Variational Optimality of Föllmer Processes in Generative Diffusions

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, and provides a conditional-expectation representation of the Föllmer drift that enables simulation-free estimation from data. 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. We provide numerical experiments to illustrate the impact of path-space variational optimality of Föllmer's processes in probabilistic forecasting and data assimilation applications.