# Martin boundary of Brownian motion on Gromov hyperbolic metric graphs

## TL;DR

This paper proves that for certain Gromov hyperbolic metric graphs with a group acting geometrically, the Martin boundary coincides with the geometric boundary for all spectral parameters up to the bottom of the spectrum.

## Contribution

It establishes the equality of Martin and geometric boundaries for a class of hyperbolic metric graphs under group actions, extending known results to a broader setting.

## Key findings

- Martin boundary equals geometric boundary for all λ in [0, λ₀]
- Results apply to graphs with infinitely many boundary points
- Includes the case at the bottom of the spectrum λ₀

## Abstract

Let $\widetilde{X}$ be a locally finite complete Gromov hyperbolic metric graph with the geometric boundary consisting of infinitely many points. Suppose that there is a discrete subgroup of the isometry group $Iso(\widetilde{X})$ acting geometrically on $\widetilde{X}$. The $\lambda$-Martin boundary is the boundary of the image of an embedding from $\widetilde{X}$ to the space of $\lambda$-superharmonic functions. We show that the $\lambda$-Martin boundary coincides with the geometric boundary for any $\lambda \in [0, \lambda_0],$ in particular at the bottom of the spectrum $\lambda_0$.