Stochastic analysis on large scale interacting systems and its applications

2014 Fiscal Year Final Research Report

• Project/Area Number: 22244007
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: The University of Tokyo
• Principal Investigator: FUNAKI Tadahisa 東京大学, 大学院数理科学研究科, 教授 (60112174)
• Co-Investigator(Kenkyū-buntansha): OSADA Hirofumi 九州大学, 大学院数理学研究院, 教授 (20177207) ; MATANO Hiroshi 東京大学, 大学院数理科学研究科, 教授 (40126165) ; HIGUCHI Yasunari 神戸大学, 理学部, 教授 (60112075) ; OTOBE Yoshiki 信州大学, 理学部, 准教授 (30334882)
• Co-Investigator(Renkei-kenkyūsha): TANEMURA Hideki 千葉大学, 理学部, 教授 (40217162) ; CHIYONOBU Taizo 関西学院大学, 理工学部, 教授 (50197638) ; KUMAGAI Takashi 京都大学, 大学院理学研究科, 教授 (90234509) ; HANDA Kenji 佐賀大学, 理工学部, 教授 (10238214) ; YOSHIDA Nobuo 名古屋大学, 大学院多元数理科学研究科, 教授 (40240303) ; SUGIURA Makoto 琉球大学, 理学部, 准教授 (70252228) ; ICHIHARA Naoyuki 青山学院大学, 理工学部, 准教授 (70452563) ; NISHIKAWA Takao 日本大学, 理工学部, 准教授 (10386005) ; SAKAGAWA Hironobu 慶應義塾大学, 理工学部, 准教授 (60348810) ; XIE Bin 信州大学, 理学部, 准教授 (50510038)
• Project Period (FY): 2010-04-01 – 2015-03-31
• Keywords: 確率論 / 解析学 / 統計力学 / 数理物理 / 関数方程式論 / 応用数学

Outline of Final Research Achievements

We studied invariant measures of KPZ equation which describes a growth of interfaces with fluctuations. This stochastic partial differential equation involves a diverging term which makes difficult to give a mathematical meaning to it. We discussed the non-equilibrium fluctuation problem for the dynamics of two-dimensional Young diagrams and derived a stochastic partial differential equation under a scaling limit. The method of the hydrodynamic limit is applied to a system of creatures with an effect of self-organized aggregation and established a link between macroscopic and microscopic descriptions. We proved unique existence of strong solutions of infinite dimensional stochastic differential equations and rigidity for Airy point process and Ginibre point process, which appear in the theory of dynamic random matrices. Furthermore, we studied percolations, nonlinear diffusion equations, stochastic partial differential equations with stable noises and others.

Free Research Field

確率論