Martin boundary of Brownian motion on Gromov hyperbolic metric graphs

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 $\lambda_0$.