Martin compactification for discrete potential theory and the mean value property (Q1904130)

Let \(h\) be a positive measurable function on a domain \(X\) in \(\mathbb{R}^n\). It is deduced, from many interesting results proved here, that if \(h\) satisfies a restricted mean-value property (i.e. the mean-value property at each point \(x\in X\) is satisfied for just one ball), then \(h\) is harmonic. Some related earlier investigations used probabilistic methods. Here the authors make use of a Martin-type compactification in the context of discrete potential theory.