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.
"Lazy random walks and optimal transport on graphs." Ann. Probab. 44 (3) 1864 - 1915, May 2016. https://doi.org/10.1214/15-AOP1012