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

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.