Conditions for zero-drift implying distributional match

Determine conditions on the anti-symmetric, kernelized drifting field V_{p,q}(x) used in Drifting Models under which the zero-drift equilibrium V_{p,q}(x)=0 for all x implies that the generated pushforward distribution q equals the target data distribution p. Specify assumptions on the interaction kernel, its normalization, and any non-degeneracy requirements that guarantee q=p when the drift vanishes, to establish theoretical identifiability of the equilibrium.

Background

Drifting Models define an anti-symmetric drifting field V_{p,q}(x) to evolve the generator’s pushforward distribution q toward the data distribution p during training, with the property that q=p implies V=0. Although the paper gives heuristics and sufficient conditions for specific kernelized constructions, a general converse—ensuring that vanishing drift forces distributional equality—remains theoretically unresolved.

Clarifying the conditions under which V→0 guarantees q→p is central to understanding the identifiability and convergence behavior of this training-time evolution paradigm, and would provide a theoretical foundation comparable to those established for diffusion- or flow-based models.