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May 2016 Lazy random walks and optimal transport on graphs
Christian Léonard
Ann. Probab. 44(3): 1864-1915 (May 2016). DOI: 10.1214/15-AOP1012


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.


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Christian Léonard. "Lazy random walks and optimal transport on graphs." Ann. Probab. 44 (3) 1864 - 1915, May 2016.


Received: 1 November 2013; Revised: 1 February 2015; Published: May 2016
First available in Project Euclid: 16 May 2016

zbMATH: 06603564
MathSciNet: MR3502596
Digital Object Identifier: 10.1214/15-AOP1012

Primary: 60J27 , 65K10

Keywords: $\Gamma$-convergence , discrete metric graph , Displacement interpolation , Entropy minimization , Optimal transport , Random walks , Schrödinger problem

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 3 • May 2016
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