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arXiv · 1704.04014

Time-dependent focusing Mean-Field Games: the sub-critical case

Marco Cirant·Daniela Tonon
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Abstract

We consider time-dependent viscous Mean-Field Games systems in the case of local, decreasing and unbounded coupling. These systems arise in mean-field game theory, and describe Nash equilibria of games with a large number of agents aiming at aggregation. We prove the existence of weak solutions that are minimisers of an associated non-convex functional, by rephrasing the problem in a convex framework. Under additional assumptions involving the growth at infinity of the coupling, the Hamiltonian, and the space dimension, we show that such minimisers are indeed classical solutions by a blow-up argument and additional Sobolev regularity for the Fokker-Planck equation. We exhibit an example of non-uniqueness of solutions. Finally, by means of a contraction principle, we observe that classical solutions exist just by local regularity of the coupling if the time horizon is short.

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Marco Cirant, Daniela Tonon. 2017-04-13. Time-dependent focusing Mean-Field Games: the sub-critical case. https://arxiv.org/abs/1704.04014

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