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題名 暫現狀態下具長域隨機漫步在整數晶格點的格林函數與容積的漸近行為
Asymptotic Behaviors of the Green Function and Capacity for Transient State Random Walks with Long-Range Interactions
作者 陳濬程
Chen, Jiun-Cheng
貢獻者 陳隆奇
Chen, Lung-Chi
陳濬程
Chen, Jiun-Cheng
關鍵詞 長域隨機漫步
格林函數
容積
Green Function
Capacity
Long-Range Random Walk
日期 2021
上傳時間 1-Oct-2021 10:05:32 (UTC+8)
摘要 在整數晶格 Zd 上的隨機漫步 S_n^x = x + X1 + X2 +...+ Xn，Xi, i = 1, 2, · · · , n 皆獨立且具有相同分佈 D(x)。此論文，我們假設 D(x) 在 Zd 空間情形下擁有對稱性且當 |x| → ∞ 時，遞減速率為 |x|−d−α，其中 α ∈ (0, ∞) \\ {2} 且 d > α ∧ 2。本文主要是探討此具長域隨機漫步下的一些 漸近行為。第一個主要結果在於獲得此模型之格林函數的漸近行為，此外 我們還得到主要項係數及其收斂速度;第二個主要結果在討論容積的漸近 行為，並且進一步得到在長域隨機漫步下的 Wiener’s Criterion。
Let S_n^x = x + X1 + X2 +...+ Xn are independent identically distributed random vectors with distribution D(x). In the thesis, we suppose that the distribution D(x) is symmetric on Zd and the rate of decayisoforder|x|−d−α as|x|→∞withα∈(0,∞)\\{2}andd>α∧2,where a ∧ b = min {a, b}. The purpose of the thesis is to investigate asymptotic behaviors of the long-range random walk. First of all, we get the asymptotic behavior of the Green function. Moreover, we obtain the coefficient of the main term and its rates of convergence. Secondly, we discuss the asymptotic behavior of the capacity for the long-range random walk. Moreover, we derive the Wiener’s Criterion for the long-range random walk.
參考文獻 [1] Béla Bollobás. Random Graphs. Cambridge University Press, Cambridge, 2001.
[2] Maury Bramson, Ofer Zeitouni, and Martin P. W. Zerner. Shortest spanning trees and a counterexample for random walks in random environments. The Annals of Probability, 34(3):821 – 856, 2006.
[3] George Green. An Essay on the Application of mathematical Analysis to the theories of Electricity and Magnetism. Nottingham, July 1828.
[4] N. Jain and S. Orey. On the range of random walk. Israel Journal of Mathematics, 6(4): 373–380, 1968.
[5] Gregory F. Lawler. Intersections of random walks / Gregory F. Lawler. Probability and its applications. Birkhäuser, Boston, 1991.
[6] KARL PEARSON. The problem of the random walk. Nature, 72(1865):294–294, 1905.
[7] Serguei Popov. Two-Dimensional Random Walk: From Path Counting to Random Interlacements. Institute of Mathematical Statistics Textbooks. Cambridge University Press, 2021.
[8] C E Soteros and S G Whittington. The statistical mechanics of random copolymers. Journal of Physics A: Mathematical and General, 37(41):R279–R325, sep 2004.
[9] Michel Talagrand. Spin glasses : a challenge for mathematicians : cavity and mean field models / Michel Talagrand. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, v. 46. Springer, New York, 2003.
[10] Robert Brown F.R.S. Hon. M.R.S.E. & R.I. Acad. V.P.L.S. 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.
描述 碩士
國立政治大學
應用數學系
108751012
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0108751012
資料類型 thesis
dc.contributor.advisor 陳隆奇zh_TW
dc.contributor.advisor Chen, Lung-Chien_US
dc.contributor.author (Authors) 陳濬程zh_TW
dc.contributor.author (Authors) Chen, Jiun-Chengen_US
dc.creator (作者) 陳濬程zh_TW
dc.creator (作者) Chen, Jiun-Chengen_US
dc.date (日期) 2021en_US
dc.date.accessioned 1-Oct-2021 10:05:32 (UTC+8)-
dc.date.available 1-Oct-2021 10:05:32 (UTC+8)-
dc.date.issued (上傳時間) 1-Oct-2021 10:05:32 (UTC+8)-
dc.identifier (Other Identifiers) G0108751012en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/137293-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用數學系zh_TW
dc.description (描述) 108751012zh_TW
dc.description.abstract (摘要) 在整數晶格 Zd 上的隨機漫步 S_n^x = x + X1 + X2 +...+ Xn，Xi, i = 1, 2, · · · , n 皆獨立且具有相同分佈 D(x)。此論文，我們假設 D(x) 在 Zd 空間情形下擁有對稱性且當 |x| → ∞ 時，遞減速率為 |x|−d−α，其中 α ∈ (0, ∞) \\ {2} 且 d > α ∧ 2。本文主要是探討此具長域隨機漫步下的一些 漸近行為。第一個主要結果在於獲得此模型之格林函數的漸近行為，此外 我們還得到主要項係數及其收斂速度;第二個主要結果在討論容積的漸近 行為，並且進一步得到在長域隨機漫步下的 Wiener’s Criterion。zh_TW
dc.description.abstract (摘要) Let S_n^x = x + X1 + X2 +...+ Xn are independent identically distributed random vectors with distribution D(x). In the thesis, we suppose that the distribution D(x) is symmetric on Zd and the rate of decayisoforder|x|−d−α as|x|→∞withα∈(0,∞)\\{2}andd>α∧2,where a ∧ b = min {a, b}. The purpose of the thesis is to investigate asymptotic behaviors of the long-range random walk. First of all, we get the asymptotic behavior of the Green function. Moreover, we obtain the coefficient of the main term and its rates of convergence. Secondly, we discuss the asymptotic behavior of the capacity for the long-range random walk. Moreover, we derive the Wiener’s Criterion for the long-range random walk.en_US
dc.description.tableofcontents 致謝 i
中文摘要 ii
Abstract iii
Contents iv
1 Introduction and The Main Results.............. 1
1.1 Introduction.................................... 1
1.2 Main Results ................................... 2
2 Proof of Theorem 1.2.4............................ 6
2.1 Propositions of Green function.......................... 6
2.2 Proof of Theorem 1.2.4.............................. 8
3 Proof of Theorem 1.2.5............................13
3.1 PropositionsofCapacity ............................. 13
3.2 Proof of Theorem 1.2.5.............................. 16
4 Proof of Theorem 1.2.6.................20
Appendix A.............................. 23
Bibliography..............................28		zh_TW
dc.format.extent 456588 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0108751012en_US
dc.subject (關鍵詞) 長域隨機漫步zh_TW
dc.subject (關鍵詞) 格林函數zh_TW
dc.subject (關鍵詞) 容積zh_TW
dc.subject (關鍵詞) Green Functionen_US
dc.subject (關鍵詞) Capacityen_US
dc.subject (關鍵詞) Long-Range Random Walken_US
dc.title (題名) 暫現狀態下具長域隨機漫步在整數晶格點的格林函數與容積的漸近行為zh_TW
dc.title (題名) Asymptotic Behaviors of the Green Function and Capacity for Transient State Random Walks with Long-Range Interactionsen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) [1] Béla Bollobás. Random Graphs. Cambridge University Press, Cambridge, 2001.
[2] Maury Bramson, Ofer Zeitouni, and Martin P. W. Zerner. Shortest spanning trees and a counterexample for random walks in random environments. The Annals of Probability, 34(3):821 – 856, 2006.
[3] George Green. An Essay on the Application of mathematical Analysis to the theories of Electricity and Magnetism. Nottingham, July 1828.
[4] N. Jain and S. Orey. On the range of random walk. Israel Journal of Mathematics, 6(4): 373–380, 1968.
[5] Gregory F. Lawler. Intersections of random walks / Gregory F. Lawler. Probability and its applications. Birkhäuser, Boston, 1991.
[6] KARL PEARSON. The problem of the random walk. Nature, 72(1865):294–294, 1905.
[7] Serguei Popov. Two-Dimensional Random Walk: From Path Counting to Random Interlacements. Institute of Mathematical Statistics Textbooks. Cambridge University Press, 2021.
[8] C E Soteros and S G Whittington. The statistical mechanics of random copolymers. Journal of Physics A: Mathematical and General, 37(41):R279–R325, sep 2004.
[9] Michel Talagrand. Spin glasses : a challenge for mathematicians : cavity and mean field models / Michel Talagrand. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, v. 46. Springer, New York, 2003.
[10] Robert Brown F.R.S. Hon. M.R.S.E. & R.I. Acad. V.P.L.S. 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.		zh_TW
dc.identifier.doi (DOI) 10.6814/NCCU202101556en_US