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.