Representation of Markov chains by random maps: existence and regularity conditions
Jürgen Jost, Martin Kell, and Christian S. Rodrigues
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Submission date: 06. Aug. 2012
published in: Calculus of variations and partial differential equations, 54 (2015) 3, p. 2637-2655
DOI number (of the published article): 10.1007/s00526-015-0878-2
MSC-Numbers: 49N60, 37C05, 37H10, 37C40, 49K45
Keywords and phrases: Markov chain, random dynamics, random maps, optimal transport, random diffeomorphisms, optimal coupling
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We systematically investigate the problem of representing Markov chains by families of random maps, and what 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 show how to construct random diffeomorphisms from a given Markov chain.