I have a question about Markov processes.


Let $\mathbb{M}=(X\_t,P\_x)$ be a Markov process on a locally compact separable metric measure space $(E,\mu)$. 


$\mathbb{M}$ is called Feller process if its semigroup $\{p\_{t}\}\_{t>0}$ satisfies the following: for all $t>0$,
\begin{align\*}
(0)\quad p\_{t}(C\_{\infty}(E)) \subset C\_{\infty}(E),
\end{align\*}
where $ C\_{\infty}(E)$ is the set of continuous functions which vanish at infinity.


If we know $\mathbb{M}$ has the following property:
\begin{align\*}
(1)\quad\lim\_{r \to \infty}\sup\_{x \in E}P\_{x}(X\_{t} \in E \setminus B(x,r))=0,
\end{align\*}
we can prove $p\_{t}f$ vanishes at infinity for all $f \in C\_{c}(E)$. Indeed, for all $f \in C\_{c}(E)$, we have


\begin{align\*}
&|p\_{t}f(x)| \le E\_{x}[|f(X\_t)|] \\
&=\int\_{E \cap B(x,r)}p\_{t}(x,y)|f(y)|\,\mu(dy)+\int\_{E \setminus B(x,r)}p\_{t}(x,y)|f(y)|\,\mu(dy) \\
&\le \sup\_{y \in E \cap B(x,r)}|f(y)|+\|f\|\_{\infty} \sup\_{x \in E}P\_{x}(X\_{t} \in E \setminus B(x,r))\end{align\*}
Then, letting $x \to \infty$ and then $r \to \infty$, we obtain the assertion.


**A sufficient condition for (1)**


Let us consider the case $\mathbb{M}$ is a diffusion process on a Euclidean domain $D$. If the transition density $p\_{t}(x,y)$ of $\mathbb{M}$ has the following estimate:
\begin{align\*}
(2) \quad p\_{t}(x,y) \le a\_{1}e^{t} t^{-d/2} \exp(-|x-y|^2/a\_{2}t)\quad (a\_1,a\_2 \text{ are some constants indep of $t,x,y$}),
\end{align\*}
we can prove (1). 


**My question**


I am interested in the property (1) of reflecting Brownian motions on smooth domains. These processes are generated by the following classical Dirichlet form:
\begin{align\*}
(3)\quad\mathcal{E}(f,g)=\frac{1}{2}\int\_{D}(\nabla f, \nabla g)\,dx,\quad f,g \in H^{1}(D).
\end{align\*}


When the boundary of $D$ is sufficiently smooth,
it is known that (3) is regular on $\bar{D}$ and we can construct a processes $(\{X\_t\},\{P\_x\})$ whose Dirichlet form is (3). Furthermore, $(\{X\_t\},\{P\_x\})$ solves the following Skorohod SDE:
\begin{align\*}
X\_{t}=x+B\_{t}+\int\_{0}^{t}n(X\_s)dL\_s,
\end{align\*}
where $B\_t$ is the $d$-dim B.M. and $n$ is the inward unit normal on $\partial D$ and $\{L\_t\}$ is boundaly local time.


If transition density of $X$ has a estimate like (2) and $D$ is bounded, we can compute expectation of $L\_t$. 



How to prove Feller property without using heat kernel estimates
In the case of a diffusion, (1) is implied for example by having bounded coefficients. This follows immediately from applying BDG to $X\_t-x$ and doesn't require (2) which is much harder to get. 
Note by the way that (1) itself is quite a bit overkill since it rules out the OU process, which is the prototypical example of a Feller process, even in your "strong" sense.


I also want to point out that the definition of "Feller property" isn't consistent across the literature. For many authors, mapping $C\_b$ (continuous bounded functions) into itself is sufficient for a Markov process to be called "Feller". Mapping $C\_\infty$ to itself rules out simple examples like $dx = -x^3\,dt + dW$.


