Stochastic processes on disordered media -- discrete models and their scaling limits

2017 Fiscal Year Final Research Report

• Project/Area Number: 25247007
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kyoto University
• Principal Investigator: Kumagai Takashi 京都大学, 数理解析研究所, 教授 (90234509)
• Co-Investigator(Kenkyū-buntansha): 相川 弘明 北海道大学, 理学研究院, 教授 (20137889) ; 竹田 雅好 東北大学, 理学研究科, 教授 (30179650) ; 日野 正訓 京都大学, 情報学研究科, 准教授 (40303888) ; 舟木 直久 東京大学, 大学院数理科学研究科, 教授 (60112174) ; 木上 淳 京都大学, 情報学研究科, 教授 (90202035)
• Co-Investigator(Renkei-kenkyūsha): SHIGEKAWA Ichiro 京都大学, 大学院理学研究科, 教授 (00127234) ; KOTANI Motoko 東北大学, 大学院理学研究科, 教授 (50230024) ; SHIRAI Tomoyuki 九州大学, 大学院マス・フォア・インダストリ研究所, 教授 (70302932) ; FUKUSHIMA Ryoki 京都大学, 数理解析研究所, 准教授 (60527886)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Keywords: 確率論 / 数理物理 / 解析学 / 複雑系 / 統計力学 / 熱方程式

Outline of Final Research Achievements

We studied dynamics on random media and their scaling limits in a systematical way. Our major achievements are as follows: i) We proved convergence of Markov chains on random conductances with boundaries under a wide framework. The convergence is with probability one with respect to the randomness of the media. ii) We proved sub-sequential convergence of the random walk on 2-dimensional uniform spanning tree, which is a random media whose scaling limit is conformal invariant, and gave detailed estimates of the heat kernel for the limiting process. iii) We proved stability of heat kernel estimates for symmetric jump processes on metric measure spaces. This was one of the major open problems in the area for more than 10 years.

Free Research Field

確率論