6 research outputs found
Representation of Markov chains by random maps: existence and regularity conditions
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.Comment: 22 pages, several changes from the previous version including
extended discussion of many detail