The Jordan–Kinderlehrer–Otto (JKO) scheme is a procedure for finding time-discrete approximations to solutions of diffusion equations using variational problems involving the optimal transportation
problem. On the other hand, the Harnack inequality is a well-known inequality in the regularity theory of
second-orderellipticandparabolicequations.Inthispaper,weestablishalinkbetweenHarnackinequalities
andtheJKOschemebyestablishingaversionoftheHarnackinequalityforthetime-discreteapproximations
mentioned above in the case of the heat equation on the flat torus.

• A Harnack inequality for the Jordan–Kinderlehrer–Otto scheme