Variational Mean Field Games

Short course in the CIME school on Mean Field Games

This course, part of the CIME school on Mean Field Games held in Cetraro from June 10 to June 15, 2019, contains four lectures.

Program

  • 1) (10/6, afternoon) Introduction to the variational formulation of some MFG.
    Deterministic MFG with local penalization of the density. Eulerian and Lagrangian formulations of the optimization problem. Convex duality.
  • 2) (11/6, morning) Optimality conditions and Lagrangian equilibria; regularity via duality.
  • 3) (13/6, morning) Tools from optimal transport and further regularity for density-penalized MFG.
  • 4) (14/6, morning) Density-constrained Mean Field Games.

    References:

    A general reference for variational MFG is this survey
    The pathwise optimality conditions giving rise to Lagrangian equilibria (lecture 2) are described (in a more complicated framework) in Section 7 of this paper
    Regularity by duality (lecture 2) is detailed in this paper
    L∞ estimates for density-penalized MFG (lecture 3) are detailed in this paper
    Sobolev and L∞ estimates for pressure in density-constrained MFG (lecture 4) are detailed in this paper

    The lecture notes of this course will be based on these references.