Existence of weak solutions to time-dependent mean-field games

Ferreira, Rita; Gomes, Diogo A.; Tada, Teruo

doi:10.48550/arxiv.2001.03928

Cited by 3 publications

(3 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Now, using the concept of the weak solutions for the monotone MFGs developed in the series of papers [23,24,25,10], we define the weak solutions to (43).…”

Section: Convergencementioning

confidence: 99%

Discrete Approximation Of Stationary Mean Field Games

Bakaryan¹,

Gomes²,

Morgado³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we focus on stationary (ergodic) mean-field games (MFGs). These games arise in the study of the long-time behavior of finite-horizon MFGs. Motivated by a prior scheme for Hamilton-Jacobi equations introduced in Aubry-Mather's theory, we introduce a discrete approximation to stationary MFGs. Relying on Kakutani's fixed-point theorem, we prove the existence and uniqueness (up to additive constant) of solutions to the discrete problem. Moreover, we show that the solutions to the discrete problem converge, uniformly in the nonlocal case and weakly in the local case, to the classical solutions of the stationary problem.

show abstract

“…Now, using the concept of the weak solutions for the monotone MFGs developed in the series of papers [23,24,25,10], we define the weak solutions to (43).…”

Section: Convergencementioning

confidence: 99%

Discrete Approximation Of Stationary Mean Field Games

Bakaryan¹,

Gomes²,

Morgado³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…This notion mimics the notion of solutions to the weak variational inequality associated with a monotone operator (see [16]). This approach based on monotonicity for solving MFGs was introduced in [9] and further developed in [10,11].…”

Section: Weak Solutionsmentioning

confidence: 99%

A Potential Approach for Planning Mean-Field Games in One Dimension

Bakaryan¹,

Ferreira²,

Gomes³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

This manuscript discusses planning problems for first-and second-order onedimensional mean-field games (MFGs). These games are comprised of a Hamilton-Jacobi equation coupled with a Fokker-Planck equation. Applying Poincaré's Lemma to the Fokker-Planck equation, we deduce the existence of a potential. Rewriting the Hamilton-Jacobi equation in terms of the potential, we obtain a system of Euler-Lagrange equations for certain variational problems. Instead of the mean-field planning problem (MFP), we study this variational problem. By the direct method in the calculus of variations, we prove the existence and uniqueness of solutions to the variational problem. The variational approach has the advantage of eliminating the continuity equation.We also consider a first-order MFP with congestion. We prove that the congestion problem has a weak solution by introducing a potential and relying on the theory of variational inequalities. We end the paper by presenting an application to the onedimensional Hughes' model.

show abstract

“…Here, u represents the value function of a typical agent and m the distribution of the agents. Under mild condition on the problem data, the existence of weak solutions of (1.1)-(1.2) is addressed in [1] and in [3] using monotonicity methods. Regarding classical solutions, it is proved in [5,6,7,8] that (1.1)-(1.2) has a unique classical solution under suitable conditions on the problem data.…”

Section: Introductionmentioning

confidence: 99%

Uniform estimates for the planning problem with potential

Bakaryan,

Ferreira,

Gomes

2020

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we study a priori estimates for a first-order mean-field planning problem with a potential. In the theory of mean-field games (MFGs), a priori estimates play a crucial role to prove the existence of classical solutions. In particular, uniform bounds for the density of players' distribution and its inverse are of utmost importance.Here, we investigate a priori bounds for those quantities for a planning problem with a non-vanishing potential. The presence of a potential raises non-trivial difficulties, which we overcome by exploring a displacement-convexity property for the mean-field planning problem with a potential together with Moser's iteration method. We show that if the potential satisfies a certain smallness condition, then a displacement-convexity property holds. This property enables L q bounds for the density. In the one-dimensional case, the displacement-convexity property also gives L q bounds for the inverse of the density. Finally, using these L q estimates and Moser's iteration method, we obtain L ∞ estimates for the density of the distribution of the players and its inverse.

show abstract

Existence of weak solutions to time-dependent mean-field games

Cited by 3 publications

References 26 publications

Discrete Approximation Of Stationary Mean Field Games

Discrete Approximation Of Stationary Mean Field Games

A Potential Approach for Planning Mean-Field Games in One Dimension

Uniform estimates for the planning problem with potential

Contact Info

Product

Resources

About