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

Jürgen Jost, Martin Kell, Christian S. Rodrigues

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.

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

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.

Paper Structure

This paper contains 10 sections, 21 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Some notation and definitions
  3. Markov chains and random maps
  4. On optimal transport
  5. Representation by measurable random maps
  6. Representation by continuous random maps
  7. Regularity of densities
  8. Measures on bundles
  9. Representation by random maps with higher regularity
  10. Conditions for a representation by random diffeomorphisms

Key Result

Theorem \oldthetheorem

Fed69. Let $M$ be a locally compact metric space, $\mu$ a Borel measure on $M$, and $N$ a separable metric space. Let $f : M \to N$ be a measurable map. Consider $A \subset M$ a measurable set with finite measure. Then for each $\delta > 0$ there is a closed set $K \subset A$, with $\mu(A \backslash

Theorems & Definitions (44)

  • Theorem \oldthetheorem
  • Definition 1
  • Theorem \oldthetheorem: Existence of optimal coupling Vil09
  • Proposition \oldthetheorem: Solution of the Monge problem
  • proof
  • Corollary \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • proof
  • Remark 1
  • ...and 34 more