Volume 27, 2021
Special issue in honor of Enrique Zuazua's 60th birthday
|Number of page(s)||40|
|Published online||03 August 2021|
Long time behavior and turnpike solutions in mildly non-monotone mean field games
Dipartimento di Matematica, Università di Padova,
Via Trieste 63,
2 Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy.
* Corresponding author: email@example.com
Accepted: 3 July 2021
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.
Mathematics Subject Classification: 49J20 / 35B40
Key words: Mean field games / turnpike property / long time behavior / ergodic limit
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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