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

Ziyang Liu, Nikos Zygouras

TL;DR

This work analyzes intermittency in the Critical $2d$ Stochastic Heat Flow by studying the $h$-th moments of the mass assigned to shrinking balls. It represents these moments via collision diagrams linked to a delta-Bose gas regularisation and employs a replica/multiplier framework to derive sharp upper bounds, showing growth like $(\log \tfrac{1}{\varepsilon})^{ {h \choose 2} }$ up to sub-logarithmic corrections. A complementary lower-bound argument uses the Gaussian Correlation Inequality and known second-moment asymptotics to establish matching order for the growth, confirming a strong intermittency signal and suggesting potential logarithmic multifractality of the mass distribution. The results illuminate the correlation structure of Brownian collisions under critical delta interactions and pave the way for further study of phase transitions and fractional-mmoment behavior in this singular SPDE setting.

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 $ε$ and we determine that its ratio to the Lebesgue volume is of order $(\log\tfrac{1}ε)^{h\choose 2}$ up to possible lower order corrections.

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

TL;DR

This work analyzes intermittency in the Critical Stochastic Heat Flow by studying the -th moments of the mass assigned to shrinking balls. It represents these moments via collision diagrams linked to a delta-Bose gas regularisation and employs a replica/multiplier framework to derive sharp upper bounds, showing growth like up to sub-logarithmic corrections. A complementary lower-bound argument uses the Gaussian Correlation Inequality and known second-moment asymptotics to establish matching order for the growth, confirming a strong intermittency signal and suggesting potential logarithmic multifractality of the mass distribution. The results illuminate the correlation structure of Brownian collisions under critical delta interactions and pave the way for further study of phase transitions and fractional-mmoment behavior in this singular SPDE setting.

Abstract

The Critical Stochastic Heat Flow (SHF) is a measure valued stochastic process on 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 -th moment of the mass that it assigns to shrinking balls of radius and we determine that its ratio to the Lebesgue volume is of order up to possible lower order corrections.

Paper Structure

This paper contains 6 sections, 13 theorems, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Auxiliary results on moments of the Critical 2d SHF
  3. Upper bound
  4. Combinatorial coeffiients
  5. Final step
  6. Lower bound

Key Result

Theorem 1.1

Let $\beta_N$ be as in (beta) for some fixed $\vartheta\in{\mathbb R}$ and $( {\mathcal{Z}} _{N; \,s,t}^\beta(\mathrm{d} x, \mathrm{d} y) )_{0\le s<t<\infty}$ be defined as in preSHF. Then, as $N\rightarrow\infty$, the process of random measures $({\mathcal{Z}} _{N; s,t}^{\beta}(\mathrm{d} x,\mathrm named the Critical 2d Stochastic Heat Flow.

Figures (2)

  • Figure 1: This picture supplies a diagrammatic representation of the moment formula \ref{['mom-formula']}, more precisely of the term corresponding to $m=3$. The wiggle lines between points $(a_r,x_r)$ and $(b_r,y_r)$ are given weight $G_\vartheta(b_r-a_r) g_{\frac{b_r-a_r}{4}}(y_r-x_r)$, representing the total collision time of Brownian motions $B^{(i_r)} , B^{(j_r)}$ with a critically scaled attractive potential. Pairs $\{i_r,j_r\}$ above wiggle lines indicate the indices of the pair of Brownian motions involved in the collisions. Solid lines between points $(a_r, x_r)$ and $(a_{\mathsf p(i_r)}, y_{\mathsf p(i_r)})$ are weighted by the heat kernel $g_{\frac{a_{r}-b_{\mathsf p(j_{r})}}{2}} (x_{r}- y_{\mathsf p(j_{r})})$.
  • Figure 2: This figure shows a diagrammatic representation of formula \ref{['mom-formula-g']}. The laces and wiggle lines are assigned weights similarly to the assignments in Figure \ref{['fig:CK2']}.

Theorems & Definitions (13)

  • Theorem 1.1: CSZ23a
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 3.3: Integration of the replica variables
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • ...and 3 more