I have a technical question on a continuity of green function.

**Setting**

Let $E$ be a locally compact separable metric space and $m$ a locally finite measure on $E$.

Let $X=(\{X_t\}_{t \ge 0},\{P_x\}_{x \in E})$ be a **transient** diffusion process. We assume that there exists a continuous function $p_{t}(x,y):(0,\infty) \times E \times E \to [0,\infty)$ such that
\begin{align*}
E_{x}[f(X_t)]:=\int_{E}p_{t}(x,y)\,dm(y),\quad t>0,\ x,y \in E
\end{align*}

**My question**

Then, under what conditions, can we show that $g(x,y)=\int_{0}^{\infty}p_{t}(x,y)\,dt$ is a continuous function on $E \times E$ off the diagonal.

**Aim**

I am interested in the case that $X$ is a part process of a recurrent process. For example,

- $E=\bar{\mathbb{D}} \setminus K$, where $\mathbb{D}$ is the unit disk of $\mathbb{R}^2$ and $K \subset \mathbb{D}$ is a closed ball,
- $X$ is a part process on $\bar{\mathbb{D}} \setminus K$ of a reflected Brownian motion on $\bar{D}$.

In this situation, $X$ possesses a continuous heat kernel $p_{t}(x,y)$. However, I do not even know whether $x \mapsto g(x,y)$ is locally bounded on $E \setminus \{y\}$.

- If $K$ is the closed ball centered at the origin with radius $r_0<1$, it is known that there exists $C \in (0,\infty)$ such that
\begin{align*}
(\ast) \quad g(x,y) \le C-2\log |x-y|,\quad m \otimes m\text{-a.e. on } E \times E
\end{align*}
However, I do not know whether $(\ast)$ holds for
**any**$(x,y) \in E \times E$. If $(\ast)$ holds for**any**$(x,y) \in E \times E$, $x \mapsto g(x,y)$ is locally bounded on $E \setminus \{y\}$. - There is a function $G(x,y)$ which is continuous on $E \times E$ off the diagonal. It holds that $g(x,y)=G(x,y),\quad m\otimes m\text{-a.e.}$

I think that the continuity of green functions is a fundamental problem, but is there a standard proof method?