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Deterministic Limit of Mean Field Games Associated with Nonlinear Markov Processes

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Abstract

The paper is concerned with the deterministic limit of mean field games with a nonlocal coupling. It is assumed that the dynamics of mean field games are given by nonlinear Markov processes. This type of games includes stochastic mean field games as well as mean field games with finite state space. We consider the limiting deterministic mean field game within the framework of minimax approach. The concept of minimax solutions is close to the probabilistic formulation. In this case the Hamilton–Jacobi equation is considered in the minimax/viscosity sense, whereas the flow of probabilities is determined by the probability on the set of solutions of the differential inclusion associated with the Hamilton–Jacobi equation such that those solutions are viable in the graph of the minimax solution. The main result of the paper is the convergence (up to subsequence) of the solutions of the mean field games to the minimax solution of a deterministic mean field game in the case when the underlying dynamics converge to the deterministic evolution.

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Acknowledgements

The author also would like to thank the anonymous referee for her/his valuable comments and suggestions.

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Authors and Affiliations

  1. Krasovskii Intitute of Mathematics and Mechanics UrB RAS, 16, S. Kovalevskaya, Yekaterinburg, Russia

    Yurii Averboukh

  2. Ural Federal University, 19, Mira, Yekaterinburg, Russia

    Yurii Averboukh

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  1. Yurii Averboukh
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Correspondence to Yurii Averboukh.

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Averboukh, Y. Deterministic Limit of Mean Field Games Associated with Nonlinear Markov Processes. Appl Math Optim 81, 711–738 (2020). https://doi.org/10.1007/s00245-018-9486-9

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