Mean field games (MFGs) and mean field control (MFC) provide a mathemat-
ical framework for modeling large controlled interacting particle systems, with
applications ranging from financial mathematics to biological neural networks. In
the first part of this talk, I will give an accessible introduction to MFGs and MFC.
I will then show how the infinite-dimensional case arises naturally in applications,
highlight its specific analytical challenges, and provide an overview of my recent
results.
In the second part of the talk, I will detail how we address these challenges in
recent joint work with A. Święch, where we investigate an MFG posed in a Hilbert
space and driven by degenerate noise. We study the associated MFG system
consisting of a Hamilton–Jacobi–Bellman (HJB) equation for the value function
coupled with a nonlinear Fokker–Planck (FP) equation for the distribution of the
particles, both governed by a degenerate Kolmogorov operator. This degeneracy
introduces significant analytical challenges. In particular, the HJB equation is
treated in the viscosity sense, while the FP equation is interpreted in a suitable
weak formulation. Proving the uniqueness of solutions to the FP equation becomes
particularly delicate. Under appropriate structural assumptions, we establish the
well-posedness of the MFG system.