**PROBABILITY, ANALYSIS, AND APPLICATIONS (PAA) WORKSHOP**

# Abstracts

## Time-Warping Invariants and the Iterated-Sums Signature of a Time Series

*by Prof. Dr. Joscha Diehl*

*In Data Science, one is often confronted with a time series representing measurements of some quantity of interest. Usually, in a first step, features of the time series need to be extracted. These are numerical quantities that aim to succinctly describe the data. In many applications, these features are required to satisfy some invariance properties. In this lecture series, I focus on time-warping invariants, i.e. the invariance to running the time series at different speeds. The resulting features correspond to certain iterated sums of the increments of the time series. I present these invariant features in a (Hopf) algebraic framework and develop some of their basic properties.*

## An Introduction to Random Interlacements and related topics

*by Prof. Dr. Alexander Drewitz*

*In these lectures we will give a gentle introduction to the model of random interlacements that has been introduced by Sznitman [Szn10] in 2007. *

*The model has originally been motivated by questions about the disconnection of discrete cylinders and tori by the trace of simple random walk, as well as by related problems that have been investigated in the theoretical physics literature. Intuitively, random interlacements is a random subset, which appears as the limiting distribution of the trace of simple random walk on a large torus when it runs up to times proportional to the volume. It serves as a model for corrosion and in addition gives rise to interesting and challenging percolation problems through its vacant set (i.e., the complement of random interlacements). *

*We will start with giving a short review of basic potential theory and then motivate how the model appears as the limiting distribution of simple random walk on the torus. Next, to obtain a better feeling for random interlacements, we will compare it to the well-known model of Bernoulli percolation. We will then survey further important results and tools for the investigation of random interlacements, such as the non-trivial percolation phase transition for its vacant set as well as “decoupling inequalities,” of which stronger and stronger versions have been proven over the last couple of years. *

*If time admits we will sketch some more recent developments and connections of random interlacements to other fields such as Markovian loop soups, the Gaussian free field, and isomorphism theorems. *

*An accompanying textbook is [DRS14], a preliminary version of which is available at http://www.mi.uni-koeln.de/~drewitz/SpringerRI.pdf*

**References:**

[DRS14] Alexander Drewitz, Bal´azs R´ath, and Art¨em Sapozhnikov. An introduction to random interlacements. SpringerBriefs in Mathematics. Springer, Cham, 2014.

[Szn10] Alain-Sol Sznitman. Vacant set of random interlacements and percolation. Ann. of Math. (2), 171(3):2039–2087, 2010.

## Micro-macro phase transitions in coagulating particle systems

*by Prof. Dr. Wolfgang König*

*We consider two types of particle systems with an interaction under which particles of a given mass tend to connect to each other to form larger particles: a coagulation particle system and the Bose gas. The two mechanisms are a priori pretty different, but it turns out that the large-system analysis of the joint distribution of all the particle sizes at a given time gives rise to rather similar formulas. In particular, we will encounter similar, but different, phase transitions from entirely microscopic particles to the first appearance of a macroscopic particle. For the coagulation system, this phase transition is called a gelation transition, and for the Bose gas, it is the famous Bose-.Einstein condensation.*

## Local Large Deviation and Deviation Principles For The Signal -To-Noise and Interference Ratio Graph Models

*by Prof. Kwabena Doku-Amponsah*

## Mean Field Stochastic Differential Equations

**by Prof. Dr. Dirk Becherer**

*This lecture will focus on an introduction to the theory of Mean Field Stochastic Differential Equations and Applications”.*

## Continuum Percolation in Random Environment

**by Dr. Benedikt Jahnel**

*In this talk, I will first introduce the basic setting of continuum percolation for Poisson point processes. Motivated by applications in telecommunications, a refined modeling approach will then be discussed, which allows to consider random environments. Finally, I present sufficient conditions on the environment such that nontrivial sub- and supercritical regimes for percolation exist.*

## Skorokhod embedding solutions to the peacock problem.

**By Dr. Antoine-Marie Bogso**

*The peacock problem is to construct as explicitely as possible a martingale with the same one-dimensional marginals as a given peacock process. The existence of such a martingale is granted by the Kellerer’s celebrated theorem. One interesting approach to solve the peacock problem is to apply Skorokhhod embedding methods. In particular, Azema-Yor and Root Skorokhod embedding algorithms provides solution to the peacock problem in many cases. These solutions as well as some open questions are presented.*

**References**

[1] F. Hirs h, C. Profeta, B. Roynette, and M. Yor. Peacocks and associated martingales. Bocconi-Springer, vol 3, 2011.

[2] Hobson, D. The Skorokhod Embedding Problem and Model-Independent Bounds for Option Prices In Paris-Princeton Lectures on Mathematical Finance 2010, pp. 267-318. Springer, Berlin, Heidelberg, 2011.

[3] H. G. Kellerer. Markov-Komposition und eine Anwendung auf Martingale. Math. Ann., 198:99 122, 1972.

## Multi-population Potts model: Existence of free Energy and Phase Diagram

**by Dr. Alex Opoku**

*In this talk we will discuss multi-population Potts model. This model results from coupling together two or more Potts models. The motivation for studying such class of statistical mechanical models stems from their potential application to discrete choice with social interaction when there are more than two alternatives to choose from.*

*We will present results on the existence and variational formula for the energy of the model. Some results on the phase diagram of the model will also be discussed.*