Abstract
We prove existence of solutions for a class of systems of subelliptic PDEs arising from mean field game systems with Hörmander diffusion. These results are motivated by the feedback synthesis mean field game solutions and the nash equilibria of a large class of N-player differential games.
Highlights
In this paper we consider a class of systems of degenerate elliptic PDEs of Hörmander type arising from certain ergodic differential games, from the mean field game (MFG) theory of Lasry and Lions [43,44,45]
Lasry and Lions have shown that for a large class of differential games the limiting model reduces to a Hamilton–Jacobi–Bellman equation for the optimal value function of the typical agent coupled with a continuity equation for the density of the typical optimal dynamic, the so-called mean field game equations
We show uniqueness for both the systems under standard monotonicity assumptions. Those results are applied to the feedback synthesis of MFG solutions and of Nash equilibria of a large class of N -player differential games
Summary
In this paper we consider a class of systems of degenerate elliptic PDEs of Hörmander type arising from certain ergodic differential games, from the mean field game (MFG) theory of Lasry and Lions [43,44,45] These systems have been introduced to model differential games with a large number of players or agents with dynamics described by controlled diffusion processes, under simplifying features such as homogeneity of the agents and a coupling of mean field type. Lasry and Lions have shown that for a large class of differential games (either deterministic or stochastic) the limiting model reduces to a Hamilton–Jacobi–Bellman equation for the optimal value function of the typical agent coupled with a continuity (or Fokker–Planck) equation for the density of the typical optimal dynamic, the so-called mean field game equations.
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