On the structure of stationary flat processes

The paper contains some results related to the fundamental question of Davidson whether all rotationally stationary line processes in the plane which have a.s. no parallel lines are Cox (i.e. doubly stochastic Poisson processes). This problem is shown to be equivalent to the corresponding one for stationarity under translations only. The partial solutions by Papangelou are improved in various directions, and they are further extended to the case of marked k-dimensional flats (hyperplanes) in R ^d for arbitrary k and d with 0<k<d. It turns out that the main result of Papangelou carries over to the case k d/2, while the opposite case seems to require stronger regularity assumptions. In the former case, stationarity is typically needed in 2(d–k) directions only. The present treatment (like the one of Papangelou) proceeds in two steps, in proving first that sufficiently smooth stationary random measures are invariant, and second that point processes without parallel atoms and with invariant conditional intensities are Cox. In the final section, some related problems are discussed which provide some further insight into the structure of the basic Davidson problem (which remains open).