## Wasserstein geometry of Gaussian measures.(English)Zbl 1245.60013

In this paper the geometric structure of Gaussian measures is considered. The space $$\mathcal{N}^d$$ of Gaussian measures on $$\mathbb{R}^d$$ is a space of finite dimension, and this allows to write down the explicit Riemannian metric, which, in turn, induces the $$L^2$$-Wasserstein distance function.
After introducing the $$L^2$$-Wasserstein geometry, results concerning the $$L^2$$-Wasserstein geometry on $$\mathcal{N}^d$$ are determined, and the $$L^2$$-Wasserstein metric is analyzed. Then the completion $$\overline{\mathcal{N}_0^d}$$ of the space $$\mathcal{N}_0^d$$ (consisting of Gaussian measures with mean $$0$$) is studied as an Alexandrov space. Stratification and tangent cones are investigated. In particular, it turns out that the singular set is stratified according to the dimension of the support of the Gaussian measures, providing an explicit nontrivial example of Alexandrov space with extremal sets.

### MSC:

 60D05 Geometric probability and stochastic geometry 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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### References:

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