Seminar Supervisor

Prof. Dr. Evgeny Spodarev

Seminar Advisor

Dr. Julian Grote

Date and Place

Preliminary discussion on Monday, October 22, 2018, 2-4pm, Helmholtzstraße 18, Room 1.20.

Depending on the number of participants the seminar may take place as a block seminar or in weekly meetings.

Prerequisites

The level of difficulty in this seminar is varying between the different topics. The audience is at least supposed to be familiar with basic probability. There are a few talks that require the lecture Stochastics II.

Intended Audience

Bachelor and Master Students majoring in any mathematical course of studies.

Content

The Poisson Process: 

In this seminar, we study the Poisson process, a core object in modern probability, focusing on recent developments in its stochastic analysis. We start by developing basic results and properties of the (Poisson) point process like the Mecke equation and the Mapping theorem. Then, we extend our remit to topics in stochastic geometry, which is concerned with mathematical models for random geometric structures. The Poisson process is fundamental to stochastic geometry, and the application areas discussed in this seminar comprise Voronoi tessellations, stable allocations, hyperplane processes, the Boolean model and Gilbert graphs. Our treatment requires a sound knowledge of measure-theoretic probability theory. However, specific knowledge of stochastic processes is not assumed. Since the focus is always on the probabilistic structure, technical issues of measure theory are kept in the background, whenever possible.

List of (possible) talks

1. Point processes: Fundamentals, Campbell's formula and distribution of a point process.

2. The Poisson point process: Denition, existence and Laplace functional.

3. The Mecke equation: The (multivariate) Mecke equation and factorial moment measures.

4. Mappings, markings and thinnings: The Mapping, Marking and Thinning Theorem.

5. Stationary point processes: The pair correlation function and a spatial ergodic theorem.

6. The Palm distribution: The Mecke-Slivnyak Theorem and an application to Voronoi tessellations.

7. The extra head problem and balanced allocations: The Point-Optimal Gale-Shapley algorithm

and allocations with large appetite.

8. Stable allocations: The Site-Optimal Gale-Shapley algorithm and uniqueness of stable allocations.

9. Poisson integrals: The Wiener-Ito integral and Poisson hyperplane processes.

10. Fockspace representation: Dierence operators, the Poincare inequality, chaos expansion.

11. Covariance identities: Mehler's formula and covariance identities.

12. Normal approximation: Stein's method and the second Poincare inequality.

13. The Boolean model and Gilbert graphs: Capacity functional, volume fraction and covering property, the Gilbert graph.

14. Normal approximation in the Boolean model: Central limit theorem for the volume and additive functionals.

Registration

To register for the seminar, please send an e-mail to julian.grote(at)uni-ulm.de before September 30, 2018.

Criteria to pass the seminar

Each student is supposed to give a talk and to attend the seminar on a regular basis. Those who give a (good) talk and attend the seminar regularly will pass the seminar. Talks will be held in German or in English.

Schedule

TBA

Literature:

[1] Chiu, S.N., Stoyan, D., Kendall,W.S. and Mecke, J.: Stochastic Geometry and its Applications. Wiley, Chichester (2013).

[2] Daley, D.J. and Vere-Jones, D.: An Introduction to the Theory of Point Processes. Volume I: Elementary Theory and Methods. Springer (2003)

[3] Daley, D.J. and Vere-Jones, D.: An Introduction to the Theory of Point Processes. Volume II: General Theory and Structure. Springer (2008).

[4] Kallenberg, O.: Foundations of Modern Probability. Springer (2003).

[5] Kingman, J.F.C.: Poisson Processes. Oxford University Press (1993).

[6] Last, G. and Penrose, M.: Lectures on the Poisson process. Cambridge University Press (2017).

[7] Reiss, R.D.: A Course on Point Processes. Springer (1993).

[8] Schmidt, V.: Raumliche Statistik. Lecture Notes (Vorlesungsunterlagen, Universität Ulm, Sommersemester 2011).