Separately superharmonic functions in product networks
Tom 113 / 2015
Annales Polonici Mathematici 113 (2015), 209-241 MSC: Primary 31C20; Secondary 31C10, 32U05. DOI: 10.4064/ap113-3-1
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 $X\times Y$.