On the structure of stationary flat processes. II

Abstract

The present paper continues the work by Davidson, Krickeberg, Papangelou, and the author on proving, under weakest possible assumptions, that a stationary random measure η or a simple point process ξ on the space of k-flats in R d is a.s. invariant or a Cox process respectively. The problems for ξ and η are related by the fact that ξ is Cox whenever the Papangelou conditional intensity measure ζ of (a thinning of) ξ is a.s. invariant. In particular, η is shown to be a.s. invariant, whenever it is absolutely continuous with respect to some fixed measure μ and has no (so called) outer degeneracies. When k=d−2≧2, no absolute continuity is needed, provided that the first moments exist and that η has no inner degeneracies either. Under a certain regularity condition on ξ, it is further shown that ξ and ζ are simultaneously non-degenerate in either sense.