Abstract
We systematically investigate the problem of representing Markov chains by families of random maps, and which regularity of these maps can be achieved depending on the properties of the probability measures. Our key idea is to use techniques from optimal transport to select optimal such maps. Optimal transport theory also tells us how convexity properties of the supports of the measures translate into regularity properties of the maps via Legendre transforms. Thus, from this scheme, we cannot only deduce the representation by measurable random maps, but we can also obtain conditions for the representation by continuous random maps. Finally, we present conditions for the representation of Markov chain by random diffeomorphisms.
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Acknowledgments
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 267087. M.K. was supported by the International Max Planck Research School “Mathematics in the Sciences”. We would like to thank the anonymous referee for the careful reading and constructive comments, which have contributed to substantially improve the presentation of this manuscript.
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Communicated by L. Ambrosio.
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Jost, J., Kell, M. & Rodrigues, C.S. Representation of Markov chains by random maps: existence and regularity conditions. Calc. Var. 54, 2637–2655 (2015). https://doi.org/10.1007/s00526-015-0878-2
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DOI: https://doi.org/10.1007/s00526-015-0878-2