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.