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SUMMARY:Variational Mean Field Games - Filippo Santambrogio (Université P
 aris Sud)
DTSTART:20170403T140000Z
DTEND:20170403T150000Z
UID:TALK69625@talks.cam.ac.uk
CONTACT:Mikaela Iacobelli
DESCRIPTION:I will give a brief introduction to the the emerging topic of 
 Mean Field Games\, introduced some years ago (by two groups at the same ti
 me: Huang\, Malhamé and Caines\, but also Lasry and Lions\, who then devo
 ted 6 subsequent editions of his course at Collège de France to its mathe
 matical analysis) as a model for the equilibrium of a population of agents
  each selecting his own optimal paths\, according to a criterion which inv
 olves the density of the other agents\, in the form of a congestion cost. 
 This gives rise to a coupled system of PDEs\, a continuity equation where 
 the density moves according to the gradient of a value function\, and a Ha
 milton-Jacobi equation solved by the value function\, where the density al
 so appears.\nI will mainly deal with the case where this equilibrium probl
 em may be seen as optimality conditions of a convex variational problem\, 
 and give the main results in this framework. In particular\, I will presen
 t some easy but recent regularity results\, as well as the connection with
  optimal transport theory. These results are also needed to make rigorous 
 the connection between optimality and equilibrium and allow for an interes
 ting extension where instead of penalizing higher densities we insert a de
 nsity constraint.\nGlobally\, the talk will be based on papers in collabor
 ations with P. Cardaliaguet\, A. Mészáros\, J.-D. Benamou\, G. Carlier\,
  A. Prosinski and H. Lavenant.
LOCATION:CMS\, MR13
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