Regularity of Gaussian white noise on the d-dimensional torus

In this paper we prove that a Gaussian white noise on the $d$-dimensional torus has paths in the Besov spaces $B^{ -d/2}_{ p,\infty}(\T^d)$ with $p\in [1, \infty)$. This result is shown to be optimal in several ways. We also show that Gaussian white noise on the $d$-dimensional torus has paths in a the Fourier-Besov space $\hat{ b}^{ -d/p}_{ p,\infty}(\T^d)$. This is shown to be optimal as well.

Categories: math.PR math.AP math.FA