I have a question about Feller processes.


In this paper ["On the doubly Feller property of resolvent"](https://projecteuclid.org/download/pdfview_1/euclid.kjm/1492194850) the transition semigroup $(P\_t)\_{t \ge 0}$ of a Hunt process (on a metric space $E$) is said to have the Feller property if the following two conditions are satisfied. 


1. For every $f \in C\_{\infty}(E)$ and $t >0$, we have $P\_{t}f \in C\_{\infty}(E)$.
2. For every $f \in C\_{\infty}(E)$ and $x \in E$, $\lim\_{t \to 0}P\_{t}f(x)=f(x)$.


Here, $C\_{\infty}(E)$ is the family of continuous functions on $E$ vanishing at infinity.


**My question**


I think the transition semigroup $(P\_t)\_{t \ge 0}$ of a Hunt process $X$ always satisfies the condition 2. Since $f(X\_t) \to f(X\_0)$ as $t \to 0$ from the right continuity of the sample path of $X$. This implies
\begin{align\*}
\lim\_{t \to 0}P\_{t}f(x)=\lim\_{t \to 0}E\_{x}[f(X\_t)]=E\_{x}[f(X\_0)]=E\_{x}[f(x)]=f(x)
\end{align\*}
by the dominated convergence theorem and the normal property of $X$.


Why they need the condition 2?



On necessity of Feller property
The Feller property is *sufficient* (bot not necessary) for the existence of a Hunt process associated with $(P\_t)$. As you observe, condition (2) in the Feller property is (in isolation) also a necessary consequence of the Hunt property. 


