For functions of measures on Riemannian manifolds, the intrinsic and Lions derivatives coincide and equal the gradient of the extrinsic derivative, giving a direct limit formula for computing them.
Gradient Flows in Metric Spaces and in the Spaces of Probability Measures
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Derivative Formulas in Measure on Riemannian Manifolds
For functions of measures on Riemannian manifolds, the intrinsic and Lions derivatives coincide and equal the gradient of the extrinsic derivative, giving a direct limit formula for computing them.