Many studies have been conducted on flows of probability measures, often in terms of gradient flows. We utilize a generalized notion of derivatives with respect to time to model the instantaneous evolution of empirically observed one-dimensional distributions that vary over time and develop consistent estimates for these derivatives. Employing local Fr\'echet regression and working in local tangent spaces with regard to the Wasserstein metric, we derive the rate of convergence of the proposed estimators. The resulting time dynamics are illustrated with time-varying distribution data that include yearly income distributions and the evolution of mortality over calendar years.

翻译：对概率计量的流量进行了许多研究,往往以梯度流量为基准。我们利用对衍生物的普遍概念,用时间来模拟经验观测到的一维分布的瞬间演变,这些分布随着时间的变化而变化,并对这些衍生物得出一致的估计。我们利用当地的Fr\'echet回归,在瓦塞尔斯坦指标方面在当地相近的空间工作,我们得出了拟议估算器的趋同率。由此产生的时间动态用时间变化的分布数据来说明,这些数据包括年收入分配和历年死亡率的演变。