Let (X, d, m) be a geodesic metric measure space. Consider a geodesic mu(t) in the L-2-Wasserstein space. Then as s goes to t, the support of mu(s) and the support of mu(t) have to overlap, provided an upper bound on the densities holds. We give a more precise formulation of this self-intersection property. Given a geodesic of (X, d, m) and t is an element of[0, 1], we consider the set of times for which this geodesic belongs to the support of mu(t). We prove that t is a point of Lebesgue density 1 for this set, in the integral sense. Our result applies to spaces satisfying CD(K, infinity). The non-branching property is not needed.

Self-intersection of optimal geodesics

Cavalletti, Fabio;
2014

Abstract

Let (X, d, m) be a geodesic metric measure space. Consider a geodesic mu(t) in the L-2-Wasserstein space. Then as s goes to t, the support of mu(s) and the support of mu(t) have to overlap, provided an upper bound on the densities holds. We give a more precise formulation of this self-intersection property. Given a geodesic of (X, d, m) and t is an element of[0, 1], we consider the set of times for which this geodesic belongs to the support of mu(t). We prove that t is a point of Lebesgue density 1 for this set, in the integral sense. Our result applies to spaces satisfying CD(K, infinity). The non-branching property is not needed.
46
June
653
656
http://dx.medra.org/doi:10.1112/blms/bdu026
https://arxiv.org/abs/1211.6547
Cavalletti, Fabio; Huesmann, M.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/44682
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