# First variation formula in Wasserstein spaces over compact Alexandrov spaces

created by gigli on 09 Feb 2010
modified on 02 Mar 2012

[BibTeX]

Accepted Paper

Inserted: 9 feb 2010
Last Updated: 2 mar 2012

We extend results proven by the second author for nonnegatively curved Alexandrov spaces to general compact Alexandrov spaces X with curvature bounded below: the gradient flow of a geodesically convex functional on the quadratic Wasserstein space $(\mathscr P(X),W_2)$ 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.