Uniform-in-time mean-field limit for the kinetic Schrödinger–Lohe equation

Establish a uniform-in-time mean-field limit between the finite \(N\)-oscillator Schrödinger–Lohe dynamics and the kinetic Schrödinger–Lohe equation, extending the finite-time convergence result proved for arbitrary initial data.

Background

The paper proves a quantitative $2$-Wasserstein fluctuation estimate of order N−1/2N^{-1/2} over every fixed finite time interval. The authors explicitly note that this estimate does not provide convergence uniformly for all times. A uniform-in-time result would require controlling the long-time growth of the stability constants or exploiting additional dissipative or synchronization structure, and would strengthen the mean-field theory substantially.

References

Moreover, note that the proposed mean-field limit is valid only in a finite-time interval for any initial data. So the extension of this finite-time mean-field limit to a uniform mean-field limit might be also an interesting open problem.

— The mean-field limit of the Schrödinger-Lohe model and emergent dynamics  (2609.03848 - Golse et al., 3 Sep 2026) in Section 7, Conclusion