Potential theory on trees and mutliplication operators

Abstract

The article surveys a number of potential theory results in the discrete setting of trees and in an application to complex analysis. On trees for which the associated random walk is recurrent, we discuss Riesz decomposition, flux, a type of potential called H-potential, and present a new result dealing with the boundary behaviour of H-potentials on a specific recurrent homogeneous tree. On general trees we discuss Brelot structures and their classification. On transient homogeneous trees we discuss clamped and simplysupported biharmonic Green functions.We also describe an application of potential theory, namely a certain minimum principle for multiply superharmonic functions, that is used to prove a result concerning the norm of a class of multiplication operators from H ∞ (D) to the Bloch space on D, where D is a bounded symmetric domain. The proof of the minimum principle involves the use of the Cartan-Brelot topology.2010 Mathematics Subject Classification. Primary 31C20; Secondary 31A30, 31C05, 31D05.