Exploration for Large Interacting Systems of Agents

Introduction

This ERC project addresses mathematical theories and numerical tools for mean-field models, which are used to describe the statistical state of a population. Mathematically, such models are often seen as time-dependent systems whose state variable takes values in a space of probability measures.

Focus of the Project

One of the key features of the ERC project is its focus on mean-field models for which the state of the population is itself random. Primarily, we aim to consider dynamics with values in the space of probability measures that are subject to an external noise, which is often called a common noise.

Main Hypothesis

Our main hypothesis may be formulated as follows: It is possible to construct a common noise that would force mean-field dynamics to explore the space of probability measures in an efficient manner, very much in the spirit of a Brownian motion in the Euclidean setting.

Research Axes

This hypothesis will be studied along the following three axes:

1. Exploration is used for forcing uniqueness and stability of solutions to Fokker-Planck and related McKean-Vlasov equations and also to mean-field models of rational agents like mean-field control problems or mean-field games.
2. Exploration is used for the numerical analysis and the statistical learning of mean-field models featuring some optimisation criteria.
3. Exploration can be subject to a more irregular noise than a Brownian motion in the spirit of rough path theory.

Expected Applications

Expected applications are in probability theory, partial differential equations, control and game theory, optimisation, mathematical modelling, and numerical methods at the interface with machine learning.