First order periodic Mean Field Games in Heisenberg group

Paola Mannucci (1) , Claudio Marchi (2, 1) , Nicoletta Tchou (3, 4)

1 Dipartimento di Matematica "Tullio Levi-Civita"
2 DEI - Department of Information Engineering [Padova]
3 UR - Université de Rennes
4 IRMAR - Institut de Recherche Mathématique de Rennes

In this paper we study evolutive first order Mean Field Games in the Heisenberg group H 1 ; in particular, the state of each agent can move only along "horizontal" trajectories which are given in terms of the vector fields generating H 1 while the kinetic part of the cost depends only on the horizontal velocity. In this case, the Hamiltonian is not coercive in the gradient term and the coefficient of the first order term in the continuity equation has a quadratic growth at infinity. The main results of this paper are two: the former is to establish the existence of a weak solution to the Mean Field Game system while the latter is to prove that this solution is a mild solution in the sense introduced by Cannarsa and Capuani [19] for state-constrained Mean Field Games. Roughly speaking, this property means that, for a.e. starting state, the agents follow optimal trajectories of the optimal control problem associated to the Hamilton-Jacobi equation given by the evolution of the population. In order to prove these results, we need some properties which have their own interest: some uniqueness results for a Fokker-Planck equation of the second order with respect to the vector fields and a probabilistic representation of the solution to the continuity equation.

Mots clés: Mean Field Games; first order Hamilton-Jacobi equations; continuity equa- tion; Fokker-Planck equation; noncoercive Hamiltonian; Heisenberg group; degenerate op- timal control problem 2010 AMS Subject classification: 35F50; 35Q91; 49K20; 49L25

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Paola Mannucci, Claudio Marchi, Nicoletta Tchou. First order periodic Mean Field Games in Heisenberg group. 2020. ⟨hal-02969837v1⟩