Abstract
Let
T be a tree rooted at
e endowed with a nearest-neighbor transition probability that yields a recurrent random walk. We show that there exists a function
K biharmonic off
e whose Laplacian has potential theoretic importance and, in addition, has the following property: Any function
f on
T which is biharmonic outside a finite set has a representation, unique up to addition of a harmonic function, of the form
f
=
β
K
+
B
+
L
, where
β a constant,
B is a biharmonic function on
T, and
L is a function, subject to certain normalization conditions, whose Laplacian is constant on all sectors sufficiently far from the root. We obtain a characterization of the functions biharmonic outside a finite set whose Laplacian has 0 flux similar to one that holds for a function biharmonic outside a compact set in
R
n
for
n
=
2
,
3
, and 4 proved by Bajunaid and Anandam. Moreover, we extend the definition of flux and, under certain restrictions on the tree, we characterize the functions biharmonic outside a finite set that have finite flux in this extended sense.