Continuity equation on metric spaces via measure-valued derivations and BV-Wasserstein curves

Abstract

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 ℝⁿ, 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 ℝⁿ).

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BibTeX

@misc{abedi2026continuity,
  title = {{Continuity equation on metric spaces via measure-valued derivations and BV-Wasserstein curves}},
  author = {Ehsan Abedi and Zhenhao Li and Timo Schultz},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.28586},
  eprint = {2608.28586},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28586},
}

RIS

TY  - UNPB
TI  - Continuity equation on metric spaces via measure-valued derivations and BV-Wasserstein curves
AU  - Ehsan Abedi
AU  - Zhenhao Li
AU  - Timo Schultz
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.28586
UR  - https://arxiv.org/abs/2608.28586
AB  - 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 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 ℝ^n).
ER  -

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