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Separately superharmonic functions in product networks

Volume 113 / 2015

Victor Anandam Annales Polonici Mathematici 113 (2015), 209-241 MSC: Primary 31C20; Secondary 31C10, 32U05. DOI: 10.4064/ap113-3-1

Abstract

Let $X\times Y$ be the Cartesian product of two locally finite, connected networks that need not have reversible conductance. If $X,Y$ represent random walks, it is known that if $X\times Y$ is recurrent, then $X,Y$ are both recurrent. This fact is proved here by non-probabilistic methods, by using the properties of separately superharmonic functions. For this class of functions on the product network $X\times Y$, the Dirichlet solution, balayage, minimum principle etc. are obtained. A unique integral representation is given for any function that belongs to a restricted subclass of positive separately superharmonic functions in $X\times Y$.

Authors

  • Victor AnandamInstitute of Mathematical Sciences
    Chennai, Tamil Nadu, India 600 113
    e-mail

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