Weak solutions to Fokker-Planck equations and Mean Field Games

Weak solutions to Fokker-Planck equations and Mean Field Games
Alessio Porretta
Università di Roma Tor Vergata, Roma, Italia
Mean Field Games theory, developed by J.-M. Lasry and P.-L. Lions ([1], [2]), describes - in a mean field continuum
limit - the behavior of large number of identical agents, each one controlling a dynamics and taking decisions in terms
of the density distribution of co-agents. At equilibrium, the model is described by a coupled system of a Bellman
equation (describing the optimization made by the generic player) and a Fokker-Planck equation, describing the
evolution of the density of players. The simplest model is
⎧
⎪
⎪−u t − ∆u + H(x, Du) = F(x, m)
⎨
⎪
m − ∆m − div(mH p (x, Du)) = 0
⎪
⎩ t
in (0, T) × Ω
in (0, T) × Ω ,
(1)
usually complemented with suitable boundary and initial/terminal conditions.
We discuss the uniqueness of weak solutions to (1), which requires new results on Fokker-Planck equations with
L 2 drifts as well as the use of renormalized solutions, a fundamental tool so much appreciated by Dominique and
deeply investigated in his research. Results to appear in [3], [4].
References
[1] J.-M. Lasry and P.-L. Lions. Jeux à champ moyen. I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris, 343 (2006),
no. 9, 619–625.
[2] J.-M. Lasry and P.-L. Lions. Jeux à champ moyen. II. Horizon fini et contròle optimal. C. R. Math. Acad. Sci.
Paris, 343 (2006), no. 10, 679–684.
[3] A. Porretta. On the planning problem for a class of Mean Field Games. C. R. Math. Acad. Sci. Paris, to appear.
[4] A. Porretta. Weak solutions to Fokker-Planck equations and Mean Field Games, in preparation.
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