Prove that the only nonnegative superharmonic functions in R are the constants and use this to find g?(x) when are for a 0 0 for z 0 Solution Let h be a super harmonic function, i , j two states, ( X 1 , X 2 ) a coupling as in statement, and t the coupling time. Without loss of generality, we can assume that X 1 n = X n , for n t . Then, with E denoting the expectation on the probability space where the coupling happens, for each n 2 N , we have h ( j ).

1. Prove that the only nonnegative superharmonic functions in R are the constants and use this to
find g?(x) when are for a 0 0 for z 0
Solution
Let
h
be a super harmonic function,
i
,
j
two states,
(
X
1
,
X
2
)
a coupling
as in statement, and
t
the coupling time. Without loss of generality,
we can assume that
X
1
n
=
X
n
, for
n
t
. Then, with
E

2. denoting the
expectation on the probability space where the coupling happens, for
each
n
2
N
, we have
h
(
j
)