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).
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