Spatial stochastics develops mathematical methods for the analysis, statistics and simulation of random spatial phenomena. It has applications in physics, materials science, medicine and mobile telecommunications, for example. Basic models of spatial stochastics are random measures, random (e.g. Gaussian) fields, (geometric) point processes and random mosaics. Random point processes and measures are a focus of the research area. This involves, for example, invariance properties of the characteristics of stationary random measures on homogeneous or more general spaces that are subject to a group action. Stochastic geometry is the central focus of the research group. The focus of interest is the modeling and analysis of point processes of convex (and more general) sets and of geometrically defined random measures. Some members of the research group deal with convex and integral geometry, a very important mathematical base of stochastic geometry. Here, for example, curvature and support measures of compact sets, additive functionals (e.g. tensor evaluations) and the associated integral geometric formulas are investigated.

For further motivation and lectures on topics of the working group see under Teaching in the AG

The subject of stochastic geometry is the mathematical modeling and analysis of random spatial geometric structures. Basic examples of such structures are Voronoi mosaics generated by point processes, random systems of overlapping and non-overlapping convex bodies (Boolean models, systems of hard spheres, packings) or the excursion and level sets of Gaussian random fields. Stochastic geometry uses and develops a wide range of mathematical techniques from convex and integral geometry, probability theory, fractal geometry and geometric measure theory.

There are numerous interesting applications of stochastic geometry, e.g. in physics (physical properties of disordered systems), materials science (statistical modeling of microstructures), medicine (stereological analysis of slices of spatial fiber processes), astronomy (distribution of galaxies) and mobile telecommunications (hierarchical networks).

The research group in Karlsruhe is particularly involved in projects on random mosaics, Boolean models, curvature measures and their applications, random polytopes, contact distributions of random sets and geometric point processes.

Point processes and random measures are ubiquitous in modern probability theory and its applications. Examples are arrival and departure processes of stochastic networks, geometric point processes of particles and planes, but also the local time of Brownian motion and its inverse. One focus of our research is the interplay between invariant transports of random measures and Palm's measures. The underlying principles of stationary random measures are very general and can be applied to state spaces subject to the action of a group. Special subjects of investigation are the mass transport principle, mass stationarity of random measures and invariance properties of Brownian motion and other random processes and fields.

Another focus of our research is the Fock space representation of functionals of general Poisson processes based on iterated difference operators. This representation has interesting consequences for chaos decomposition and martingale theory of Poissonian functionals. Currently, we use the Fock space representation as a suitable approach to a general theory for the perturbation of Poissonian processes and its applications to Levy processes and continuous percolation.

Convex geometry and integral geometry are closely related to various mathematical fields such as functional analysis, optimization and discrete geometry, in particular to stochastic geometry. Classical integral geometry investigates integral means of geometrically relevant functionals with respect to the full group of motions. The stochastic modeling of current application problems motivates the investigation of more general group operations. This leads in particular to translational integral geometry. In the context of convex geometry, these developments are reflected in the investigation of new geometric functionals. A central topic of the working group is geometric and functional inequalities and their stability. Another focus is the analysis of inverse geometric problems. Information about a geometric object is often available in the form of an integral transformation, as projection or section information. An important task is therefore the determination and reconstruction of information about the underlying object. Geometric results are required in a number of investigations of the Stochastic Geometry group, in particular in the analysis of geometric point processes, random mosaics and random polytopes.