Harmonic Functions and Potentials on Finite or Infinite Networks (Lecture Notes of the Unione Matematica Italiana Book 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. Front Matter....Pages i-x Laplace Operators on Networks and Trees....Pages 1-20 Potential Theory on Finite Networks....Pages 21-44 Harmonic Function Theory on Infinite Networks....Pages 45-90 Schrödinger Operators and Subordinate Structures on Infinite Networks....Pages 91-108 Polyharmonic Functions on Trees....Pages 109-132 Back Matter....Pages 133-141