Spatial generalization of BMAPs with finite state space (Q1600576)

A stepped Markov process with the set of all finite integer-valued measures (point fields) on \(\mathbb R^d\) \((d\geq 2)\) as the state space is considered. The intensity of the sequence of jump times of the process is determined by some auxiliary Markov process with a finite state space. At each jump time a random finite number of batches of points is being added. These batches positions and values are distributed randomly in \(\mathbb R^d\) according to some given conditional transition probability. It can be proved that the \(K\)-vector of numbers of points hitting at the fixed measurable sets \((S_1,\dots,S_K)\) \((S_i\cap S_j=\emptyset\), \(\bigcup S_i=\mathbb R^d)\) varies like a stepped Markov process with the state space \(Z^K_+\), and the set of all such processes determines the initial Markov process. The distribution of the value of the initial process at a fixed time in terms of such \(K\)-vector processes is expressed in the form of an operator exponent. Formulae both for multi-dimensional generating functions of these distributions, and for the Laplace transformation in time of this generating function are derived.