I am interested in the question whether a Feller process retains the Feller property when we kill it upon entering a closed subset of the state space.
Setup
Let $S$ be a locally compact topological Hausdorff space and $X = (X_t)$ a strong Markov process on some probability space $(\Omega, \mathcal{A}, \mathbb{P})$ with values in the one point compactification $S \cup \{\vartheta\}$ of $S$. For $f \in \mathcal{B}_b$ bounded and (Borel-)measurable and $t>0$ denote by $$ P_t f(x) = \mathbb{E}_x[f(X_t)] $$ the semigroup of $X$.
Write $C(S)$ for the continuous functions $S \to \mathbb R$ and $C_\infty(S) \subset C(S)$ for the continuous functions vanishing at infinity and $C_b(S)$ for the bounded continous functions.
I am working with the following definition of the Feller property:
(F1) $\lim_{t \to 0} P_t f(x) = f(x)$ for all $x \in S$ and $f \in C_\infty(S)$,
(F2) $P_t f \in C_\infty(S)$ for all $f \in C_\infty(S)$,
(F3) $P_t f \in C_b(S)$ for all $f \in \mathcal{B}_b(S)$.
We say that a Markov process $X$ is Feller if its semigroup $(P_t)_t$ satisfies (F1) and (F2). We say that $X$ is strongly Feller if $(P_t)_t$ satisfies (F1) and (F3) and we say that $X$ is doubly Feller if $(P_t)_t$ satisfies (F1), (F2) and (F3).
We always assume that we are working with a modification of a Feller process that has cadlag paths.
Now suppose $A \subset S$ is closed and write $$ \tau_A := \inf\{t>0\ \mid\ X_t \in A\} $$ for the first hitting time of $A$.
We define the killed process, more specifically the process killed upon hitting $A$, $X^A$ by $$ X_t^A = X_t, \quad t< \tau_A \qquad X_t^A = \vartheta, \quad t \geq \tau_A. $$ Observe that the new point at infinity (or cemetery point) $\vartheta$ may not coincide with the original one.
I can show that $X^A$ is again a strong Markov process with values in $D=S\setminus A$ and semigroup $$ P_t^A f(x) = \mathbb{E}_x\left[ f(X_t^A) \right] = \mathbb{E}_x[f(X_t);\ t < \tau_A]. $$
Note that $D$ is open and therefore locally compact and regular in the sense that $\mathbb{P}_x( X_0 = x) = 1$ for all $x \in D$. Further note that we can identify $$ C_\infty(D) = \{f \in C_\infty(S)\ \mid \ f|_A = 0\}. $$ On the other hand, every $f \in C_\infty(D)$ or $f \in \mathcal{B}_b(D)$ is naturally extended to $C_\infty(S)$ and $\mathcal{B}_b(S)$, respectively, by setting $f=0$ on $A$.
Question
Is it true that $X^A$ is again Feller if $X$ is Feller? Is it true that $X^A$ is again strongly Feller if $X$ is strongly Feller?
Literature
Strangely enough I could not find many results in this direction in the literature. Maybe I was searching for the wrong keywords? What I found however is a paper by Chung (1986) https://doi.org/10.1142/9789812833860_0051 where he proves that the killed process is again doubly Feller, when $X$ is doubly Feller. The proof relies on the observation that for every $f \in \mathcal{B}_b(D)$, $$ P_t^A f(x) = \mathbb{E}_x[ \psi_s(X_s);\ s < \tau_B], $$ where $$ \psi_s(x) = \mathbb{E}_x[f(X_{t-s});\ t-s < \tau_A]. $$ Then, $\psi_s \in \mathcal{B}_b(S)$ and by the strong Feller property $P_s \psi_s \in C_b(D)$ and we can deduce that for all $x \in D$, $$ \left| P_t^A f(x) - P_s \psi_s(x) \right| \leq P_x(\tau_A \leq s) ||\psi_s||. $$ Since the right hand side converges uniformly on compacta to $0$ as $s \to 0$, we can conclude that $P_t^A f \in C_b(D)$.
Detailed Question
I am trying to show that the simple Feller property (F2) is retained by the killed process, i.e. that $P_t^A: C_\infty(D) \to C_\infty(D)$. One possible way would be to go the analytical way and use Dirichlet forms (there are results on part processes in the theory of Dirichlet forms, namely Theorem 3.3.9 in the book by Chen and Fukushima https://www.jstor.org/stable/j.ctt7s6w6) or Hille-Yosida.
I am trying to find a more probabilistic argument. Inspired by Chung's argument I am trying to approximate $P_t^A$ by $C_\infty(D)$-functions.
I would be very indebted if anyone can give me a hint in the right direction. Either an idea how to proof the statement, a reference or a counter example would be extremely helpful.
Thank you!
Edit: Clarification about cadlag paths.