M
x, ℓ
ǫ
′
are defined to be constant on [0, T ]\[τ
ℓ
, τ
ℓ+1
]. Thus, it is enough to consider the case where [s, t] intersects [
τ
ℓ
, τ
ℓ+1
]. Setting ˆ s
i
:= s ∨ τ
ℓ
and ˆt := t ∧ τ
ℓ+1
we have |ˆt − ˆs| ≤ |t − s|. By
the definition of M
x, ℓ
ǫ and M
x, ℓ
ǫ
′
we have E
∆ M
x, ℓ
t
ǫ, ǫ
′
− ∆M
x, ℓ
s
ǫ, ǫ
′ p
≤c E
Z
ˆt ˆ
s
b
1 m
ℓ
X
r
x
ǫ
− b
1 m
ℓ
X
r
x
ǫ
′
d r
p
+ c E
Z
ˆt ˆ
s
σ
1 m
ℓ
X
r
x
ǫ
− σ
1 m
ℓ
X
r
x
ǫ
′
d w
r p
. By the uniform Lipschitz continuity of b
m
and Proposition 3.2 the first term can be estimated by c
|t − s|
p
E
sup
r ∈[ˆs,ˆt]
kX
r
x
ǫ
− X
r
x
ǫ
′
k
p
≤ c |ǫ − ǫ
′
|
p
|t − s|
p
. For the second term we get the following estimate by Burkholder’s inequality, the uniform Lipschitz
continuity of σ
m
and again by Proposition 3.2: c E
sup
r ∈[ˆs,ˆt]
Z
r ˆ
s
σ
1 m
ℓ
X
r
x
ǫ
− σ
1 m
ℓ
X
r
x
ǫ
′
d w
r p
≤c E
Z
ˆt ˆ
s
kX
r
x
ǫ
− X
r
x
ǫ
′
k
2
d r
p 2
≤c |t − s|
p 2
E
sup
r ∈[ˆs,ˆt]
kX
r
x
ǫ
− X
r
x
ǫ
′
k
p
≤c |ǫ − ǫ
′
|
p
|t − s|
p 2
and we obtain the desired estimate. We apply now Kolmogorov’s continuity theorem, in particular the version for double parameter random fields in Theorem 1.4.4 in [17], which implies that for
any given δ
1
∈ 0, 1 and δ
2
∈ 0,
1 2
there exists a modification of the random field M
x, ℓ
ǫ
t, ǫ
satisfying 3.2 for some random constant K = K ω, δ
1
, δ
2
, T . Step 2: The existence of a modification shown in Step 1 immediately implies that a.s. 3.2 holds
for all s, t ∈ [0, T ] ∩ Q and ǫ, ǫ
′
∈ [a
x
, b
x
] ∩ Q. The claim follows if ∆M
x, ℓ
t
ǫ, ǫ
′
− ∆M
x, ℓ
s
ǫ, ǫ
′
is pathwise continuous in s and t as well as in
ǫ and ǫ
′
. It is enough to show the continuity of M
x, ℓ
t
ǫ in t and
ǫ. The continuity in t is obvious and for every ǫ such that x
ǫ
∈ B
∆
T
x ∩ G we get by an application of Itô’s formula as in 2.2
M
x, ℓ
t
ǫ = u
1 m
ℓ
X
t
x
ǫ
− u
1 m
ℓ
X
τ
ℓ
x
ǫ
− L
t
x
ǫ
, where the right hand side is continuous in
ǫ by Proposition 3.2 and Lemma 3.4.
3.2 Convergence of Minimum Times
In this section we investigate the behaviour of the local time, when the starting point x of X x has been perturbed by a small
ǫ. To that purpose we shall transform the process locally into a process 858
on the halfspace as indicated in Section 2.3, which allows us to use Skorohod’s Lemma to compute the local time in terms of the time when the continuous martingale M
x, ℓ
attains its minimum. As a result we shall obtain that for
ǫ tending to zero the minimum time of M
x, ℓ
converges almost surely faster than polynomially to the minimum time of M
x, ℓ
. We fix from now on an arbitrary T
0. In the following let A
n
be the family of connected components of [0, T ]
\C. Then, A
n
is open and recall that for every ℓ the mapping t 7→ [M
x, ℓ
]
t
is continuous and increasing on [ τ
ℓ
, τ
ℓ+1
. Thus, for every n we may choose a random q
n
∈ A
n
a follows: Let ℓ be such that inf A
n
∈ [τ
ℓ
, τ
ℓ+1
, then we choose q
n
∈ [τ
ℓ
, τ
ℓ+1
∩ A
n
such that [M
x, ℓ
]
q
n
∈ Q. In order to compute the local time lx, recall that on every interval [
τ
ℓ
, τ
ℓ+1
, ℓ ≥ 0, lx is carried by the set of times t, when u
1 m
ℓ
X
t
x = 0. Therefore, we can apply Skorohod’s Lemma see e.g. Lemma VI.2.1 in [23] to equation 2.4 to obtain
L
t
x = −u
1 m
ℓ
X
τ
ℓ
x − inf
τ
ℓ
≤s≤t
M
x, ℓ
s +
, t
∈ [τ
ℓ
, τ
ℓ+1
. Fix any q
n
inf C and ℓ such that q
n
∈ [τ
ℓ
, τ
ℓ+1
. Since u
1 m
ℓ
X
rq
n
x = 0 and t 7→ L
t
x is non-decreasing, we have for all
τ
ℓ
≤ s ≤ rq
n
: M
x, ℓ
rq
n
= −u
1 m
ℓ
X
τ
ℓ
x − L
rq
n
x ≤ −u
1 m
ℓ
X
τ
ℓ
x − L
s
x = −u
1 m
ℓ
X
s
x + M
x, ℓ
s
≤ M
x, ℓ
s
. Moreover, Lx is constant on [rq
n
, t] for all t ∈ A
n
∩ [τ
ℓ
, τ
ℓ+1
, so that L
t
x = L
rq
n
x = h
−u
1 m
ℓ
X
τ
ℓ
x − M
x, ℓ
rq
n
i
+
, t
∈ A
n
∩ [τ
ℓ
, τ
ℓ+1
. 3.3
Note that M
x, ℓ
rq
n
≤ M
x, ℓ
s
for all s ∈ [τ
ℓ
, q
n
]. Further, with probability one we have that rq
n
is the unique time in [
τ
ℓ
, q
n
], when M
x, ℓ
attains its minimum. Analogously we compute the local time of the process with perturbed starting point. For fixed v
∈ R
d
we set x
ǫ
:= x + ǫv, ǫ ∈ R, where |ǫ| is
always supposed to be sufficiently small, such that x
ǫ
lies in G. Furthermore, there exists a random ∆
n
0 such that for all ǫ ∈ −∆
n
, ∆
n
we have X
t
x
ǫ
∈ U
m
ℓ
for all t ∈ [τ
ℓ
, q
n
] cf. Remark 3.3. As above we obtain for such
ǫ: L
q
n
x
ǫ
= L
r
ǫ
q
n
x
ǫ
= h
−u
1 m
ℓ
X
τ
ℓ
x
ǫ
− M
x, ℓ
r
ǫ
q
n
ǫ i
+
, 3.4
where r
ǫ
q
n
is defined similarly as rq
n
. In particular, M
x, ℓ
r
ǫ
q
n
ǫ ≤ M
x, ℓ
s
ǫ for all s ∈ [τ
ℓ
, q
n
].
Lemma 3.6. For all n we have r
ǫ
q
n
→ rq
n
a.s. for ǫ → 0. Proof. Consider some q
n
and let ℓ be such that q
n
∈ [τ
ℓ
, τ
ℓ+1
. We fix now a typical ω such that rq
n
is the unique time in [τ
ℓ
, q
n
], when M
x, ℓ
attains its minimum and such that Lemma 3.5 holds. For every sequence
ǫ
k k
converging to zero we can extract a subsequence of r
ǫ
k
q
n
, still denoted by r
ǫ
k
q
n
, converging to some ˆrq
n
. By construction we have M
x, ℓ
r
ǫk
q
n
ǫ
k
≤ M
x, ℓ
rq
n
ǫ
k
859
for every k. Note that on one hand the right hand side converges to M
x, ℓ
rq
n
as k → ∞ by Lemma 3.5.
On the other hand the left hand side converges to M
x, ℓ
ˆrq
n
, since M
x, ℓ
r
ǫk
q
n
ǫ
k
− M
x, ℓ
ˆrq
n
≤ M
x, ℓ
r
ǫk
q
n
ǫ
k
− M
x, ℓ
r
ǫk
q
n
+ M
x, ℓ
r
ǫk
q
n
− M
x, ℓ
ˆrq
n
, where the first term tends to zero for k
→ ∞ by Lemma 3.5 and the second term by the continuity of M
x, ℓ
. Thus, M
x, ℓ
ˆrq
n
≤ M
x, ℓ
rq
n
. Since rq
n
is unique time in [τ
ℓ
, q
n
], when M
x, ℓ
attains its minimum, this implies ˆrq
n
= rq
n
.
Lemma 3.7. Let W
t t
≥0
be a Brownian motion on Ω, F , P. For all T 0, let ϑ : Ω → [0, T ] be the
random variable such that a.s. W
ϑ
W
s
∀s ∈ [0, T ]\{ϑ}. Then,
lim inf
s →ϑ
W
s
− W
ϑ
p |s − ϑ| h|s − ϑ|
≥ 1 a.s., 3.5
for every function h on [0, ∞ satisfying 0 ht ↓ 0 as t ↓ 0 and
R
r
ht
d t t
∞ for some r 0.
Proof. It suffices to consider the case T = 1. We recall the following path decomposition of a Brownian motion, proven in [10]. Denoting by M , ˆ
M two independent copies of the standard Brownian meander see [23], we set for all r
∈ 0, 1, V
r
t := −
p r M 1 +
p r M
r −t
r
, t
∈ [0, r] −
p r M 1 +
p 1
− r ˆ M
t −r
1 −r
, t
∈ r, 1] Let now
τ, M , ˆ M be an independent triple, such that
τ has the arcsine law. Then, V
τ d
= W . This formula has the following meaning:
τ is the unique time in [0, 1], when the path attains minimum −
p τM 1. The path starts in zero at time t = 0 and runs backward the path of M on [0, τ] and
then it runs the path of ˆ M . Moreover, it was proved in [14] that the law of the Brownian meander
is absolutely continuous w.r.t. the law of the three-dimensional Bessel process R
t t
≥0
on the time interval [0, 1] starting in zero. We recall that a.s.
lim inf
t →0
R
t
p t ht
≥ 1 for every function h satisfying the conditions in the statement see [15], p. 164. Since the same
asymptotics hold for the Brownian meander at zero, the claim follows. In the next proposition we will apply Lemma 3.7 to the Brownian motions B
x, ℓ
defined in Section 2.3. More precisely, Lemma 3.7 gives that a.s. for every
ℓ and every nonnegative q ∈ Q the following holds: If q
≤ [M
x, ℓ
]
τ
ℓ+1
, denoting by ϑ
ℓ q
the unique time when B
x, ℓ
attains its minimum over [0, q], B
x, ℓ
satisfies the asymptotic behaviour stated in 3.5 at ϑ
ℓ q
.
860
Proposition 3.8. Let δ 0 be arbitrary. Then, for all n we have
|r
ǫ
q
n
− rq
n
|
δ
ǫ −→ 0 a.s. as ǫ → 0.
Proof. First we fix 0 δ
′
1. By construction we have for every q
n
and ℓ such that q
n
∈ [τ
ℓ
, τ
ℓ+1
and for ǫ small enough,
M
x, ℓ
r
ǫ
q
n
ǫ ≤ M
x, ℓ
rq
n
ǫ. Since M
x, ℓ
t
ǫ = M
x, ℓ
t
+ ∆M
x, ℓ
t
ǫ, 0 for every t ∈ [τ
ℓ
, q
n
] with ∆M
x, ℓ
t
ǫ, 0 as in Lemma 3.5, this can be rewritten as
M
x, ℓ
r
ǫ
q
n
− M
x, ℓ
rq
n
≤ ∆M
x, ℓ
rq
n
ǫ, 0 − ∆M
x, ℓ
r
ǫ
q
n
ǫ, 0, which implies
M
x, ℓ
r
ǫ
q
n
− M
x, ℓ
rq
n
|r
ǫ
q
n
− rq
n
|
1−δ
′
2
1l
{r
ǫ
q
n
6=rq
n
}
≤ ∆ M
x, ℓ
rq
n
ǫ, 0 − ∆M
x, ℓ
r
ǫ
q
n
ǫ, 0 |r
ǫ
q
n
− rq
n
|
1−δ
′
2
1l
{r
ǫ
q
n
6=rq
n
}
. 3.6
Recall that M
x, ℓ
·
= B
x, ℓ
[M
x, ℓ
]
.
, where B
x, ℓ
is a ˜ P
ℓ
x-Brownian motion see 2.5 and B
x, ℓ
attains its minimum over
0, [M
x, ℓ
]
q
n
at time
ϑ
ℓ [M
x, ℓ
]
qn
= [M
x, ℓ
]
rq
n
. Note that q
n
has been chosen such that [M
x, ℓ
]
q
n
∈ Q. Hence, applying Lemma 3.7 with ht = t
δ
′
2
it follows that a.s. M
x, ℓ
r
ǫ
q
n
− M
x, ℓ
rq
n
= B
x, ℓ
[M
x, ℓ
]
rǫ qn
− B
x, ℓ
[M
x, ℓ
]
rqn
≥
1 2
[M
x, ℓ
]
r
ǫ
q
n
− [M
x, ℓ
]
rq
n
1+δ
′
2
=
1 2
Z
r
ǫ
q
n
rq
n
kσ
1 m
ℓ
X
r
xk
2
d r
1+δ
′
2
for all ǫ ∈ −∆
n
, ∆
n
for some positive ∆
n
. Since kσ
1 m
ℓ
X
rq
n
xk
2
= k∇u
1 m
l
X
rq
n
xk
2
= knX
rq
n
xk
2
= 1, we have by Lemma 3.6, possibly after choosing a smaller ∆
n
, that kσ
1 m
ℓ
X
r
xk
2
is bounded away from zero uniformly in r between rq
n
and r
ǫ
q
n
. Thus, M
x, ℓ
r
ǫ
q
n
− M
x, ℓ
rq
n
≥ c
1
r
ǫ
q
n
− rq
n 1+δ
′
2
and we derive from 3.6 that a.s. c
1
r
ǫ
q
n
− rq
n δ
′
≤ sup
s,t ∈[0,T ]
s 6=t
∆ M
x, ℓ
t
ǫ, 0 − ∆M
x, ℓ
s
ǫ, 0 |t − s|
1−δ
′
2
≤ K |ǫ|
1 −δ
′
for some random constant K = K ω, δ
′
, T , where we have used Lemma 3.5. Hence, for every δ 0
we have |ǫ|
−1
|r
ǫ
q
n
− rq
n
|
δ
≤ K
δ
for some random constant K
δ
. In particular, since |r
ǫ
q
n
− rq
n
|
δ
ǫ ≤ K
δ2
|r
ǫ
q
n
− rq
n
|
δ2
, the claim follows by Lemma 3.6.
861
Corollary 3.9. For any n and let ℓ be such that q
n
∈ [τ
ℓ
, τ
ℓ+1
. Then, i
1 ǫ
M
x, ℓ
rq
n
ǫ − M
x, ℓ
r
ǫ
q
n
ǫ −→ 0 a.s. as ǫ → 0,
ii
1 ǫ
l
rq
n
x
ǫ
− l
r
ǫ
q
n
x
ǫ
−→ 0 a.s. as ǫ → 0. Proof. For arbitrary
δ ∈ 0,
1 2
we have 1
ǫ M
x, ℓ
rq
n
ǫ − M
x, ℓ
r
ǫ
q
n
ǫ ≤ 1l
{r
ǫ
q
n
6=rq
n
}
|r
ǫ
q
n
− rq
n
|
1 2 −
δ
ǫ M
x, ℓ
rq
n
ǫ − M
x, ℓ
r
ǫ
q
n
ǫ |r
ǫ
q
n
− rq
n
|
1 2
−δ
≤ |r
ǫ
q
n
− rq
n
|
1 2 −
δ
ǫ sup
s,t ∈[0,T ]
s 6=t
M
x, ℓ
t
ǫ − M
x, ℓ
s
ǫ |t − s|
1 2
−δ
. Since
sup
s 6=t
M
x, ℓ
t
ǫ − M
x, ℓ
s
ǫ |t − s|
1 2
−δ
≤ sup
s 6=t
∆ M
x, ℓ
t
ǫ, 0 − ∆M
x, ℓ
s
ǫ, 0 |t − s|
1 2
−δ
+ sup
s 6=t
M
x, ℓ
t
− M
x, ℓ
s
|t − s|
1 2
−δ
, is a.s bounded by a random constant due to Lemma 3.5 and due to the fact that M
x, ℓ
is Hölder continuous of order
1 2
− δ, we obtain i from Proposition 3.8. ii follows from i. Indeed, by Proposition 3.2 and Lemma 3.6 we have for
ǫ sufficiently small that l
rq
n
x
ǫ
= l
r
ǫ
q
n
x
ǫ
= 0 if q
n
inf C and l
rq
n
x
ǫ
, l
r
ǫ
q
n
x
ǫ
0 if q
n
inf C. In the first case ii is trivial and the latter case we have by 3.4
l
rq
n
x
ǫ
− l
r
ǫ
q
n
x
ǫ
= L
rq
n
x
ǫ
− L
r
ǫ
q
n
x
ǫ
= M
x, ℓ
r
ǫ
rq
n
ǫ − M
x, ℓ
r
ǫ
q
n
ǫ ≤ M
x, ℓ
rq
n
ǫ − M
x, ℓ
r
ǫ
q
n
ǫ, 3.7
where we have used the fact that M
x, ℓ
ǫ attains its minimum over [τ
ℓ
, q
n
] at time r
ǫ
q
n
and its minimum over [
τ
ℓ
, rq
n
] at time r
ǫ
rq
n
, respectively.
3.3 Proof of the Differentiability
