1.2. Boundary value problems 19
The operator L is often called the (discrete) Laplacian. We let Sn be
a simple random walk in Zd. Then we can write
LF (x) = E[F (S1) F (S0) | S0 = x].
We say that F is (discrete) harmonic at x if LF (x) = 0; this is an
example of a mean-value property. The corresponding boundary value
problem we will state is sometimes called the Dirichlet problem for
harmonic functions.

The term linear operator is often used for a linear function whose
domain is a space of functions. In our case, the domain is the space of functions
on the finite set A which is isomorphic to
where K = #(A). In this case a
linear operator is the same as a linear transformation from linear algebra. We
can think of Q and L as K × K matrices. We can write Q = [Q(x, y)]x,y∈A
where Q(x, y) = 1/(2d) if |x y| = 1 and otherwise Q(x, y) = 0. Define
Qn(x, y) by
= [Qn(x, y)]. Then Qn(x, y) is the probability that the
random walk starts at x, is at site y at time n, and and has not left the set A
by time n.
Dirichlet problem for harmonic functions. Given a set A Zd
and a function F : ∂A R find an extension of F to A such that F
is harmonic in A, i.e.,
(1.9) LF (x) = 0 for all x A.
For the case d = 1 and A = {1,...,N 1}, we were able to guess
the solution and then verify that it is correct. In higher dimensions,
it is not so obvious how to give a formula for the solution. We will
show that the last proof for d = 1 generalizes in a natural way to
d 1. We let TA = min{n 0 : Sn A}.
Theorem 1.5. If A
is finite, then for every F : ∂A R, there
is a unique extension of F to A that satisfies (1.9). It is given by
F0(x) = E[F (STA ) | S0 = x] =
P{STA = y | S0 = x} F (y).
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