Discrete versions of the transport equation and the Shepp-Olkin conjecture

TL;DR

This paper develops a framework for discrete transport problems for integer-valued variables, introducing a discrete Benamou-Brenier formula and applying it to prove a case of the Shepp-Olkin entropy conjecture.

Contribution

It introduces a novel discrete transport framework with a new form of weighted log-concavity, leading to a proof of the monotone case of the Shepp-Olkin conjecture.

Findings

Established a discrete gradient flow framework for integer-valued variables.

Formulated a discrete Benamou-Brenier formula.

Proved the monotone case of the Shepp-Olkin entropy concavity conjecture.

Abstract

We introduce a framework to consider transport problems for integer-valued random variables. We introduce weighting coefficients which allow us to characterize transport problems in a gradient flow setting, and form the basis of our introduction of a discrete version of the Benamou-Brenier formula. Further, we use these coefficients to state a new form of weighted log-concavity. These results are applied to prove the monotone case of the Shepp-Olkin entropy concavity conjecture.