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*} 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

Can we prove (1) without using heat kernel estimates like (2)?

If you know another way please tell me.

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 $L_t$.