First Variation Formula in Wasserstein Spaces over Compact Alexandrov Spaces
Printed: Dec 2012
We extend results proved by the second author (Amer. J. Math., 2009)
for nonnegatively curved Alexandrov spaces
to general compact Alexandrov spaces $X$ with curvature bounded
The gradient flow of a geodesically convex functional on the quadratic Wasserstein
space $(\mathcal P(X),W_2)$ satisfies the evolution variational inequality.
Moreover, the gradient flow enjoys uniqueness and contractivity.
These results are obtained by proving a first variation formula for
the Wasserstein distance.
Alexandrov spaces, Wasserstein spaces, first variation formula, gradient flow
53C23 - Global geometric and topological methods (a la Gromov); differential geometric analysis on metric spaces
28A35 - Measures and integrals in product spaces
49Q20 - Variational problems in a geometric measure-theoretic setting
58A35 - Stratified sets [See also 32S60]