Prove that the only nonnegative superharmonic functions in R

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

)