Point processes with Papangelou conditional intensity: from the Skorohod integral to the Dirichlet form (Q2866534)

Consider a Riemannian manifold \(X\) endowed with a \(\sigma\)-finite diffuse Radon measure \(\sigma\), the set \(\Gamma_X\) of locally finite subsets (``configurations'') of \(X\), a measurable intensity \(\pi: X\times\Gamma_X\to [0,\infty]\), and a probability measure \(\mu\) on \(\Gamma_X\), such that NEWLINE\[NEWLINE\int_{\Gamma_X}\;\sum_{x\in\gamma} \varphi(x,\gamma\setminus\{x\})\mu(d\gamma)= \iint_{X\times\Gamma_X} \varphi(x,\gamma) \pi(x,\gamma)\mu(d\gamma)\,\sigma(dx)NEWLINE\]NEWLINE for any test-function \(\varphi\) on \(X\times\Gamma_X\).NEWLINENEWLINE Such \(\mu\) can for example be absolutely continuous with respect to a Poisson measure (point process) or to a Gibbsian measure. The basic example is that of a Poisson measure, for which \(\pi\equiv 1\).NEWLINENEWLINE This article extends previous articles by \textit{S. Albeverio} et al. [J. Funct. Anal. 154, No.~2, 444--500 (1998; Zbl 0914.58028)], \textit{Z.-M. Ma} and \textit{M. Röckner} [Osaka J. Math. 37, No.~2, 273--314 (2000; Zbl 0968.58028)] to the case of more general intensities \(\pi\) and measures \(\mu\).NEWLINENEWLINE Thus, following an essentially classical procedure, the author successively constructs the Dirichlet form associated to \(\mu\), computes its generator, and proves the existence and uniqueness of the corresponding diffusion process. Meanwhile, gradients, Skorohod integral and integration by parts formulas are considered and analyzed.NEWLINENEWLINE Then some point dynamics is shown to approach the diffusion process and, finally, examples of pairwise interaction point processes are presented.