Interacting Particle Systems and Their Large Scale Behavior II
This is a continuation of my course given in the last semester (Oct 2022-Jan 2023). In the last semester, we started with a quick introduction to modern probability theory and stochastic analysis including strong law of large numbers, central limit theorem, continuous time martingales, Brownian motion, Poisson point processes. Then, we briefly discussed the construction and equilibrium states of interacting particle systems such as exclusion process (Kawasaki dynamics), zero-range process, Glauber dynamics. The core of the course was the study of hydrodynamic large space-time scaling limits of interacting particle systems. We explained two methods: the entropy method (GPV method) and the relative entropy method (due to H.T. Yau). Important notions and tools are local ergodicity, one block estimate, two blocks estimate, local equilibrium states, entropy inequality, large deviation principle. We also discussed linear and nonlinear (KPZ) fluctuations via Boltzmann-Gibbs principle. We touched non-gradient models. Finally, we briefly gave some ideas to derive an interface motion from interacting particle systems.
The course in this semester gives some applications and extensions of the methods explained in the last semester. More details are found in the syllabus.
The course in this semester gives some applications and extensions of the methods explained in the last semester. More details are found in the syllabus.
Lecturer
Date
22nd March ~ 19th June, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Wednesday | 10:40 - 12:15 | A3-1-103 | ZOOM 01 | 928 682 9093 | BIMSA |
Prerequisite
It is desirable that the audience had attended my course in the last semester. But, at the beginning of the course, I will try to summarize the methods and results which I explained in the last semester.
Syllabus
We first give a quick survey of the course in the last semester. Then, we discuss some applications and extensions of these methods. In particular, we discuss derivation of motion by mean curvature in phase separation phenomena, derivation of free boundary problem describing segregation of species, Boltzmann-Gibbs principle and discrete Schauder estimate.
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.
[9] T. Funaki, P. van Meurs, S. Sethuraman and K. Tsunoda, Motion by mean curvature from Glauber-Kawasaki dynamics with speed change, J. Statis. Phys., 190 (2023), Article no. 45, 1-30.
[10] T. Funaki, P. van Meurs, S. Sethuraman and K. Tsunoda, Constant-speed interface flow from unbalanced Glauber-Kawasaki dynamics, arXiv:2210.03857.
[11] P. El Kettani, T. Funaki, D. Hilhorst, H. Park and S. Sethuraman, Singular limit of an Allen-Cahn equation with nonlinear diffusion, Tunisian J. Math., 4 (2022), 719-754.
[12] T. Funaki and S. Sethuraman, Schauder estimate for quasilinear discrete PDEs of parabolic type, arXiv:2112.13973.
[13] C. Bernardin, T. Funaki and S. Sethuraman, Derivation of coupled KPZ-Burgers equation from multi-species zero-range processes, Ann. Appl. Probab., 31 (2021), 1966-2017.
[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.
[9] T. Funaki, P. van Meurs, S. Sethuraman and K. Tsunoda, Motion by mean curvature from Glauber-Kawasaki dynamics with speed change, J. Statis. Phys., 190 (2023), Article no. 45, 1-30.
[10] T. Funaki, P. van Meurs, S. Sethuraman and K. Tsunoda, Constant-speed interface flow from unbalanced Glauber-Kawasaki dynamics, arXiv:2210.03857.
[11] P. El Kettani, T. Funaki, D. Hilhorst, H. Park and S. Sethuraman, Singular limit of an Allen-Cahn equation with nonlinear diffusion, Tunisian J. Math., 4 (2022), 719-754.
[12] T. Funaki and S. Sethuraman, Schauder estimate for quasilinear discrete PDEs of parabolic type, arXiv:2112.13973.
[13] C. Bernardin, T. Funaki and S. Sethuraman, Derivation of coupled KPZ-Burgers equation from multi-species zero-range processes, Ann. Appl. Probab., 31 (2021), 1966-2017.
Audience
Graduate
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Funaki Tadahisa was a professor at University of Tokyo (1995-2017) and at Waseda University (2017-2022) in Japan. His research subject is probability theory mostly related to statistical physics, specifically interacting systems and stochastic PDEs. He was a president of Mathematical Society of Japan (2013-2015), and was an invited sectional lecturer at ICM 2022.