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
B
m
f
·, uζ
k
∗ v + P
d i=
1
f
i
·, uζ
k
∗ u
x
i
+ f
d+ 1
·, u
, g
k
u, v = 1
B
m
g
·, uζ
k
∗ v + P
d i=
1
g
i
·, uζ
k
∗ u
x
i
+ g
d+ 1
·, u
ξ, ξ ∈ H
µ
.
Lemma 9.2. For every k ∈ N, there exists a completely filtered stochastic basis Ω
k
, F
k
, P
k
with a spatially homogeneous
F
k t
-Wiener process W
k
with spectral measure µ and an F
k t
-adapted process z
k
= u
k
, v
k
with H -continuous paths such that ν is the law of z
k
0 under P
k
and z
k
t = S
m t
z
k
0 + Z
t
S
m t
−s
f
k
z
k
s ds + Z
t
S
m t
−s
g
k
z
k
s dW
k s
, t
≥ 0. Moreover, for every p
∈ [2, ∞, there exists a constant K
1 p
,l
= K
1 l
,m,g, f ,p, ν,c,a
such that E
k
sup
s ∈[0,l]
kz
k
sk
2p H
≤ K
1 p
,l
, k
, l ∈ N
9.2 1058
and, if q ∈ 1, ∞ and γ 0 are such that γ +
1 q
1 2
then there exists a constant K
2 q
,l
= K
2 l
,m, γ,q, f ,g,a
2,m
,c, ν
such that E
k
kv
k
tk
2q C
γ
[0,l];W
−1,2
≤ K
2 q
,l
, k
, l ∈ N.
9.3 Proof.
The mappings f
k
: H → H and g
k
: H → L
2
H
µ
, H are Lipschitz on bounded sets and
have at most linear growth hence there exists a completely filtered stochastic basis Ω
k
, F
k
, P
k
with a spatially homogeneous
F
k t
-Wiener process W
k
with spectral measure µ and an F
k t
-adapted process z
k
with H -continuous paths such that ν is the law of z
k
0 under P
k
and z
k
t = S
m t
z
k
0 + Z
t
S
m t
−s
f
k
z
k
s ds + Z
t
S
m t
−s
g
k
z
k
s dW
k s
, t
≥ 0 by e.g. [11] extended in the sense of Theorem 12.1 in Chapter V.2.12 in [39] whose generalization
to SPDE is possible and can be proved in the same way as in [39] since
〈G
m
z , z
〉
Dom I −A
m 1
2
⊕L
2
≤ 1
2 kzk
2 Dom I
−A
m 1
2
⊕L
2
, z
∈ Dom G
m
hence the square norm of the local solution z
k 2
Dom I −A
m 1
2
⊕L
2
cannot explode in finite time and so z
k
is a global solution in the sense of 4.3 by the Chojnowska-Michalik theorem see [9] or Theorem 12 in [33].
By Proposition 8.1 applied on T 0, x = 0, lr = log1 + r
p
for p ∈ [2, ∞, F y = | y|
2
2, λ
= sup
w ∈R
d
kaφ
m
wk
1 2
, with the notation F
T
= F
m ,
λ ,0,T
and
F
∞
u, v = 1
2 Z
R
d
d
X
i= 1
d
X
l= 1
a
il
◦ φ
m
¬ u
x
i
, u
x
l
¶
R
n
+ |v|
2 R
n
+ |u|
2 R
n
d y ,
there is l
F
T
t, z
k
t ≤ lF
T
0, z
k
0 + M
T ,k
t − 1
2 〈M
T ,k
〉t
+
c
2
2 Z
t
pp − 1F
p −2
T
s, z
k
s
1 + F
p T
s, z
k
s kv
k
sk
2 L
2
B
T −λ0s
kg·, u
k
s, v
k
s, ∇u
k
sk
2 L
2
B
m
∩B
T −λ0s
ds +
c
2
2 Z
t
l
′
F
T
s, z
k
skg·, u
k
s, v
k
s, ∇u
k
sk
2 L
2
B
m
∩B
T −λ0s
ds +
Z
t
l
′
F
T
s, z
k
skv
k
sk
L
2
B
T −λ0s
ku
k
sk
L
2
B
T −λ0s
ds +
Z
t
l
′
F
T
s, z
k
skv
k
sk
L
2
B
T −λ0s
k f ·, u
k
s, v
k
s, ∇u
k
sk
L
2
B
m
∩B
T −λ0s
ds
≤ lF
∞
z
k
0 + M
T ,k
t − 1
2 〈M
T ,k
〉t +
K Z
t
1 + F
2 ∞
z
k
s
1 + F
2 T
s, z
k
s ds + K
Z
t
1 + F
∞
z
k
s
1 + F
T
s, z
k
s ds
1059
for t ∈ [0, T λ
] by Lemma 3.3 where K depends only on p, c, m, a, g, f and
M
T ,k
t = p Z
t
F
p −1
T
s, z
k
s
1 + F
p T
s, z
k
s ¬
v
k
s, g·, u
k
s, v
k
s, ∇u
k
s dW
k s
¶
L
2
B
m
∩B
T −λ0s
for t ∈ [0, T λ
]. Thus, letting T → ∞, we obtain l
F
∞
z
k
t ≤ lF
∞
z
k
0 + M
k
t − 1
2 〈M
k
〉t + 2K t for t
∈ R
+
by the Lebesgue dominated convergence theorem and a convergence result for stochastic integrals e.g. Proposition 4.1. in [33] where
M
k
t = p
Z
t
F
p −1
∞
z
k
s
1 + F
p ∞
z
k
s ¬
v
k
s, g·, u
k
s, v
k
s, ∇u
k
s dW
k s
¶
L
2
B
m
, t
∈ R
+
, 〈M
k
〉t = p
2
X
l
Z
t
F
2p −1
∞
z
k
s
[1 + F
p ∞
z
k
s]
2
¬ v
k
s, g·, u
k
s, v
k
s, ∇u
k
se
l
¶
2 L
2
B
m
ds ≤ p
2
c
2
Z
t
F
2p −1
∞
z
k
s
[1 + F
p ∞
z
k
s]
2
v
k
s
2 L
2
B
m
g·,u
k
s, v
k
s, ∇u
k
s
2 L
2
B
m
ds ≤ K
Z
t
F
2p −1
∞
z
k
s
[1 + F
p ∞
z
k
s]
2
1 + F
∞
z
k
s
2
ds ≤ K
3
t ,
t ∈ R
+
. Hence, applying the exponential on both sides, we get
sup
s ∈[0,t]
F
p ∞
z
k
t ≤ e
2K t
[1 + F
p ∞
z
k
0] sup
s ∈[0,t]
e
M
k
t−
1 2
〈M
k
〉t
, t
∈ R
+
so E
k
sup
s ∈[0,t]
F
p ∞
z
k
t ≤ e
2K t
¦ E
k
[1 + F
p ∞
z
k
0]
2
©
1 2
¨ E
k
sup
s ∈[0,t]
e
2M
k
s−〈M
k
〉s
«
1 2
≤ 2K
1
e
2K t
¦ E
k
e
2M
k
t−〈M
k
〉t
©
1 2
≤ 2K
1
e
2K+
K3 2
t
¦ E
k
e
2M
k
t−2〈M
k
〉t
©
1 2
≤ 2K
1
e
2K+
K3 2
t
by the Doob maximal inequality for martingales and the Novikov criterion, where K
2 1
= E
k
[1 + F
p ∞
z
k
0]
2
= Z
H
[1 + F
p ∞
z]
2
d ν ∞.
Since F
1 2
∞
is an equivalent norm on H , we have proved 9.2.
Next, by the Chojnowska-Michalik theorem see [9] or Theorem 12 in [33] P
k
Z
t
u
k
s ds ∈ Dom A
m
= 1,
t ∈ R
+
1060
and v
k
t = v
k
0 + A
m
Z
t
u
k
s ds + Z
t
f
2 k
z
k
s ds + Z
t
g
2 k
z
k
s dW
k s
9.4 v
k
t = v
k
0 + A
m
I
1
t + I
2
t + I
3
t almost surely for every t
∈ R
+
, where f
2 k
and g
2 k
are the second components of f
k
and g
k
, respectively, and the integrals converge in L
2
. We get that E
k
kv
k
k
2q C
γ
[0,l];W
−1,2
≤ 4
2q −1
h E
k
kv
k
0k
2q L
2
+ E
k
kA
m
I
1
k
2q C
γ
[0,l];W
−1,2
i +
4
2q −1
h E
k
kI
2
k
2q C
γ
[0,l];W
−1,2
+ E
k
kI
3
k
2q C
γ
[0,l];W
−1,2
i ≤ 4
2q −1
h E
k
kv
k
0k
2q L
2
+ c
a
2,m
E
k
kI
1
k
2q C
γ
[0,l];W
1,2
i +
4
2q −1
h E
k
kI
2
k
2q C
γ
[0,l];L
2
+ E
k
kI
3
k
2q C
γ
[0,l];L
2
i
≤ c
a, γ,q,l, f ,g,m
