  We consider marked point processes on the d-dimensional euclidean space,
defined in terms of a quasilocal specification based on marked Poisson point
processes. We investigate the possibility of constructing absolutely-summable
Hamiltonians in terms of hyperedge potentials in the sense of Georgii et al.
These potentials are a natural generalization of physical multi-body potentials
which are useful in models of stochastic geometry. We prove that such
representations can be achieved, under appropriate locality conditions of the
specification. As an illustration we also provide such potential
representations for the Widom-Rowlinson model under independent spin-flip
time-evolution. Our paper draws a link between the abstract theory of point
processes in infinite volume, the study of measures under transformations, and
statistical mechanics of systems of point particles.
