38 1. Random Walk and Discrete Heat Equation
(Although (1.29) gives this only for |x| = 1, we can apply the estimate
O(|x|) times to get this estimate.) By letting R we see that
F (x) = F (0). Since this is true for every x, F must be constant.
1.5.1. Exterior Dirichlet problem. Consider the following prob-
lem. Suppose A is a cofinite subset of
Zd,
i.e., a subset such that
Zd
\ A is finite. Suppose F :
Zd
\ A R is given. Find all bounded
functions on
Zd
that are harmonic on A and take on the boundary
value F on
Zd
\ A. If A =
Zd,
then this was answered at the end
of the last section; the only possible functions are constants. For the
remainder of this section we assume that A is nontrivial, i.e.,
Zd
\ A
is nonempty.
For d = 1, 2, there is, in fact, only a single solution. Suppose
F is such a function with L = sup |F (x)| ∞. Let Sn be a simple
random walk starting at x Zd, and let T = TA be the first time n
with Sn A. If d 2, we know that the random walk is recurrent
and hence T with probability one. As done before, we can see
that Mn = F (Sn∧T ) is a martingale and hence
F (x) = M0 = E[Mn] = E[F (ST ) 1{T n}] + E [F (Sn) 1{T n}] .
The monotone convergence theorem tells us that
lim
n→∞
E[F (ST ) 1{T n}] = E[F (ST )].
Also,
lim
n→∞
|E [F (Sn) 1{T n}]| lim
n→∞
L P{T n} = 0.
Therefore,
F (x) = E[F (ST ) | S0 = x],
which is exactly the same solution as we had for bounded A.
If d 3, there is more than one solution. In fact,
f(x) = P{TA = | S0 = x},
is a bounded function that is harmonic in A and equals zero on
Zd
\
A. The next theorem shows that this is essentially the only new
function that we get. We can interpret the theorem as saying that
the boundary value determines the function if we include as a
boundary point.
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