Analysis of branching process with interacting particle systems

• Project/Area Number: 26800051
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Nagoya University (2015-2017); University of Tsukuba (2014)
• Principal Investigator: Nakashima Makoto 名古屋大学, 多元数理科学研究科, 准教授 (60635902)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000); Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2016: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2015: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2014: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
• Keywords: 統計力学 / ディレクティドポリマー / 相転移 / 臨界点 / ピニング模型 / 分枝ランダムウォーク / 分枝過程 / ランダム環境 / ピニング / KPZ方程式 / 自由エネルギー / ランダムウォークピンニング模型 / ランダム環境中

Outline of Final Research Achievements

Asymptotics of logarithms of the total number of branching random walks in random environment are controlled by the free energy of directed polymers in random environment.
We gave estimates of free energies for directed polymers in random environment in 1 and 2 dimensions.
Also, we study the random walk pinning model to analyze the critical points of directed polymers in random environment in higher dimension. We found that two critical points coincide with each other and we proved the central limit theorem and invariance principle in the subcritical phase for dimension d\geq 3.

Research Products

• [Journal Article] The free energy of the random walk pinning model (2018)
  • Author(s): Nakashima Makoto
  • Journal Title: Stochastic Processes and their Applications Volume: 128 Issue: 2 Pages: 373-403
  • DOI: 10.1016/j.spa.2017.04.015
  • Peer Reviewed
• [Journal Article] Branching random walks in random environment and Super-Brownian motion in random environment (2015)
  • Author(s): Makoto Nakashima
  • Journal Title: Annales de l’Institut Henri Poincare Volume: 印刷中
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] A remark on the bound for the free energy of directed polymers in random environment in 1+2 dimension (2014)
  • Author(s): Makoto Nakashima
  • Journal Title: Journal of Mathematical Physics Volume: 55 Issue: 9 Pages: 1-14
  • DOI: 10.1063/1.4895760
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] ランダム媒質中のディレクティドポリマー (2018)
  • Author(s): 中島誠
  • Organizer: 2018 年確率論早春セミナー
  • Invited
• [Presentation] ランダム環境中の１次元ディレクティドポリマーの自由エネルギーの高温での挙動について (2017)
  • Author(s): 中島誠
  • Organizer: 新潟確率論ワークショップ
  • Place of Presentation: 新潟大学ときめいとキャンパス
  • Year and Date: 2017-03-16
• [Presentation] Branching random walks in random environment (2017)
  • Author(s): Makoto Nakashima
  • Organizer: Stochastic processes and their applications 2017
  • Int'l Joint Research / Invited
• [Presentation] Asymptotics of the free energy of 1+1 directed polymers in random environment at high temperature (2017)
  • Author(s): Makoto Nakashima
  • Organizer: Large scale properties of partial differential equations with random coefficients
  • Int'l Joint Research / Invited
• [Presentation] On the asymptotics of the free energy of directed polymers in random environment in 1+1 dimension (2017)
  • Author(s): Makoto Nakashima
  • Organizer: Japanese-German Open Conference on Stochastic Analysis 2017
  • Int'l Joint Research / Invited
• [Presentation] On the asymptotics of the free energy of directed polymers in random environment in 1+1 dimension (2017)
  • Author(s): Makoto Nakashima
  • Organizer: 16th International symposium "Stochastic Analysis on Large Scale Interacting Systems",
  • Int'l Joint Research
• [Presentation] On the asymptotics of the free energy of directed polymers on hierarchical lattice (2016)
  • Author(s): 中島誠
  • Organizer: 15th Stochastic Analysis on Large Scale Interacting Systems
  • Place of Presentation: 東京大学
  • Year and Date: 2016-11-02
  • Int'l Joint Research
• [Presentation] Phase transitions of random walk pinning model (2015)
  • Author(s): 中島誠
  • Organizer: Stochastic Analysis on Large Scale Interacting Systems
  • Place of Presentation: 京都大学
  • Year and Date: 2015-10-26
  • Int'l Joint Research
• [Presentation] Phase transitions of random walk pinning model (2015)
  • Author(s): 中島誠
  • Organizer: Stochastic Analysis and Applications
  • Place of Presentation: 東北大学
  • Year and Date: 2015-08-31
  • Int'l Joint Research / Invited
• [Presentation] Critical points on directed polymers in random environment (2015)
  • Author(s): 中島誠
  • Organizer: New Trends in Stochastic Analysis
  • Place of Presentation: 国際高等研究所
  • Year and Date: 2015-06-06
  • Int'l Joint Research / Invited
• [Presentation] 生物人口模型から現れる確率偏微分方程式 (2015)
  • Author(s): 中島誠
  • Organizer: HMAセミナー冬の研究会2015
  • Place of Presentation: 広島大学
  • Year and Date: 2015-01-09
  • Invited
• [Presentation] Branching random walks in random environment and stochastic heat equation, Stochastic Analysis on Large Scale Interacting Systems (2014)
  • Author(s): Makoto Nakashima
  • Organizer: Stochastic Analysis on Large Scale Interacting Systems
  • Place of Presentation: 東京大学
  • Year and Date: 2014-11-07
  • Invited
• [Presentation] Stochastic heat equation arising from a certain branching systems in random environment (2014)
  • Author(s): Makoto Nakashima
  • Organizer: 2014年度確率解析シンポジウム
  • Place of Presentation: 東北大学
  • Year and Date: 2014-10-14
  • Invited
• [Presentation] 1+2次元ランダム媒質中のディレクティドポリマーの自由エネルギーの高温度での評価 (2014)
  • Author(s): 中島誠
  • Organizer: 日本数学会2014年度秋季総合分科会
  • Place of Presentation: 広島大学
  • Year and Date: 2014-09-26
• [Remarks] Makoto Nakashima's website
  • URL: http://www.math.nagoya-u.ac.jp/~nakamako/index.html