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Journal of the London Mathematical Society Advance Access originally published online on January 8, 2007
Journal of the London Mathematical Society 2007 75(1):1-17; doi:10.1112/jlms/jdl001
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© 2007 London Mathematical Society

A global Riesz decomposition theorem on trees without positive potentials

Joel M. Cohen

University of Maryland
College Park
MD 20742
USA
jcohen{at}umd.edu

Flavia Colonna and David Singman

George Mason University
Fairfax
VA 22030
USA
fcolonna{at}gmu.edu
dsingman{at}gmu.edu

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.

1991 Mathematics Subject Classification 31C05, 05C05(primary), 60J45(secondary).

Received May 12, 2005; revised February 13, 2006; published online January 8, 2007.


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