Poisson processes: Stochastic Analysis, Malliavin-Stein Method, and Stochastic Geometry

Poisson processes are an important class of stochastic processes featuring in many branches of probability theory. They play a central role in stochastic geometry, where one is often interested in random geometric structures constructed from underlying Poisson processes. Examples include random tessellations, geometric random graphs, Boolean models, and random polytopes.

At the beginning of this course, Poisson processes and their basic properties as well as some models from stochastic geometry are introduced. Thereafter, the Malliavin calculus for Poisson functionals (i.e. random variables depending on an underlying Poisson process) is considered. Combining Malliavin calculus with Stein’s method, one obtains general bounds for the normal approximation of Poisson functionals, in particular second order Poincaré inequalities. These results are applied to several examples from stochastic geometry. Moreover, central limit theorems for so-called stabilising functionals are derived. Finally, several examples for Poisson and Poisson process approximation are discussed.

In this course, a general theory for Poisson functionals is developed on the one hand. On the other hand, these results are applied to models from stochastic geometry. Apart from a solid background in probability theory, no pre-knowledge is assumed.