Geodesic
A survey on dynamical transport distances
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XX, Tome 390 (2011), pp. 5-51

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In this paper we review some transport models based on the continuity equation, starting with the so-called Benamou–Brenier formula, which is nothing but a fluid mechanics reformulation of the Monge–Kantorovich problem with cost $c(x,y)=|x-y|^2$. We discuss some of its applications (gradient flows, sharp functional inequalities …), as well as some variants and generalizations to dynamical transport problems, where interaction effects among mass particles are considered. 
@article{ZNSL_2011_390_a0,
     author = {L. Brasco},
     title = {A survey on dynamical transport distances},
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     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_390_a0/}
}
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L. Brasco. A survey on dynamical transport distances. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XX, Tome 390 (2011), pp. 5-51. http://geodesic.mathdoc.fr/item/ZNSL_2011_390_a0/