32 1. Random Walk and Discrete Heat Equation
Example 1.11. Suppose that A is the “discrete ball” of radius r
about the origin,
A = {x
Zd
: |x| r}.
Then every y ∂A satisfies r |y| r + 1. Suppose we start the
random walk at the origin. Then,
r2
E[TA] (r +
1)2.
For any y A, let Vy denote the number of visits to y before
leaving A,
Vy =
TA−1
n=0
1{Sn = y} =

n=0
1{Sn = y, TA n}.
Here we again use the indicator function notation. Note that
E[Vy | S0 = x] =

n=0
P{Sn = y, TA n | S0 = x} =

n=0
pn(x, y; A).
This quantity is of sufficient interest that it is given a name. The
Green’s function GA(x, y) is the function on A × A given by
GA(x, y) = E[Vy | S0 = x] =

n=0
pn(x, y; A).
We define GA(x, y) = 0 if x A or y A. The Green’s function
satisfies GA(x, y) = GA(y, x). This is not immediately obvious from
the first equality but follows from the symmetry of pn(x, y; A). If we
fix y A, then the function f(x) = GA(x, y) satisfies the following:
Lf(y) = −1,
Lf(x) = 0, x A \ {y},
f(x) = 0, x ∂A.
Note that
TA =
y∈A
Vy,
and hence
E[TA | S0 = x] =
y∈A
GA(x, y).
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