We introduce an abstract framework for the study of general mean field games and mean field control problems. Given a multiagent system, its macroscopic description is provided by a time-depending probability measure, where at every instant of time the measure of a set represents the fraction of (microscopic) agents contained in it. The trajectories available to each of the microscopic agents are affected also by the overall state of the system. By using a suitable concept of random lift of set valued maps, together with fixed point arguments, we are able to derive properties of the macroscopic description of the system from properties of the set valued map expressing the admissible trajectories for the microscopical agents. The techniques used can be applied to consider a broad class of dependence between the trajectories of the single agent and the state of the system. We apply the results in the case in which the admissible trajectories of the agents are the minimizers of a suitable integral functional depending also from the macroscopic evolution of the system.
Bibliographical noteKAUST Repository Item: Exported on 2023-08-31
Acknowledgements: Open access funding provided by Università degli Studi di Verona within the CRUI-CARE Agreement. Rossana Capuani has received funding from the Italian Ministry of University and Research according to D.M. 1062/2021 PON Ricerca e Innovazione 2014-2020, Asse IV Istruzione e ricerca per il recupero - Azione IV.4 - Dottorati e contratti di ricerca su tematiche dell’innovazione, Azione IV.6 Contratti di ricerca su tematiche Green. Antonio Marigonda has been partially supported by INdAM-GNAMPA Project 2023 GNAMPA CUP_E53C22001930001 entitled “Mean field games methods for mobility and sustainable development”.
ASJC Scopus subject areas
- Statistics and Probability
- Numerical Analysis
- Applied Mathematics
- Geometry and Topology