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Journal Articles Nonlinear Differential Equations and Applications Year : 2022

## Second order local minimal-time Mean Field Games

Romain Ducasse
• Function : Author
• PersonId : 1023113
Guilherme Mazanti
Filippo Santambrogio
• Function : Author
• PersonId : 911311

#### Abstract

The paper considers a forward-backward system of parabolic PDEs arising in a Mean Field Game (MFG) model where every agent controls the drift of a trajectory subject to Brownian diffusion, trying to escape a given bounded domain $\Omega$ in minimal expected time. Agents are constrained by a bound on the drift depending on the density of other agents at their location. Existence for a finite time horizon $T$ is proven via a fixed point argument, but the natural setting for this problem is in infinite time horizon. Estimates are needed to treat the limit $T\to\infty$, and the asymptotic behavior of the solution obtained in this way is also studied. This passes through classical parabolic arguments and specific computations for MFGs. Both the Fokker--Planck equation on the density of agents and the Hamilton--Jacobi--Bellman equation on the value function display Dirichlet boundary conditions as a consequence of the fact that agents stop as soon as they reach $\partial\Omega$. The initial datum for the density is given, and the long-time limit of the value function is characterized as the solution of a stationary problem.

#### Domains

Mathematics [math] Analysis of PDEs [math.AP]

### Dates and versions

hal-02616965 , version 1 (25-05-2020)
hal-02616965 , version 2 (29-01-2021)
hal-02616965 , version 3 (25-01-2022)

### Identifiers

• HAL Id : hal-02616965 , version 3
• ARXIV :
• DOI :

### Cite

Romain Ducasse, Guilherme Mazanti, Filippo Santambrogio. Second order local minimal-time Mean Field Games. Nonlinear Differential Equations and Applications, 2022, 29, Paper No. 37, 32 pp. ⟨10.1007/s00030-022-00767-2⟩. ⟨hal-02616965v3⟩

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