Calculus of Variations and Geometric Measure Theory

N. Gigli - S. I. Ohta

First variation formula in Wasserstein spaces over compact Alexandrov spaces

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


Accepted Paper

Inserted: 9 feb 2010
Last Updated: 2 mar 2012

Journal: Canad. Math. Bull.
Year: 2010


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.

Keywords: Wasserstein distance, Gradient Flow, Alexandrov space