holds for every k ∈ N, R ∈ [0, r
m
j
] and t ∈ [0, T λ] by 11.10. Applying Fatou’s lemma on the LHS of 11.12 and the Lebesgue dominated convergence theorem on the RHS of 11.12, we obtain,
letting j → ∞,
E ¨
h E
R
π
R
z0 sup
r ∈[0,t]
L
k
Gr, zr
« ≤ 4e
ρ t
Z
H
l oc
h E
R
π
R
zL
k
G0, z dΘ
for every t ∈ [0, T λ] and m ∈ N by 11.1, 11.9 and ii, iii in Section 11.3 as L
k
is a bounded nondecreasing continuous and eventually constant function. Since
E
R
◦ π
R
: H
l oc
→ H
l oc
converges uniformly to identity on
H
l oc
as R → ∞, we get
E ¨
h z0 sup
r ∈[0,t]
L
k
Gr, zr
« ≤ 4e
ρ t
Z
H
l oc
hzL
k
G0, z dΘ 11.13
for every t ∈ [0, T λ] and k ∈ N by the Lebesgue dominated convergence theorem. Consequently,
11.13 holds also for h = 1
K
where K is closed in H
l oc
, whence also for every F
σ
-set and every Borel set K
⊆ H
l oc
by regularity of Θ = P [z0 ∈ ·] Remark 11.1. The claim now follows from
Fatou’s lemma when letting k → ∞, applied on the LHS, since L
k
≤ L for every k ∈ N, applied on the RHS.
11.7 Martingale property
Let us remind the reader that the integrals in the following Proposition converge by the assumption v in Section 11.1 and by 11.11.
Proposition 11.9. Let ϕ ∈ D. Then
〈vt, ϕ〉 = 〈v0, ϕ〉 +
Z
t
〈ur, A ϕ〉 d r +
Z
t
〈 f ·, ur, vr, ∇ur, ϕ〉 d r
+ Z
t
〈g·, ur, vr, ∇ur dW
r
, ϕ〉
holds a.s. for every t ≥ 0 where W was defined in Corollary 11.6.
Proof. Let k
∈ N, let ϕ ∈ D have support in B
k
and, throughout this proof, consider only j ∈ N such
that r
m
j
≥ T
k
, i.e. j ≥ j
for some j and it holds that
k ≤ T
k
≤ r
m
j
≤ T
r
m j
≤ m
j
, j
≥ j .
Fixing 0 ≤ s t ≤ k, we consider the sequence ϕ
i
from Corollary C.1. Let also J ∈ N, 0 ≤ s
1
≤
1079
· · · ≤ s
J
≤ s, let H : R
2 J
×J
× R
dim H
µ
J
× R
N +
→ [0, 1] be a continuous function and define X
1 j
= E
r
m j
u
m
j
s
i
∧ r
m
j
E
r
m j
v
m
j
s
i
∧ r
m
j
, ϕ
i
1
+
L
2
i ,i
1
≤J
X
2 j
= W
m
j
s
1
e
l l
, . . . , W
m
j
s
J
e
l l
,
F
m
j
·, E
r
m j
u
m
j
L
1
B
Tρ
ρ∈N
X
j
= X
1 j
, X
2 j
X
j
=
〈z
j
s
i
, ϕ
i
1
〉
L
2
i ,i
1
≤J
, β
j
s
1
, . . . , β
j
s
J
,
F
m
j
·, u
j L
1
B
Tρ
ρ∈N
X =
〈zs
i
, ϕ
i
1
〉
L
2
i ,i
1
≤J
, βs
1
, . . . , βs
J
, ν for j
≥ j . If
h
δ
: R
+
→ [0, 1] 11.14
is any continuous function with support in [0, δ] such that h
δ
= 1 on [0, δ2] then we also define continuous mappings
d
j q
: CR
+
; H → R
u, v 7→ h
δ
˜ F
m
j
0, u0, v0 〈vq, ϕ〉 − 〈v0, ϕ〉
− h
δ
˜ F
m
j
0, u0, v0 Z
q
〈ur, A ϕ〉 d r − h
δ
˜ F
m
j
0, u0, v0 Z
q
〈 f
m
j
·, ur, vr, ∇ur, ϕ〉 d r D
j ,l
q
: CR
+
; H → R
u, v 7→ h
δ
˜ F
m
j
0, u0, v0 Z
q
〈g
m
j
·, ur, vr, ∇ure
l
, ϕ〉 d r
D
j q
: CR
+
; H → R
u, v 7→ h
2 δ
˜ F
m
j
0, u0, v0 X
l
Z
q
〈g
m
j
·, ur, vr, ∇ure
l
, ϕ〉
2
d r for q
∈ [0, k], j ≥ j and l indexing the ONB e
l
in H
µ
that satisfy |d
j q
z| + |D
j ,l
q
z| + |D
j q
z| ≤ K1
[˜ F
m j
0,z0≤δ]
[1 + sup
r ∈[0,k]
˜ F
m
j
r, zr] 11.15
for q ∈ [0, k], z ∈ CR
+
; H , j ≥ j
, l up to dim H
µ
and for some K = K
d ,k,
κ,a,ϕ,c
as k f
m
j
·, ur, vr, ∇urk
L
1
B
k
≤ 8κ
1 2
Leb
d
B
k
+ κ[1 + ˜ F
m
j
r, zr] 11.16
kg
m
j
·, ur, vr, ∇urk
L
2
B
k
≤ 5κ
1 2
˜ F
1 2
m
j
r, zr
1080
holds for every r ∈ [0, k] where ˜F
m
= ˜ F
m λ
Tk
,0,T
k
is the conic energy function for ˜ F
m
w, y = F
m
w, y + | y|
2
2 defined as in 2.2. Also, for every p 0, there exist constants K
p
depending also on d, k,
κ, a, ϕ and c such that
E sup
q ∈[0,k]
h |d
j q
z