Separately superharmonic functions in product networks

Victor Anandam¹

¹ Institute of Mathematical Sciences Chennai, Tamil Nadu, India 600 113

Annales Polonici Mathematici, Tome 113 (2015) no. 3, pp. 209-241

Résumé

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 $X\times Y$.

Détail

DOI : 10.4064/ap113-3-1

Keywords: times cartesian product locally finite connected networks have reversible conductance represent random walks known times recurrent recurrent proved here non probabilistic methods using properties separately superharmonic functions class functions product network times dirichlet solution balayage minimum principle etc obtained unique integral representation given function belongs restricted subclass positive separately superharmonic functions times