Continuity of harmonic functions

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.

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!

• It was very helpful! Thank you for teaching me. Feb 22 '19 at 16:04