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A global Riesz decomposition theorem on trees without positive potentials. (English) Zbl 1129.31004

J. Lond. Math. Soc., II. Ser. 75, No. 1, 1-17 (2007); corrigendum ibid. 83, No. 3, 810 (2011).
The authors define in section 2 a non-negative unbounded subharmonic function \(H\) harmonic except at \(e\) which plays a key role in the study of potential theory on recurrent trees. In section 3 an explicit construction of the \(H\)-potentials with one-point harmonic support is given, while in section 4 an explicit formula for the flux of a function \(s\) which is superharmonic outside a finite set in terms of its Laplacian is also provided. In section 5 it is shown that a function of finite flux is the sum of a harmonic function and the difference of two \(H\)-potentials. Finally, in section 6 the authors provide a description of the connections between \(H\)-potentials and the potentials developed for Markov chains by other authors.

MSC:

31C05 Harmonic, subharmonic, superharmonic functions on other spaces
05C05 Trees
31C20 Discrete potential theory
60J45 Probabilistic potential theory
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