Continuity of green functions 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?
 A: (A comment too long for a comment).

I am not an expert, but I guess an answer will depend on how specific you want it to be. If this is about reflecting Brownian motions, then we know a lot, including heat kernel bounds for nice enough domains; see Bass–Hsu, Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains, Ann. Probab. 19 (1991) and Yang–Zhang, Estimates of Heat Kernels with Neumann Boundary Conditions, Potential Anal. 38 (2013). If I understand correctly, in particular the Green function (away from the diagonal) is smooth in the interior of the domain, and Hölder continuous at the boundary (if the domain is Lipschitz).

For more general diffusions (or differential operators), one often knows "a priori" supremum bounds for and Hölder continuity results for harmonic functions. This should also give the desired continuity of the Green function. To my understanding, this is often parallel to heat kernel bounds, which are either assumed (in the general setting of metric measure spaces) or proved (for, say, certain elliptic operators on $\mathbb{R}^n$, or on smooth manifolds).
