I have been studying the construction of the Martin boundary on a discrete set $X$ admitting an irreducible transient random walk $(X,P)$ from Wolfgang Woess' book titled "Random Walks on Infinte Graphs and Groups", where I came across the Poisson-Martin representation theorem which has been stated below:

For every positive harmonic function $h$ on a $X$, there is a positive Borel measure $\nu^h$ on the Martin boundary $\mathcal{M}$ of $X$ such that $$h(x)=\int_{\mathcal{M}} K(x,.) \mathrm{d} \nu^h$$ where $K(.,.)$ denotes the Martin kernel corresponding to $(X, P)$.

I also read about the Martin boundary on the Encyclopedia of Mathematics, where it is stated that for certain choices of $X$ (say $X$ is a bounded domain in $\mathbb{R}^n$ or a Green space), the Borel measure $\nu^h$ is actually a Radon measure. One question that intrigues me is whether we can always expect the Borel measure $\nu^h$ occurring in the above theorem to be a Radon measure? If not, can we expect that $\nu^h$ is a finite measure on $\mathcal{M}$? Is it finite on any compact subset of $\mathcal{M}$? Any help or reference is deeply appreciated. Thanks in advance!


1 Answer 1


There is one more ingredient in the definition of the representing measure $\nu^h$ missing in your description. This is a reference point $o$ from the state space (more generally, a probability measure on the state space), so that it would be more appropriate to use the notation $\nu_o^h$. The functions $\phi$ representing the points of the Martin boundary are normalized by requiring that $\phi(o)=1$. [In a more invariant language, in order to use the Choquet theory for the convex cone of positive superharmonic functions one has to fix a convex compact base of that cone.] It implies that the representing measure $\nu^h_o$ is actually finite, and its mass is $h(o)$.


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