Lipschitz regularity for viscous Hamilton-Jacobi equations with LpL^p terms

We provide Lipschitz regularity for solutions to viscous time-dependent Hamilton-Jacobi equations with right-hand side belonging to Lebesgue spaces. Our approach is based on a duality method, and relies on the analysis of the regularity of the gradient of solutions to a dual (Fokker-Planck) equation. Here, the regularizing effect is due to the non-degenerate diffusion and coercivity of the Hamiltonian in the gradient variable.Comment: 31 page

A policy iteration method for Mean Field Games

The policy iteration method is a classical algorithm for solving optimal control problems. In this paper, we introduce a policy iteration method for Mean Field Games systems, and we study the convergence of this procedure to a solution of the problem. We also introduce suitable discretizations to numerically solve both stationary and evolutive problems. We show the convergence of the policy iteration method for the discrete problem and we study the performance of the proposed algorithm on some examples in dimension one and two

Lipschitz regularity for viscous Hamilton-Jacobi equations with Lp terms

We provide Lipschitz regularity for solutions to viscous time-dependent Hamilton-Jacobi equations with right-hand side belonging to Lebesgue spaces. Our approach is based on a duality method, and relies on the analysis of the regularity of the gradient of solutions to a dual (Fokker-Planck) equation. Here, the regularizing effect is due to the non-degenerate diffusion and coercivity of the Hamiltonian in the gradient variable

Mean Field Games Systems under Displacement Monotonicity

In this note we prove the uniqueness of solutions to a class of mean field games systems subject to possibly degenerate individual noise. Our results hold true for arbitrary long time horizons and for general nonseparable Hamiltonians that satisfy a so-called displacement mono-tonicity condition. This monotonicity condition that we propose for nonseparable Hamiltonians is sharper and more general than the one proposed in the work [W. Gangbo et al., Ann. Probab., 50 (2022), pp. 2178-2217]. The displacement monotonicity assumptions imposed on the data actually provide not only uniqueness, but also the existence and regularity of the solutions. Our analysis uses elementary arguments and does not rely on the well-posedness of the corresponding master equations

On monotonicity conditions for mean field games

In this paper we propose two new monotonicity conditions that could serve as sufficient conditions for uniqueness of Nash equilibria in mean field games. In this study we aim for unconditional uniqueness that is independent of the length of the time horizon, the regularity of the starting distribution of the agents, or the regularization effect of a non-degenerate idiosyncratic noise. Through a rich class of simple examples we show that these new conditions are not only in dichotomy with each other, but also with the two widely studied monotonicity conditions in the literature, the Lasry–Lions monotonicity and displacement monotonicity conditions

On the existence and uniqueness of solutions to time-dependent fractional MFG

We establish existence and uniqueness of solutions to evolutive fractional mean field game systems with regularizing coupling for any order of the fractional Laplacian s 08 (0,1). The existence is addressed via the vanishing viscosity method. In particular, we prove that in the subcritical regime s > 1/2 the solution of the system is classical, while if s 64 1/2, we find a distributional energy solution. To this aim, we develop an appropriate functional setting based on parabolic Bessel potential spaces. We show uniqueness of solutions both under monotonicity conditions and for short time horizons