Second order mean field games with degenerate diffusion and local coupling

By Pierre Cardaliaguet, P. J. Graber, Alessio Porretta, Daniela Tonon. 2014

184 citations recorded by the source corpus.

We analyze a (possibly degenerate) second order mean field games system of partial differential equations. The distinguishing features of the model considered are (1) that it is not uniformly parabolic, including the first order case as a possibility, and (2) the coupling is a local operator on the density. As a result we look for weak, not smooth, solutions. Our main result is the existence and uniqueness of suitably defined weak solutions, which are characterized as minimizers of two optimal control problems. We also show that such solutions are stable with respect to the data, so that in particular the degenerate case can be approximated by a uniformly parabolic (viscous) perturbation.

Subfields: Applied Mathematics, Finance, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/8534be619aaed2bef69afebf58fefa9516ea7d42

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Citations in the corpus

Regularity for second order stationary mean-field games (2015, background): “Well-posedness in the class of weak solutions have been recently investigated in [24] and [4].”
Stable solutions in potential mean field game systems (2016, background): “We can then follow the standard arguments for convex functionals (see for instance [6]): the problem of minimizing J̃ on E is the dual problem (in the sense of the Fenchel-Rockafellar duality theorem) of the problem inf u∈C(2) {”
Time dependent mean-field games in the superquadratic case (2013, background): “We observe that for this choice of g, our problems admit a variational structure that was explored in [12] in order to construct weak solutions.”
Mean field games with congestion (2017, background): “We recall a lemma, whose proof can be found in [10, 3], which establishes continuity for subsolutions in a weaker sense.”
Linear Quadratic Mean Field Type Control and Mean Field Games with Common Noise, with Application to Production of an Exhaustible Resource (2016, background): “Many researchers have used this insight to generate results concerning existence and uniqueness [15, 17, 18, 53, 19] as well as computation of mean field Nash equilibria [7].”
Some remarks on mean field games (2018, background): “This approach is particularly successful to find weak solutions of (1) [11].”
Optimal stopping in mean field games, an obstacle problem approach (2017, background): “This approach was presented in [19] and has been well studied since (see [3,6,8] for example).”
Wellposedness of Second Order Master Equations for Mean Field Games with Nonsmooth Data (2019, unknown): “3) in Porretta [61] and Cardaliaguet-Graber-Porretta-Tonon [17].”
Existence and Uniqueness of Solutions for Bertrand and Cournot Mean Field Games (2015, background): “There have been a number of existence and uniqueness theorems proved when V [m] depends on m both locally [23, 5, 4, 8, 7, 14, 15] and non-locally [6].”