Representation of Markov chains via random maps

In the theory of random perturbations two of the main modelling objects are random maps and Markov chains. It is easy to show that every random map driven by a Markov process can represented by a Markov chain. The opposite is less clear. Previous approaches assumed smooth and sometimes even uniform noise, and implicitly required parallelizability of the tangent space. In this talk we attack this problem translating it into the language of optimal transport theory. Via some well-known lifts to trivial vector bundles we show that the problem can be reduced to study parametrized families of measure in $\mathbb{R}^n$. An existence proof for measurable random maps directly follows from the existence of transport maps and their stability properties. Using some more recent regularity results, it can be shown that a Markov chain satisfying some mild assumptions can be represented by a continuous random map (Joint work with Jost, and Rodrigues).