Variational Optimality of Föllmer Processes in Generative Diffusions
Yifan Chen, Eric Vanden-Eijnden
TL;DR
This work formulates a variational view of generative diffusions that transport a point mass to a target distribution by fixing time-marginal evolution via stochastic interpolants. By allowing a posteriori tuning of the diffusion coefficient, the authors derive a KL-based criterion whose minimizer yields a Föllmer process relative to a reference diffusion determined solely by the interpolation schedules. This yields a new variational characterization of Föllmer processes, linking them to Schrödinger bridges and stochastic control without fixing the reference dynamics a priori. A key result is schedule invariance: once the diffusion is optimally tuned, the path-space KL divergence depends only on score estimation accuracy across noise scales, making different interpolation schedules statistically equivalent in this variational sense. The framework provides a principled approach to design and analyze diffusion-based samplers for probabilistic forecasting and data assimilation, with simulation-free drift estimation and robust marginal matching.
Abstract
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.