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A martingale-type of characterisation of the Gaussian free field and fractional Gaussian free fields

Juhan Aru, Guillaume Woessner

TL;DR

The paper develops martingale-type characterisations for the continuum Gaussian free field and fractional Gaussian free fields by connecting them to (fractional) stochastic heat equations. It constructs a resampling dynamics based on random ball updates, proves its convergence to the (fractional) SHE, and uses stationarity of the limiting SPDE to identify the target fields as the unique invariant laws. The GFF is characterised by a martingale-type decomposition on every ball, with a harmonic part and a zero-mean, zero-boundary remainder whose second-moment structure and scaling match the Green function; the approach extends to FGFs with $0<\alpha<1$, where the nonlocal fractional Laplacian necessitates a fractional mean-property and a fractional Poisson extension. The results unify the martingale framework with potential-theoretic and SPDE tools, providing robust characterisations of universality classes and linking local and nonlocal Gaussian fields to their natural stochastic dynamics. This has implications for understanding universality in discrete height functions and related Gaussian fields via continuous-time, ball-based resampling mechanisms.

Abstract

We establish a martingale-type characterisations for the continuum Gaussian free field (GFF) and for fractional Gaussian free fields (FGFs), using their connection to the stochastic heat equation and to fractional stochastic heat equations. The main theorem on the GFF generalizes previous results of similar flavour and the characterisation theorems on the FGFs are new. The proof strategy is to link the resampling dynamics coming from a martingale-type of decomposition property to the stationary dynamics of the desired field, i.e. to the (fractional) stochastic heat equation.

A martingale-type of characterisation of the Gaussian free field and fractional Gaussian free fields

TL;DR

The paper develops martingale-type characterisations for the continuum Gaussian free field and fractional Gaussian free fields by connecting them to (fractional) stochastic heat equations. It constructs a resampling dynamics based on random ball updates, proves its convergence to the (fractional) SHE, and uses stationarity of the limiting SPDE to identify the target fields as the unique invariant laws. The GFF is characterised by a martingale-type decomposition on every ball, with a harmonic part and a zero-mean, zero-boundary remainder whose second-moment structure and scaling match the Green function; the approach extends to FGFs with , where the nonlocal fractional Laplacian necessitates a fractional mean-property and a fractional Poisson extension. The results unify the martingale framework with potential-theoretic and SPDE tools, providing robust characterisations of universality classes and linking local and nonlocal Gaussian fields to their natural stochastic dynamics. This has implications for understanding universality in discrete height functions and related Gaussian fields via continuous-time, ball-based resampling mechanisms.

Abstract

We establish a martingale-type characterisations for the continuum Gaussian free field (GFF) and for fractional Gaussian free fields (FGFs), using their connection to the stochastic heat equation and to fractional stochastic heat equations. The main theorem on the GFF generalizes previous results of similar flavour and the characterisation theorems on the FGFs are new. The proof strategy is to link the resampling dynamics coming from a martingale-type of decomposition property to the stationary dynamics of the desired field, i.e. to the (fractional) stochastic heat equation.
Paper Structure (30 sections, 30 theorems, 122 equations, 1 figure)

This paper contains 30 sections, 30 theorems, 122 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The case of the Gaussian free field
  3. The case of fractional Gaussian fields
  4. The case of the GFF
  5. Preliminaries
  6. Preliminaries on the stochastic heat equation
  7. Some simple properties of the field $h^D$
  8. The two-point function
  9. Definition of the dynamics and proof of Theorem \ref{['mainthm']}
  10. The map $x \to \varphi^{B(x,\varepsilon)}$ and an approximation of $\Delta$
  11. An approximation of the Laplacian
  12. The martingales associated to $h^\varepsilon$ and their convergence
  13. The martingale central limit theorem
  14. The convergence of $h^\varepsilon_t$ as $\varepsilon \to 0$
  15. Tightness of the family $(h^\varepsilon)_{\varepsilon>0}$
  16. ...and 15 more sections

Key Result

Theorem 1.4

Let $d\geq 2$ and $D$ be a regular domain of $\mathbb{R}^d$. Suppose that a random distribution $h^D\in \mathcal{S}_D$ satisfies the following properties Then there exists $c\geq 0$ such that $c\,h^D$ is a GFF.

Figures (1)

  • Figure 1: Figure of the situation for $d=2$ and $r=0.6$. The left hand side set of \ref{['eq_figure']} is in blue, the sets $S^{1/2}(\omega,1+t)$ for $t\in \{0;0.2;0.4;0.6\}$ are shown in green

Theorems & Definitions (65)

  • Definition 1.1
  • Definition 1.2: Martingale-type decomposition
  • Definition 1.3: Zero boundary conditions
  • Theorem 1.4
  • Remark 1.5
  • Definition 1.6
  • Definition 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 55 more