First variation formula in Wasserstein spaces over compact Alexandrov spaces

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 below. The gradient flow of a geodesically convex functional on the quadratic Wasserstein space (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. © Canadian Mathematical Society 2011.