The Annals of Probability

Lazy random walks and optimal transport on graphs

Christian Léonard

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Abstract

This paper is about the construction of displacement interpolations of probability distributions on a discrete metric graph. Our approach is based on the approximation of any optimal transport problem whose cost function is a distance on a discrete graph by a sequence of entropy minimization problems under marginal constraints, called Schrödinger problems, which are associated with random walks. Displacement interpolations are defined as the limit of the time-marginal flows of the solutions to the Schrödinger problems as the jump frequencies of the random walks tend down to zero. The main convergence results are based on $\Gamma$-convergence of entropy minimization problems.

As a by-product, we obtain new results about optimal transport on graphs.

Article information

Source
Ann. Probab., Volume 44, Number 3 (2016), 1864-1915.

Dates
Received: November 2013
Revised: February 2015
First available in Project Euclid: 16 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1463410034

Digital Object Identifier
doi:10.1214/15-AOP1012

Mathematical Reviews number (MathSciNet)
MR3502596

Zentralblatt MATH identifier
06603564

Subjects
Primary: 60J27: Continuous-time Markov processes on discrete state spaces 65K10: Optimization and variational techniques [See also 49Mxx, 93B40]

Keywords
Displacement interpolation discrete metric graph optimal transport Schrödinger problem random walks entropy minimization $\Gamma$-convergence

Citation

Léonard, Christian. Lazy random walks and optimal transport on graphs. Ann. Probab. 44 (2016), no. 3, 1864--1915. doi:10.1214/15-AOP1012. https://projecteuclid.org/euclid.aop/1463410034


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