# Harmonic Functions and Potentials on Finite or Infinite Networks

Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 12)

Advertisement

Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 12)

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.

31C20; 31D05; 30F20; 31A30; 15A09 Discrete Laplace and Schrödinger operators Discrete harmonic functions and potentials Flux in parabolic networks Ploypotentials in hyperbolic networks Subordinate harmonic structures

- DOI https://doi.org/10.1007/978-3-642-21399-1
- Copyright Information Springer-Verlag Berlin Heidelberg 2011
- Publisher Name Springer, Berlin, Heidelberg
- eBook Packages Mathematics and Statistics
- Print ISBN 978-3-642-21398-4
- Online ISBN 978-3-642-21399-1
- Series Print ISSN 1862-9113
- Buy this book on publisher's site