We consider a continuous, infinitely divisible random field in $\mathbb{R}^d$ given as an integral of a kernel function with respect to a Lévy basis with convolution equivalent Lévy measure. For a large class of such random fields we compute the asymptotic probability that the supremum of the field exceeds the level $x$ as $x\to\infty$. Our main result is that the asymptotic probability is equivalent to the right tail of the underlying Lévy measure.

Keywords: Asymptotic supremum; convolution equivalence; infinite divisibility; Lévy-based modelling