  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.
