Diogo A. Gomes

Effective Hamiltonians and Averaging¶for Hamiltonian Dynamics II

Abstract We extend to time-dependent Hamiltonians some of the PDE methods from our previous paper [E-G1], and in particular the theory of “effective Hamiltonians” introduced by Lions, Papanicolaou & Varadhan [L-P-V]. These PDE techniques augment the variational approach of Mather [Mt1,Mt2,Mt3,Mt4,M-F] and the weak KAM methods of Fathi [F1,F2,F3,F4,F5].We also provide a weak interpretation of adiabatic invariance of the action and suggest a formula for the Berry-Hannay geometric phase in terms of an effective Hamiltonian.

A stochastic analogue of Aubry-Mather theory

In this paper, we discuss a stochastic analogue of Aubry-Mather theory in which a deterministic control problem is replaced by a controlled diffusion. We prove the existence of a minimizing measure (Mather measure) and discuss its main properties using viscosity solutions of Hamilton-Jacobi equations. Then we prove regularity estimates on viscosity solutions of the Hamilton-Jacobi equation using the Mather measure. Finally, we apply these results to prove asymptotic estimates on the trajectories of controlled diffusions and study the convergence of Mather measures as the rate of diffusion vanishes.

Time-Dependent Mean-Field Games in the Subquadratic Case

In this paper we consider time-dependent mean-field games with subquadratic Hamiltonians and power-like local dependence on the measure. We establish existence of classical solutions under a certain set of conditions depending on both the growth of the Hamiltonian and the dimension. This is done by combining regularity estimates for the Hamilton-Jacobi equation based on the Gagliardo-Nirenberg interpolation inequality with polynomial estimates for the Fokker-Planck equation. This technique improves substantially the previous results on the regularity of time-dependent mean-field games.

Continuous Time Finite State Mean Field Games

In this paper we consider symmetric games where a large number of players can be in any one of d states. We derive a limiting mean field model and characterize its main properties. This mean field limit is a system of coupled ordinary differential equations with initial-terminal data. For this mean field problem we prove a trend to equilibrium theorem, that is convergence, in an appropriate limit, to stationary solutions. Then we study an N+1-player problem, which the mean field model attempts to approximate. Our main result is the convergence as N→∞ of the mean field model and an estimate of the rate of convergence. We end the paper with some further examples for potential mean field games.

Computing the Effective Hamiltonian Using a Variational Approach

A numerical method for homogenization of Hamilton-Jacobi equations is presented and implemented as an L∞calculus of variations problem. Solutions are found by solving a nonlinear convex optimization problem. The numerical method is shown to be convergent and error estimates are provided. Several examples are worked in detail, including the cases of non-strictly convex Hamiltonians and Hamiltonians for which the cell problem has no solution.

Time-Dependent Mean-Field Games with Logarithmic Nonlinearities

In this paper, we prove the existence of classical solutions for time-dependent mean-field games with a logarithmic nonlinearity and subquadratic Hamiltonians. Because the logarithm is unbounded from below, this nonlinearity poses substantial mathematical challenges that have not been addressed in the literature. Our result is proven by recurring to a delicate argument which combines Lipschitz regularity for the Hamilton--Jacobi equation with estimates for the nonlinearity in suitable Lebesgue spaces. Lipschitz estimates follow from an application of the nonlinear adjoint method. These are then combined with a priori bounds for solutions of the Fokker--Planck equation and a concavity argument for the nonlinearity.

A-priori estimates for stationary mean-field games

In this paper we establish a new class of a-priori estimates for stationary mean-field games which have a quasi-variational structure. In particular we prove

Mather measures selected by an approximation scheme

W^{1,2}

AUBRY-MATHER MEASURES IN THE NONCONVEX SETTING ∗

estimates for the value function

THE MATHER MEASURE AND A LARGE DEVIATION PRINCIPLE FOR THE ENTROPY PENALIZED METHOD

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