Integration by parts formulae for point processes (Q2702410)

\textit{S. Albeverio, Yu. G. Kondratiev} and \textit{M. Röckner} [J. Funct. Anal. 157, No. 1, 242-291 (1998; Zbl 0931.58019)] investigated the geometrical and analytical structures of a configuration space over a Riemannian manifold \(X\). The configuration space consists of all locally finite subsets of \(X\) and may be identified with the set of counting measures on \(X\). The authors proved that for a large class of potentials the canonical Gibbs measures can be completely characterized by an integration by parts formula. NEWLINENEWLINENEWLINEIn the paper under review the author uses results from the theory of Gibbsian point processes to sharpen the results obtained in the above mentioned paper. Based on the characterization of Gibbsian point processes by absolute continuity properties of the related so-called reduced Campbell measures the author is able to show the interesting result that all point processes with an integration by parts formula fulfil the condition \(\Sigma_\lambda^c\) introduced by \textit{A. Wakolbinger} and \textit{G. Eder} [Math. Nachr. 116, 209-232 (1984; Zbl 0583.60043)]. Consequently, Gibbs measures give rise to an integration by parts formula with respect to the canonical gradient of the conditional energy.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00049].