Regularity of Gaussian white noise on the $d$-dimensional torus
Volume 95 / 2011
Banach Center Publications 95 (2011), 385-398
MSC: Primary: 60G15; Secondary: 46E35, 60H40, 60G17.
DOI: 10.4064/bc95-0-24
Abstract
In this paper we prove that a Gaussian white noise on the $d$-dimensional torus has paths in the Besov spaces $B^{-d/2}_{p,\infty}(\mathbb T^d)$ 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 the Fourier–Besov space $\hat{b}^{-d/p}_{p,\infty}(\mathbb T^d)$. This is shown to be optimal as well.
