We study dynamic measure transport for generative modelling in the setting of a stochastic process $X_\bullet$ whose marginals interpolate between a source distribution $P_0$ and a target distribution $P_1$ while remaining independent, i.e., when $(X_0,X_1)\sim P_0\otimes P_1$. Conditional expectations of this process $X_\bullet$ define an ODE whose flow map transports from $P_0$ to $P_1$. We discuss when such a process induces a \emph{straight-line flow}, namely one whose pointwise acceleration vanishes and is therefore exactly integrable by any first-order method. We first develop multiple characterizations of straightness in terms of PDEs involving the conditional statistics of the process. Then, we prove that straightness under endpoint independence exhibits a sharp dichotomy. On one hand, we construct explicit, computable straight-line processes for arbitrary Gaussian endpoints. On the other hand, we show straight-line processes do not exist for targets with sufficiently well-separated modes. We demonstrate this through a sequence of increasingly general impossibility theorems that uncover a fundamental relationship between the sample-path behavior of a process with independent endpoints and the space-time geometry of this process' flow map. Taken together, these results provide a structural theory of when straight generative flows can, and cannot, exist.

翻译：我们研究生成建模中的动态测度输运，其背景为随机过程 $X_\bullet$，该过程的边缘分布在保持独立性的前提下插值于源分布 $P_0$ 与目标分布 $P_1$ 之间，即当 $(X_0,X_1)\sim P_0\otimes P_1$ 时。该过程 $X_\bullet$ 的条件期望定义了一个常微分方程（ODE），其流映射实现了从 $P_0$ 到 $P_1$ 的输运。我们探讨了何时此类过程能诱导出 \emph{直线流}，即其逐点加速度为零，从而可通过任意一阶方法精确积分。我们首先在涉及过程条件统计量的偏微分方程（PDE）框架下，发展了直线性的多重刻画。继而证明：在端点独立条件下，直线性呈现出尖锐的二分性。一方面，我们为任意高斯端点显式构造了可计算的直线过程；另一方面，我们证明对于具有充分分离模式的靶分布，直线过程不存在。通过一系列逐步泛化的不可能性定理，我们揭示了具有独立端点的过程样本路径行为与其流映射时空几何之间的基本关系。综合而言，这些结果为生成流何时能或不能实现直线性提供了结构性理论。