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MiS Preprint

Representation of Markov chains by random maps: existence and regularity conditions

Jürgen Jost, Martin Kell and Christian S. Rodrigues


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.

Aug 6, 2012
Aug 13, 2012
MSC Codes:
49N60, 37C05, 37H10, 37C40, 49K45
Markov chain, random dynamics, random maps, optimal transport, random diffeomorphisms, optimal coupling

Related publications

2015 Repository Open Access
Jürgen Jost, Martin Kell and Christian S. Rodrigues

Representation of Markov chains by random maps : existence and regularity conditions

In: Calculus of variations and partial differential equations, 54 (2015) 3, pp. 2637-2655