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Kinetic‐type mean field games with non‐separable local Hamiltonians
Journal article   Peer reviewed

Kinetic‐type mean field games with non‐separable local Hamiltonians

David M. Ambrose, Megan Griffin-Pickering and Alpár R. Mészáros
Journal of the London Mathematical Society, v 111(6), e70202
Jun 2025
DOI: https://doi.org/10.1112/jlms.70202
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https://doi.org/10.1112/jlms.70202View
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Abstract

We prove well‐posedness of a class of kinetic‐type mean field games (MFGs), which typically arise when agents control their acceleration. Such systems include independent variables representing the spatial position as well as velocity. We consider non‐separable Hamiltonians without any structural conditions, which depend locally on the density variable. Our analysis is based on two main ingredients: an energy method for the forward–backward system in Sobolev spaces, on the one hand, and on a suitable vector field method to control derivatives with respect to the velocity variable, on the other hand. The careful combination of these two techniques reveals interesting phenomena applicable for MFGs involving general classes of drift‐diffusion operators and non‐linearities. While many prior existence theories for general MFGs systems take the final datum function to be smoothing, we can allow this function to be non‐smoothing, that is, also depending locally on the final measure. Our well‐posedness results hold under an appropriate smallness condition, assumed jointly on the data.

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