Random Measures, Point Processes, and Stochastic Geometry

This book is centered on the mathematical analysis of random structures embedded in the Euclidean space or more general topological spaces, with a main focus on random measures, point processes, and stochastic geometry. Such random structures have been known to play a key role in several branches of natural sciences (cosmology, ecology, cell biology) and engineering (material sciences, networks) for several decades. Their use is currently expanding to new fields like data sciences. The book was designed to help researchers finding a direct path from the basic definitions and properties of these mathematical objects to their use in new and concrete stochastic models. The theory part of the book is structured to be self-contained, with all proofs included, in particular on measurability questions, and at the same time comprehensive. In addition to the illustrative examples which one finds in all classical mathematical books, the document features sections on more elaborate examples which are referred to as models}in the book. Special care is taken to express these models, which stem from the natural sciences and engineering domains listed above, in clear and self-contained mathematical terms. This continuum from a comprehensive treatise on the theory of point processes and stochastic geometry to the collection of models that illustrate its representation power is probably the main originality of this book. The book contains two types of mathematical results: (1) structural results on stationary random measures and stochastic geometry objects, which do not rely on any parametric assumptions; (2) more computational results on the most important parametric classes of point processes, in particular Poisson or Determinantal point processes. These two types are used to structure the book. The material is organized as follows. Random measures and point processes are presented first, whereas stochastic geometry is discussed at the end of the book. For point processes and random measures, parametric models are discussed before non-parametric ones. For the stochastic geometry part, the objects as point processes are often considered in the space of random sets of the Euclidean space. Both general processes are discussed as, e.g., particle processes, and parametric ones like, e.g., Poisson Boolean models of Poisson hyperplane processes. We assume that the reader is acquainted with the basic results on measure and probability theories. We prove all technical auxiliary results when they are not easily available in the literature or when existing proofs appeared to us not sufficiently explicit. In all cases, the corresponding references will always be given.