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A global Riesz decomposition theorem on trees without positive potentials
Author(s) -
Cohen Joel M.,
Colonna Flavia,
Singman David
Publication year - 2007
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdl001
Subject(s) - subharmonic function , mathematics , potential theory , markov chain , harmonic function , random walk , harmonic , connection (principal bundle) , function (biology) , pure mathematics , set (abstract data type) , decomposition , finite set , simple (philosophy) , statistical physics , discrete mathematics , mathematical analysis , quantum mechanics , physics , computer science , statistics , geometry , evolutionary biology , biology , programming language , philosophy , epistemology , ecology
We study the potential theory of trees with nearest‐neighbor transition probability that yields a recurrent random walk and show that, although such trees have no positive potentials, many of the standard results of potential theory can be transferred to this setting. We accomplish this by defining a non‐negative function H , harmonic outside the root e and vanishing only at e , and a substitute notion of potential which we call H ‐ potential . We define the flux of a superharmonic function outside a finite set of vertices, give some simple formulas for calculating the flux and derive a global Riesz decomposition theorem for superharmonic functions with a harmonic minorant outside a finite set. We discuss the connection of the H ‐potentials with other notions of potentials for recurrent Markov chains in the literature.