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

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.