A distributional approach to multiple stochastic integrals and transformations of the Poisson measure. II
The authors develop an analysis on the space of configurations over the space \(G=S\times{\mathbb R}\), where \(S\) is a complete separable metric space. On this configuration space they consider the Poisson measure \(P\) with intensity measure \(\pi\times\gamma\), where \(\pi\) is a finite measure on \(S\) and \(\gamma\) is a \(\sigma\)-finite measure on \(\mathbb R\) with a singularity at the origin. Let \({\mathcal X}={\mathcal X}(G)\) be the space of locally finite configurations on a metric space \(G\). Let \(F\) be a measurable mapping \(F:{\mathcal X}\to{\mathcal X}\), and \(PF^{-1}\) be the corresponding transformation of the measure \(P\). This transformation is ``nonpoissonian''. This means that the ``new'' position of each point depends not only on the ``old'' position but also on other points of the configuration. The authors find conditions such that the measure \(PF^{-1}\) is absolutely continuous with respect to the measure \(P\) and get then the expression for the corresponding Radon-Nikodym derivative \(\frac{dPF^{-1}}{dP}\). For part I, cf. Acta Appl. Math. 94, No. 1, 1--19 (2006; Zbl 1107.28010).
scientific article; zbMATH DE number 5306232
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