Direct extension principle for tangent measure fields (small-time limit tangency)

Establish that, for any probability measure μ in the 2-Wasserstein space, the limit (in the narrow topology or the Wasserstein metric on the tangent bundle) of geodesic velocities issued from μ as the geodesic time parameter decreases to 0 remains in the geometric tangent cone Tan_μ. Concretely, prove that taking optimal transport plans that induce geodesics at time τ>0 and letting τ→0 yields a limiting measure field that is tangent to μ, thereby providing a direct extension result for tangent measure fields analogous to the extension principle available for optimal transport plans.

Background

The paper proves restriction and decomposition results for centred tangent and solenoidal cones in Wasserstein spaces. A key technical tool is an extension lemma for optimal transport plans: given a decomposition μ=(1−λ)μ1+λμ2, an optimal plan between μ1 and a compactly supported target can be extended to an optimal plan for μ, enabling restriction and extension arguments for solenoidal fields.

By contrast, obtaining a comparable direct extension for tangent measure fields is obstructed by the behavior of geodesic velocities as the time parameter approaches zero: limits in the narrow or Wasserstein topology may fail to remain tangent. The authors therefore avoid a direct extension by first handling solenoidal fields via optimal plans and then mirroring the result to tangent fields through duality. A direct proof ensuring small-time limits remain tangent would close this gap.