Spatial stochastics develops mathematical methods for the analysis, statistics and simulation of random spatial phenomena. It has, for example, applications in physics, materials science, medicine, or mobile telecommunications.

Basic models of spatial stochastics are random measures, random (e.g. Gaussian) fields, (geometric) point processes and random tessellations. An essential part of the research is concerned with random point processes and random measures. In this context, for instance, invariance properties of the characteristics of stationary random measures are explored on homogeneous or on more general spaces which are subject to a group action. Stochastic geometry constitutes a central part of the research area. The focus of interest is the modeling and analysis of distributional properties of point processes, of convex (and more general) sets as well as of geometrically defined measures.

Some members of the research unit also deal with convex and integral geometry which is a strong pillar of stochastic geometry. Here curvature and support measures of compact sets, additive functionals (such as tensor valuations) and related integral-geometric formulas are explored.

For some further motivation and information about lectures on topics of the research group, see Courses on Spatial Statistics.

Stochastic geometry is concerned with the mathematical modeling and the 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 or level sets of Gaussian random fields. Stochastic geometry uses and developes a wide range of mathematical techniques from convex and integral geometry, probability theory, fractal geometry and geometric measure theory.

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

The research group in Karlsruhe is particularly concerned with projects related to 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 all areas of modern probability and its applications. Examples are arrival and departure processes in stochastic (fluid) networks, geometric point processes of particles and flats, as well as local time of Brownian measures and its inverse. One focus of our research is on the interplay between balancing invariant transports of random measures and Palm measures. The underlying theory of stationarity is quite general and applies to random measures on a state space that is subject to a group operation. Specific topics of research 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 some interesting consequences for chaos expansion and martingale theory of Poisson functionals. Currently we are using the Fock space representation as convenient aproach to perturbation theory for Poisson processes and its applications to Levy processes and continuum percolation.

Convex and Integral Geometry are connected to various mathematical disciplines such as functional analysis, optimization and discrete geometry, and in particular to Stochastic Geometry. Classical Integral Geometry investigates integral averages of geometrically relevant functionals with respect to the full motion group. The stochastic modelling of problems coming from applications motivates the study of more general groups of transformations. In particular, this leads to translative integral geometry. In the context of Convex Geometry, these developments correspond to the investigation of new geometric functionals. A central topic in the research of this workgroup are geometric and functional inequalities and related stability results. Another key aspect is the analysis of inverse geometric problems. Information about a geometric object is often available only in the form of an integral transform, or in terms of information about projections or sections of the object. Therefore an important task is the determination and reconstruction of information about the original object itself. Geometric results are also required in investigations of the group in the context of Stochastic Geometry, in particular in the analysis of geometric point processes, random tessellations and random polytopes.