Poisson cluster measures: Quasi-invariance, integration by parts and equilibrium stochastic dynamics

Abstract

The distribution μ cl of a Poisson cluster process in X = R d (with i.i.d. clusters) is studied via an auxiliary Poisson measure on the space of configurations in X = n X n , with intensity measure defined as a convolution of the background intensity of cluster centres and the probability distribution of a generic cluster. We show that the measure μ cl is quasi-invariant with respect to the group of compactly supported diffeomorphisms of X and prove an integration-by-parts formula for μ cl . The corresponding equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms.

Gibbs cluster measures on configuration spaces

“…However, developing the differential analysis on configuration spaces in the spirit of [2,3] demands that the measure μ cl be supported on the proper configuration space Γ X ; conditions for the latter to be true are also of general interest. We shall address this issue in Section 2.4 below for the Gibbs CPPs (see [9] for the case of the Poisson CPPs).…”

“…In our earlier papers [8,9], a similar analysis was developed for a different class of random spatial structures, namely Poisson cluster point processes, featured by spatial grouping ("clustering") of points around the background random (Poisson) configuration of invisible "centers". Cluster models are well known in the general theory of random point processes [12,13] and are widely used in numerous applications ranging from neurophysiology (nerve impulses) and ecology (spatial aggregation of species) to seismology (earthquakes) and cosmology (constellations and galaxies); see [9,12,13] for some references to original papers.…”

“…Cluster models are well known in the general theory of random point processes [12,13] and are widely used in numerous applications ranging from neurophysiology (nerve impulses) and ecology (spatial aggregation of species) to seismology (earthquakes) and cosmology (constellations and galaxies); see [9,12,13] for some references to original papers.…”

“…Our technique in [8,9] was based on the representation of a given Poisson cluster measure on the configuration space Γ X as the projection image of an auxiliary Poisson measure on a more complex configuration space Γ X over the disjoint-union space X := n X n , with "droplet" pointsȳ ∈ X representing individual clusters (of variable size). The principal advantage of this construction is that it allows one to apply the well-developed apparatus of Poisson measures to the study of the Poisson cluster measure.…”

Push-Forward Measures on Configuration Spaces: Integration by Parts and Log-Sobolev Inequality

“…[2,3] and references given there. Another important class of measures-the cluster point processes in X -have been considered from this point of view in [4][5][6]. For general definitions and notions of the point processes theory see e.g.…”