Random valuations

Abstract

A valuation is a finitely additive function on the family of compact convex sets in $\mathbb{R}^d$. We study non-negative infinitely divisible random valuations, with particular emphasis on monotone, $\sigma$-continuous models with independent increments along nested families. After separating the deterministic part, we show that the L\'evy measure of such a valuation is generated by pairs $(F,r)$, where $F$ is a non-empty closed convex set and $r>0$, with each pair contributing $r\mathbf{1}_{F\cap K=\emptyset}$. This yields a Poisson representation and an equivalent formulation through a pure-jump completely random measure on the space of closed convex sets. For stationary valuations, we derive a cylinder-Grassmannian representation of the L\'evy measure. In the stationary isotropic case, we obtain a McMullen-type decomposition, at the level of one-dimensional distributions, into independent components stable under dilation of the argument.