Overview

Random walks, Markov chains and electrical networks serve as an introduction to the study of real-valued functions on finite or infinite graphs, with appropriate interpretations using probability theory and current-voltage laws. The relation between this type of function theory and the (Newton) potential theory on the Euclidean spaces is well-established. The latter theory has been variously generalized, one example being the axiomatic potential theory on locally compact spaces developed by Brelot, with later ramifications from Bauer, Constantinescu and Cornea. A network is a graph with edge-weights that need not be symmetric. This book presents an autonomous theory of harmonic functions and potentials defined on a finite or infinite network, on the lines of axiomatic potential theory. Random walks and electrical networks are important sources for the advancement of the theory.

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9783642213984

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From the reviews: In this book a potential-theoretic style of the theory is built into the framework of finite or infinite networks. The motivation of the book is to build a function theory on networks reflecting ideas of potential theory on locally compact spaces. ... The book is written in a reader-friendly way and contains various potential-theoretic results ... . (Sirkka-Liisa Eriksson, Zentralblatt MATH, Vol. 1239, 2012)

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