Stochastic evolution in random environment and its phase transition

• Project/Area Number: 17K05295
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Nagoya University
• Principal Investigator: Yoshida Nobuo 名古屋大学, 多元数理科学研究科, 教授 (40240303)
• Project Period (FY): 2017-04-01 – 2024-03-31
• Project Status: Completed (Fiscal Year 2023)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2021: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: random walk / perturbation / sparse potential / directed polymer / random environment / branchig random walk / branching random walk / 確率成長 / ランダム環境 / 相転移

Outline of Final Research Achievements

We investigated the period group of the characteristic function of a random variable. We proved that　the random variable is non-degenerate if and only if the period group is a lattice. We considered symmetric irreducible random walk and investigated its perturbation.

Academic Significance and Societal Importance of the Research Achievements

確率変数の特性関数という極めて基本的な対象に対し重要な新知見を得たことは意義深い．
ランダムウォークの摂動は広い応用を持ち，そのスペクトルの理解はランダムウォークの長時間挙動を理解する上で重要である．摂動がコンパクト作用素の場合の解析は，本質的スペクトルが摂動により不変であることから比較的容易であるが，非コンパクト摂動の場合は本質的スペクトルが摂動により変化するため同じ手法は通用しない．このような状況の中で，解析を実行し，コンパクト作用素の場合と遜色のない結果を得た点は意義深い．

Research Products

• [Journal Article] The Period Group of a Characteristic Function (2022)
  • Author(s): Ryoki Fukushima, Makoto Nakashima, Nobuo Yoshida
  • Journal Title: Real Anal. Exchange Volume: 47 (2) Issue: 2 Pages: 323-332
  • DOI: 10.14321/realanalexch.47.2.1633525535
  • Peer Reviewed / Open Access