Regularisation of cylindrical Lévy processes in Besov spaces

We quantify the irregularity of a given cylindrical Lévy process $L$ in $L^2(\mathbb R^d)$ by determining the range of weighted Besov spaces $B$ in which $L$ has a regularised version $Y$, that is, a stochastic process $Y$ in the classical sense with values in $B$. Our approach is based on characterising Lévy measures on Besov spaces. As a by-product, we determine those Besov spaces $B$ for which the embedding of $L^2(\mathbb R^d)$ into $B$ is $0$-Radonifying and $p$-Radonifying for $p \gt 1$.