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February 2023 Lyapunov exponents of the SHE under general initial data
Promit Ghosal, Yier Lin
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 59(1): 476-502 (February 2023). DOI: 10.1214/22-AIHP1253


We consider the (1+1)-dimensional stochastic heat equation (SHE) with multiplicative white noise and the Cole-Hopf solution of the Kardar–Parisi–Zhang (KPZ) equation. We show an exact way of computing the Lyapunov exponents of the SHE for a large class of initial data which includes any bounded deterministic positive initial data and the stationary initial data. As a consequence, we derive exact formulas for the upper tail large deviation rate functions of the KPZ equation for general initial data.

Nous considérons l’équation de chaleur stochastique (SHE) de dimension (1+1) avec bruit blanc multiplicatif et la solution de Cole-Hopf de l’équation de Kardar–Parisi–Zhang. Nous montrons une manière exacte de calculer les exposants de Lyapunov du SHE pour une grande classe de données initiales qui inclut toutes les données initiales positives déterministes bornées et les données initiales stationnaires. En conséquence, nous déduisons des formules exactes pour les fonctions de taux de grande déviation de la queue supérieure de l’équation KPZ pour les données initiales générales.

Funding Statement

YL was partially supported by the Fernholz Foundation’s “Summer Minerva Fellow” program and also received summer support from Ivan Corwin’s NSF grant DMS-1811143, DMS-1664650.


PG and YL would like to thank Ivan Corwin, Sayan Das, Shalin Parekh for helpful conversations, and three anonymous referees for helpful comments.


Download Citation

Promit Ghosal. Yier Lin. "Lyapunov exponents of the SHE under general initial data." Ann. Inst. H. Poincaré Probab. Statist. 59 (1) 476 - 502, February 2023.


Received: 23 July 2020; Revised: 9 January 2022; Accepted: 4 February 2022; Published: February 2023
First available in Project Euclid: 16 January 2023

Digital Object Identifier: 10.1214/22-AIHP1253

Primary: 60F10 , 60H15
Secondary: 60H40

Keywords: KPZ equation , large deviation , Lyapunov exponents , Stochastic heat equation

Rights: Copyright © 2023 Association des Publications de l’Institut Henri Poincaré


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Vol.59 • No. 1 • February 2023
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