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
1 + E
k
sup
s ∈[0,t]
kz
k
sk
2q H
≤ K
2 l
by the inequality 18 in the proof of Lemma 4 in [29] and 9.2.
Lemma 9.3. The sequence of processes z
k
constructed in Lemma 9.2 is tight in the space Z = C
w
R
+
; W
1,2 l oc
× C
w
R
+
; L
2 l oc
. Proof.
It holds, by the Chojnowska-Michalik theorem see [9] or Theorem 12 in [33], that u
k
t = u
k
0 + Z
t
v
k
s ds, t
∈ R
+
9.5 almost surely where the integral converges in L
2
. So, if γ ∈ 0, 1 then
ku
k
k
C
γ
[0,l];L
2
≤ 1 + l sup
s ∈[0,l]
kz
k
sk
H
. Hence, if we fix
ǫ 0, q ∈ 1, ∞ and γ 0 such that γ +
1 q
1 2
and we assume a
l
2 + l
4
l
ǫ K
1 q
,l
+ K
2 q
,l
1 2q
, there is
P
k
sup
s ∈[0,l]
ku
k
sk
W
1,2
B
l
+ ku
k
sk
C
γ
[0,l];L
2
B
l
a
l
≤
≤ P
k
sup
s ∈[0,l]
kz
k
sk
H
a
l
2 + l
≤ 2 + l
a
l 2q
E
k
sup
s ∈[0,l]
kz
k
sk
2q H
≤ ǫ
4
l
and P
k
sup
s ∈[0,l]
kv
k
sk
L
2
B
l
+ kv
k
sk
C
γ
[0,l];W
−1,2 l
a
l
≤
1061
≤ P
k
sup
s ∈[0,l]
kz
k
sk
H
a
l
2
+ P
k
kv
k
sk
C
γ
[0,l];W
−1,2 l
a
l
2
≤ 2
a
l 2q
E
k
sup
s ∈[0,l]
kz
k
sk
2q H
+ E
k
kv
k
sk
2q C
γ
[0,l];W
−1,2
≤
ǫ 4
l
, the sets
C
1
= ¦
h ∈ C
w
R
+
; W
1,2 l oc
: khk
L
∞
0,l;W
1,2
B
l
+ khk
C
γ
[0,l];L
2
B
l
≤ a
l
© C
2
= n
h ∈ C
w
R
+
; L
2 l oc
: khk
L
∞
0,l;L
2
B
l
+ khk
C
γ
[0,l];W
−1,2 l
≤ a
l
o are such that C
1
× C
2
is compact in Z by Corollary B.2 and P
k
[z
k
∈ C
1
× C
2
] ≥ 1 − ǫ. We may now proceed to the proof of Lemma 9.1. Fixing an ONB e
l
in H
µ
, by the Jakubowski- Skorokhod theorem A.1 applied to the
Z × C R
+
; R
dim H
µ
-valued sequence z
k
, W
k
e
l l
k
, there exists
• a subsequence k
j
, a probability space Ω, F , P with • CR
+
; H -valued random variables Z
j
, j ∈ N,
• a Z -valued random variable Z, • C
R
+
; R
dim H
µ
-valued random variables β
j
, j ∈ N and β
such that i the law of z
k
j
, W
k
j
e
l l
under P
k
j
coincides with the law of Z
j
, β
j
under P on BCR
+
; H ⊗ BC
R
+
; R
dim H
µ
ii Z
j
, β
j
converges in Z × C R
+
; R
dim H
µ
to Z, β on Ω. Remark
9.4. We point out for completeness that tightness of the sequence z
k
, W
k
e
l l
k
in Z ×
C R
+
; R
dim H
µ
follows from Lemma 9.3 and from the fact that all W
k
e
l l
have the same Radon law on the Polish space C
R
+
; R
dim H
µ
for every k ∈ N. Consequently, the sequence W
k
e
l l
is tight in C
R
+
; R
dim H
µ
. Remark
9.5. It should be also noted that the random variables Z
j
are Z -valued by the Jakubowski-
Skorokhod theorem. However, since z
k
j
and Z
j
have the same law on Z and z
k
j
are CR
+
; H -
valued, we conclude that Z
j
may be assumed to be CR
+
; H -valued satisfying the property i
above without loss of generality as CR
+
; H is a measurable subset of Z by Corollary A.2.
Lemma 9.6. If p ∈ [2, ∞ then
E sup
s ∈[0,l]
kZ
j
sk
2p H
≤ K
1 p
,l
, j
, l ∈ N
9.6 E sup
s ∈[0,l]
kZsk
2p H
≤ K
1 p
,l
, j
, l ∈ N
9.7 where K
1 p
,l
is the same constant as in 9.2. 1062
Proof. The mapping
CR
+
; H → R
+
: z 7→ sup
s ∈[0,l]
kzsk
2p H
is continuous hence Borel measurable and so E sup
s ∈[0,l]
kZ
j
sk
2p H
= E
k
j
sup
s ∈[0,l]
kz
k
j
sk
2p H
≤ K
1 p
,l
, j
, l ∈ N
by the property i and E sup
s ∈[0,l]
kZsk
2p H
≤ lim inf
j →∞
E sup
s ∈[0,l]
kZ
j
sk
2p H
≤ K
1 p
,l
, j
, l ∈ N
by the Fatou lemma and the property ii.
Corollary 9.7. The process Z has weakly continuous paths in H a.s.
Lemma 9.8. It holds that
Ut = U 0 +
Z
t
V s ds ,
t ∈ R
+
almost surely where the integral converges in L
2
. Proof.
The mapping CR
+
; H → R : z 7→ 〈ut, ϕ〉 − 〈u0, ϕ〉 −
Z
t
〈vs, ϕ〉 ds is continuous hence Borel measurable for every
ϕ ∈ D and so 〈U
j
t, ϕ〉 = 〈U
j
0, ϕ〉 + Z
t
〈V
j
s, ϕ〉 ds, t
∈ R
+
holds almost surely for every j ∈ N by the property i and 9.5. Letting j → ∞, we get
〈Ut, ϕ〉 = 〈U0, ϕ〉 + Z
t
〈V s, ϕ〉 ds, t
∈ R
+
P-a.s. by the property ii. The result now follows from 9.7 and density of D in L
2
. If we define the complete filtration
F
t
= σ σ Zs, βs : s ∈ [0, t] ∪ {N ∈ F : PN = 0}
, t
∈ R
+
then the following results.
Lemma 9.9. The processes β
1
, β
2
, β
3
, . . . are independent standard F
t
-Wiener processes.
1063
Proof. Let us consider the sequence
ϕ
i
from Corollary C.1, let 0 ≤ s t, J ∈ N, 0 ≤ s
1
≤ · · · ≤ s
J
≤ s
, let h : R
2 J
×J
× R
dim H
µ
J
→ [0, 1] and h
1
: R
dim H
µ
→ [0, 1] be continuous functions and define X
j
=
〈z
k
j
s
i
, ϕ
i
1
〉
L
2
i ,i
1
≤J
, W
k
j
s
1
e
l l
, . . . , W
k
j
s
J
e
l l
X
j
=
〈Z
j
s
i
, ϕ
i
1
〉
L
2
i ,i
1
≤J
, β
j
s
1
, . . . , β
j
s
J
X =
〈Zs
i
, ϕ
i
1
〉
L
2
i ,i
1
≤J
, βs
1
, . . . , βs
J
for j ∈ N. Then, for every j ∈ N,
Z
Ω
k j
h X
j
h
1
W
k
j
t
e
l
− W
k
j
s
e
l l
dP
k
j
= Z
Ω
k j
h X
j
dP
k
j
Z
Ω
k j
h
1
W
k
j
t
e
l
− W
k
j
s
e
l l
dP
k
j
by P
k
j
-independence of σW
k
j
t
ξ − W
k
j
s
ξ
ξ∈H
µ
and F
k
j
s
. So, by the property i, E
¦ h
X
j
h
1
β
j
t − β
j
s ©
= E h X
j
E h
1
β
j
t − β
j
s, j
∈ N whence
E h
X h
1
βt − βs = E h
X E h
1
βt − βs by the property ii and we conclude that
E ¦
1
F
s
h
1
βt − βs ©
= P F
s
E h
1
βt − βs holds for every F
s
∈ F
s
whenever s t, i.e. σβt − βs is P-independent from F
s
. Since β
j 1
t − β
j 1
s, . . . , β
j l
t − β
j l
s and W
k
j
t
e
1
− W
k
j
s
e
1
, . . . , W
k
j
t
e
l
− W
k
j
s
e
l
have the normal centered distribution with covariance t
− sI
l
for every j, l ∈ N and 0 ≤ s t by the property i
preceding Remark 9.4 and the fact that W
k
j
are cylindrical Wiener processes on H
µ
by Section 3, we conclude that
β
1
t − β
1
s, . . . , β
l
t − β
l
s has the normal centered distribution with covariance t − sI
l
as well, as β
j
→ β on Ω by the property ii preceding Remark 9.4. The proof of Lemma 9.9 is thus complete.
Corollary 9.10. Let e
l
be the previously fixed ONB in H
µ
. Then 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.
Lemma 9.11.
Proof. Fix
ϕ ∈ D and define the continuous operators d
k t
: CR
+
; H → R : z 7→ 〈vt, ϕ〉 − 〈v0, ϕ〉 −
Z
t
h 〈ur, A
m
ϕ〉 + 〈 f
2 k
zr, ϕ〉 i
d r D
k ,l
t
: CR
+
; H → R : z 7→
Z
t
〈g
2 k
zre
l
, ϕ〉 d r
D
k t
: CR
+
; H → R : z 7→
X
l
Z
t
〈g
2 k
zre
l
, ϕ〉
2
d r 1064
where f
2 k
and g
2 k
are the second components of f
k
and g
k
, respectively. Then, fixing 0 ≤ s t
and with the notation of the proof of Lemma 9.9, E h
X
j
n d
k
j
t
Z
j
− d
k
j
s
Z
j
o = E
k
j
h X
j
n d
k
j
t
z
k
j
− d
k
j
s
z
k
j
o = 0
9.8 E h
X
j
n d
k
j
t
Z
j
β
j l
t − D
k
j
,l t
Z
j
− d
k
j
s
Z
j
β
j l
s + D
k
j
,l s
Z
j
o =
9.9 = E
k
j
h X
j
n d
k
j
t
z
k
j
W
k
j
t
e
l
− D
k
j
,l t
z
k
j
− d
k
j
s
z
k
j
W
k
j
s
e
l
+ D
k
j
,l s
z
k
j
o = 0
E h X
j
n d
k
j
t
Z
j 2
− D
k
j
t
Z
j
− d
k
j
s
Z
j 2
+ D
k
j
s
Z
j
o =
9.10 = E
k
j
h X
j
n d
k
j
t
z
k
j
2
− D
k
j
t
z
k
j
− d
k
j
s
z
k
j
2
+ D
k
j
s
z
k
j
o = 0
by the property i since, by 9.4, d
k
j
t
z
k
j
= Z
t
g
2 k
j
z
k
j
s dW
k
j
s
, t
∈ R
+
is an L
2
Ω
k
j
-integrable martingale in L
2
by 9.2 and Lemma 3.3, and the integrals expectations in 9.8-9.10 converge by 9.2 and 9.6. Since
sup
j ∈N
E h
|d
k
j
r
Z
j
|
q
+ |D
k
j
,l r
Z
j
|
q
+ |D
k
j
r
Z
j
|
q
i ∞
for every r ∈ R
+
, l ∈ N and q 0 by 9.6, we get
E h X
d
t
− d
s
= 0 E h
X ¨
d
t
β
l
t − d
s
β
l
s − Z
t s
¬ g
2
Zre
l
, ϕ
¶ d r
« = 0
E h X
d
t 2
− d
s 2
− X
l
Z
t s
¬ g
2
Zre
l
, ϕ
¶
2
d r = 0
by the property ii where d
t
= 〈V t, ϕ〉 − 〈V 0, ϕ〉 −
Z
t
〈Ur, A
m
ϕ〉 + 〈 f
2
Zr, ϕ〉
d r f
2
z = 1
B
m
f ·, u, v, ∇u
g
2
z = 1
B
m
g ·, u, v, ∇u.
In particular, the processes d
, d
· β
l
− Z
·
¬ g
2
Zre
l
, ϕ
¶ d r
, d
2
− X
l
Z
·
¬ g
2
Zre
l
, ϕ
¶
2
d r are
F
t
-martingales hence the quadratic variation ®
d −
Z
·
¬ g
2
Zr dW
r
, ϕ
¶ ¸
= 0 1065
and so 〈V t, ϕ〉 = 〈V 0, ϕ〉 +
Z
t
〈Ur, A
m
ϕ〉 + 〈 f
2
Zr, ϕ〉
d r + Z
t
¬ g
2
Zr dW
r
, ϕ
¶ .
Thus Z is a solution of 9.1. Moreover, by the Chojnowska-Michalik theorem see [9] or Theorem 13 in [33],
Zt = S
m
Z 0 +
Z
t
S
m t
−s
1
B
m
f Zs
ds + Z
t
S
m t
−s
1
B
m
gZs
dW
s
holds a.s. for every t ∈ R
+
, hence paths of Z are H -continuous almost surely.
10 General growth + Local space case
In this section, we use the existence result for the localized equation 9.1 and mimic the com- pactness method of the previous section based on the local energy estimates, tightness of an
approximating sequence of solutions, convergence to a limit on another probability space due to the Jakubowski-Skorokhod theorem and final identification of the limit with a solution. The
construction-approximation procedure is, however, much more refined this time.
Lemma 10.1. Let κ ∈ R
+
. Then there exists a constant ρ ∈ R
+
depending only on κ and c see
Section 2 such that the following holds: If f
i
, g
i
: R
d
× R
n
→ R
n ×n
for i ∈ {0, . . . , d} and f
d+ 1
, g
d+ 1
: R
d
× R
n
→ R
n
are measurable functions satisfying the assumptions of Lemma 9.1, m ∈ N, z is an
H -continuous solution of 9.1, L : R
+
→ R
+
is a continuous nondecreasing function in C
2
0, ∞ satisfying 5.4 with
κ, if T 0 and x ∈ R
d
satisfy Bx , T
⊆ B
m
, if F : R
d
× R
n
→ R
+
satisfies the assumptions a-c in Proposition 8.1 and, for every y
∈ R
n
, there is | f
w, y|
2
+ |g w, y|
2
≤ κ 10.1
n
X
j= 1
n
X
k= 1
a
−
1 2
w
f
1 jk
w, y ..
. f
d jk
w, y
2
R
d
+ a
−
1 2
w
g
1 jk
w, y ..
. g
d jk
w, y
2
R
d
≤ κ
10.2
|g
d+ 1
w, y|
2
+ |∇
y
F w , y + f
d+ 1
w, y|
2
≤ κFw, y 10.3
for a.e. w ∈ Bx, T , if λ satisfies 5.3 and F = F
λ,x,T
is the conic energy function for F defined as in 2.2 then
E ¨
1
Ω
sup
r ∈[0,t]
L
Fr, zr
« ≤ 4e
ρ t
E ¦
1
Ω
L
F0, z
© 10.4
holds for every t ∈ [0, T λ] and every Ω
∈ F .
Proof. Define l
ǫ
r = logǫ + Lǫ + r for r ∈ R
+
and
ǫ 0, put et = F
m ,
λ,x,T
t, zt for t
∈ [0, T λ] and write shortly f z = f
·, u, v, ∇u, gz = g
·, u, v, ∇u for
z = u , v
∈ H
l oc
. 1066
Then, by Proposition 8.1, l
ǫ
et ≤ l
ǫ
e0 + M
ǫ
t − 1
2 〈M
ǫ
〉t +
1 2
X
l
Z
t
L
′′
ǫ + es ǫ + Lǫ + es
〈vs, gzse
l
〉
2 L
2
Bx,T −λs
ds +
c
2
2 X
l
Z
t
L
′
ǫ + es ǫ + Lǫ + es
kgzsk
2 L
2
Bx,T −λs
ds +
Z
t
L
′
ǫ + es ǫ + Lǫ + es
〈vs, ∇
y
F ·, us + f zs〉
L
2
Bx,T −λs
ds ≤ l
ǫ
e0 + ρκt + M
ǫ
t − 1
2 〈M
ǫ
〉t, t
∈ [0, T λ] almost surely by 8.4 and Lemma 3.3 where
ρκ = 12c
2
κ + 4κ + 1κ and M
ǫ
t = Z
t
L
′
ǫ + es ǫ + Lǫ + es
〈vs, gzs dW
s
〉
L
2
Bx,T −λs
, t
∈ [0, T λ] as, for every s
∈ [0, T λ], kgzsk
2 L
2
Bx,T −λs
≤ 3 g
·, usvs
2 L
2
Bx,T −λs
+ 3 g
d+ 1
·, us
2 L
2
Bx,T −λs
+ 3
d
X
i= 1
g
i
·, usu
x
i
s
2 L
2
Bx,T −λs
≤ 12κes and, analogously,
∇
y
F ·, us + f zs
2 L
2
Bx,T −λs
≤ 12κes. Hence, almost surely,
L ǫ + et ≤ e
ρκt
[ǫ + Lǫ + e0] e
M
ǫ
t−
1 2
〈M
ǫ
〉t
, t
∈ [0, T λ]. Since
〈M
ǫ
〉t = X
l
Z
t
L
′
ǫ + es ǫ + Lǫ + es
2
〈vs, gzse
l
〉
2 L
2
Bx,T −λs
ds ≤ 24κ
3
c
2
t ,
t ∈ [0, T λ],
there is E sup
s ∈[0,t]
¦
1
Ω ∩[e0≤δ]
L ǫ + es
© ≤ e
ρκt
E sup
s ∈[0,t]
Y
1,1
s 10.5
≤ e
ρκt
E sup
s ∈[0,t]
[Y
1 2
,
1 2
s]
2
≤ 4e
ρκt
E [Y
1 2
,
1 2
t]
2
= 4e
ρκt
E n
Y
1,1
te
1 4
〈M
ǫ
〉t
o ≤ 4e
[6κ
3
c
2
+ρκ]t
E Y
1,1
t =
4e
[6κ
3
c
2
+ρκ]t
E ¦
1
Ω ∩[e0≤δ]
[ǫ + Lǫ + e0] ©
1067
by the Doob maximal inequality for submartingales where Y
α,β
t = 1
Ω ∩[e0≤δ]
[ǫ + Lǫ + e0]
α
e
β M
ǫ
t−
β2 2
〈M
ǫ
〉t
, t
∈ [0, T ] is a martingale for every
α, β 0 by the Novikov criterion. Now, we get the claim by letting ǫ ↓ 0 Fatou’s lemma on the left hand side and Lebesgue’s dominated convergence theorem on the right
hand side and δ ↑ ∞ Levi’s theorem on the left hand side in 10.5.
With the notation λ
T
defined in 2.3, given r 0, let T
r
be the smallest radius of the base of a backward cone
{t, x ∈ R
+
× R
d
: |x| + tλ
T
≤ T } that contains houses the cylinder [0, r]
× B
r
and, given m ∈ N, let r
m
be the radius of the largest cylinder [0, r]
× B
r
for which the radius of the housing backward cone {t, x ∈ R
+
× R
d
: |x| + tλ
T
r
≤ T
r
} is not larger than m, i.e. T
r
≤ m. We can define these radii by T
r
= inf T
0 : T
1 + λ
T
≥ r ,
r
m
= sup {r 0 : T
r
≤ m}. 10.6
Remark 10.2. Observe that T
r
∞ for every r 0 and r
m
∈ 0, m] satisfy r
m
↑ ∞ by 2.1. Given r
0, let use define extension operators E
r
ϕx = ϕx, |x| r
E
r
ϕx = −η
r
xϕP
r
x, |x| r 10.7
E
∗ r
ψx = ψx − r
2d
|x|
2d
η
r
P
r
xψP
r
x, |x| r
for ϕ : B
r
→ R
n
and ψ : R
d
→ R
n
where η
r
x = ηxr and η is a smooth [0, 1]-valued function such that
ηx = 1 if |x| ≤ 1 and ηx = 0 if |x| ≥ 2 and P
r
x = r
2
|x|
−2
x .
Lemma 10.3. For every p ∈ [1, ∞], the operator
• E
r
maps L
p
B
r
continuously to L
p
R
d
, • E
∗ r
maps L
p
R
d
continuously to L
p
B
r
, • E
∗ r
maps W
1,2
R
d
continuously to W
1,2
B
r
, kE
r
k
L L
p
B
r
,L
p
+ kE
∗ r
k
L L
p
,L
p
B
r
≤ c
d ,p
, kE
∗ r
k
L W
1,2
,W
1,2
B
r
≤ c
d
1 + 1
r and
Z
R
d
E
r
ϕ, ψ
R
n
d x = Z
B
r
¬ ϕ, E
∗ r
ψ ¶
R
n
d x hold for every
ϕ ∈ L
p
B
r
, ψ ∈ L
q
R
d
whenever p, q ∈ [1, ∞] are Hölder conjugate exponents. 1068
Lemma 10.4. Let κ ∈ R
+
, R 0, δ 0, d
∗
=
d 2
+ 1, γ 0, p ∈ 1, ∞ such that γ +
2 p
1 2
. Then there exists a constant
˜ ρ ∈ R
+
depending only on R, d, p, κ, r
, δ, γ and c see Section 2 and
10.6 such that the following holds. If f
i
, g
i
: R
d
× R
n
→ R
n ×n
and f
d+ 1
, g
d+ 1
: R
d
× R
n
→ R
n
are measurable functions satisfying the assumptions of Lemma 9.1, m
∈ N, z is an H -continuous solution of 9.1, F
: R
d
× R
n
→ R
+
satisfies the assumptions a-c in Proposition 8.1 and, for every y ∈ R
n
, the inequalities 10.1-10.3 hold for a.e. w
∈ B
T
R ∧rm
. If, for every y ∈ R
n
, | f
d+ 1
w, y| ≤ κFw, y for a.e.
w ∈ B
r
m
∧R
10.8 and
F
r
= F
λ
Tr
,0,T
r
is the conic energy function defined as in 2.2 for the function F then E
1
[F
rm∧R
0,z0≤δ]
kE
r
m
v · ∧ r
m
k
p C
γ
[0,R],W
−d∗,2 R
≤ ˜ ρ
where the space W
−d
∗
,2 R
is defined in Appendix B. Proof.
There is kE
r
m
h k
W
−d∗,2 R
≤ c
1
khk
L
2
B
rm∧R
, h
∈ L
2 l oc
kE
r
m
h k
W
−d∗,2 R
≤ c
2
khk
L
1
B
rm∧R
, h
∈ L
1 l oc
kE
r
m
A
m
h k
2 W
−d∗,2 R
≤ c
2 3
Q
r
m
∧R
h, h, h
∈ W
2,2 l oc
10.9 where
c
1
= 1, c
2
= k ⊆ k
L W
d∗,2
,L
∞
, c
3
= λ
R
if R ≤ r,
c
1
= kE
r
k
L L
2
B
r
,L
2
, c
2
= k ⊆ k
L W
d∗,2
,L
∞
kE
∗ r
k
L L
∞
,L
∞
B
r
, c
3
= λ
r
kE
∗ r
k
L W
1,2
;W
1,2
B
r
if R r and
Q
δ
h
1
, h
2
=
d
X
i= 1
d
X
j= 1
Z
B
δ
a
i j
® ∂ h
1
∂ x
i
, ∂ h
2
∂ x
j
¸
R
n
d w .
Observe that max {c
1
, c
2
, c
3
} can be dominated by a constant c that depends only on d, R and r by
Lemma 10.3. Let us prove just the third inequality in case R r as the other cases are straightfor-
ward. For let ϕ ∈ W
d
∗
,2 R
. Then E
r
A
m
h ,
ϕ =
Z
R
d
E
r
A
m
h ,
ϕ
R
n
d w = Z
B
r
A
m
h ,
ψ
R
n
d w =
d
X
i= 1
d
X
j= 1
Z
B
r
∂
∂ x
j
¨
a
i j
∂ h ∂ x
i
, ψ
R
n
«
− a
i j
∂ h ∂ x
i
, ∂ ψ
∂ x
i R
n
d w
= −
d
X
i= 1
d
X
j= 1
Z
B
r
a
i j
∂ h ∂ x
i
, ∂ ψ
∂ x
i R
n
d w
1069
where ψ = E
∗ r
ϕ ∈ W
1,2
B
r
by Lemma 10.3. So |
E
r
A
m
h ,
ϕ | ≤ λ
r
Q
1 2
r
h, hkψk
W
1,2
B
r
≤ λ
r
Q
1 2
r
h, hkE
∗ r
k
L W
1,2
,W
1,2
B
r
kϕk
W
1,2
≤ c
3
kϕk
W
d∗,2
. If we define the processes
I
1
t = v
0, I
2
t = R
t
us ds ,
I
3
t = R
t
1
B
m
f ·, zs, ∇us ds, I
4
t = R
t
1
B
m
g ·, zs, ∇us dW
s
where the integral I
2
converges in W
1,2
and the integrals I
3
, I
4
converge in L
2
then P [I
2
t ∈ W
2,2
] = 1, P [vt = I
1
t + A
m
I
2
t + I
3
t + I
4
t] = 1, t
∈ R
+
by the Chojnowska-Michalik theorem see [9] or Theorem 13 in [33]. Since E
r
m
◦ A
m
can be extended to a linear continuous operator from W
1,2
to W
−d
∗
,2 R
by 10.9, E
r
m
vt = E
r
m
I
1
t + E
r
m
A
m
I
2
t + E
r
m
I
3
t + E
r
m
I
4
t, t
∈ R
+
in W
−d
∗
,2 R
a.s. Since k f ·, zt, ∇utk
L
1
B
rm∧R
≤ c
d
R
d 2
f ·, utvt +
d
X
i= 1
f
i
·, utu
x
i
t
L
2
B
rm∧R
+ k f
d+ 1
·, utk
L
1
B
rm∧R
≤ c
d
R
d 2
1 + 2
1 2
κ
1 2
F
1 2
r
m
∧R
t, zt + κF
