Random walks and Kuramochi boundaries of infinite networks
\textit{A. Ancona}, \textit{R. Lyons} and \textit{Y. Peres} [Ann. Probab. 27, No. 2, 970--989 (1999; Zbl 0945.60063)] proved a result on the Royden decomposition; namely, they proved that a function of finite Dirichlet energy on a transient network converges along the random walk almost surely and in \(L^2\). It is introduced an admissible weight \(r\) on the set of edges of a given graph \(G= (V,E)\), which is a positive function satisfying a boundary condition. This, in turn, provides a distance on \(V\), called the geodesic distance. Furthermore, it is introduced the Kuramochi compactification of a connected infinite network \((V,E,r)\). The present paper deals with the Kuramochi boundary of a transient network. The author proves that the random walk converges almost surely to a random variable valued in the harmonic boundary. Furthermore, it is shown that a function of finite Dirichlet energy converges along the random walk to a random variable, almost surely and in \(L^2\).
scientific article; zbMATH DE number 6159566