Amongst the main concerns of Dynamics one wants to decide whether
asymptotic states are robust under random perturbations. Considering
the iteration of \(f\colon M \to M\), such randomness are
represented by a family \(\{p_{\varepsilon}( \, \cdot \, |x)\}\) of
Borel probability measures, such that every \(p_{\varepsilon}( \,
\cdot \, |x)\) is supported inside the \(\varepsilon\)-neighbourhood
of \(f(x)\). Alternatively, the orbit is given by the iteration
\(x_{j} = g_{j} \circ \cdots \circ g_{1}(x_{0})\), where each
measurable \(g_{j}\) is picked at random \(\varepsilon\)-close from
the original map \(f\). Endowing the collection of maps
\(\{g_{j}\}\) with a probability distribution \(\nu_{\varepsilon}\),
we say that the sequence of random maps is a representation of that
Markov chain if for every Borel subset \(U\), \(p_{\varepsilon}(U|x)
= \nu_{\varepsilon}(\{g : g(x) \in U \})\). In this talk 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. From this scheme, we not only deduce
the representation by measurable and continuous random maps, but
also obtain conditions for the to construct random diffeomorphisms
from a given Markov chain. This is a joint work with Jost, and Kell.
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