Splitting‐characterizations of the Papangelou process

Abstract

For point processes we establish a link between integration-by-parts-and splitting-formulas which can also be considered as integration-by-parts-formulas of a new type. First we characterize finite Papangelou processes in terms of their splitting kernels. The main part then consists in extending these results to the case of infinitely extended Papangelou and, in particular, Pólya and Gibbs processes. and denoted also by P ε z, . Here ε ∈ {−1, +1}. This class of processes has its origin in [26]. Finally the class of Gibbs processes G(V, ) is of fundamental importance. Such processes are, in general not uniquely, specified by some reference measure ∈ M and some abstract Boltzmann factor V (x, μ) by which one understands a non-negative, measurable function having the property V (x, ν + κ) = V (x, ν)V (x, κ), x ∈ X, ν, κ ∈ M ·· .

General thinning characterizations of distributions and point processes

“…Let T be the independent q-thinning matrix for some q ∈ (0, 1). The results of [12] show that if ν is a Poisson, binomial or negative binomial distribution, then also the rows of ϒ and therefore also those of Q are of the same distribution apart from a shift in case of Q. Moreover, if T and Q are fixed that way, they determine a unique distribution and the related splitting equations are equivalent to integration-by-parts formulas for the distributions.…”

“…Let g be a non-negative, measurable function. Successive application of the disintegra-tion (12) and change of the order of integration yields…”

“…The expectation on the right hand side requires in fact the knowledge of the distribution of N and the sampling rule; whereas the expectation on the left hand side requires the law of N * and the conditional law of N * given N * with the law of N being only implicit in the law of N * . Such a situation was discussed in the context of point processes with an independent sampling mechanism, see e.g [5,10,11,12]: Suppose that N is a point process realizing a possibly infinite number of points in some Polish space. Colour these points independently of each other with probability q red and with probability 1 − q blue.…”

“…[16]), which are given by an integration-by-parts formula. In [12], it was shown that a splitting equation of type (1) with the independent splitting is equivalent to an integration-by-parts formula, which is known to be equivalent to the Dobrushin-Lanford-Ruelle equations [13]. Thus, the static splitting mechanism is related to a reversibility condition for a spatial birth-and-death process.…”

Conditioned Point Processes with Application to Lévy Bridges

“…A generalisation of this equation in Q, where the measure ρ(dγ)∆ γ (dµ) is replaced by a more complicated one, is the subject of a recent study, see e.g. [12].…”