## Advanced Studies in Pure Mathematics

### On Wasserstein geometry of Gaussian measures

Asuka Takatsu

#### Abstract

The space of Gaussian measures on a Euclidean space is geodesically convex in the $L^2$-Wasserstein space. This is a finite dimensional manifold since Gaussian measures are parameterized by means and covariance matrices. By restricting to the space of Gaussian measures inside the $L^2$-Wasserstein space, we manage to provide detailed descriptions of the $L^2$-Wasserstein geometry from a Riemannian geometric viewpoint. We obtain a formula for the sectional curvatures of the space of Gaussian measures, which is written out in terms of the eigenvalues of the covariance matrix.

#### Article information

Dates
Received: 15 January 2009
Revised: 1 May 2009
First available in Project Euclid: 24 November 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1543086332

Digital Object Identifier
doi:10.2969/aspm/05710463

Mathematical Reviews number (MathSciNet)
MR2648273

Zentralblatt MATH identifier
1206.60016

#### Citation

Takatsu, Asuka. On Wasserstein geometry of Gaussian measures. Probabilistic Approach to Geometry, 463--472, Mathematical Society of Japan, Tokyo, Japan, 2010. doi:10.2969/aspm/05710463. https://projecteuclid.org/euclid.aspm/1543086332