then
F
j
= F
j m
, λ,x,T
∈ C
1,2
[0, T λ] × W
2,2
⊕ W
2,2
as
kϕk
L
2
∂ B y,r
≤ 2kϕk
L
2
k∇ϕk
L
2
, ϕ ∈ W
1,2
holds for every y ∈ R
d
and r
0. We may thus apply the Ito formula see [11] on LF
j
z
ǫ
to obtain
L F
j
t, z
ǫ
t − LF
j
r, z
ǫ
r = − λ
Z
t r
L
′
F
j
s, z
ǫ
skF
j
·, u
ǫ
sk
L
1
∂ Bx,T −λs
ds −
λ 2
Z
t r
L
′
F
j
s, z
ǫ
s
d
X
i= 1
d
X
k= 1
a
ik
◦ φ
m
∂ u
ǫ
s ∂ x
i
, ∂ u
ǫ
s ∂ x
k R
n
L
1
∂ Bx,T −λs
ds −
λ 2
Z
t r
L
′
F
j
s, z
ǫ
s v
ǫ
s
2 L
2
∂ Bx,T −λs
ds +
Z
t r
L
′
F
j
s, z
ǫ
s Z
Bx ,T
−λs d
X
i= 1
d
X
k= 1
a
ik
◦ φ
m
∂ u
ǫ
s ∂ x
i
, ∂ v
ǫ
s ∂ x
k R
n
ds +
Z
t r
L
′
F
j
s, z
ǫ
s n¬
v
ǫ
s, A
m
u
ǫ
s + α
ǫ
s + ∇
y
F
j
·, u
ǫ
s ¶
L
2
Bx,T −λs
o ds
+ Z
t r
L
′
F
j
s, z
ǫ
s ¦
v
ǫ
s, β
ǫ
s dW
s L
2
Bx,T −λs
© ds
+ 1
2 Z
t r
L
′
F
j
s, z
ǫ
skβ
ǫ
sk
2 L
2
U,L
2
Bx,T −λs
ds +
1 2
X
l
Z
t r
L
′′
F
j
s, z
ǫ
s v
ǫ
s, β
ǫ
se
l 2
L
2
Bx,T −λs
ds 8.5
for every 0 ≤ r t ≤ T a.s. We may, in fact, find the functions F
j
satisfying i-iii even so that iv F
j
w, · → Fw, · uniformly on compacts in R
n
, for every w ∈ R
d
, v
∇
y
F
j
w, · → ∇
y
F w ,
· uniformly on compacts in R
n
, for every w ∈ R
d
, vi and
sup F
j
w, y 1 +
| y|
2
+ ∇
y
F
j
w, y 1 +
| y| :
|w| ≤ r, y ∈ R
n
, j ∈ N
∞, r
0. Thus
lim
j →∞
F
j
t, z
ǫ
t, ω = Ft, z
ǫ
t, ω, t
∈ [0, T λ], sup
¦ F
j
t, z
ǫ
t, ω : t ∈ [0, T λ], j ∈ N ©
∞, lim
j →∞
k∇
y
F
j
·, u
ǫ
t, ω − ∇
y
F ·, u
ǫ
t, ωk
L
2
Bx,T −λt
= 0, t
∈ [0, T λ], sup
¦ k∇
y
F
j
·, u
ǫ
t, ωk
L
2
Bx,T −λt
: t ∈ [0, T λ], j ∈ N
© ∞
1056
for every ω ∈ Ω. Moreover, by the Gauss theorem,
Z
Bx ,T
−λs
d
X
i= 1
d
X
k= 1
a
ik
◦ φ
m
∂ u
ǫ
s ∂ x
i
, ∂ v
ǫ
s ∂ x
k R
n
+ v
ǫ
s, A
m
u
ǫ
s
R
n
ds
=
=
d
X
i= 1
d
X
k= 1
Z
∂ Bx,T −λs
a
ik
◦ φ
m
v
ǫ
, ∂ u
ǫ
∂ x
i R
n
y
k
− x
k
T − λs
d y
≤ Z
∂ Bx,T −λs
|v
ǫ
|
n
X
l= 1
ka ◦ φ
m
∇u
l ǫ
k
2
1 2
d y
≤ λ Z
∂ Bx,T −λs
|v
ǫ
|
n
X
l= 1
ka
1 2
◦ φ
m
∇u
l ǫ
k
2
1 2
d y
= λ
Z
∂ Bx,T −λs
|v
ǫ
|
d
X
i= 1
d
X
k= 1
a
ik
◦ φ
m
∂ u
ǫ
∂ x
i
, ∂ u
ǫ
∂ x
k R
n 1
2
d y ≤
λ 2
kv
ǫ
k
2 L
2
∂ Bx,T −λs
+ λ
2
d
X
i= 1
d
X
k= 1
a
ik
◦ φ
m
∂ u
ǫ
∂ x
i
, ∂ u
ǫ
∂ x
k R
n
L
1
∂ Bx,T −λs
8.6 so, after applying 8.6 and letting j
→ ∞ in 8.5, L
Ft, z
ǫ
t − LFr, z
ǫ
r ≤ +
Z
t r
L
′
Fs, z
ǫ
s n¬
v
ǫ
s, α
ǫ
s + ∇
y
F ·, u
ǫ
s ¶
L
2
Bx,T −λs
o ds
+ Z
t r
L
′
Fs, z
ǫ
s ¦
v
ǫ
s, β
ǫ
s dW
s L
2
Bx,T −λs
© ds
+ 1
2 Z
t r
L
′
Fs, z
ǫ
skβ
ǫ
sk
2 L
2
U,L
2
Bx,T −λs
ds +
1 2
X
l
Z
t r
L
′′
Fs, z
ǫ
s v
ǫ
s, β
ǫ
se
l 2
L
2
Bx,T −λs
ds 8.7
for every 0 ≤ r t ≤ T a.s. by the Lebesgue dominated convergence theorem and the convergence
theorem for stochastic integrals see e.g. Proposition 4.1 in [33]. Now, since
sup
ǫ1
kǫ
2
ǫ − A
m −2
k
L L
2
∞ and lim
ǫ→∞
kǫ
2
ǫ − A
m −2
ϕ − ϕk
L
2
= 0, ϕ ∈ L
2
, there is
lim
ǫ→∞
sup
t ∈[0,T λ]
kz
ǫ
t, ω − zt, ωk
W
1,2
⊕L
2
+ sup
t ∈[0,T λ]
|Ft, z
ǫ
t, ω − Ft, zt, ω|
= 0
sup {kz
ǫ
t, ωk
W
1,2
⊕L
2
+ Ft, z
ǫ
t, ω : ǫ 0, t ∈ [0, T λ]} ∞ for every
ω ∈ Ω so we get the result from 8.7 by the Lebesgue dominated convergence theorem and a convergence result for stochastic integrals e.g. Proposition 4.1 in [33].
1057
9 Linear growth + Global space case
We first prove existence of weak solutions for a localized equation with regular nonlinearities. The proof is based on a compactness method: local energy estimates yield tightness of an approximating
sequence of solutions. This sequence converges, on another probability space, to a limit due to the Jakubowski-Skorokhod theorem and finally, it is shown, that this limit is the desired weak solution
of the localized equation 9.1.
Lemma 9.1. Let
µ be a finite spectral measure on R
d
, let m ∈ N, let ν be a Borel probability measure
supported in a ball in H , let f
i
, g
i
: R
d
× R
n
→ R
n ×n
for i ∈ {0, . . . , d}, f
d+ 1
, g
d+ 1
: R
d
× R
n
→ R
n
be measurable functions such that sup
¦ | f
i
w, y| + |g
i
w, y| : |w| ≤ r, y ∈ R
n
, i ∈ {0, . . . , d}
© ∞
sup ¨
| f
d+ 1
w, y| + |g
d+ 1
w, y| 1 +
| y| :
|w| ≤ r, y ∈ R
n
« ∞,
sup ¨
| f
i
w, y
1
− f
i
w, y
2
| | y
1
− y
2
| :
|w| ≤ r, y
1
6= y
2
∈ R
n
, i ∈ {0, . . . , d + 1}
« ∞
sup ¨
|g
i
w, y
1
− g
i
w, y
2
| | y
1
− y
2
| :
|w| ≤ r, y
1
6= y
2
∈ R
n
, i ∈ {0, . . . , d + 1}
« ∞
hold for every r 0. Then there exists a completely filtered stochastic basis Ω, F , F
t
, P with a spatially homogeneous
F
t
-Wiener process W with the spectral measure µ and an F
t
-adapted process z with continuous paths in
H which is a solution of the equation u
t t
= A
m
u + 1
B
m
f ·, u, u
t
, ∇u + 1
B
m
g ·, u, u
t
, ∇u ˙
W 9.1
in the sense of Section 4 with the notation 4.1, 4.1, ν is the law of z0 and B
m
is the open centered ball in R
d
with radius m. The proof of Lemma 9.1 will be carried out in a sequence of lemmas. For, let us introduce the
mappings f
k
: H → H and g
k
: H → L
2
H
µ
, H defined by
f
k
u, v = 1
