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Kengo Terai

Publications and source records attributed to Kengo Terai.

6 recordsLinked to original sources
arXiv2026

Remarks on selection problems for first-order discounted mean field games

We study selection problems for the vanishing discount limit of first-order stationary mean field games with local coupling. The associated ergodic problem may admit several value functions for the same density and ergodic constant. We decompose the state space into the positive-density region, the essential zero-density interior, and a residual set, and show that possible non-uniqueness of selected value functions can occur only on the gap part of the residual set. If the gradients of selected value functions coincide on this gap residual set, then the selected value function is unique up to additive constants; under compactness and stability assumptions, this yields convergence of the whole normalized discounted family. We show that the gap residual set is null for one-dimensional problems and for a specific Hamiltonian in the multidimensional setting, and hence obtain convergence results in these cases.

math.AP↗
arXiv2021

On weak solutions to first-order discount mean field games

In this paper, we establish the existence and uniqueness of weak solutions to first-order discount mean field games and a stability result to give the existence for the ergodic problem. We show an example to illustrate the multiplicity of weak solutions to the ergodic problem. With this motivation, we address a selection condition, which is a necessary condition that any limit of solutions under subsequence satisfies. As an application, we show a nontrivial example to get the convergence of weak solutions.

math.AP↗
arXiv2019

The selection problem for some first-order stationary mean-field games

Here, we study the existence and the convergence of solutions for the vanishing discount MFG problem with a quadratic Hamiltonian. We give conditions under which the discounted problem has a unique classical solution and prove convergence of the vanishing-discount limit to a unique solution up to constants. Then, we establish refined asymptotics for the limit. When those conditions do not hold, the limit problem may not have a unique solution and its solutions may not be smooth, as we illustrate in an elementary example. Finally, we investigate the stability of regular weak solutions and address the selection problem. Using ideas from Aubry-Mather theory, we establish a selection criterion for the limit.

math.AP↗
arXiv2019

Uniqueness structure of weakly coupled systems of ergodic problems of Hamilton-Jacobi equations

Here, we address a uniqueness structure of viscosity solutions for ergodic problems of weakly coupled Hamilton-Jacobi systems. In particular, we study comparison principle with respect to generalized Mather measures as a generalization of the result proved by Mitake and Tran, which addressed the case of a single equation. To get the main result, it is important to construct Mather measures effectively. We overcome this difficulty by nonlinear adjoint methods.

math.AP↗
arXiv2015

Existence of positive solutions for an approximation of stationary mean-field games

Here, we consider a regularized mean-field game model that features a low-order regularization. We prove the existence of solutions with positive density. To do so, we combine a priori estimates with the continuation method. In contrast with high-order regularizations, the low-order regularizations are easier to implement numerically. Moreover, our methods give a theoretical foundation for this approach.

math.AP↗