22 1. Random Walk and Discrete Heat Equation

Theorem 1.6. Suppose A is a proper subset of Zd such that for all

x ∈ Zd,

lim

n→∞

P{TA n | S0 = x} = 0.

Suppose F : ∂A → R is a bounded function. Then there is a unique

bounded extension of F to A that satisfies (1.9). It is given by

F0(x) = E[F (STA ) | S0 = x] =

y∈∂A

P{STA = y | S0 = x} F (y).

1.3. Heat equation

We will now introduce a mathematical model for heat flow. Let A be a

finite subset of

Zd

with boundary ∂A. We set the temperature at the

boundary to be zero at all times and, as an initial condition, set the

temperature at x ∈ A to be pn(x). At each integer time unit n, the

heat at x at time n is spread evenly among its 2d nearest neighbors.

If one of those neighbors is a boundary point, then the heat that goes

to that site is lost forever. A more probabilistic view of this is given

by imagining that the temperature in A is controlled by a very large

number of “heat particles”. These particles perform random walks on

A until they leave A at which time they are killed. The temperature

at x at time n, pn(x) is given by the density of particles at x. Either

interpretation gives a difference equation for the temperature pn(x).

For x ∈ A, the temperature at x is given by the amount of heat going

in from neighboring sites,

pn+1(x) =

1

2d

|y−x|=1

pn(y).

If we introduce the notation ∂npn(x) = pn+1(x) − pn(x), we get the

heat equation

(1.12) ∂npn(x) = Lpn(x), x ∈ A,

where L denotes the discrete Laplacian as before. The initial temper-

ature is given as the initial condition

(1.13) p0(x) = f(x), x ∈ A.

We rewrite the boundary condition as

(1.14) pn(x) = 0, x ∈ ∂A.