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,