們也能求得逃脫原點的機率。再者，比較三角點陣與正方點陣，我們觀察到兩者在逃脫大小圓的機率行為是沒有差別的。最後，我們介紹了有關調和測度與容度，這些工具可以將我們的結果延伸至逃脫任意有限集合，我們也介紹些定理證明調和測度是為從無窮遠處開始到入口點的機率，並一樣討論其收斂速度。 In this thesis, we will introduce the simple random walk on the triangular lattice. We first introduce the potential kernel function a(x) for x ∈ Z2. We conclude that a(x) ≈ ln ∥x∥ as ∥x∥ → ∞. Moreover, the rate of convergence is discussed too. Besides, let Sn be the simple random walk on the triangular lattice. We observe that a(Sn) is a martingale without visiting the origin. We set our Sn starting at the point between two circle, B(r) and B(R) with r < R. Using the optional stopping theorem, we make the connection between a(·) and escaping probability from two circle. Moreover, as R → ∞, we find that the probability that visiting B(R) first is O(1/ lnR). In the specific case, we can also find the probability that escaping from the origin. Futhermore, compare triangular lattice with the square lattice, we observe that there is no difference between them in the behavior of escaping from circle. Finally, we introduce the concept of harmonic measure and capacity. These can extend our results to calculate the probability of escaping from any finite set. We also introduce some theorem to prove that the harmonic measure is the probability of entrance point starting at infinity and also discuss the rate of convergence.
參考文獻:	[1] Robert Brown. Xxvii. a brief account of microscopical observations made in the months of june, july and august 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. The philosophical magazine, 4(21):161–173, 1828. [2] Monroe D Donsker. An invariance principle for certain probability linit theorems. AMS, 1951. [3] Albert Einstein et al. On the motion of small particles suspended in liquids at rest required by the molecularkinetic theory of heat. Annalen der physik, 17(549560): 208, 1905. [4] Yasunari Fukai and Kôhei Uchiyama. Potential kernel for two dimensional random walk. The Annals of Probability, 24(4):1979–1992, 1996. [5] Takashi Hara, Gordon Slade, and Remco van der Hofstad. Critical two point functions and the lace expansion for spread out highdimensional percolation and related models. The Annals of Probability, 31(1):349–408, 2003. [6] Gregory F Lawler and Vlada Limic. Random walk: a modern introduction, volume 123. Cambridge University Press, 2010. [7] Paul Lévy. Propriétés asymptotiques des sommes de variables aléatoires indépendantes ou enchaînées. J. Math, 14(4), 1935. [8] Karl Pearson. The problem of the random walk. Nature, 72(1867):342–342, 1905. [9] Georg Pólya. Über eine aufgabe der wahrscheinlichkeitsrechnung betreffend die irrfahrt im straßennetz. Mathematische Annalen, 84(1):149–160, 1921. [10] Serguei Popov. Two dimensional Random Walk: From Path Counting to Random Interlacements, volume 13. Cambridge University Press, 2021. [11] Frank Spitzer. Principles of random walk, volume 34. Springer Science & Business Media, 2001.
描述:	碩士國立政治大學應用數學系 108751011
資料來源:	http://thesis.lib.nccu.edu.tw/record/#G0108751011
数据类型:	thesis
DOI:	10.6814/NCCU202200379
显示于类别:	[應用數學系] 學位論文

文件中的档案:

档案	描述	大小	格式	浏览次数
101101.pdf		1245Kb	Adobe PDF2	157	检视/开启

在政大典藏中所有的数据项都受到原著作权保护.

社群 sharing

著作權政策宣告 Copyright Announcement

1.本網站之數位內容為國立政治大學所收錄之機構典藏，無償提供學術研究與公眾教育等公益性使用，惟仍請適度，合理使用本網站之內容，以尊重著作權人之權益。商業上之利用，則請先取得著作權人之授權。
The digital content of this website is part of National Chengchi University Institutional Repository. It provides free access to academic research and public education for non-commercial use. Please utilize it in a proper and reasonable manner and respect the rights of copyright owners. For commercial use, please obtain authorization from the copyright owner in advance.

2.本網站之製作，已盡力防止侵害著作權人之權益，如仍發現本網站之數位內容有侵害著作權人權益情事者，請權利人通知本網站維護人員(nccur@nccu.edu.tw)，維護人員將立即採取移除該數位著作等補救措施。
NCCU Institutional Repository is made to protect the interests of copyright owners. If you believe that any material on the website infringes copyright, please contact our staff(nccur@nccu.edu.tw). We will remove the work from the repository and investigate your claim.

DSpace Software Copyright © 2002-2004 MIT & Hewlett-Packard / Enhanced by NTU Library IR team Copyright © - 回馈