2021/01/25 by Marco Cirant, Cirant, Marco, Alessio Porretta +1
Economics, Econometrics and Finance · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2101.09965
openalex publication_date 2021/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 in (0,T) towards the system in (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.