Decomposable Product Systems Associated to Non-stationary Poisson Processes

Abstract

Abstract Let $P$ be a closed convex cone in $\mathbb {R}^{d}$, which we assume is pointed and spanning, i.e, $P \cap -P=\{0\}$ and $P-P=\mathbb {R}^{d}$. We demonstrate that, when $d \geq 2$, in contrast to the one-parameter situation, Poisson processes on $\mathbb {R}^{d}$, with intensity measure absolutely continuous with respect to the Lebesgue measure, restricted to $P$-invariant closed subsets, provide us with a source of examples of decomposable $E_0$-semigroups that are not always CCR flows.