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.
"Discrete versions of the transport equation and the Shepp–Olkin conjecture." Ann. Probab. 44 (1) 276 - 306, January 2016. https://doi.org/10.1214/14-AOP973