3.3.3 Spatially ergodic measures
The σ -field of shift-invariant events is
S
: = {A ∈
B
K : T
−1 i
A = A ∀i ∈ }.
3.3.19 A probability measure µ on K
is spatially ergodic if for every A ∈
S
either µ
A = 1 or µA = 0. We state the following standard ergodic theorem in L
2
without proof see [24].
Lemma 3.3.2 For n
= 1, 2, . . ., let p
n
: → [0, ∞ be functions satisfying
i
p
n
i = 1 and
lim
n →∞
k
| p
n
i − k − p
n
j − k| = 0
∀i, j ∈ . 3.3.20
Let X = X
i i
∈
be a family of K -valued random variables with shift-invariant ergodic law
L
X . If E[X ]
= θ, then lim
n →∞
E θ
−
i
p
n
i X
i 2
= 0. 3.3.21
In our case, probability distributions p
n
satisfying 3.3.20 will arise in the follow- ing way.
Lemma 3.3.3 Let P : [0,
∞ × →
R
be as in 3.3.10. Then for any i, j ∈ :
lim
t →∞
k
|P
t
i − k − P
t
j − k| = 0.
3.3.22
Proof of Lemma 3.3.3: We use the Ornstein coupling [25]. To see how this works for random walks on arbitrary Abelian groups, let
⊂ be a set such that a
S
k 0 for each k
∈ and such that of each k ∈ with a
S
k 0, either k or −k but
not both is in . By irreducibility, we can decompose j − i as
j − i =
k ∈
nkk, 3.3.23
where nk ∈
Z
and only a finite number of nk’s are non-zero. We may couple two random walks starting in points i and j in such a way that they always make
a jump of size k or −k at the same time. They choose k or −k independently of
each other, until the walk starting in j has made nk more of these jumps than the walk starting in i . After that, they choose either both k or both
−k. This coupling is obviously successful and Lemma 3.3.3 now follows easily.
3.3.4 Proof of Theorem 3.1.3
The proof consists of several steps. X
∞ is an invariant law: By this we mean that there exists a shift-invariant
solution X
∞
to the martingale problem for the operator A in 3.1.10 such that
L
X
∞
t = X ∞
∀t ≥ 0. 3.3.24
To see this, define solutions to the martingale problem for A by X
n
t : = X t
n
+ t, 3.3.25
where t
n
is some sequence tending to infinity. By Lemma 3.2.2 we can find a subsequence X
nk
that converges weakly to some solution X
∞
to the martingale problem for A. Now
L
X
∞
t = lim
n →∞
L
X t
n
+ t =
L
X ∞
∀t ≥ 0, 3.3.26
where the limit denotes weak convergence of probability measures on K . It is
easy to see that X
∞
is shift-invariant.
Recurrent a
S
, P[X
i
∞ ∈ ∂
w
K
∀i ∈ ] = 1: Let us write
CovX
∞ i
t, X
∞ j
t = C
∞ t
j − i
3.3.27 for covariances belonging to the process X
∞
constructed above. We can apply Lemma 3.1.4 to this process. Lemma 3.3.1 now leads to the representation
C
∞ t
i −
j
P
t
j − iC
∞
j = 2
t
P
s
− iE[trwX
∞
t − s]ds. 3.3.28
By the compactness of the state space K , the left-hand side of 3.3.28 is bounded. The right-hand side is equal to
2E[tr wX ∞]
t
P
s
− ids. 3.3.29
By the recurrence of the random walk with kernel a
S
, the integral in 3.3.29 di- verges as t tends to infinity, and therefore 3.3.28 can only hold if
E[tr wX ∞] = 0.
3.3.30
Kategori