We survey here some recent studies concerning what we call mean-field models by analogy with Statistical Mechanics and Physics. More precisely, we present three examples of our mean-field approach to modelling in Economics and Finance (or other related subjects...). Roughly speaking, we are concerned with situations that involve a very large number of “rational players” with a limited information (or visibility) on the “game”. Each player chooses his optimal strategy in view of the global (or macroscopic) informations that are available to him and that result from the actions of all players. In the three examples we mention here, we derive a mean-field problem which consists in nonlinear differential equations. These equations are of a new type and our main goal here is to study them and establish their links with various fields of Analysis. We show in particular that these nonlinear problems are essentially well-posed problems i.e., have unique solutions. In addition, we give various limiting cases, examples and possible extensions. And we mention many open problems.

Mean Field Games

Regularity for a more general class of quasilinear elliptic equations

: It is demonstrated the local C(1 + Alpha) nature of weak solutions of elliptic equations of the type (1.1) in the introduction under the degeneracy (or singularity) assumptions (A sub 1)-(A sub 3).

C1 + α local regularity of weak solutions of degenerate elliptic equations

Jeux à champ moyen. I – Le cas stationnaire

Jeux à champ moyen. II – Horizon fini et contrôle optimal

Introduction.- General Presentation of Mean Field Control Problems.- Discussion of the Mean Field game.- Discussion of the Mean Field Type Control.- Approximation of Nash Games with a large number of players.- Linear Quadratic Models.- Stationary Problems- Different Populations.- Nash differential games with Mean Field effect.

https://sri-uq.kaust.edu.sa/Documents/pdf%20seminars/Bensoussan_KAUST_Feb.14.pdf

Mean Field Games and Mean Field Type Control Theory

General Technical Results- General Regularity Results- Nonlinear Elliptic Systems Arising from Stochastic Games- Nonlinear Elliptic Systems Arising from Ergodic Control- Harmonic Mappings- Nonlinear Elliptic Systems Arising from the Theory of Semiconductors- Stationary Navier-Stokes Equations- Strongly Coupled Elliptic Systems- Dual Approach to Nonlinear Elliptic Systems- Nonlinear Elliptic Systems Arising from plasticity Theory

Regularity Results for Nonlinear Elliptic Systems and Applications

We deal with a system of partial differential equations describing a steady motion of an incompressible fluid with shear-dependent viscosity and present a new global existence result for $ p>\frac{2d}{d+2} $. Here p is the coercivity parameter of the nonlinear elliptic operator related to the stress tensor and d is the dimension of the space. Lipschitz test functions, a subtle splitting of the level sets of the maximal functions for the velocity gradients, and a decomposition of the pressure are incorporated to obtain almost everywhere convergence of the velocity gradients.

Jens Frehse

Papers

Mean Field Games and Mean Field Type Control Theory

Regularity Results for Nonlinear Elliptic Systems and Applications

On analysis of steady flows of fluids with shear-dependent viscosity based on the lipschitz truncation method ∗

An existence result for fluids with shear dependent viscosity — Steady flows

The Master equation in mean field theory