# Some Geometric Calculations on Wasserstein Space

@article{Lott2006SomeGC, title={Some Geometric Calculations on Wasserstein Space}, author={John Lott}, journal={Communications in Mathematical Physics}, year={2006}, volume={277}, pages={423-437} }

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

## 135 Citations

### Geometry on the Wasserstein Space Over a Compact Riemannian Manifold

- MathematicsActa Mathematica Scientia
- 2021

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.

### An intrinsic parallel transport in Wasserstein space

- Mathematics
- 2017

If M is a smooth compact connected Riemannian manifold, let P(M) denote the Wasserstein space of probability measures on M. We describe a geometric construction of parallel transport of some tangent…

### A Geometric Perspective on Regularized Optimal Transport

- MathematicsJournal of Dynamics and Differential Equations
- 2018

We present new geometric intuition on dynamical versions of regularized optimal transport. We introduce two families of variational problems on Riemannian manifolds which contain analogues of the…

### A Geometric Perspective on Regularized Optimal Transport

- MathematicsJournal of Dynamics and Differential Equations
- 2018

We present new geometric intuition on dynamical versions of regularized optimal transport. We introduce two families of variational problems on Riemannian manifolds which contain analogues of the…

### On tangent cones in Wasserstein space

- Mathematics
- 2014

If M is a smooth compact Riemannian manifold, let P(M) denote the Wasserstein space of probability measures on M. If S is an embedded submanifold of M, and $\mu$ is an absolutely continuous measure…

### Wasserstein Riemannian geometry of Gaussian densities

- MathematicsInformation Geometry
- 2018

The Wasserstein distance on multivariate non-degenerate Gaussian densities is a Riemannian distance. After reviewing the properties of the distance and the metric geodesic, we present an explicit…

### A geometric study of Wasserstein spaces: Euclidean spaces

- Mathematics
- 2010

We study the Wasserstein space (with quadratic cost) of Euclidean spaces as an intrinsic metric space. In particular we compute their isometry groups. Surprisingly, in the case of the line, there…

### The Dirichlet-Ferguson Diffusion on the Space of Probability Measures over a Closed Riemannian Manifold

- Mathematics
- 2018

We construct a recurrent diffusion process with values in the space of probability measures over a closed Riemannian manifold of arbitrary dimension. The process is associated with the Dirichlet…

### Optimal transport and large number of particles

- Mathematics
- 2013

We present an approach for proving uniqueness of ODEs in the Wasserstein space. We give an overview of basic tools needed to deal with Hamiltonian ODE in the Wasserstein space and show various…

### Differential Geometric Heuristics for Riemannian Optimal Mass Transportation

- Mathematics
- 2009

We give an account on Otto’s geometrical heuristics for realizing, on a compact Riemannian manifold M, the L 2 Wasserstein distance restricted to smooth positive probability measures, as a Riemannian…

## References

SHOWING 1-10 OF 27 REFERENCES

### The Convenient Setting of Global Analysis

- Mathematics
- 1997

Introduction Calculus of smooth mappings Calculus of holomorphic and real analytic mappings Partitions of unity Smoothly realcompact spaces Extensions and liftings of mappings Infinite dimensional…

### LOCAL LIE ALGEBRAS

- Mathematics
- 1976

In this article we investigate the structure of local Lie algebras with a one-dimensional fibre. We show that all such Lie algebras are essentially exhausted by the classical examples of the…

### Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems

- Mathematics
- 2008

Let M denote the space of probability measures on R^D endowed with the Wasserstein metric. A differential calculus for a certain class of absolutely continuous curves in M was introduced by Ambrosio,…

### Topics in Optimal Transportation

- Mathematics
- 2003

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

### THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION

- Mathematics
- 2001

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…

### An introduction to symplectic topology

- Mathematics
- 1991

Proposition 1.4. (1) Any symplectic vector space has even dimension (2) Any isotropic subspace is contained in a Lagrangian subspace and Lagrangians have dimension equal to half the dimension of the…

### Gradient Flows: In Metric Spaces and in the Space of Probability Measures

- Mathematics
- 2005

Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence…

### Ricci curvature for metric-measure spaces via optimal transport

- Mathematics
- 2004

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…

### Contractions in the 2-Wasserstein Length Space and Thermalization of Granular Media

- Mathematics
- 2006

An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow-through model representing the continuum limit of a gas of particles…