14 1. Random Walk and Discrete Heat Equation
Proof. Let A be the event that infinitely many of E1,E2,... occur.
For each integer N, A AN where AN is the event that at least one
of the events EN , EN+1,... occurs. Then,
P(A) P(AN ) = P

n=N
En

n=N
P(En);
but

P(En) implies
lim
N→∞

n=N
P(En) = 0.
Hence, P(A) = 0.
As an example, consider the simple random walk in
Zd,d
3 and
let En be the event that Sn = 0. Then, the estimates of the previous
section show that

n=1
P(En) ∞,
and hence with probability one, only finitely many of the events En
occur. This says that with probability one, the random walk visits
the origin only finitely often.
1.2. Boundary value problems
1.2.1. One dimension: Gambler’s ruin. Suppose N is a positive
integer and a random walker starts at x {0, 1,...,N}. Let Sn
denote the position of the walker at time n. Suppose the walker stops
when the walker reaches 0 or N. To be more precise, let
T = min {n : Sn = 0 or N} .
Then the position of the walker at time n is given by
ˆ
S
n
= Sn∧T
where n T means the minimum of n and T . It is not hard to see
that with probability one, T ∞, i.e., eventually the walker will
reach 0 or N and then stop. Our goal is to try to figure out which
point it stops at. Define the function F : {0,...,N} [0, 1] by
F (x) = P{ST = N | S0 = x}.
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