We study dynamic measure transport for generative modeling: specifically, flows induced by stochastic processes that bridge a specified source and target distribution. The conditional expectation of the process' velocity defines an ODE whose flow map achieves the desired transport. We ask \emph{which processes produce straight-line flows} -- i.e., flows whose pointwise acceleration vanishes and thus are exactly integrable with a first-order method? We provide a concise PDE characterization of straightness as a balance between conditional acceleration and the divergence of a weighted covariance (Reynolds) tensor. Using this lens, we fully characterize affine-in-time interpolants and show that straightness occurs exactly under deterministic endpoint couplings. We also derive necessary conditions that constrain flow geometry for general processes, offering broad guidance for designing transports that are easier to integrate.

翻译：本文研究用于生成建模的动态测度传输问题：具体而言，我们关注由连接指定源分布与目标分布的随机过程所诱导的流。该过程速度的条件期望定义了一个常微分方程（ODE），其流映射实现了所需的传输。我们探究\emph{何种过程能产生直线流}——即其逐点加速度为零，从而可通过一阶方法精确积分的流？我们提出了一个简洁的偏微分方程（PDE）特征描述，将直线性表征为条件加速度与加权协方差（雷诺）张量散度之间的平衡关系。基于此视角，我们完整刻画了时间仿射插值过程，并证明直线性仅在确定性端点耦合条件下精确成立。同时，我们推导了约束一般过程流几何结构的必要条件，为设计更易积分的传输方案提供了广泛的理论指导。