22 1. Random Walk and Discrete Heat Equation
Theorem 1.6. Suppose A is a proper subset of Zd such that for all
x Zd,
lim
n→∞
P{TA n | S0 = x} = 0.
Suppose F : ∂A R is a bounded function. Then there is a unique
bounded extension of F to A that satisfies (1.9). It is given by
F0(x) = E[F (STA ) | S0 = x] =
y∈∂A
P{STA = y | S0 = x} F (y).
1.3. Heat equation
We will now introduce a mathematical model for heat flow. Let A be a
finite subset of
Zd
with boundary ∂A. We set the temperature at the
boundary to be zero at all times and, as an initial condition, set the
temperature at x A to be pn(x). At each integer time unit n, the
heat at x at time n is spread evenly among its 2d nearest neighbors.
If one of those neighbors is a boundary point, then the heat that goes
to that site is lost forever. A more probabilistic view of this is given
by imagining that the temperature in A is controlled by a very large
number of “heat particles”. These particles perform random walks on
A until they leave A at which time they are killed. The temperature
at x at time n, pn(x) is given by the density of particles at x. Either
interpretation gives a difference equation for the temperature pn(x).
For x A, the temperature at x is given by the amount of heat going
in from neighboring sites,
pn+1(x) =
1
2d
|y−x|=1
pn(y).
If we introduce the notation ∂npn(x) = pn+1(x) pn(x), we get the
heat equation
(1.12) ∂npn(x) = Lpn(x), x A,
where L denotes the discrete Laplacian as before. The initial temper-
ature is given as the initial condition
(1.13) p0(x) = f(x), x A.
We rewrite the boundary condition as
(1.14) pn(x) = 0, x ∂A.
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