Alessio Porretta

Long time average of mean field games

We consider a model of mean field games system defined on a time interval

Holder estimates for degenerate elliptic equations with coercive Hamiltonians

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Long time average of mean field games with a nonlocal coupling

and investigate its asymptotic behavior as the horizon

On the Planning Problem for the Mean Field Games System

T

Long Time versus Steady State Optimal Control

tends to infinity. We show that the system, rescaled in a suitable way, converges to a stationary ergodic mean field game. The convergence holds with exponential rate and relies on energy estimates and the Hamiltonian structure of the system.

Symmetry of large solutions of nonlinear elliptic equations in a ball

We prove a priori estimates and regularity results for some quasi-linear degenerate elliptic equations arising in optimal stochastic control problems. Our main results show that strong coerciveness of gradient terms forces bounded viscosity subsolutions to be globally Holder continuous, and solutions to be locally Lipschitz continuous. We also give an existence result for the associated Dirichlet problem.

Elliptic equations with vertical asymptotes in the nonlinear term

We study the long time average, as the time horizon tends to infinity, of the solution of a mean field game system with a nonlocal coupling. We show an exponential convergence to the solution of the associated stationary ergodic mean field game. Proofs rely on semiconcavity estimates and smoothing properties of the linearized system.

THE BOUNDARY BEHAVIOR OF BLOW-UP SOLUTIONS RELATED TO A STOCHASTIC CONTROL PROBLEM WITH STATE CONSTRAINT

We consider the planning problem for a class of mean field games, consisting in a coupled system of a Hamilton–Jacobi–Bellman equation for the value function u and a Fokker–Planck equation for the density m of the players, whereas one wishes to drive the density of players from the given initial configuration to a target one at time T through the optimal decisions of the agents. Assuming that the coupling F(x,m) in the cost criterion is monotone with respect to m, and that the Hamiltonian has some growth bounded below and above by quadratic functions, we prove the existence of a weak solution to the system with prescribed initial and terminal conditions m0, m1 (positive and smooth) for the density m. This is also a special case of an exact controllability result for the Fokker–Planck equation through some optimal transport field.

Separable p-harmonic functions in a cone and related quasilinear equations on manifolds

This paper analyzes the convergence of optimal control problems for an evolution equation in a finite time-horizon

Remarks on Long Time Versus Steady State Optimal Control

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