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](http://www.ams.org/journals/proc/2009-137-10/S0002-9939-09-09945-6/S0002-9939-09-09945-6.pdf).


**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.



Continuity of harmonic functions
I do not know if a general answer to your question is known. This is an extended comment about a rather specific case.




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In his article [*Doubly-Feller Process with Multiplicative Functional*](https://link.springer.com/chapter/10.1007/978-1-4684-6748-2_4), K.-L. Chung proved that if $X$ is both Feller and strong-Feller, then the process $X$ killed when it exists a set is Feller and strong-Feller, too.


If $H\_t := h(X\_{t \wedge \tau\_U})$ is a martingale, then
$$ h(x) = \mathbb{E}^x H\_0 = \mathbb{E}^x H\_t = \mathbb{E}^x(h(X\_t) \mathbb{1}\_{t < \tau\_U}) + \mathbb{E}^x(h(X\_{\tau\_U}) \mathbb{1}\_{t \geqslant \tau\_U}). $$
By Chung's result, for each $t > 0$ the first summand in the right-hand side is a continuous function of $x \in U$. If we are able to prove that the second summand converges to zero as $t \to 0^+$ uniformly on compacts, then continuity of $h$ on $U$ follows.


If, for example, $h$ is a bounded function, then clearly the second summand is no greater than $\|h\|\_\infty \mathbb{P}^x(t \geqslant \tau\_U)$, which indeed converges to zero uniformly on compacts by Lemma 2 in Chung's paper.




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For more general results, one may like to see at the [list of articles](https://scholar.google.com/scholar?cites=13396002259052176146) which cite Chung's work — there are not so many of them!


