We consider mean field game systems in time-horizon (0, T), where the individual cost functional depends locally on the density distribution of the agents, and the Hamiltonian is locally uniformly convex. We show that, even if the coupling cost functions are mildly non-monotone, then the system is still well posed due to the effect of individual noise. The rate of anti-monotonicity (i.e. the aggregation rate of the cost functions) which can be afforded depends on the intensity of the diffusion and on global bounds of solutions. We give applications to either the case of globally Lipschitz Hamiltonians or the case of quadratic Hamiltonians and couplings having mild growth. Under similar conditions, we investigate the long time behavior of solutions and we give a complete description of the ergodic and long term properties of the system. In particular we prove: (i) the turnpike property of solutions in the finite (long) horizon (0, T), (ii) the convergence of the system from (0, T) towards (0, ∞), (iii) the vanishing discount limit of the infinite horizon problem and the long time convergence towards the ergodic stationary solution. This way we extend previous results which were known only for the case of monotone and smoothing couplings; our approach is self-contained and does not need the use of the linearized system or of the master equation.
Keywords: Mean field games, turnpike property, long time behavior, ergodic limit
@article{COCV_2021__27_1_A88_0,
author = {Cirant, Marco and Porretta, Alessio},
editor = {Buttazzo, G. and Casas, E. and de Teresa, L. and Glowinski, R. and Leugering, G. and Tr\'elat, E. and Zhang, X.},
title = {Long time behavior and turnpike solutions in mildly non-monotone mean field games},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021077},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021077/}
}
TY - JOUR AU - Cirant, Marco AU - Porretta, Alessio ED - Buttazzo, G. ED - Casas, E. ED - de Teresa, L. ED - Glowinski, R. ED - Leugering, G. ED - Trélat, E. ED - Zhang, X. TI - Long time behavior and turnpike solutions in mildly non-monotone mean field games JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021077/ DO - 10.1051/cocv/2021077 LA - en ID - COCV_2021__27_1_A88_0 ER -
%0 Journal Article %A Cirant, Marco %A Porretta, Alessio %E Buttazzo, G. %E Casas, E. %E de Teresa, L. %E Glowinski, R. %E Leugering, G. %E Trélat, E. %E Zhang, X. %T Long time behavior and turnpike solutions in mildly non-monotone mean field games %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021077/ %R 10.1051/cocv/2021077 %G en %F COCV_2021__27_1_A88_0
Cirant, Marco; Porretta, Alessio. Long time behavior and turnpike solutions in mildly non-monotone mean field games. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 86. doi: 10.1051/cocv/2021077
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