The zero–one law for Besov regularity of periodic Lévy noises

Let $w$ be a periodic noise on $\mathbb{T}^d$, let $\tau\bin\tyu0005\mathbb{R}$, and let $0<p,q\textless\textless=\infty$. The zero–one law. For every such noise and every such Besov parameters,

$$\mathbb{P}\left(w\in B_{p,q}^{\tau}(\mathbb{T}^d)\right)=0\quad\text{or}\quad 1.$$

The preceding results establish smoothness bounds for periodic Lévy noises, but do not exhaust their Besov regularity. This conjecture proposes that membership in any specified Besov space is an almost-sure event or an almost-sure non-event.