Geodesic
A description of transport cost for signed measures
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XX, Tome 390 (2011), pp. 147-181

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In this paper we develop the analysis of [3] about the extension of the optimal transport framework to the space of real measures. The main motivation comes from the study of nonpositive solutions to some evolution PDEs. Although a canonical optimal transport distance does not seem to be available, we may describe the cost for transporting signed measures in various ways and with interesting properties. 
@article{ZNSL_2011_390_a5,
     author = {E. Mainini},
     title = {A description of transport cost for signed measures},
     journal = {Zapiski Nauchnykh Seminarov POMI},
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     volume = {390},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_390_a5/}
}
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E. Mainini. A description of transport cost for signed measures. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XX, Tome 390 (2011), pp. 147-181. http://geodesic.mathdoc.fr/item/ZNSL_2011_390_a5/