Variational Optimality of Follmer Processes in Generative Diffusions

arXiv:2602.10989v2 Announce Type: replace-cross 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 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 Follmer 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 Follmer processes, complementing classical formulations via Schrodinger 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. We provide numerical experiments to illustrate the impact of path-space variational optimality of Follmer's processes in probabilistic forecasting and data assimilation applications.

Source: arXiv cs.LG | 2026-05-19