Minimal fine limits on trees

Abstract

Let $T$ be the set of vertices of a tree. We assume that the Green function is finite and $G(s,t) \rightarrow 0$ as $|s| \rightarrow \infty$ for each vertex $t$. For $v$ positive superharmonic on $T$ and $E$ a subset of $T$, the reduced function of $v$ on $E$ is the pointwise infimum of the set of positive superharmonic functions that majorize $v$ on $E$. We give an explicit formula for the reduced function in case $E$ is finite as well as several applications of this formula. We define the minimal fine filter corresponding to each boundary point of the tree and prove a tree version of the Fatou-Naïm-Doob limit theorem, which involves the existence of limits at boundary points following the minimal fine filter of the quotient of a positive superharmonic by a positive harmonic function. We deduce from this a radial limit theorem for such functions. We prove a growth result for positive superharmonic functions from which we deduce that, if the trees has transition probabilities all of which lie between $\delta$ and $1/2-\delta$ for some $\delta \in (0,1/2)$ (for example homogeneous trees with isotropic transition probabilities), then any real-valued function on T which has a limit at a boundary point following the minimal fine filter necessarily has a nontangential limit there. We give an example of a tree for which minimal fine limits do not imply nontangential limits, even for positive superharmonic functions. Motivated by work on potential theory on halfspaces and Brelot spaces, we define the harmonic fine filter corresponding to each boundary point of the tree. In contrast to the classical setting, we are able to show that it is the same as the minimal fine filter.