Journal
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
Volume 48, Issue 4, Pages 2869-2911Publisher
SIAM PUBLICATIONS
DOI: 10.1137/15M1041420
Keywords
dissipation distance; geodesic curves; cone space; optimal transport; Onsager operator; reaction-diffusion equations
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Funding
- Einstein Foundation Berlin via ECMath/MATHEON project [SE2]
- DFG via project C5 (Scaling cascades in complex systems) [CRC 1114]
- ERC under AdG AnaMultiScale [267802]
- MIUR for the project Calculus of Variations
- European Research Council (ERC) [267802] Funding Source: European Research Council (ERC)
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We discuss a new notion of distance on the space of finite and nonnegative measures on Omega subset of R-d, which we call the Hellinger-Kantorovich distance. It can be seen as an inf-convolution of the well-known Kantorovich-Wasserstein distance and the Hellinger-Kakutani distance. The new distance is based on a dynamical formulation given by an Onsager operator that is the sum of a Wasserstein diffusion part and an additional reaction part describing the generation and absorption of mass. We present a full characterization of the distance and some of its properties. In particular, the distance can be equivalently described by an optimal transport problem on the cone space over the underlying space Omega. We give a construction of geodesic curves and discuss examples and their general properties.
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