1.4. Expected time to escape 33
Theorem 1.12. Suppose A is a bounded subset of Zd, and g : A R
is a given function. Then the unique function F : A R satisfying
F (x) = 0, x ∂A,
LF (x) = −g(x), x A,
is
(1.22) F (x) = E


TA−1
j=0
g(Sj) | S0 =
x⎦

=
y∈A
g(y) GA(x, y).
We have essentially already proved this. Uniqueness follows from
the fact that if F, F1 are both solutions, then F F1 is harmonic in
A with boundary value 0 and hence equals 0 everywhere. Linearity
of L shows that
(1.23) L


y∈A
g(y) GA(x,
y)⎦

=
y∈A
g(y) LGA(x, y) = −g(x).
The second equality in (1.22) follows by writing
TA−1
j=0
g(Sj) =
TA−1
j=0 y∈A
g(y) 1{Sj = y}
=
y∈A
g(y)
TA−1
j=0
1{Sj = y}
=
y∈A
g(y) Vy.
We can consider the Green’s function as a matrix or operator,
GAg(x) =
x∈A
GA(x, y) g(y).
Then (1.23) can be written as
−LGAg(x) = g(x),
or GA =
[−L]−1.
For this reason the Green’s function is often referred
to as the inverse of (the negative of) the Laplacian.
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