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I’m trying to understand if, when I move the marginals of a Wasserstein geodesic along a contractive flow, the geodesic between the new probability measures is “near” to the geodesic connecting the ...

I am reading this paper "A JKO splitting scheme for Kantorovich-Fisher-Rao gradient flows" (https://arxiv.org/abs/1602.04457) and in the proof of Proposition 2.2, basically, if the measure ...