Global-in-Time Well-Posedness for Heterogeneous Mean Field Games

Establish global-in-time well-posedness for the heterogeneous mean field game system—defined by coupled HJB–FPK equations under Lipschitz continuity of drifts in state and mean-field arguments, bounded nondegenerate diffusion coefficients, and compact state space—thereby extending local well-posedness results from time horizons T ≤ δ0 to arbitrary time horizons T.

Background

The paper’s finite-horizon analysis relies on local well-posedness of the HMFG system, ensuring stability and bounded constants over short horizons. Some bounds inherit dependence on the time horizon T, and controlling these for arbitrary T would require global-in-time well-posedness.

The authors explicitly note that extending their results beyond the local horizon demands global well-posedness for HMFG, which remains largely open in the literature. Resolving this would provide theoretical guarantees for long-horizon HMFG models, including those with LEO backhaul dynamics.

References

For $T \leq \delta_0$ (local well-posedness horizon), $e{L_b T}$ remains bounded; extending to arbitrary $T$ requires global well-posedness (still largely open; cf.).

— Heterogeneous Mean Field Game Framework for LEO Satellite-Assisted V2X Networks  (2604.00621 - Sun et al., 1 Apr 2026) in Footnote to Theorem 1 (Error Decomposition), Section 4.3

Extending the population equilibrium construction to the infinite horizon would require substantially stronger a priori estimates and a compactness argument as T\longrightarrow+\infty. In particular, one would need estimates on the population distributions that are uniform with respect to the population horizon on every fixed compact time interval, together with appropriate stability properties for the value functions and the associated optimal feedbacks on increasingly long horizons. The discussion at the end of Section~\ref{sec:MFG-system} outlines a possible compactness approach based on a sequence of finite-horizon equilibria, but a complete global-in-time existence theory remains an open problem.

— Mean field game model for human capital accumulation under budget constraints  (2608.24417 - Arjmand, 25 Aug 2026) in Section 4, Discussion and economic interpretation; see also the concluding discussion of Section 3