KPZ equation and interacting particle systems
KPZ (Kardar--Parisi--Zhang) equation has recently attracted considerable attention in the probability community. It is a singular stochastic partial differential equation that describes the fluctuations of growing interfaces. Since the equation is ill-posed in the classical sense, it requires a suitable renormalization to be given a rigorous meaning.
The goal of this course is to derive KPZ equation and coupled KPZ equation from microscopic interacting particle systems. We begin with a brief review of the basic concepts and tools of stochastic analysis, including martingales, Brownian motion, stochastic differential equations, and martingale problems. We also recall some fundamental results on interacting particle systems.
The goal of this course is to derive KPZ equation and coupled KPZ equation from microscopic interacting particle systems. We begin with a brief review of the basic concepts and tools of stochastic analysis, including martingales, Brownian motion, stochastic differential equations, and martingale problems. We also recall some fundamental results on interacting particle systems.
讲师
日期
2026年09月14日 至 12月15日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周一 | 13:30 - 15:05 | Shuimo | - | - | - |
| 周二 | 09:50 - 11:25 | Shuimo | - | - | - |
修课要求
It is desirable that the audience is familiar with some tools in stochastic analysis such as martingales and stochastic differential equations, cf. [1], [2].
课程大纲
The course covers the following topics.
(1) Introduction: KPZ equation, heuristic derivation of KPZ equation (following the original KPZ paper, 1986), reason for KPZ equation to attract an attention, ill-posedness, renormalization, Cole-Hopf solution, multiplicative linear stochastic heat equation, Ito's formula, KPZ equation from interacting particle systems
(2) Preliminaries from stochastic analysis: Brownian motion, space-time Gaussian white noise, additive linear stochastic partial differential equations, finite-dimensional stochastic differential equations and their invariant/reversible measures, martingales
(3) Invariant measures of KPZ equation (Funaki-Quastel, 2015)
(4) Coupled KPZ equation by paracontrolled calculus (Funaki-Hoshino 2017)
(5) Coupled KPZ equation from multi-species zero-range process: independent particle systems, single species zero-range process, n-species zero-range process, hydrodynamic limit, linear fluctuation, nonlinear fluctuation=KPZ limit (Bernardin-Funaki-Sethuraman 2021)
(6) Decoupleability of coupled KPZ equation
(1) Introduction: KPZ equation, heuristic derivation of KPZ equation (following the original KPZ paper, 1986), reason for KPZ equation to attract an attention, ill-posedness, renormalization, Cole-Hopf solution, multiplicative linear stochastic heat equation, Ito's formula, KPZ equation from interacting particle systems
(2) Preliminaries from stochastic analysis: Brownian motion, space-time Gaussian white noise, additive linear stochastic partial differential equations, finite-dimensional stochastic differential equations and their invariant/reversible measures, martingales
(3) Invariant measures of KPZ equation (Funaki-Quastel, 2015)
(4) Coupled KPZ equation by paracontrolled calculus (Funaki-Hoshino 2017)
(5) Coupled KPZ equation from multi-species zero-range process: independent particle systems, single species zero-range process, n-species zero-range process, hydrodynamic limit, linear fluctuation, nonlinear fluctuation=KPZ limit (Bernardin-Funaki-Sethuraman 2021)
(6) Decoupleability of coupled KPZ equation
参考资料
[1] J-F. Le Gall, Brownian motion, martingales, and stochastic calculus, Springer, 2013.
[2] I. Karatzas and S.E. Shreve, Brownian motion and stochastic calculus, Springer, 1991.
[3] C. Kipnis and C. Landim, Scaling limits of interacting particle systems, Springer, 1999.
[4] T.M. Liggett, Interacting particle systems, Springer, 1985.
[5] T.M. Liggett, Stochastic interacting systems: contact, voter and exclusion processes, Springer, 1999.
[6] T. Funaki and J. Quastel, KPZ equation, its renormalization and invariant measures, Stoch. PDE: Anal. Comp., 3 (2015), 159--220.
[7] T. Funaki, Infinitesimal invariance for the coupled KPZ equations, S\'eminaire de Probabilit\'es XLVII, Lect. Notes Math., 2137, Springer (2015), 37--47.
[8] T. Funaki and M. Hoshino, A coupled KPZ equation, its two types of approximations and existence of global solutions, J. Funct. Anal., 273 (2017), 1165--1204.
[9] T. Funaki, Invariant measures in coupled KPZ equations, Stochastic Dynamics Out of Equilibrium, Institut H. Poincar\'e (2017), Springer 2019, 560--568.
[10] T. Funaki, Hydrodynamic limit for exclusion processes, Commun. Math. Stat., 6 (2018), 417-480.
[11] 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.
[12] T. Funaki, Y. Nishijima and H. Suda, Stochastic eight-vertex model, its invariant measures and KPZ limit, J. Statis. Phys., 184 (2021), Article no. 11, 1--30.
[13] Boliang Fu, T. Funaki, S. Sethuraman, S. Venkataramani, Coupled KPZ equations and their decoupleability, arXiv:2508.04637.
[14] T. Funaki, L^p-Boltzmann--Gibbs principle via Littlewood-Paley--Stein inequality, arXiv:2512.05687.
[2] I. Karatzas and S.E. Shreve, Brownian motion and stochastic calculus, Springer, 1991.
[3] C. Kipnis and C. Landim, Scaling limits of interacting particle systems, Springer, 1999.
[4] T.M. Liggett, Interacting particle systems, Springer, 1985.
[5] T.M. Liggett, Stochastic interacting systems: contact, voter and exclusion processes, Springer, 1999.
[6] T. Funaki and J. Quastel, KPZ equation, its renormalization and invariant measures, Stoch. PDE: Anal. Comp., 3 (2015), 159--220.
[7] T. Funaki, Infinitesimal invariance for the coupled KPZ equations, S\'eminaire de Probabilit\'es XLVII, Lect. Notes Math., 2137, Springer (2015), 37--47.
[8] T. Funaki and M. Hoshino, A coupled KPZ equation, its two types of approximations and existence of global solutions, J. Funct. Anal., 273 (2017), 1165--1204.
[9] T. Funaki, Invariant measures in coupled KPZ equations, Stochastic Dynamics Out of Equilibrium, Institut H. Poincar\'e (2017), Springer 2019, 560--568.
[10] T. Funaki, Hydrodynamic limit for exclusion processes, Commun. Math. Stat., 6 (2018), 417-480.
[11] 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.
[12] T. Funaki, Y. Nishijima and H. Suda, Stochastic eight-vertex model, its invariant measures and KPZ limit, J. Statis. Phys., 184 (2021), Article no. 11, 1--30.
[13] Boliang Fu, T. Funaki, S. Sethuraman, S. Venkataramani, Coupled KPZ equations and their decoupleability, arXiv:2508.04637.
[14] T. Funaki, L^p-Boltzmann--Gibbs principle via Littlewood-Paley--Stein inequality, arXiv:2512.05687.
听众
Advanced Undergraduate
, Graduate
, 博士后
, Researcher
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笔记公开
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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。