Inserted: 4 jun 2018
Last Updated: 23 jul 2018
We present a pathwise proof of the HWI inequality which is based on entropic interpolations rather than displacement ones. Unlike the latter, entropic interpolations are regular both in space and time. Consequently, our approach is closer to the Otto-Villani heuristics, presented in the first part of the article $$, than the original rigorous proof presented in the second part of $$.