The Annals of Probability

Lazy random walks and optimal transport on graphs

Christian Léonard

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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

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