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

where B is a Gaussian field on [0, 1] × S1 whose behavior in time is that of fractional Brownian motion (fBm) with any parameter H ∈ (0, 1), and whose behavior in space is homogeneous, and can be completely arbitrary within that restriction. By “regularity theory” for a Gaussian field Y we mean a characterization of almostsure modulus of continuity for Y that can be written using information about Y ’s covariance. We seek necessary and sufficient conditions whenever possible, hence the use of the word “characterization”. By “spatial regularity theory” for the stochastic heat equation, we mean a characerization of the almost-sure modulus of continuity for the equation’s solution in its space parameter x ∈ S1, that can be written using information about the spatial covariance of the equation’s data (additive Gaussian fractional noise ∂B/∂t), or that can be formulated in exact relation to the data’s almost sure modulus of continuity in x. Let us be more specific about the distinction between the various characterizations. Let Y (x) := (I −∆x) B (1, x). This defined a homogeneous Gaussian field on S1. We can also abusively use the notation Y for the random field Y := (I −∆)−H B on [0, 1]×S1, which can be called the “2H-fractional spatial antiderivative” of B. It is well understood (for the Brownian case, see [19], or more recently [16], [17], [12]) that in our one-dimensional situation, B does not need to be a bonafide function in x for the SHE (1) to have a solution. In fact only Y needs to be a bonafide function; in [15] it is shown that this is a necessary and sufficient condition even in the fractional Brownian case. Once a condition for existence is given, it is natural to seek conditions for regularity. We consider two types of conditions for guaranteeing/characterizing the fact that the solution X of the SHE (1) admits a given fixed function f as an almost-sure uniform modulus of continuity: