MEAN FIELD GAMES AND OPTIMAL TRANSPORT

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- Lecturers: Fabio Bagagiolo (University of Trento) and Antonio Marigonda (University of Verona)
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- Period: January 30 - April 3 2019
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- Venue: Trento and Verona
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- Hours: first five lessons in Trento from January 30 to February 27 on Wednesdays 14:00-17:00 in Seminar Room Mathematics (CHANGE: FIRST LESSON in room A220, 
10:00-13:00). Second 
five lessons from March 6 to 
April 3 on Wednesdays in Verona.  
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- CONTENTS: 
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In the Mean Field Games models (MFG), in the view of an optimization criterion, many (even infinitely many) agents individually take their decisions, 
being anyway influenced by the behavior of all other agents. Each single agent has only the perception of the average behavior of the others. This fact brings to 
a limit model given by two partial differential equations: one (Hamilton-Jacobi) for the optimal behavior of the single agent, the other one (transport, 
Fokker-Planck) for the evolution of the distribution of the population. MFG is a recent subject of research and may suitably describe: crowd/opinion/electrical 
grid dynamics, financial markets, and many others. Optimal Transport theory (OT) deals with the optimal transportation/allocation of resources, and was formalized 
firstly by G. Monge in 1781 and developed by L. Kantorovich in 1942-1948, More recently, optimal transport methods have earned an increasing importance, both from 
the point of view of the applications and from a theoretical point of view. From a modeling point of view, optimal transport theory is useful to model complex 
systems, where the number of particles is so large that only a statistical description is viable (as in statistical mechanics).
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- - MFG (Bagagiolo): Optimal control problems: Dynamic Programming and Hamilton-Jacobi equations. The deterministic and the stochastic cases. Equilibrium in 
non-cooperative games. Mean field games: from N players to infinitely many players. The mean field games system of PDEs. Switching mean field games and mean field 
games on networks.
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- - - M. Bardi & I. Capuzzo Dolcetta: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhauser, 1997
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- - - A. Bressan: Noncooperative Differential Games: a Tutorial (from the website of the author)
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- - - P. Cardaliaguet: Notes on Mean Field Games (from the website of the author)
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- - - L. C. Evans: Partial Differential Equations, AMS, 1998.
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- - OT (Marigonda): Review on measure theory; The optimal transport problem in the Monge's formulation; Relaxation of the optimal transport problem: Kantorovich's 
formulation; Kantorovich Duality and its consequences; Special costs: |x-y|, h(|x-y|) with h strictly convex.; Benamou-Brenier's Dynamical formulation of the 
optimal transport problem; The Wasserstein space and its differential structure; Applications.
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- - - L. Ambrosio, N. Gigli and G. Savare', Gradient flows in metric spaces and in the spaces of probability measures. Lectures in Mathematics, ETH Zurich, 
Birkhauser, 
2005
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- - - F. Santambrogio. Optimal Transport for Applied Mathematicians, Progress in Nonlinear Differential Equations and Their Applications, Vol. 87, Birkhauser 
Basel, 2015.
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- - - C. Villani. Topics in Optimal Transportation. Graduate Studies in Mathematics, AMS, 2003.