Integral and differential characterizations of the Gibbs process

where M denotes the set of all non-negative integer-valued Radon measures in X and H the set of all measurable, non-negative functions on X x M. E(x IJ.t) is the 'energy of x given the configuration u:' and p is some Radon measure in X. In the case U == 0 (i.e. E ( ./ . ) == 0, the case of the Poisson point process) this equation reduces to an integral equation due to Mecke [1]. We also show the equivalence of this equation to the equilibrium equations due to Ruelle [2]. Some easy corollaries show that this new integral equation is an effective tool in analysing Gibbs processes. In particular we get a differential characterization of Gibbs processes in terms of their Palm measures, which generalizes a recent result of Georgii [3]. In the case U == 0 this reduces to characterizations of the Poisson process via Palm measures due to Ambartzumian, Jagers, Mecke and Sliwnjak. Finally we point out some relations of this integral to the Kirkwood-Salsburg equations.