相互作用粒子系统及其大尺度行为
I explain our recent results on the derivation of interface motion such as motion by mean curvature or free boundary problem from particle systems in some details. The core of these results was presented in my talk at vICM2022 [5], [6].
讲师
日期
2022年10月12日 至 2023年01月09日
网站
修课要求
It is desirable that the audience is familiar with Modern Probability Theory and some tools in Stochastic Analysis such as martingales and stochastic differential equations. But I will try to briefly explain these in my course. For example, Parts I and II of my course given at Yau Mathematical Sciences Center from March to June, 2022 fit to this purpose; see slides of Lect-1 to Lect-20 posted on the web page of YMSC.
课程大纲
The course consists of the following three parts.
(1) Interacting particle systems, cf. [1], [2], [3]
Exclusion process (Kawasaki dynamics), Zero-range process, Glauber dynamics, Basic facts and tools, Quick introduction to stochastic analysis
(2) Hydrodynamic scaling limit and fluctuation limit, cf. [1], [4]
Entropy method, One block estimate, Two blocks estimate, Equilibrium fluctuation, Boltzmann-Gibbs principle, Relative entropy method
(3) Applications and extensions of these methods, cf. [5], [6], [7], [8]
Derivation of motion by mean curvature in phase separation phenomena, Derivation of free boundary problem describing segregation of species, Boltzmann-Gibbs principle revisited, Discrete Schauder estimate
(1) Interacting particle systems, cf. [1], [2], [3]
Exclusion process (Kawasaki dynamics), Zero-range process, Glauber dynamics, Basic facts and tools, Quick introduction to stochastic analysis
(2) Hydrodynamic scaling limit and fluctuation limit, cf. [1], [4]
Entropy method, One block estimate, Two blocks estimate, Equilibrium fluctuation, Boltzmann-Gibbs principle, Relative entropy method
(3) Applications and extensions of these methods, cf. [5], [6], [7], [8]
Derivation of motion by mean curvature in phase separation phenomena, Derivation of free boundary problem describing segregation of species, Boltzmann-Gibbs principle revisited, Discrete Schauder estimate
参考资料
[1] C. Kipnis and C. Landim, Scaling limits of interacting particle systems, Springer, 1999. xvi+442 pp.
[2] T.M. Liggett, Interacting particle systems, Springer, 1985, xv+488 pp.
[3] T.M. Liggett, Stochastic interacting systems: contact, voter and exclusion processes, Springer, 1999. xii+332 pp.
[4] T. Funaki, Hydrodynamic limit for exclusion processes, Commun. Math. Stat. 6 (2018), 417—480.
[5] T. Funaki, Hydrodynamic limit and stochastic PDEs related to interface motion, talk at vICM2022, video available at https://www.youtube.com/watch?v=Af9qN7wz4fM
[6] T. Funaki, Ibid., ICM2022 Proceedings, EMS Press.
[7] A. De Masi, T. Funaki, E. Presutti and M. E. Vares, Fast-reaction limit for Glauber-Kawasaki dynamics with two components, ALEA, Lat. Am. J. Probab. Math. Stat. 16 (2019), 957—976.
[8] P. El Kettani, T. Funaki, D. Hilhorst, H. Park and S. Sethuraman, Mean curvature interface limit from Glauber+zero-range interacting particles, Commun. Math. Phys, 394 (2022), 1173-1223.
[2] T.M. Liggett, Interacting particle systems, Springer, 1985, xv+488 pp.
[3] T.M. Liggett, Stochastic interacting systems: contact, voter and exclusion processes, Springer, 1999. xii+332 pp.
[4] T. Funaki, Hydrodynamic limit for exclusion processes, Commun. Math. Stat. 6 (2018), 417—480.
[5] T. Funaki, Hydrodynamic limit and stochastic PDEs related to interface motion, talk at vICM2022, video available at https://www.youtube.com/watch?v=Af9qN7wz4fM
[6] T. Funaki, Ibid., ICM2022 Proceedings, EMS Press.
[7] A. De Masi, T. Funaki, E. Presutti and M. E. Vares, Fast-reaction limit for Glauber-Kawasaki dynamics with two components, ALEA, Lat. Am. J. Probab. Math. Stat. 16 (2019), 957—976.
[8] P. El Kettani, T. Funaki, D. Hilhorst, H. Park and S. Sethuraman, Mean curvature interface limit from Glauber+zero-range interacting particles, Commun. Math. Phys, 394 (2022), 1173-1223.
听众
Graduate
视频公开
公开
笔记公开
公开
语言
英文
讲师介绍
Funaki Tadahisa曾任东京大学教授，后任早稻田大学教授，2022年加入北京雁栖湖应用数学研究院任研究员。2007年获得日本数学会秋季奖，2022年国际数学家大会受邀报告人，曾担任日本数学会理事长。他的主要研究与统计物理学有关概率论，特别是相互作用系统和随机偏微分方程，而随着几个菲尔兹奖被授予这些领域，其重要性也在逐步增加。