Let $(\mu\_n)\_{n=1}^{\infty}$ be a sequence in $\mathcal{P}\_1(X)$ for some compact metric space $(X,d)$. Suppose that there is a weakly-continuous function $F:\mathcal{P}\_1(X)\rightarrow \mathcal{P}\_1(X)$ satisfying:
$$
\mu\_{n+1} = F(\mu\_n) \qquad \forall n=2,\dots
.
$$
Then:


1. Must there exist a Markov process $(X\_n)\_{n=1}^{\infty}$ and a reference Borel probability measure $\mathbb{P}$ on $(X,\mathcal{B}(X))$ such that:
$$
\mu\_n = \mathbb{P}\left(X\_n \in \cdot\right), \qquad \boldsymbol{(0)}
$$
2. If the answer is "yes" to (1); can there also be a non-Markovian process $(X\_n)\_{n=1}^{\infty}$ on $(X,\mathcal{B}(X))$ such that $\boldsymbol{(0)}$ holds?



Do measure-valued dynamical systems correspond to marginals of Markov processes?
No. Any Markov operator is contracting in the total variation norm, whereas your function $F$ is subject to a much weaker condition of weak continuity. It is easy to construct a counterexample. For instance, take $X$ to be the two point set $\{0,1\}$, then the probability measures on $X$ are parameterized by a single parameter $t=\mu(1)\in [0,1]$. Take for $F$ any continuous function $[0,1]\to[0,1]$ which is not 1-Lipschitz.


