44 1. Random Walk and Discrete Heat Equation

are given functions. Show that there is a unique function pn(x),n =

0, 1, 2,...,x ∈ A satisfying the following:

pn(x) = F (x), x ∈ ∂A,

∂pn(x) = LF (x), x ∈ A.

Show that p(x) = limn→∞ pn(x) exists and describe the limit func-

tion p.

Exercise 1.13. Prove (1.26) and (1.28).

Exercise 1.14. Find the analogue of the formula (1.25) for the d-

dimensional cube

A = {(x1,...,xd) ∈

Zd

: xj = 1,...,N − 1}

Exercise 1.15. Suppose F is a harmonic function on Zd such that

lim

|x|→∞

|F (x)|

|x|

= 0.

Show that F is constant.

Exercise 1.16. The relaxation method for solving the Dirichlet prob-

lem is the following. Suppose A is a bounded subset of Zd and

F : ∂A → R is a given function. Define the functions Fn(x),x ∈ A as

follows:

Fn(x) = F (x) for all n if x ∈ ∂A.

F0(x),x ∈ A, is defined arbitrarily,

and for n ≥ 0,

Fn+1(x) =

1

2d

|x−y|=1

Fn(y), x ∈ A.

Show that for any choice of initial function F0 on A,

lim

n→∞

Fn(x) = F (x), x ∈ A,

where F is the solution to the Dirichlet problem with the given bound-

ary value. (Hint: Compare this to Exercise 1.12.)

Exercise 1.17. Let Sn denote a d-dimensional simple random walk

and let Rn,...,Rn

1 d

denote the number of steps taken in each of

the d components. Show that for all n 0, the probability that

R2n,...,R2n 1 d are all even is 2−(d−1).