Integral and Differential Characterizations of the GIBBS Process
By NGUYEN XUAN XANH and HANS ZESSIN of Bielefeld (Eingegangen am 1.11.1976) We consider general GIBBS processes, i.e. point processes in a locally compact HAUSDORFF topological space whose conditional probabilities in bounded volumes are given by the so-called grand canonical GIBBS distribution. For a general class of interactions U we characterize GIBBS processes by means of an integral equation, which in the case U=O (the case of the POISSON point process) reduces to an integral equation due to MECKE [ll], and show the equivalence of this equation to the DLR-equilibrium equation due to DOBRUSHIN [ 1 -41, LANFORD/ RUELLE [ 5 ] , RUELLE [6]. Some easy corollaries show that this integral equation is an effective tool in analyzing the GIBBS process. In particular we get a differential characterization of GIBBS processes in terms of their PALM measures which has been recently obtained by GEORGII [7] (and which in the case U=O reduces to characterizations of the POISSON process via PALM measures due to AMBART-ZUMIAN [ O ] , JAGERS [8], MECKE [ l l ] and SLIWNJAK [16]).
- Preliminsries. Let X be a locally compact second countable HAUSDORFF topological space and B the a-field of BOREL sets in X, i.e. the a-field generated