18 1. Random Walk and Discrete Heat Equation
Considering the case F (0) = 0,F (N) = 1 gives P{ST = N | S0 =
x} = x/N and for more general boundary conditions,
F (x) = F (0) +
x
N
[F (N) F (0)].
One nice thing about the last proof is that it was not necessary
to have already guessed the linear functions as solutions. The proof
produces these solutions.
1.2.2. Higher dimensions. We will generalize this result to higher
dimensions. We replace the interval {1,...,N} with an arbitrary
finite subset A of Zd. We let ∂A be the (outer) boundary of A defined
by
∂A = {z
Zd
\ A : dist(z, A) = 1},
and we let A = A ∂A be the “closure” of A.
Figure 3. The white dots are A and the black dots are ∂A

The term closure may seem strange, but in the continuous analogue,
A will be an open set, ∂A its topological boundary and
A = A ∂A its
topological closure.
We define the linear operators Q, L on functions by
QF (x) =
1
2d
y∈Zd,|x−y|=1
F (y),
LF (x) = (Q I)F (x) =
1
2d
y∈Zd,|x−y|=1
[F (y) F (x)].
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