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

TL;DR

This paper explores conditions under which Markov chains can be represented by random maps with various regularity levels, using optimal transport techniques to connect measure properties with map regularity.

Contribution

It introduces a framework leveraging optimal transport to determine existence and regularity of random map representations of Markov chains based on measure properties.

Findings

Existence of measurable random map representations for Markov chains.

Conditions for continuous random map representations based on measure convexity.

Criteria for representing Markov chains by random diffeomorphisms.

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.