We investigate contraction of the Wasserstein distances in Euclidean spaces under Gaussian smoothing. It is well known that the heat semigroup is exponentially contractive with respect to the Wasserstein distances on manifolds of positive curvature; however, on flat Euclidean space---where the heat semigroup corresponds to smoothing the measures by Gaussian convolution---the situation is more subtle. We obtain precise asymptotics for the 2-Wasserstein distance under the action of the Euclidean heat semigroup, and show that, in contrast to the positively curved case, the contraction rate is always polynomial, with exponent depending on the moment sequences of the measures. We establish similar results for other p-Wasserstein distances as well as some f-divergences. This is a joint work with Jonathan Niles-Weed.
Graduated from NYU Shanghai in 2017 with a B.S. in Honors Mathematics, Hong-Bin Chen started his PhD studies at the Courant Institute of Mathematical Sciences, NYU. His primary advisor is Prof. Yuri Bakhtin and his secondary advisor is Prof. Jean-Christophe Mourrat. His research mainly focuses on random dynamics and probabilistic models.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai