Mean field games systems of first order

Cardaliaguet, Pierre; Graber, P. Jameson

doi:10.1051/cocv/2014044

Cited by 107 publications

(168 citation statements)

References 22 publications

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“…The first one has to do with the regularity of the value function

ϕ

. Indeed, as understood in the works by Cardaliaguet and collaborators , a solution of a Hamilton–Jacobi equation exhibits regularity as soon as the right‐hand side is bounded from below and its positive part lies in

L^{1 + d / 2 + ε}

. In the aforementioned articles, such an assumption on the right‐hand side was obtained by assuming a moderate growth on the penalization of congestion in the primal problem.…”

Section: Introductionmentioning

confidence: 99%

New estimates on the regularity of the pressure in density‐constrained mean field games

Lavenant

Santambrogio

2019

J. London Math. Soc.

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We consider variational mean field games (MFGs) endowed with a constraint on the maximal density of the distribution of players. Minimizers of the variational formulation are equilibria for a game where both the running cost and the final cost of each player are augmented by a pressure effect, that is, a positive cost concentrated on the set where the density saturates the constraint. Yet, this pressure is a priori only a measure and regularity is needed to give a precise meaning to its integral on trajectories. We improve, in the limited case where the Hamiltonian is quadratic, which allows to use optimal transport techniques after time‐discretization, the results obtained in a paper of the second author with Cardaliaguet and Mészáros. We prove H1 and L∞ regularity under very mild assumptions on the data, and explain the consequences for the MFG, in terms of the value function and of the Lagrangian equilibrium formulation.

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“…The first one has to do with the regularity of the value function

ϕ

L^{1 + d / 2 + ε}

. In the aforementioned articles, such an assumption on the right‐hand side was obtained by assuming a moderate growth on the penalization of congestion in the primal problem.…”

Section: Introductionmentioning

confidence: 99%

New estimates on the regularity of the pressure in density‐constrained mean field games

Lavenant

Santambrogio

2019

J. London Math. Soc.

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“…At this point, let us remark that in our existence proof for the dual problem, we require a joint condition on the order of growth of H in the momentum variable and the order of growth of f in the second variable (similarly as in [Car15,CG15]). This is mainly due to the lack of enough summability on m. It is worth mentioning that L ∞ estimates on m would allow us to drop this joint condition on H and f .…”

mentioning

confidence: 99%

“…Moreover, in contrast to [Car15,CG15] and [CGPT15], we also address questions of regularity of weak solutions, based on techniques developed in the recent work [GM18] by the first two authors. The inspiration for these results comes from the alternative interpretation of the planning problem in terms of optimal mass transport.…”

mentioning

confidence: 99%

The planning problem in mean field games as regularized mass transport

Graber

Mészáros²,

Silva

et al. 2019

Calc. Var.

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In this paper, using variational approaches, we investigate the first order planning problem arising in the theory of mean field games. We show the existence and uniqueness of weak solutions of the problem in the case of a large class of Hamiltonians with arbitrary superlinear order of growth at infinity and local coupling functions. We require the initial and final measures to be merely summable. At the same time (relying on the techniques developed recently in [GM18]), under stronger monotonicity and convexity conditions on the data, we obtain Sobolev estimates on the solutions of the planning problem both for space and time derivatives.The data consist of probability measures m 0 , m T ∈ P(T d ), a fixed time horizon T > 0, a coupling function f : T d × [0, +∞) → R and a Hamiltonian H :Our aim is to find conditions on the data for which weak solutions to (1.1) can be shown to exist and are unique.

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“…If one extends the problem min A to a larger class of admissible functions, then the notion of solution that we find should be weakened. We refer the reader to [6] for a survey about these notions, or to [10,12] for the original papers. In particular we underline that the solutions to the MFG system would involve a BV function u, and the HJ equation would become an inequality in the region {m = 0}.…”

Section: Preliminary Results On Dualitymentioning

confidence: 99%

Global-in-time regularity via duality for congestion-penalized Mean Field Games

Prosinski

Santambrogio

2017

Stochastics

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After a brief introduction to one of the most typical problems in Mean Field Games, the congestion case (where agents pay a cost depending on the density of the regions they visit), and to its variational structure, we consider the question of the regularity of the optimal solutions. A duality argument, used for the first time in a paper by Y. Brenier on incompressible fluid mechanics, and recently applied to MFG with density constraints, allows to easily get some Sobolev regularity, locally in space and time. In the paper we prove that a careful analysis of the behaviour close to the final time allows to extend the same result including t = T .

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Mean field games systems of first order

Cited by 107 publications

References 22 publications

New estimates on the regularity of the pressure in density‐constrained mean field games

New estimates on the regularity of the pressure in density‐constrained mean field games

The planning problem in mean field games as regularized mass transport

Global-in-time regularity via duality for congestion-penalized Mean Field Games

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