Uniform-in-time regularity on Wasserstein space for infinite-horizon analysis

Establish uniform-in-time regularity for the corresponding Fokker–Planck partial differential equation on the Wasserstein space (W2, P2) to extend finite-time quantitative results for the dynamical interacting particle system to the infinite-horizon setting.

Background

The authors discuss an interacting particle system approximation of the mean-field Langevin dynamics and note that a version of their approximation result can be proven in this dynamical setup for finite horizons, citing existing work.

They state that extending these results to infinite horizons necessitates uniform-in-time regularity of the associated PDE on Wasserstein space. This requirement is not established in the paper and is explicitly left for future research.