# nLab Wasserstein metric

### Context

#### Measure and probability theory

measure theory

probability theory

# Contents

## Idea

A Wasserstein metric is a certain metric over a space of probability measures on a measurable space $X$.

By (JKO) the heat flow?/diffusion equation? on $X$ is the gradient flow of the Boltzman-Shannon entropy functional with respect to the Wasserstein metric.

The Wasserstein metric does not seem to arise from a Riemann metric tensor. A detailed discussion of the relevant gradient flows in non-smooth metric spaces is in (AGS).

The characterization of heat flow as the gradient flow of Shannon-entropy is due to

• R. Jordan, D. Kinderlehrer, F. Otto, The variational formulation of the Fokker-Planck equation , SIAM J. Math. Anal. 29 (1998), no. 1, 1-17.(pdf)

The analog of this for finite probability spaces is discussed in

• Jan Maas, Gradient flows of the entropy for finite Markov chains (pdf)

A comprehensive discussion of the corresponding gradient flows is in

• l Ambrosio, N. Gigli, G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Second edition. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008. x+334 pp. (pdf of toc and introduction) MR2009h:49002
• Luigi Ambrosio, Nicola Gigli, Construction of the parallel transport in the Wasserstein space, Methods Appl. Anal. 15 (2008), no. 1, 1–29, MR2010c:49082
Revised on July 5, 2011 14:11:58 by Zoran Škoda (161.53.130.104)