34 1. Random Walk and Discrete Heat Equation
If d 3, then the expected number of visits to a point is finite
and we can define the (whole space) Green’s function
G(x, y) = lim
A↑Zd
GA(x, y) = E

n=0
1{Sn = y} | S0 = x
=

n=0
P{Sn = y | S0 = x}.
It is a bounded function. In fact, if τy denotes the smallest n 0
such that Sn = y, then
G(x, y) = P{τy | S0 = x} G(y, y)
= P{τy | S0 = x} G(0, 0) G(0, 0) ∞.
The function G is symmetric and satisfies a translation invariance
property: G(x, y) = G(0,y x). For fixed y, f(x) = G(x, y) satisfies
Lf(y) = −1, Lf(x) = 0, x = y, f(x) 0 as x ∞.
1.5. Space of harmonic functions
If d = 1, the only harmonic functions f : Z R are the linear
functions f(x) = ax + b. This follows since Lf(x) = 0 implies
f(x + 1) = 2f(x) f(x 1), f(x 1) = 2f(x) f(x + 1).
If f(0),f(1) are given, then the value of f(x) for all other x is deter-
mined uniquely by the equations above. In other words, the space of
harmonic functions is a vector space of dimension 2.
For d 1, the space of harmonic functions on Zd is still a vector
space, but it is infinite dimensional. Let us consider the case d = 2.
For every positive number t and real r let
(1.24) f(x1,x2) = ft,r(x1,x2) =
erx1
sin(tx2).
Using the sum rule for sine, we get
Lf(x1,x2) =
1
2
f(x1,x2) [cosh(r) + cos(t) 2].
If we choose r such that
cosh(r) + cos(t) = 2,
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