Any BV-in-time curve of probability measures with finite first moment and its singular flux admit a probabilistic representation by reparametrized Lipschitz curves, and by augmented BV curves with explicit jump transitions.
Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent
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The superposition principle for the continuity equation with singular flux
Any BV-in-time curve of probability measures with finite first moment and its singular flux admit a probabilistic representation by reparametrized Lipschitz curves, and by augmented BV curves with explicit jump transitions.