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Harmonic functions on cartesian products of trees with finite graphs
Ist Teil von
Journal of functional analysis, 1991-12, Vol.102 (2), p.379-400
Ort / Verlag
Orlando, FL: Elsevier Inc
Erscheinungsjahr
1991
Link zum Volltext
Quelle
Elsevier Journal Backfiles on ScienceDirect (DFG Nationallizenzen)
Beschreibungen/Notizen
Let
G
be a graph which is the Cartesian product of an infinite, locally finite tree
T
and a finite, connected graph
A
. On
G
, consider a stochastic transition operator
P
giving rise to a transient random walk and such that positive transitions occur only along the edges of
G
. We construct a matrix-valued kernel on
T
, which extends naturally in the second variable to the space of ends Ω of
T
. This kernel is used to derive a unique integral representation over Ω of all—not necessarily positive—functions on
G
which are harmonic with respect to
P
. We explain the relation with the Martin boundary and the positive harmonic functions and, as a particular case, we show what happens when
A
arises from a finite abelian group and
P
is compatible with the structure of
A
.