Variational Optimality of Föllmer Processes in Generative Diffusions

We construct and analyze generative diffusions that transport a point mass to a prescribed target distribution over a finite time horizon using the stochastic interpolant framework. The drift is expressed as a conditional expectation that can be estimated from independent samples without simulating stochastic processes. We show that the diffusion coefficient can be tuned \emph{a~posteriori} without changing the time-marginal distributions. Among all such tunings, we prove that minimizing the impact of estimation error on the path-space Kullback–Leibler divergence selects, in closed form, a Föllmer process – a diffusion whose path measure minimizes relative entropy with respect to a reference process determined by the interpolation schedules alone. This yields a new variational characterization of Föllmer processes, complementing classical formulations via Schrödinger bridges and stochastic control. We further establish that, under this optimal diffusion coefficient, the path-space Kullback–Leibler divergence becomes independent of the interpolation schedule, rendering different schedules statistically equivalent in this variational sense.

💡 Research Summary

The paper studies the problem of transporting a point mass at the origin to an arbitrary target distribution μ★ on ℝ^d using continuous‑time stochastic dynamics. The authors adopt the stochastic interpolant framework: they fix two smooth schedules β(t) and σ(t) satisfying β(0)=0, β(1)=1, σ(0)>0, σ(1)=0, β̇>0 and σ̇<0, and define the interpolant Iₜ = β(t)·x★ + σ(t)·Wₜ, where x★∼μ★ and Wₜ is a standard Wiener process independent of x★. By construction I₀=0 and I₁=x★, so the interpolant bridges the Dirac source and the target. Although Iₜ is not Markovian, Gyöngy’s mimicking theorem guarantees the existence of a Markov diffusion Xₜ with the same one‑time marginals.

The drift of the mimicking diffusion admits a closed‑form conditional‑expectation representation: bₜ(x) = E