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.

翻译：我们基于随机插值框架构建并分析了一类生成扩散过程，该过程能在有限时间范围内将点质量传输至预设目标分布。漂移项被表达为条件期望，可直接从独立样本中估计而无需模拟随机过程。我们证明扩散系数可在不改变时间边缘分布的前提下进行后验调节。在所有此类调节中，我们通过理论证明发现：最小化估计误差对路径空间Kullback-Leibler散度的影响，能以闭合形式选择Föllmer过程——该扩散过程的路径测度在相对熵意义上最小化与参考过程的差异，而参考过程完全由插值调度方案决定。这为Föllmer过程提供了新的变分表征，补充了通过薛定谔桥和随机控制的经典表述。我们进一步证明，在此最优扩散系数下，路径空间Kullback-Leibler散度将独立于插值调度方案，使得不同调度方案在此变分意义下具有统计等价性。