Spatial Poisson Processes

Abstract

Spatial Poisson process models are popular partly because of their mathematical tractability based on a simple and powerful calculus whose main ingredients, besides Campbell’s formula, are the covariance formula and the exponential formula (also know as Campbell’s second formula). These formulas extend immediately to marked Poisson processes with independent (or location dependent) IID marks as a consequence of the fact that such marked Poisson processes can be described as unmarked Poisson processes in a higher-dimensional space. This slight modification of point of view will be applied to the study of the point processes obtained by meas of elementary transformations of a basic Poisson process, such as thinning, translation or clustering. All the results concerning Poisson processes find straightforward extensions concerning Cox processes. The calculus of spatial marked Poisson or Cox processes will be applied to a particular type of stochastic geometry (featuring the Boolean model) and to the exact sampling of cluster point processes.