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
We will use the following lemma to prove 1.16. The lemma can be generalized to symmetric Markov chains on more general graphs. However, we restrict ourselves to random walks on Z
d
, since it is enough for our purpose.
Lemma 2.3.1. Let Z
t t
≥0
, P
x
be continuous-time random walk on Z
d
with the generator: L
Z
f x = X
y ∈Z
d
a
y −x
f y − f x, where we assume that
the set {x ∈ Z
d
; a
x
6= 0} is bounded and contains a linear basis of R
d
, a
x
= a
−x
for all x ∈ Z
d
, Let v : Z
d
→ R be a function such that C
v def
= sup
x ∈Z
d
P
x
exp
Z
∞
|vZ
t
|d t
∞. Then, there exists C
∈ 0, ∞ such that sup
x ∈Z
d
P
x
exp
Z
t
vZ
u
du
f Z
t
≤ C t
−d2
X
x ∈Z
d
f x 2.13
for all t 0 and f : Z
d
→ [0, ∞ with P
x ∈Z
d
f x ∞.
Proof: We adapt the argument in [1, Lemma 3.1.3]. For a bounded function f : Z
d
→ R, we introduce
T
t
f x = P
x
exp
Z
t
vZ
u
du
f Z
t
, x
∈ Z
d
, T
h t
f =
1 h
T
t
[ f h], where hx = P
x
exp
Z
∞
vZ
t
d t
. Then, T
t t
≥0
extends to a symmetric, strongly continuous semi-group on ℓ
2
Z
d
. We now consider the measure
P
x ∈Z
d
hx
2
δ
x
on Z
d
, and denote by ℓ
p,h
Z
d
, k · k
p,h
the associated L
p
-space. Then, it is standard e.g., proofs of [2, page 74, Theorem 3.10] and [8, page 16, Proposition 3.3]
to see that T
h t
t ≥0
defines a symmetric strongly continuous semi-group on ℓ
2,h
Z
d
and that for f
∈ ℓ
2,h
Z
d
, E
h
f , f
def.
= lim
t ց0
1 t
X
x ∈Z
d
f x f − T
h t
f xhx
2
=
1 2
X
x, y ∈Z
d
a
y −x
| f y − f x|
2
hxh y. By the assumptions on a
x
, we have the Sobolev inequality: 973
1
X
x ∈Z
d
| f x|
2d d
−2
≤ c
1
1 2
X
x, y ∈Z
d
a
y −x
| f y − f x|
2
d d
−2
for all f ∈ ℓ
2
Z
d
, where c
1
∈ 0, ∞ is independent of f . This can be seen via an isoperimetric inequality [9, page 40, 4.3]. We have on the other hand that
2 1C
v
≤ hx ≤ C
v
. We see from 1 and 2 that
X
x ∈Z
d
| f x|
2d d
−2
hx
2
≤ c
2
E
h
f , f
d d
−2
for all f ∈ ℓ
2,h
Z
d
, where c
2
∈ 0, ∞ is independent of f . This implies that there is a constant C such that kT
h t
k
2 →∞,h
≤ C t
−d4
for all t 0, e.g.,[3, page 75, Theorem 2.4.2], where
k · k
p →q,h
denotes the operator norm from ℓ
p,h
Z
d
to ℓ
q,h
Z
d
. Note that kT
h t
k
1 →2,h
= kT
h t
k
2 →∞,h
by duality. We therefore have via semi-group property that
3 kT
h t
k
1 →∞,h
≤ kT
h t2
k
2 2
→∞,h
≤ C
2
t
−d2
for all t 0. Since T
t
f = hT
h t
[ f h], the desired bound 2.13 follows from 2 and 3.
3 Proof of Theorem 1.2.1 and Theorem 1.2.3
Kategori