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Short time solution to the master equation of a first order mean field game. (English) Zbl 1447.35331

Summary: The goal of this paper is to show existence of short-time classical solutions to the so called Master Equation of first order Mean Field Games, which can be thought of as the limit of the corresponding master equation of a stochastic mean field game as the individual noises approach zero. Despite being the equation of an idealistic model, its study is justified as a way of understanding mean field games in which the individual players’ randomness is negligible; in this sense it can be compared to the study of ideal fluids. We restrict ourselves to mean field games with smooth coefficients but do not impose any monotonicity conditions on the running and initial costs, and we do not require convexity of the Hamiltonian, thus extending the result of W. Gangbo and A. Święch [J. Differ. Equations 259, No. 11, 6573–6643 (2015; Zbl 1359.35221)] to a considerably broader class of Hamiltonians.

MSC:

35Q91 PDEs in connection with game theory, economics, social and behavioral sciences
35R06 PDEs with measure
35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
45K05 Integro-partial differential equations
49L99 Hamilton-Jacobi theories
49N70 Differential games and control
91A15 Stochastic games, stochastic differential games
91A23 Differential games (aspects of game theory)
91A07 Games with infinitely many players

Citations:

Zbl 1359.35221
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References:

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