2.2 Central limit theorems for Markov chains
We prepare central limit theorems for Markov chains, which is obtained by perturbation of random walks.
Lemma 2.2.1. Let Z
t t
≥0
, P
x
be a continuous-time random walk on Z
d
starting from x, with the generator
L
Z
f x = X
y ∈Z
d
a
y −x
f y − f x, where we assume that
X
x ∈Z
d
|x|
2
a
x
∞. Then, for any B
∈ σ[Z
u
; u ∈ [0, ∞], x ∈ Z
d
, and f ∈ C
b
R
d
, lim
t →∞
P
x
[ f Z
t
− mt p
t : B] = P
x
B Z
R
d
f dν, where m =
P
x ∈Z
d
x a
x
and ν is the Gaussian measure with Z
R
d
x
i
dνx = 0, Z
R
d
x
i
x
j
dνx = X
x ∈Z
d
x
i
x
j
a
x
, i, j = 1, .., d. 2.11
Proof: By subtracting a constant, we may assume that R
R
d
f dν = 0. We first consider the case that B
∈ F
s def
= σ[Z
u
; u ∈ [0, s]] for some s ∈ 0, ∞. It is easy to see from the central limit theorem for
Z
t
that for any x ∈ Z
d
, lim
t →∞
P
x
[ f Z
t −s
− mt p
t] = 0. With this and the bounded convergence theorem, we have
P
x
[ f Z
t
− mt p
t : B] = P
x
[P
Z
s
[ f Z
t −s
− mt p
t] : B] −→ 0 as t ր ∞.
Next, we take B ∈ σ[Z
u
; u ∈ [0, ∞]. For any ǫ 0, there exist s ∈ 0, ∞ and e
B ∈ F
s
such that P
x
[|1
B
− 1
e B
|] ǫ. Then, by what we already have seen, lim
t →∞
P
x
[ f Z
t
− mt p
t : B] ≤ lim
t →∞
P
x
[ f Z
t
− mt p
t : e B] +
k f kǫ = k f kǫ, where
k f k is the sup norm of f . Similarly, lim
t →∞
P
x
[ f Z
t
− mt p
t : B] ≥ −k f kǫ.
Since ǫ 0 is arbitrary, we are done.
Lemma 2.2.2. Let Z = Z
t t
≥0
, P
x
be as in Lemma 2.2.1 and and D ⊂ Z
d
be transient for Z. On the other hand, let e
Z = e Z
t t
≥0
, e P
x
be the continuous-time Markov chain on Z
d
starting from x, with the generator
L
e Z
f x = X
y ∈Z
d
e a
x, y
f y − f x,
971
where we assume that e
a
x, y
= a
y −x
if x 6∈ D ∪ { y} and that D is also transient for e
Z. Furthermore, we assume that a function v : Z
d
→ R satisfies v
≡ 0 outside D, e
P
z
exp
Z
∞
|ve Z
t
|d t
∞ for some z ∈ Z
d
. Then, for f
∈ C
b
R
d
, lim
t →∞
e P
z
exp
Z
t
ve Z
u
du
f e Z
t
− mt p
t
= e P
z
exp
Z
∞
ve Z
t
d t Z
R
d
f dν, where ν is the Gaussian measure such that 2.11 holds.
Proof: Define H
D
e Z =
inf {t ≥ 0 ; e
Z
t
∈ D}, T
D
e Z = sup
{t ≥ 0 ; e Z
t
∈ D}, e
t
= exp
Z
t
ve Z
s
ds
. Then, for s t,
e P
z
e
t
f e Z
t
− mt p
t
= e
P
z
e
t
f e Z
t
− mt p
t : T
D
e Z s
+ ǫ
s,t
= e
P
z
e
s
f e Z
t
− mt p
t : T
D
e Z s
+ ǫ
s,t
= e
P
z
h e
s
1
e Z
s
6∈D
e P
e Z
s
f e
Z
t −s
− mt p
t : H
D
e Z =
∞ i
+ ǫ
s,t
, 2.12
where |ǫ
s,t
| = e
P
z
e
t
f e Z
t
− mt p
t : T
D
e Z
≥ s
≤ k f keP
z
exp
Z
∞
|ve Z
t
|d t
: T
D
e Z
≥ s
→ 0 as s → ∞. We now observe that
e P
x
· |H
D
e Z =
∞ = P
x
· |H
D
Z = ∞ for x 6∈ D, where H
D
Z is defined similarly as H
D
e Z. Hence, for x
6∈ D and fixed s 0, we have by Lemma 2.2.1 that
lim
t →∞
e P
x
f e
Z
t −s
− mt p
t : H
D
e Z =
∞
= e P
x
[H
D
e Z =
∞] Z
R
d
f dν. Therefore,
lim
t →∞
e P
z
h e
s
1
e Z
s
6∈D
e P
e Z
s
f e
Z
t −s
− mt p
t : H
D
e Z =
∞ i
= e
P
z
h e
s
1
e Z
s
6∈D
e P
e Z
s
[H
D
e Z =
∞] i Z
R
d
f dν =
e P
z
e
s
: T
D
e Z s
Z
R
d
f dν.
972
Thus, letting t → ∞ first, and then s → ∞, in 2.12, we get the lemma.
2.3 A Nash type upper bound for the Schrödinger semi-group
Kategori