Geometry on the Wasserstein Space Over a Compact Riemannian Manifold

@article{Ding2021GeometryOT,
  title={Geometry on the Wasserstein Space Over a Compact Riemannian Manifold},
  author={Hao Ding and Shizan Fang},
  journal={Acta Mathematica Scientia},
  year={2021},
  volume={41},
  pages={1959 - 1984}
}
  • Hao DingS. Fang
  • Published 2 April 2021
  • Mathematics
  • Acta Mathematica Scientia
We revisit the intrinsic differential geometry of the Wasserstein space over a Riemannian manifold, due to a series of papers by Otto, Otto-Villani, Lott, Ambrosio-Gigli-Savaré, etc. 

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References

SHOWING 1-10 OF 27 REFERENCES

Some Geometric Calculations on Wasserstein Space

We compute the Riemannian connection and curvature for the Wasserstein space of a smooth compact Riemannian manifold.

Construction of the Parallel Transport in the Wasserstein Space

In this paper we study the problem of parallel transport in the Wasserstein spaces P2(R ). We show that the parallel transport exists along a class of curves whose velocity field is sufficiently

Transport inequalities, gradient estimates, entropy and Ricci curvature

We present various characterizations of uniform lower bounds for the Ricci curvature of a smooth Riemannian manifold M in terms of convexity properties of the entropy (considered as a function on the

Topics in Optimal Transportation

Introduction The Kantorovich duality Geometry of optimal transportation Brenier's polar factorization theorem The Monge-Ampere equation Displacement interpolation and displacement convexity Geometric

Ricci curvature for metric-measure spaces via optimal transport

We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of the

THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION

We show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural. In order to convince the reader that it is mathematically natural, we show

Derivative formulas in measure on Riemannian manifolds

We characterize the link of derivatives in measure, which are introduced in Albeverio, Kondratiev, and Röckner (C. R. Acad. Sci. Paris Sér I Math. 323 (1996) 1129–1134); Cardaliaguet (P.‐L. Lions

On the inverse implication of Brenier-Mccann theorems and the structure of (P 2 (M),W 2 )

We do three things. First, we characterize the class of measures μ ∈P2(M) such that for any other ν ∈P2(M) there exists a unique optimal transport plan, and this plan is induced by a map. Second, we