Stochastic ranking processes and their applications

• Project/Area Number: 22540147
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Tokyo Metropolitan University
• Principal Investigator: HATTORI KUMIKO 首都大学東京, 理工学研究科, 教授 (80231520)
• Co-Investigator(Renkei-kenkyūsha): HATTORI Tetsuya 慶應義塾大学, 経済学部, 教授 (10180902)
• Project Period (FY): 2010-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2014: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2013: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2012: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 確率過程 / 極限定理 / 非マルコフ過程 / ループ・イレーズド・ランダム・ウォーク / 連続極限 / 確率ランキングモデル / ロングテール / ループイレーズドウォーク / 確率論 / 偏微分方程式

Outline of Final Research Achievements

We constructed a stochastic ranking process with jump times determined by Poisson random measures and studied its infinite particle limit. The limit distribution of the scaled position and the intensity converges almost surely and the distribution function is related to a system of non-linear partial differential equations. This process has application to the sales ranks of an online bookstore, where the sales rate of each book varies with time. /We found a new method to construct a loop-erased random walk on the finite pre-Sierpinski gasket. We proved the existence of the scaling limit and obtained some sample path properties of the limit process. This method is applicable to various sorts of random walks and allows us to obtain a new family of self-avoiding walks.

Research Products

• [Journal Article] Loop-eerased random walk on the Sierpinski gasket (2014)
  • Author(s): K.Hattori, M.Mizuno
  • Journal Title: Stochastic processes and their applications Volume: 124 Issue: 1 Pages: 566-585
  • DOI: 10.1016/j.spa.2013.08.006
  • Peer Reviewed
• [Journal Article] Stochastic ranking process with time dependent intensities (2011)
  • Author(s): K.Hattori, 他5名
  • Journal Title: Tohoku Math.J. Volume: 63 Pages: 77-111
  • NAID: 110008425970
  • Peer Reviewed
• [Journal Article] Sales ranks, Burgers-like equations and least-recently-used caching (2010)
  • Author(s): K.Hattori, T.Hattori
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B21 Pages: 149-161
  • Peer Reviewed
• [Presentation] Non-Markov processes on fractals (2015)
  • Author(s): 服部久美子
  • Organizer: Bremen Winter School/Symposium on Diffusion on Fractals and Non-linear Dynamical Systems
  • Place of Presentation: ブレーメン（ドイツ）
  • Year and Date: 2015-03-26
  • Invited
• [Presentation] フラクタル上の数理 (2010)
  • Author(s): 服部久美子
  • Organizer: 日本数学会応用数学特別セッション
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-24
• [Presentation] Non-Markov processes on fractals
  • Author(s): K.Hattori
  • Organizer: RIMS研究集会「確率解析｝
  • Place of Presentation: 京都大学数理解析研究所
  • Invited
• [Presentation] The scaling limit of loop-erased random walks on fractals -- the erasing-larger-loops-first model and the uniform spanning trees
  • Author(s): K.Hattori
  • Organizer: Fractal Geometry and Stochastics V
  • Place of Presentation: タバルツ（ドイツ）
• [Presentation] Loop-erased random walk on a fractal -- a random fractal approach
  • Author(s): K.Hattori
  • Organizer: イエナ大学フラクタルコロキウム
  • Place of Presentation: イエナ大学（ドイツ）
  • Invited
• [Presentation] The scaling limit of a loop-erased random walk on the pre-Sierpinski gasket
  • Author(s): 服部久美子
  • Organizer: 8th World Congress in Probability and Statistics
  • Place of Presentation: イスタンブール（トルコ）
• [Presentation] The scaling limit of a loop-erased random walk on the Sierpinski gasket
  • Author(s): 服部久美子
  • Organizer: XVIIth International Congress on Mathmatical Physics
  • Place of Presentation: オールボー（デンマーク）
• [Presentation] The scaling limit of a loop-erased random walk on the pre-Sierpinski gasket -- the larger-loop-first model and the uniform spanning trees
  • Author(s): 服部久美子
  • Organizer: Markov chains on graphs and related topics
  • Place of Presentation: 京都
  • Invited
• [Remarks] Ｎｏｎ－Ｍａｒｋｏｖ ｐｒｏｃｅｓｓｅｓ ｏｎ ｆｒａｃｔａｌｓ
  • URL: http://www.wis-dfnd.math.uni-bremen.de/slides/Hattori_Bremen_2015.pdf