On the moments of the mass of shrinking balls under the Critical $2d$ Stochastic Heat Flow

TL;DR

This paper investigates the asymptotic behavior of the moments of the mass assigned to shrinking balls by the critical 2D stochastic heat flow, revealing a logarithmic scaling of the ratio to Lebesgue volume.

Contribution

It provides a detailed analysis of the intermittency and moment asymptotics of the critical 2D stochastic heat flow's mass on shrinking regions.

Findings

The ratio of the $h$-th moment of mass to Lebesgue volume scales as $(rac{1}{ ext{radius}})^{( ext{logarithmic factor})}$.

The moments exhibit a specific logarithmic growth rate related to the binomial coefficient ${h race 2}$.

The mass assigned to shrinking balls decays faster than Lebesgue measure, indicating singularity and intermittency.

Abstract

The Critical $2d$ Stochastic Heat Flow (SHF) is a measure valued stochastic process on $\mathbb{R}^2$ that defines a non-trivial solution to the two-dimensional stochastic heat equation with multiplicative space-time noise. Its one-time marginals are a.s. singular with respect to the Lebesgue measure, meaning that the mass they assign to shrinking balls decays to zero faster than their Lebesgue volume. In this work we explore the intermittency properties of the Critical 2d SHF by studying the asymptotics of the $h$-th moment of the mass that it assigns to shrinking balls of radius $\epsilon$ and we determine that its ratio to the Lebesgue volume is of order $(\log\tfrac{1}{\epsilon})^{{h\choose 2}}$ up to possible lower order corrections.