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
This proves that P[X ∞ ∈ ∂
w
K ] = 1 and by shift-invariance
P[X
i
∞ ∈ ∂
w
K ∀i ∈ ] = 1.
3.3.31
Recurrent a
S
, P[X
i
∞ = X
j
∞ ∀i, j ∈ ] = 1: Applying Lemma 3.1.4 to
the process X
∞
, we see that
∂ ∂
t
C
∞ t
i =
j
a
S
j − iC
∞ t
j − C
∞ t
i + 2δ
i 0
E[tr wX
∞
t]. 3.3.32
Here C
∞ t
i = C
∞
i , where we use the notation CovX
i
∞, X
j
∞ = C
∞
j − i.
3.3.33 Note that
∂ ∂
t
C
∞ t
i = 0, while E[trwX
∞
t] = 0 by 3.3.30. Inserting this
into 3.3.32, we get
j
a
S
j − iC
∞
j − C
∞
i = 0.
3.3.34
This means that C
∞
is a bounded a
S
-harmonic function. By the Choquet-Deny theorem which follows easily from Lemma 3.3.3 –see [25], Chapter II, Theorem
1.5 it follows that C
∞
is constant. We write ˜ X
i
t : = X
i
t − θ with θ as in
Lemma 3.1.4 and note that by Cauchy-Schwarz C
∞
j − i = E[ ˜X
i
∞ · ˜X
j
∞] ≤ E[| ˜X
i
∞|
2
]
1 2
E[ | ˜X
j
∞|
2
]
1 2
= E[| ˜X ∞|
2
] = C
∞
0, 3.3.35
where equality holds if and only if P[X
i
∞ = X
j
∞] = 1. This proves that P[X
i
∞ = X
j
∞ ∀i, j ∈ ] = 1. 3.3.36
Transient a
S
, P[X
i
∞ ∈ ∂
w
K ] 1
∀i ∈ : We start by noting that the ergodic-
ity of
L
X 0 implies that for each i ∈
lim
t →∞
j
P
t
j − iC
j = 0.
3.3.37
To see this, write ˜ X
i
0 : = X
i
− θ as before and note that by Lemma 3.3.2 and 3.3.3
lim
t →∞
E
j
P
t
j ˜ X
j 2
= 0. 3.3.38
Here E
j
P
t
j ˜ X
j 2
=
j k
P
t
j P
t
kE[ ˜ X
j
0 ˜ X
k
0] =
j k
P
t
j P
t
kC k
− j =
i j
P
t
j P
t
i + jC
i =
i j
P
t
j P
t
i − j
C i
=
i
P
2t
i C i ,
3.3.39
where all infinite sums are absolutely convergent and we have used that, by the symmetry of a
S
, P
t
i = P
t
−i. Formula 3.3.38 and 3.3.39 show that 3.3.37 holds for i
= 0. Using Lemma 3.3.3 we can easily generalize this to arbitrary i
∈ . By Lemma 3.1.4 and Lemma 3.3.1 we have the representation
C
t
i =
j
P
t
j − iC
j + 2
t
P
s
− iE[trwX t
− s]ds. 3.3.40 Taking the limit t
→ ∞ we get with the help of 3.3.37 that C
∞
i = lim
t →∞
2
t
P
s
− iE[trwX t
− s]ds = 2E[trwX
∞]
∞
P
t
− idt, 3.3.41
where we use the notation in 3.3.33. Let us assume for the moment that
E[tr wX ∞] = 0. Then P[X
∞ ∈ ∂
w
K ] = 1. On the other hand,
3.3.41 gives C
∞
= 0 and hence P[X ∞ = θ] = 1. This contradicts our
assumption that θ ∈ ∂
w
K and we conclude that E[tr wX ∞] 0. Therefore
P[X ∞ ∈ ∂
w
K ] 1 and the claim follows from shift-invariance.
Transient a
S
, P[X
i
∞ = X
j
∞] 1 ∀i = j ∈ : Let I
t t
≥0
be the random walk with kernel a
S
. Let τ
i
be the stopping time τ
i
: = inf{t ≥ 0 : I
t
= i} i
∈ . 3.3.42
It is easy to see that for all i ∈
∞
P
t
− idt = P
i
[τ ∞]
∞
P
t
0dt. 3.3.43
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