February 2024 Space–time fluctuation of the Kardar–Parisi–Zhang equation in d3 and the Gaussian free field
Francis Comets, Clément Cosco, Chiranjib Mukherjee
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 60(1): 82-112 (February 2024). DOI: 10.1214/22-AIHP1272


We study the solution hε of the Kardar–Parisi–Zhang (KPZ) equation in Rd×[0,) for d3:


Here β>0 is a parameter called the disorder strength, ξε=ξϕε is a spatially smoothened (at scale ε) Gaussian space–time white noise and Cε is a divergent constant as ε0. When β is sufficiently small and ε0, hε(t,x)hεst(t,x)0 in probability where hεst(t,x) is the stationary solution of the KPZ equation – more precisely, hεst(t,x) solves the above equation with a random initial condition (that is independent of the driving noise ξ) and its marginal law is constant in (ε,t,x). In the present article we quantify the rate of the above convergence in this regime and show that the fluctuations (ε1d2[hε(t,x)hεst(t,x)])xRd,t>0 about the stationary solution converge pointwise (with finite dimensional distributions in space and time) to a Gaussian free field convoluted with the deterministic heat equation. We also identify the fluctuations of the stationary solution itself and show that the rescaled averages Rddxφ(x)ε1d2[hεst(t,x)Ehεst(t,x)] converge to that of the stationary solution of the stochastic heat equation with additive noise, but with (random) Gaussian free field marginals (instead of flat initial condition).

Nous étudions la solution hε de l’équation de Kardar–Parisi–Zhang (KPZ) sur Rd×[0,) avec d3 :


Ici β>0 est un paramètre appelé la force du désordre, ξε=ξϕε est un bruit blanc gaussien espace-temps régularisé en espace (à l’échelle ε) et Cε est une constante qui diverge lorsque ε0. Lorsque β est suffisamment petit et ε0, hε(t,x)hεst(t,x)0 en probabilité où hεst(t,x) est la solution stationnaire de l’équation KPZ – plus précisément, hεst(t,x) est solution de l’équation ci-dessus avc une condition initiale aléatoire (laquelle est indépendante du bruit ξ) et dont la loi marginale est constante en (ε,t,x). Dans cet article nous quantifions la vitesse de la convergence ci-dessus et nous montrons que les fluctuations (ε1d2[hε(t,x)hεst(t,x)])xRd,t>0 autour de la solution stationnaire converge ponctuellement (de manière jointe pour un nombre fini de points de l’espace et du temps) vers un champ libre gaussien convolé à l’équation de la chaleur déterministe. Nous identifions également les fluctuations de l’équation stationnaire autour de sa moyenne et montrons que Rddxφ(x)ε1d2[hεst(t,x)Ehεst(t,x)] converge vers la solution stationnaire de l’équation de la chaleur avec bruit additif, dont la loi marginale est donnée par le champ libre gaussien (au lieu de la condition initiale plate).

Funding Statement

The authors were partly supported by the French Agence Nationale de la Recherche under grant ANR-17-CE40-0032.


Dedicated to the memory of Dima Ioffe


The authors would like to thank Ofer Zeitouni (Rehovot/ New York) for very useful feedback and discussions. Research of the third author is funded by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics-Geometry-Structure. The authors acknowledge the hospitality of ICTS-TIFR Bengaluru during the program Large deviation theory in statistical physics (ICTS/Prog-ldt/2017/8), where the present work was initiated. The second and the third author would like to thank the hospitality of NYU Shanghai where part of the present work was completed during the first author’s long term stay during the academic year 2018-19. Finally, we would like to thank two anonymous referees for very useful comments leading to substantial improvements of the presentation.


Download Citation

Francis Comets. Clément Cosco. Chiranjib Mukherjee. "Space–time fluctuation of the Kardar–Parisi–Zhang equation in d3 and the Gaussian free field." Ann. Inst. H. Poincaré Probab. Statist. 60 (1) 82 - 112, February 2024. https://doi.org/10.1214/22-AIHP1272


Received: 8 April 2021; Revised: 30 March 2022; Accepted: 31 March 2022; Published: February 2024
First available in Project Euclid: 3 March 2024

MathSciNet: MR4718375
Digital Object Identifier: 10.1214/22-AIHP1272

Primary: 60K35
Secondary: 35Q82 , 35R60 , 60H15 , 82D60

Keywords: Directed polymers , Edwards–Wilkinson limit , Gaussian free field , Kardar–Parisi–Zhang equation , random environment , rate of convergence , SPDE , Stochastic heat equation

Rights: Copyright © 2024 Association des Publications de l’Institut Henri Poincaré


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Vol.60 • No. 1 • February 2024
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