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

TL;DR

This paper introduces a new martingale-based characterization of the Gaussian free field and fractional Gaussian free fields, linking their properties to stochastic heat equations, which enhances understanding of their dynamics and structure.

Contribution

It provides a novel martingale-type characterization for GFF and FGFs, extending previous results for GFF and introducing new characterizations for FGFs.

Findings

Generalizes previous GFF characterization results

Introduces new characterizations for fractional GFFs

Links field dynamics to fractional stochastic heat equations

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.