Let $E$ be a locally compact separable metric with countable base. We consider a sequence of Hunt processes $\{X^{(n)}\}_{n \in \mathbb{N}}$ on $E$. That is, each $X^{(n)}=(\{X_t^{(n)}\}_{t \in [0,\infty]},\{P_x^{(n)}\}_{x \in E})$ is a cad-lag Markov process on $E$ with the strong Markov property (and the quasi-left-continuity). Let $m$ be a probability measure on $E$. For $n \in \mathbb{N}$, we write $P_{m}^{(n)}$ for the law of $X^{(n)}$ with initial distribution $m$.

We now assume the following condition.

${\rm \bf(A)}$ The laws $\{P_{m}^{(n)}\}_{n\in \mathbb{N}}$ of $\{X^{(n)}\}_{n \in \mathbb{N}}$ are tight in $D([0,\infty),E)$. Here, $D([0,\infty),E)$ is the space of $E$-valued right continuous functions on $[0,\infty)$ with finite left limits.

Under the condition ${\rm \bf(A)}$, we have a subsequential limit $(X,P)$ of $\{X^{(n)},P_m^{(n)}\}_{n \in \mathbb{N}})$. For a bounded measurable function $f\colon E \to \mathbb{R}$, we set $(P_t f)(x)=E[f(X_t) \mid X_0=x]$ for $m$-a.e. $x$. Then, each $P_t$ is extended to a contraction operator on $L^\infty(E,m)$ (the extended operator is denoted by the same symbol).

Can we show that $P_t(P_sf)=P_{t+s}f$ for every $t,s>0$ and $f \in L^\infty(E,m)$?

That is, I would like to know that $(X,P)$ is a time-homogeneous Markov process on $E.$

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