Decomposable Product Systems Associated to Non-stationary Poisson Processes

Let P be a closed convex cone in R-d, which we assume is pointed and spanning, i.e, P boolean AND -P = {0} and P - P = R-d. We demonstrate that, when d >= 2, in contrast to the one-parameter situation, Poisson processes on 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 f lows.