The Annals of Probability
- Ann. Probab.
- Volume 44, Number 3 (2016), 1864-1915.
Lazy random walks and optimal transport on graphs
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.
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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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