The critical 2d stochastic heat flow is not a Gaussian multiplicative chaos

Abstract

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.