11.3 Skorokhod representation
Since Z
m
are tight in Z by Lemma 11.3,
kF
m
·, U
m
0k
L
1
T
k
k ∈N
are tight in R
N +
where T
k
were defined in 10.6 by Remark 11.2 and P
m
[Z
m
0 ∈ ·] converge to Θ weakly on H
l oc
by Remark 11.1, i.e. Z
m
0 are tight in H
l oc
by the Prokhorov theorem, fixing an ONB e
l
in H
µ
, we may apply Theorem A.1 on the sequence Z
m
0, Z
m
, W
m
e
l l
, kF
m
·, U
m
0k
L
1
T
k
k ∈N
: Ω
m
→ H
l oc
× Z × C R
+
; R
dim H
µ
× R
N +
to claim that there exist • a probability space Ω, F , P,
• a subsequence m
j
, • CR
+
; H -valued random variables z
j
= u
j
, v
j
defined on Ω, • C
R
+
; R
dim H
µ
-valued random variables β
j
= β
j l
, β = β
l
defined on Ω, • R
N +
-valued random variable ν = ν
k k
∈N
defined on Ω,
• a Z -valued random variable z = u, v with σ-compact range defined on Ω
such that
i Z
m
j
, W
m
j
e
l l
has the same law under P
m
j
as z
j
, β
j
under P on the space BCR
+
; H × C
R
+
; R
dim H
µ
for every j ∈ N,
ii z
j
, β
j
converges to z, β on Ω in the topology of Z × C R
+
; R
dim H
µ
,
iii z
j
0 converges to z0 on Ω in H
l oc
,
iv kF
m
j
·, u
j
0k
L
1
B
Tk
converges to ν
k
for every k ∈ N on Ω.
Definition 11.4. We also define, for completness,
˜ ν
k
= ν
k
+ 1
2 Z
B
Tk
d
X
i= 1
d
X
j= 1
a
i j
®
∂ u0
∂ x
i
,
∂ u0
∂ x
j
¸
R
n
+ |u0|
2 R
n
+ |v0|
2 R
n
d x 11.8
for k ∈ N.
1076
11.4 Property of
β
Let us define F
t
= σ σν, zs, βs : s ≤ t ∪ {N ∈ F : P N = 0} ,
t ≥ 0.
Apparently, the filtration F
t
is complete. The proof of the following Lemma is analogous to the proof of Lemma 9.9.
Lemma 11.5. The processes β
1
, β
2
, β
3
, . . . are independent standard F
t
-Wiener processes.
Corollary 11.6. The cylindrical process
W
t
ξ = X
l
β
l
t〈ξ, e
l
〉
H
µ
, ξ ∈ H
µ
, t
≥ 0 is a spatially homogeneous
F
t
-Wiener process with spectral measure µ.
11.5 Property of u
Lemma 11.7. There is 〈ut, ϕ〉
L
2
= 〈u0, ϕ〉
L
2
+ Z
t
〈vs, ϕ〉
L
2
ds ,
t ≥ 0
almost surely for every ϕ ∈ D.
Proof. If
ϕ is supported in B
r
m j
and t ∈ [0, r
m
j
] then U
m
j
t, ϕ
L
2
− U
m
j
0, ϕ
L
2
= u
m
j
t, ϕ
L
2
− u
m
j
0, ϕ
L
2
= Z
t
v
m
j
s, ϕ
L
2
ds = Z
t
V
m
j
s, ϕ
L
2
ds The rest of the proof is analogous with the proof of Lemma 9.8.
11.6 Energy estimates
Lemma 11.8. Let T
0 and x ∈ R
d
, let G
m
: R
d
× R
n
→ R
+
satisfy the assumptions a-c in Proposition 8.1, let G
: R
d
×R
n
→ R
+
be a measurable function such that G
m
w, · converges to Gw, · uniformly on compact sets in R
n
for a.e. w ∈ R
d
, let ˜
κ ∈ R
+
be such that |g
d+ 1
m
w, y|
2
+ |∇
y
G
m
w, y + f
d+ 1
m
w, y|
2
≤ ˜κG
m
w, y, y
∈ R
n
, m
≥ |x| + T holds for a.e. w
∈ Bx, T , let λ satisfy 5.3, let L : R
+
→ R
+
be a continuous function in C
2
0, ∞ satisfying 5.4 with
˜ κ and define G
∗
= sup
m ∈N
G
m
. Assume that Θ
¦ u, v ∈ H
l oc
: kG
∗
·, uk
L
1
B
R
∞ ©
= 1, R
0. 11.9
1077
Then, for A ∈ BH
l oc
and with the convention 0 · ∞ = 0, E
¨
1
A
z
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