Prof. Dr. Hendrik Weber, Angewandte Mathematik Münster: Institut für Analysis und Numerik / Institut für Mathematische Stochastik

The Research Training Group is dedicated to educating mathematicians in the field of complex random systems. It provides a strong platform for the development of both industrial and academic careers for its graduate students. The central theme is a mathematically rigorous understanding of how probabilistic systems, modelled on a microscopic level, behave effectively at a macroscopic scale. A quintessential example for this RTG lies in statistical mechanics, where systems comprising an astronomical number of particles, upwards of 10^{23}, can be accurately described by a handful of observables including temperature and entropy. Other examples come from stochastic homogenisation in material sciences, from the behaviour of training algorithms in machine learning, and from geometric discrete structures build from point processes or random graphs. The challenge to understand these phenomena with mathematical rigour has been and continues to be a source of exciting research in probability theory. Within this RTG we strive for macroscopic representations of such complex random systems. It is the main research focus of this RTG to advance (tools for) both qualitative and quantitative analyses of random complex systems using macroscopic/effective variables and to unveil deeper insights into the nature of these intricate mathematical constructs. We will employ a blend of tools from discrete to continuous probability including point processes, large deviations, stochastic analysis and stochastic approximation arguments. Importantly, the techniques that we will use and the underlying mathematical ideas are universal across projects coming from completely different origin. This particular facet stands as a cornerstone within the RTG, holding significant importance for the participating students. For our students to be able to exploit the synergies between the different projects, they will pass through a structured and rich qualification programme with several specialised courses, regular colloquia and seminars, working groups, and yearly retreats. Moreover, the PhD students will benefit from the lively mathematical community in Münster with a mentoring programme and several interaction and networking activities with other mathematicians and the local industry.

Semi-linear stochastic partial differential equations: global solutions’ behaviours
Partial differential equations are fundamental to describing processes in which one variable is dependent on two or more others – most situations in real life. Stochastic partial differential equations (SPDEs) describe physical systems subject to random effects. In the description of scaling limits of interacting particle systems and in quantum field theories analysis, the randomness is due to fluctuations related to noise terms on all length scales. The presence of a non-linear term can lead to divergencies. Funded by the European Research Council, the GE4SPDE project will describe the global behaviour of solutions of some of the most prominent examples of semi-linear SPDEs, building on the systematic treatment of the renormalisation procedure used to deal with these divergencies.