I have a question about harmonic functions with respect to symmetric Markov processes.

Let $E$ be a locally compact separable metric space and $\mu$ a positive Radon measure on $E$.

Let $X=(\{X_t\}_{t \ge 0},\{P_x\}_{x \in E})$ be a $\mu$-symmetric Hunt process on $E$.

Def. A Borel measurable function $h$ on $E$ is called $X$-harmonic function if $$ \{h(X_{t \wedge \tau_U})\}_{ t \ge 0} \text{ is a uniformly integrable $P_x$-martingale }$$ for any relatively compact open subset $U \subset E$ and any $x \in U$. Here, $\tau_{U}=\inf\{t>0 \mid X_t \notin U\}$.

The above definition is essentially the same as the definition of harmonic functions adopted in this paper Ch.

**I am interested in the following question:**

Under what conditions on $X$ will $X$-harmonic functions be continuous on $E$?

If $X$ is strong Feller, the above question is true?

I would like to know if there are checkable conditions for the above question.