通过测度值导子与BV-瓦瑟斯坦曲线建立度量空间上的连续性方程
Continuity equation on metric spaces via measure-valued derivations and BV-Wasserstein curves
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中文总结 AI 辅助
本文引入度量空间上的连续性方程概念，基于测度值导子刻画配备1-瓦瑟斯坦距离的概率测度空间中的BV-曲线，其公式与Almi等人的相关工作一致，还扩展了此前的概率表示研究。
中文摘要 AI 辅助
我们引入了度量空间上连续性方程的概念，该方程能够描述关于1-瓦瑟斯坦距离绝对连续、更一般地为有界变差（BV）的概率测度曲线。此连续性方程基于测度值导子的概念，本文也发展了其基础理论。在$\boldsymbol{\text{R}}^n$上，我们的公式与Almi–Rossi–Savaré（arXiv:2506.15333）提出的带奇异通量的连续性方程一致，包括对应的极小解概念。本研究将配备（扩展）1-瓦瑟斯坦距离的概率测度空间中的BV-曲线，刻画为满足具有有限质量的测度值导子的连续性方程的曲线。为此，我们扩展了此前关于BV-曲线概率表示的工作（this http URL.(2024)63:16），并从这些表示出发，在测地度量空间上构造测度值导子（对应在$\boldsymbol{\text{R}}^n$上构造通量测度）。
英文摘要
We introduce a notion of continuity equation on metric spaces that is capable of describing curves of probability measures which are absolutely continuous, and more generally of bounded variation (BV), with respect to the 1-Wasserstein distance. This continuity equation is based on a notion of measure-valued derivations, whose basic theory is also developed in this paper. On $\mathbb{R}^n$, our formulation is consistent with the continuity equation with singular flux introduced by Almi--Rossi--Savaré (arXiv:2506.15333), including the corresponding notion of minimal solutions. In this work, we characterize BV-curves in the space of probability measures equipped with the (extended) 1-Wasserstein distance as those curves satisfying the continuity equation with a measure-valued derivation of finite mass. To this aim, we extend our previous work (Calc.Var.(2024)63:16) on probabilistic representations on BV-curves and construct from them measure-valued derivations (resp. flux measures) on geodesic metric spaces (resp. on $\mathbb{R}^n$).
