November 2023 The critical 2d stochastic heat flow is not a Gaussian multiplicative chaos
Francesco Caravenna, Rongfeng Sun, Nikos Zygouras
Author Affiliations +
Ann. Probab. 51(6): 2265-2300 (November 2023). DOI: 10.1214/23-AOP1648


The critical 2d stochastic heat flow (SHF) is a stochastic process of random measures on R2, recently constructed in (Invent. Math. 233 (2023) 325–460). We show that this process falls outside the class of Gaussian multiplicative chaos (GMC), in the sense that it cannot be realised as the exponential of a (generalised) Gaussian field. We achieve this by deriving strict lower bounds on the moments of the SHF that are of independent interest.

Funding Statement

F.C. is supported by INdAM/GNAMPA.
R.S. is supported by NUS grant R-146-000-288-114.
N.Z. is supported by EPRSC through grant EP/R024456/1.


We wish to thank Christophe Garban for asking us the question that led to the present paper and for interesting discussions during our visit to Lyon. We also thank Jeremy Clark for informing us of his ongoing work [13].

The completion of this work coincided with the passing of Francis Comets. We wish to express our deepest gratitude to Francis and dedicate this work to him for the inspiration he provided us over the years.


Download Citation

Francesco Caravenna. Rongfeng Sun. Nikos Zygouras. "The critical 2d stochastic heat flow is not a Gaussian multiplicative chaos." Ann. Probab. 51 (6) 2265 - 2300, November 2023.


Received: 1 June 2022; Revised: 1 July 2023; Published: November 2023
First available in Project Euclid: 12 November 2023

Digital Object Identifier: 10.1214/23-AOP1648

Primary: 82B44
Secondary: 35R60 , 60H15 , 82D60

Keywords: Directed polymer in random environment , Gaussian correlation inequality , Gaussian multiplicative chaos , KPZ equation , Stochastic heat equation , stochastic heat flow

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.51 • No. 6 • November 2023
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