Stochastic geometry and dynamics of infinite particle systems interacting with two-dimensional Coulomb potential

2016 Fiscal Year Final Research Report

• Project/Area Number: 24244010
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kyushu University
• Principal Investigator: Osada Hirofumi 九州大学, 数理学研究院, 教授 (20177207)
• Co-Investigator(Kenkyū-buntansha): 種村 秀紀 千葉大学, 理学(系)研究科(研究院), 教授 (40217162) ; 舟木 直久 東京大学, 数理(科)学研究科(研究院), 教授 (60112174) ; 白井 朋之 九州大学, マス・フォア・インダストリ研究所, 教授 (70302932) ; 熊谷 隆 京都大学, 数理解析研究所, 教授 (90234509)
• Co-Investigator(Renkei-kenkyūsha): KOTANI Shinichi 関西学院大学, 理工学部, 教授 (10025463) ; KATORI Makoto 中央大学, 理工学部, 教授 (60202016) ; OTOBE Yoshiki 信州大学, 理学部, 准教授 (30334882)
• Research Collaborator: SHINPDA Masato 奈良女子大学, 理学部, 教授 (50271044)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Keywords: 無限粒子系 / 確率力学 / 確率幾何 / ランダム行列 / クーロンポテンシャル / 行列式点過程 / 無限次元確率微分方程式 / 干渉ブラウン運動

Outline of Final Research Achievements

We establish a general theory for solving infinite-dimensional stochastic differential equations (ISDE) with symmetry typically appearing in statistical mechanics. In particular, we prove the pathwise uniqueness and the existence of the strong solution under a very general framework. This method is novel, and regards the tail sigma field of the configuration space as a boundary of the ISDE. Furthermore, if the tail sigma field is trivial, then a strong solution exists. If the set of probability-one events is unique, then the pathwise uniqueness of solution holds.
The method is effective for the ISDE with logarithmic interaction potentials, which appear in random matrix theory.

Free Research Field

確率論