Record: Yes

Level: Graduate

Language: English

Prerequisite

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.

Abstract

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].

Reference

[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.

Syllabus

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

Lecturer Intro

Funaki Tadahisa was a professor at University of Tokyo and then at Waseda University in Japan. His research subject is probability theory mostly related to statistical physics, specifically interacting systems and stochastic PDEs, whose importance increases as several Fields medals are given to this area.

Lecturer Email: t-funaki@g.ecc.u-tokyo.ac.jp

TA: Dr. Dingqun Deng, dingqun.deng@gmail.com