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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?
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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.

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  • $\begingroup$ Can you give a good reference to Markov operators in this context? I.e: Mapping $\mathcal{P}_1(X)$ to itself? Everything I have found in wrt $L^1$. $\endgroup$
    – ABIM
    Commented Jun 21, 2021 at 7:16

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