Interpolation of probability measures on graphs (Q2406325)

The aim of this paper is to review some results concerning the construction of interpolating families between finitely supported probability measures on a graph using the optimal transportation methods. Most of these results made the content of the following three papers previously published by the author [Potential Anal. 41, No. 3, 679--698 (2014; Zbl 1305.60014); Electron. J. Probab. 19, Paper No. 92, 29 p. (2014; Zbl 1347.49077); ``Entropy along \(W_{1;+}\)-geodesics on graphs'', Preprint, \url{arXiv:1406:5089}]. The paper under review mainly focuses on the simpler case when the underlying graph is \( \mathbb{Z}\) and explains briefly how these constructions can be extended to the general case. Section 5 is dedicated to the case of entropic interpolations, along which concavity of entropy results hold. The paper ends by explaining how ideas coming from the optimal transportation theory on \(\mathbb{Z}\) can be used to solve the Shepp-Olkin conjecture concerning the concavity of the entropy of sums of independent Bernoulli random variables. For the entire collection see [Zbl 1377.52002].