Sharp Gaussian regularity on the circle, and applications to the fractional stochastic heat equation

A sharp regularity theory is established for homogeneous Gaussian fields on the unit circle. Two types of characterizations for such a field to have a given almost-sure uniform modulus of continuity are established in a general setting. The first characterization relates the modulus to the fieldʹs canonical metric; the full force of Ferniqueʹs zero-one laws and Talagrandʹs theory of majorizing measures is required. The second characterization ties the modulus to the fieldʹs random Fourier series representation. As an application, it is shown that the fractional stochastic heat equation has, up to a non-random constant, a given spatial modulus of continuity if and only if the same property holds for a fractional antiderivative of the equationʹs additive noise; a random Fourier series characterization is also given.