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.
Lecturer
Date
14th September ~ 15th December, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday | 13:30 - 15:05 | Shuimo | - | - | - |
| Tuesday | 09:50 - 11:25 | Shuimo | - | - | - |
Prerequisite
It is desirable that the audience is familiar with some tools in stochastic analysis such as martingales and stochastic differential equations, cf. [1], [2].
Syllabus
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
Reference
[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.
Audience
Advanced Undergraduate
, Graduate
, Postdoc
, Researcher
Video Public
No
Notes Public
No
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.