Advanced Studies in Pure Mathematics

On Wasserstein geometry of Gaussian measures

Asuka Takatsu

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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

Probabilistic Approach to Geometry, M. Kotani, M. Hino and T. Kumagai, eds. (Tokyo: Mathematical Society of Japan, 2010), 463-472

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

Permanent link to this document euclid.aspm/1543086332

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 28A33: Spaces of measures, convergence of measures [See also 46E27, 60Bxx]

Wasserstein space Gaussian measures


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.

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