First-order kinetic mean-field games formally describe the Nash equilibria of deterministic differential games where agents control their acceleration, asymptotically in the limit as the number of agents tends to infinity. The known results for the well-posedness theory of mean field games with control on the acceleration assume either that the running and final costs are regularizing functionals of the density variable, or the presence of noise, i.e. a second-order system. In this talk we describe how to construct global in time weak solutions to a first order mean-field games system involving kinetic transport operators, where the costs are local (hence non-regularizing) functions of the density variable with polynomial growth. We show the uniqueness of these solutions on the support of the agent density. The heart of the analysis is to characterize solutions through two convex optimization problems in duality. We will introduce a notion of ‘reachable set’, built from the initial agent distribution, that allows us to work with initial measures with or without compact support. In this way we are able to obtain crucial estimates on minimizing sequences for merely bounded and continuous initial measures. These are then carefully combined with L1-type averaging lemmas from kinetic theory to obtain pre-compactness for the minimizing sequence. Finally, under stronger convexity and monotonicity assumptions on the data, we can prove higher-order Sobolev estimates of the solutions.
The talk is based on joint work with M. Griffin-Pickering (UCL).
This video was produced by the SITE Research Center at New York University, as part of their talk series.
