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.

翻译：我们构建并分析了利用随机插值框架在有限时间范围内将点质量传输到指定目标分布的生成扩散过程。其漂移项被表示为条件期望，可通过独立样本进行估计而无需模拟随机过程。我们证明了扩散系数可在不改变时间边际分布的情况下进行事后调整。在所有此类调整中，我们证明最小化估计误差对路径空间Kullback-Leibler散度的影响，将选择闭式解形式的Föllmer过程——该扩散的路径测度相对于由插值方案唯一确定的参考过程实现了相对熵最小化。这为Föllmer过程提供了新的变分表征，补充了通过薛定谔桥与随机控制的经典表述，并给出了Föllmer漂移的条件期望表示，使其可通过数据实现免模拟估计。进一步地，我们证明在此最优扩散系数下，路径空间Kullback-Leibler散度变得与插值方案无关，使得不同方案在此变分意义上具有统计等价性。我们通过数值实验阐明了Föllmer过程路径空间变分最优性在概率预报与数据同化应用中的影响。