Year: 2026
Type: preprint
Abstract: We study the terminal behavior of singular deterministic and stochastic dynamics driven by convex or extrapolative combinations of logarithmic heat gradients. These equations arise in exact-score guided diffusion sampling, including classifier-free guidance. The rescaled potentials converge to weighted sums of squared support distances, but extrapolative mixing can produce limiting directions that depend on probability masses and lie outside the Clarke subdifferential of the limiting potential. A componentwise chain rule gives dissipation for the limiting differential inclusion. Posterior relative entropy yields finite deterministic kinetic action in logarithmic time, while a conditional free-energy estimate bounds expected energy increments uniformly in the restart time. For compactly supported laws, these estimates give energy convergence and equilibrium localization, almost surely for each fixed stochastic initialization; extrapolation requires a lower small-ball mass bound on the reference law. For finite atomic measures, we prove that the equilibrium set is finite for arbitrary site configurations. Every deterministic trajectory therefore converges to an equilibrium, and stochastic convergence holds almost surely for each fixed initialization, including endpoints at projection interfaces. At empirical endpoints with a unique nearest atom in each active support, deterministic trajectories admit a square-root heat-time expansion with an exponentially small remainder. The exponent 1/2 is sharp when the leading coefficient is nonzero. These results provide a geometric and dynamical framework for understanding terminal convergence in guided diffusion models.
Source: arXiv (Cornell University)
Authors: Liu, Kang, Zuazua, Enrique
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7161083774