1. CMB 2011 (vol 55 pp. 723)
|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 $(\mathcal P(X),W_2)$ 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.
Keywords:Alexandrov spaces, Wasserstein spaces, first variation formula, gradient flow
Categories:53C23, 28A35, 49Q20, 58A35