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We prove the correspondence between the solutions of the sub-elliptic heat equation in a Carnot group and the gradient flows of the relative entropy functional in the Wasserstein space of probability measures on . Our result completely answers a question left open in a previous paper by N. Juillet, where the same correspondence was proved for , the -dimensional Heisenberg group.
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Luigi Ambrosio, Giorgio Stefani, Heat and entropy flows in Carnot groups. Rev. Mat. Iberoam. 36 (2020), no. 1, pp. 257–290DOI 10.4171/RMI/1129