Martin boundary of Brownian motion on hyperbolic graphs

Abstract

Let X􏰏 be a hyperbolic graph with infinitely many geometric boundary points. Suppose that there exists a discrete group Γ that acts geometrically on the graph X􏰏. We study the relation between the geometric boundary and the λ- Martin boundary of the hyperbolic graph X􏰏.<br/> First, we show the properties of the Laplacian on the graph. We observe the eigenfunctions of the Laplacian. The bottom of the spectrum of the Laplacian is positive. Using the properties, we construct the heat kernel of the graph.<br/> After we construct the heat kernel, we show the convergence of λ-Green functions and the Ancona inequality. Using the Ancona inequality, we show that the geometric boundary coincides with the λ-Martin boundary.<br/>