Some simple but challenging Markov processes

TL;DR

This paper introduces simple yet challenging Piecewise Deterministic Markov Processes, exploring their long-term behavior and mathematical properties, with applications in biology, networks, and chemistry, and relating them to open research problems.

Contribution

It provides concrete examples of PDMPs with rich mathematical structures and discusses their analysis using advanced techniques, connecting to current open problems.

Findings

Examples of PDMPs with complex long-term behavior

Application of coupling, spectral, PDE, and inequalities methods

Relation of simple models to open research questions

Abstract

In this note, we present few examples of Piecewise Deterministic Markov Processes and their long time behavior. They share two important features: they are related to concrete models (in biology, networks, chemistry,. . .) and they are mathematically rich. Their math-ematical study relies on coupling method, spectral decomposition, PDE technics, functional inequalities. We also relate these simple examples to recent and open problems.