# What are the courses Spatial Stochastics and Stochastic Geometry about?

Being familiar with the concept of a random variable (a random point in ) and a random vector (a random point in ), it is a natural question to ask how to define properly random versions of other geometric objects, e.g. a random line, a random chord in a circle, a random convex set, a random pattern of points, a randomly moved set etc., or even more complicated geometric structures. Such random geometric objects are a powerful tool to model complex structures in nature, science and engineering, in particular, in situations when their precise shape or geometry is unknown or subject to natural variations. It is then vital to understand the 'typical' behaviour of such structures, or their geometric properties 'on average'.

Some typical questions one may ask in practice: How can we estimate the amount of wood per hectare in a forest? What is the surface area of a catalytic converter? How can we estimate the volume of air in a metal foam and how can we maximize it while keeping a certain stability of the material? Complex structures like foams, cell tissues, soil or the fiber structure of paper often show a macroscopic homogeneity but strong local variations in its fine details. Therefore probabilistic models are a natural approach to describe and analyse such structures.

In the courses *Spatial Stochastics* and *Stochastic Geometry* several fundamental models for such random geometric structures are introduced and investigated. Often ** point processes**, which can be seen as random locally finite sets of points in some space, are the basis for such models. Figure 1 shows a realization of a so called ** Boolean Model**, where a random convex set is assigned to each point of the point process (here a disk with random radius).

In the second figure one can see a realization of a ** Voronoi tesselation**. To every point of the underlying point process a compact set (called cell) is associated consisting of all locations, whose distance to the point is smaller than the distance to every other point of the point process. The cells form a decomposition of the space.

A classical example of the difficulties that one faces in geometric modelling is ** Bertrand's paradox**. Here a random chord in the unit circle is considered, which is the intersection of the unit ball with a random line. We ask for the probability, that the length of this chord is at least . But what exactly is a random chord, i.e., how can it be modeled (and simulated)? As a chord is uniquely determined by its midpoint, one may, e.g., choose the midpoint uniformly from the unit circle. This gives probability 1/4. As a chord is also determined by its endpoints, one may alternatively choose two points randomly (uniformly) on the boundary of the circle, leading to probability 1/3. The question arises which model of a random chord is most reasonable? What are the resulting properties of a model? Each of the two images below shows 500 realizations of a random chord based on one of the two previously mentioned models. Observe the differences! Which model was used for which image?

The course *Spatial Stochastics* is more fundamental. It discusses in detail ** random closed sets**, **point processes** and **random measures** and studies their properties. Another important topic of the course are random functions (**random fields**), especially those, for which each value is normally distributed (**Gaussian fields**). Do such random fields always exist and how can they be characterized?

In the course *Stochastic Geometry*, random flats and randomly moved bodies are discussed, for example. Point processes and random measures are used to generate more complex geometric structures such as **germ-grain models** (for which the Boolean model is an example) and **random tesselations**. In addition, geometric properties of these models are studied employing geometric functionals such as **intrinsic volumes**. A very helpful tool provided in the course are formulas from the field of **integral geometry**. Knowledge of *Spatial Stochastics* is an advantage but is not assumed. All tools required from this course will be reviewed.

Occasionally additional courses on related topics are offered, e.g. *The Poisson process*, *Percolation* and *Convex Geometry*.