Abstract
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.
Highlights
The theory of mean field games was initiated by Lasry and Lions since 2006 [24,25,26] in order to describe Nash equilibrium configurations in multi-agents strategic interactions
When coming at mean field game systems as in (1.1), the long time behavior was investigated in several papers under the assumption that the cost functions F, G are nondecreasing in m
Let us stress that, in the aforementioned results, the admissible threshold γ of anti-monotonicity depends on the diffusivity constant κ and on the initial datum. This latter fact explains very well why the master equation is hardly usable, in this context; there is no general feedback policy of u in the space of probability measures
Summary
The theory of mean field games was initiated by Lasry and Lions since 2006 [24,25,26] in order to describe Nash equilibrium configurations in multi-agents strategic interactions. When coming at mean field game systems as in (1.1), the long time behavior was investigated in several papers under the assumption that the cost functions F, G are nondecreasing in m It is well established in the theory that this monotonicity condition gives uniqueness and stability of solutions; under the same condition the turnpike property and the convergence of solutions towards the stationary ergodic state have been first proved in [3, 4] for quadratic Hamiltonian (i.e. H(x, p) = |p|2). Let us point out that mis the unique invariant measure of problem (1.3); in this problem μ is uniquely determined (while v is unique up to addition of constants) and satisfies μ(t, x) t→→∞ m (x) uniformly in Td. To conclude, we recall that the results described in the above items and contained in Sections 4–6 were previously proved in [5] for the case of smoothing and monotone couplings F, G, using the long time convergence of the master equation. Let us point out that even the case of non local mildly non-monotone couplings could be dealt with in a similar way but we did not pursue this extension here for the sake of simplicity
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