The Institut Louis Bachelier is committed, within the framework of agreements with its clients, to contribute to fundamental exploratory and collaborative research, with the objective that this research draws avenues of scientific work in the perspective of the creation of a new Institute for fundamental and applied research in mathematics oriented towards fields likely to advance mathematical knowledge having an impact in new information and communications technologies (NTIC).
Fundamental research on MFG methods for distributed high density systems, particularly found in the Internet of Things, naturally fits into this context.
The high density inter-communication of multi-antenna transmitters and receivers leads to many questions on the statistical properties of large random matrix systems, and on the decentralized optimization of these systems.
The approximation of large random matrices by random variables in the sense of free probabilities is in the same register as the mean fields in classical probability.
It is therefore natural to constitute mathematical theoretical tools which transpose the MFG concepts into this new non-commutative framework (large random matrices, free probabilities) with very probably the same impact on quantitative modeling as “classical” MFGs (for example). example for the applications to the intercommunication systems mentioned above).
More particularly, it is first of all desired that fundamental research make it possible to build a theory of the control of systems of equations of the Fokker Planck type describing the dynamics of the spectral measurement as the limit of random scattering of large matrices (for example for Dyson Brownians with Drift), which is the exploration of a difficult and vast mathematical question that is completely new.
The IEF as a Research Foundation, provides the appropriate and necessary framework for fundamental research with the objective of advancing scientific knowledge, and has the capacity and attractiveness necessary to mobilize advanced academic skills of the highest level. necessary for this fundamental research leading to scientific publications.
The ILB therefore wishes to entrust a research mission to the IEF on “the construction of a control theory for non-linear Fokker Planck-type PDE systems associated with semi-groups appearing in random dynamics of large matrices , in particular deformations of Dyson processes ”.
“The construction of a control theory for nonlinear Fokker Planck-type PDE systems associated with semigroups occurring in random dynamics of large matrices, including deformations of Dyson processes”; to ensure the scientific publication of the results obtained: two articles published in mathematics journals, and to send the ILB a research report on the research carried out, the results obtained, the possible consequences and the expected impact.
The IEF must therefore: