The Wasserstein metric does not seem to arise from a Riemann metric tensor. A detailed discussion of the relevant gradient flows in non-smooth metric spaces is in (AGS).

The characterization of heat flow as the gradient flow of Shannon-entropy is due to

R. Jordan, D. Kinderlehrer, F. Otto, The variational formulation of the Fokker-Planck equation , SIAM J. Math. Anal. 29 (1998), no. 1, 1-17.(pdf)

The analog of this for finite probability spaces is discussed in

Jan Maas, Gradient flows of the entropy for finite Markov chains (pdf)

A comprehensive discussion of the corresponding gradient flows is in

l Ambrosio, N. Gigli, G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Second edition. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008. x+334 pp. (pdf of toc and introduction) MR2009h:49002

Luigi Ambrosio, Nicola Gigli, Construction of the parallel transport in the Wasserstein space, Methods Appl. Anal. 15 (2008), no. 1, 1–29, MR2010c:49082

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