203 research outputs found
Mean field games systems of first order
We consider a system of mean field games with local coupling in the
deterministic limit. Under general structure conditions on the Hamiltonian and
coupling, we prove existence and uniqueness of the weak solution,
characterizing this solution as the minimizer of some optimal control of
Hamilton-Jacobi and continuity equations. We also prove that this solution
converges in the long time average to the solution of the associated ergodic
problem
Second order mean field games with degenerate diffusion and local coupling
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
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