Let X×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×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×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×Y.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap113-3-1,
author = {Victor Anandam},
title = {Separately superharmonic functions in product networks},
journal = {Annales Polonici Mathematici},
volume = {113},
year = {2015},
pages = {209-241},
zbl = {1323.31014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap113-3-1}
}
Victor Anandam. Separately superharmonic functions in product networks. Annales Polonici Mathematici, Tome 113 (2015) pp. 209-241. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap113-3-1/