An Invitation to Random Schr¨odinger operators - FernUniversität in ...

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measure preserving transformations we call a set A ∈ F invariant (under {T i }) if
Ti −1 A = A for all i ∈ Z d .
A family {T i } of measure preserving transformations on a probability space (Ω, F, P)
is called ergodic (with respect to the probability measure P) if any invariant A ∈ F
has probability zero or one. A stochastic process {X i } i∈Z d is called ergodic, if
there exists an ergodic family of measure preserving transformations {T i } i∈Z d such
that X i (T j ω) = X i−j (ω).
Our main example of an ergodic stochastic process is given by independent, identically
distributed random variables X i (ω) = V ω (i) (a random potential on Z d ).
Due to the independence of the random variables the probability measure P (on
Ω = R Zd ) is just the infinite product measure of the probability measure P 0 on R
given by P 0 (M) = P(V ω (0) ∈ M). P 0 is the distribution of V ω (0).
It is easy to see that the shift operators
(T i ω) j = ω j−i
form a family of measure preserving transformations on R Zd in this case.
It is not hard to see that the family of shift operators is ergodic with respect to the
product measure P. One way to prove this is to show that
P(T −1
i
A ∩ B) → P(A) P(B) (4.2)
as || i|| ∞ → ∞ for all A, B ∈ F. This is obvious if both A and B are of the form
(4.1). Moreover, the system of sets A, B for which (4.2) holds is a σ-algebra, thus
(4.2) is true for the σ-algebra generated by sets of the form (4.1), i.e. on F.
Now let M be an invariant set. Then (4.2) (with A = B = M) gives
P(M) = P(M ∩ M) = P(T −1
i
M ∩ M) → P(M) 2
proving that M has probability zero or one.
We will need two more results on ergodicity.
PROPOSITION 4.1. Let {T i } i∈Z d be an ergodic family of measure preserving transformations
on a probability space (Ω, F, P ). If a random variable Y is invariant
under {T i } (i.e. Y (T i ω) = Y (ω) for all i ∈ Z d ) then Y is almost surely constant,
i.e. there is a c ∈ R, such that P(Y = c) = 1.
We may allow the values ±∞ for Y (and hence for c) in the above result.The proof
is not difficult (see e.g. [30]).
The final result is the celebrated ergodic theorem by Birkhoff. It generalizes the
strong law of large number to ergodic processes.
We denote by E(·) the expectation with respect to the probability measure P.
THEOREM 4.2. If {X i } i∈Z d is an ergodic process and E(|X 0 |) < ∞ then
1 ∑
lim
L→∞ (2L + 1) d X i → E(X 0 )
i∈Λ L
for P-almost all ω.