The critical 2d Stochastic Heat Flow is not a Gaussian Multiplicative Chaos

TL;DR

This paper demonstrates that the critical 2D Stochastic Heat Flow cannot be represented as Gaussian Multiplicative Chaos, revealing fundamental differences in its probabilistic structure and providing new insights into its moment behavior.

Contribution

The paper proves that the critical 2D SHF is not a GMC, establishing a key distinction and deriving lower bounds on its moments that are of independent interest.

Findings

SHF cannot be realized as exponential of Gaussian field

Derived strict lower bounds on SHF moments

Established fundamental difference from GMC

Abstract

The critical $2d$ Stochastic Heat Flow (SHF) is a stochastic process of random measures on ${\mathbb R}^2$, recently constructed in [CSZ23]. 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.