Continuity of harmonic functions

Question

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.

Answer 1

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

---

In his article Doubly-Feller Process with Multiplicative Functional, 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.

---

For more general results, one may like to see at the list of articles which cite Chung's work — there are not so many of them!