Stochastic analysis on large scale interacting systems and its development

2018 Fiscal Year Final Research Report

• Project/Area Number: 26287014
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Waseda University (2017-2018); The University of Tokyo (2014-2016)
• Principal Investigator: FUNAKI Tadahisa 早稲田大学, 理工学術院, 特任教授 (60112174)
• Co-Investigator(Kenkyū-buntansha): 長田 博文 九州大学, 数理学研究院, 教授 (20177207) ; 熊谷 隆 京都大学, 数理解析研究所, 教授 (90234509)
• Research Collaborator: MIMURA Masayasu ; MATANO Hiroshi ; OTOBE Yoshiki ; SAKAGAWA Hironobu ; XIE Bin ; SASADA Makiko
• Project Period (FY): 2014-04-01 – 2019-03-31
• Keywords: 確率論 / 解析学 / 統計力学 / 数理物理 / 関数方程式 / 応用数学

Outline of Final Research Achievements

We studied the stationary measure of the multi-component coupled KPZ(Kardar-Parisi-Zhang) equation and showed its global well-posedness by applying the paracontrolled calculus. Moreover, we derived such singular stochastic partial differential equation from a particle system with several conservation laws. We also studied the sharp interface limit for mass conserving Allen-Cahn equation with stochastic fluctuation term, the derivation of motion by mean curvature from particle systems, the motion by mean curvature perturbed by a direction-dependent noise, the conservation law with a multiplicative noise term that is a first order stochastic partial differential equation, to establish an affirmative mathematical base to the adaptive dynamics employed in the mathematical genetics, stochastic dynamics related to the random matrices, Markov chains in random environments, and others.

Free Research Field

確率論

Academic Significance and Societal Importance of the Research Achievements

KPZ方程式は無限大の繰込みを必要とする特異な確率偏微分方程式であるが、その数学的研究は Martin Hairer氏のFields賞受賞を契機として一層の注目を集めている。多成分を持ちそれらが互いにカップルしたKPZ方程式系は物理的に自然に現れるが、本研究ではそのような多成分方程式系について系統的な解析を行い、さらにその大規模相互作用系からの導出に成功した。これは意義深いことと考えている。