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Necessary and sufficient condition for the smoothness of intersection local time
of subfractional Brownian motions
Journal of Inequalities and Applications 2011, 2011:139 doi:10.1186/1029-242X-2011-139
Guangjun Shen ([email protected])
ISSN 1029-242X
Article type Research
Submission date 6 September 2011
Acceptance date 19 December 2011
Publication date 19 December 2011
Article URL http://www.journalofinequalitiesandapplications.com/content/2011/1/139
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Necessary and sufficient condition for the smoothness of
intersection local time of subfractional Brownian motions
Guangjun Shen
Department of Mathematics, Anhui Normal University,
Wuhu 241000, China
Email address: [email protected]
Abstract Let S
H

inequalities, we show that 
T
exists in L
2
if and only if Hd < 2
and it is smooth in the sense of the Meyer-Watanabe if and only if
H <
2
d+2
. As a related problem, we give also the regularity of the
intersection local time process.
2010 Mathematics Subject Classification: 60G15; 60F25;
60G18; 60J55.
Keywords: subfractional Brownian motion; intersection local time;
Chaos expansion.
1. Introduction
The intersection properties of Brownian motion paths have been in-
vestigated since the forties (see [1]), and since then, a large number
of results on intersection local times of Brownian motion have been
accumulated (see Wolpert [2], Geman et al. [3], Imkeller et al. [4], de
Faria et al. [5], Albeverio et al. [6] and the references therein). The
intersection local time of independent fractional Brownian motions has
been studied by Chen and Yan [7], Nualart et al. [8], Rosen [9], Wu and
Xiao [10] and the references therein. As for applications in physics, the
Edwards
,
model of long polymer molecules by Brownian motion paths
uses the intersection local time to model the ‘excluded volume’ effect:
different parts of the molecule should not be located at the same point
in space, while Symanzik [11], Wolpert [12] introduced the intersection

H
s

= s
2H
+ t
2H
−
1
2

(s + t)
2H
+ |t −s|
2H

(1.1)
for all s, t ≥ 0. For H =
1
2
, S
H
coincides with the Brownian motion B.
S
H
is neither a semimartingale nor a Markov process unless H = 1/2,
so many of the powerful techniques from stochastic analysis are not
available when dealing with S
H
. The sub-fBm has self-similarity and

d
, d ≥ 2, with the same indices H ∈ (0, 1).
This means that we have two d-dimensional independent centered Gauss-
ian processes S
H
= {S
H
t
, t ≥ 0} and

S
H
= {

S
H
t
, t ≥ 0} with covariance
structure given by
E

S
H,i
t
S
H,j
s

= E


−

S
H
s

dsdt, (1.3)
where δ(·) denotes the Dirac delta function. It is a measure of the
amount of time that the trajectories of the two processes, S
H
and

S
H
,
3
intersect on the time interval [0, T ]. As we pointed out, this definition is
only formal. In order to give a rigorous meaning to 
T
, we approximate
the Dirac delta function by the heat kernel
p
ε
(x) = (2πε)
−
d
2
e
−
|x|


S
H
. An interesting question is to study the behavior of 
ε,T
as ε tends
to zero.
For H =
1
2
, the process S
H
and

S
H
are Brownian motions. The inter-
section local time of independent Brownian motions has been studied
by several authors (see Wolpert [2], Geman et al. [3] and the references
therein). In the general case, that is H =
1
2
, only the collision local
time has been studied by Yan and Shen [24]. Because of interesting
properties of sub-fBm, such as short-/long-range dependence and self-
similarity, it can be widely used in a variety of areas such as signal
processing and telecommunications( see Doukhan et al. [26]). There-
fore, it seems interesting to study the so-called intersection local time
for sub-fBms, a rather special class of self-similar Gaussian processes.
The aim of this paper is to prove the existence, smoothness, regu-

(x
1
, . . . , x
k
)
is a polynomial of degree n of k variables x
1
, . . . , x
k
, then we call
p
n
(X
i
1
t
1
, . . . , X
i
k
t
k
) a polynomial functional of X with t
1
, . . . , t
k
∈ [0, T ]
and 1 ≤ i
1
, . . . , i

, then L
2
(Ω, P) is actually the direct sum of C
n
, i.e.,
L
2
(Ω, P) =
∞

n=0
C
n
. (2.1)
For F ∈ L
2
(Ω, P), we then see that there exists F
n
∈ C
n
, n = 0, 1, 2, . . . ,
such that
F =
∞

n=0
F
n
, (2.2)
This decomposition is called the chaos expansion of F . F

|
2
) < ∞},
and F ∈ L
2
(Ω, P) is said to be smooth if F ∈ U .
Now, for F ∈ L
2
(Ω, P), we define an operator Υ
u
with u ∈ [0, 1] by
Υ
u
F :=
∞

n=0
u
n
F
n
. (2.4)
Set Θ(u) := Υ
√
u
F . Then, Θ(1) = F. Define Φ
Θ
(u) :=
d
du

n=1
E(u
n
|F
n
|
2
).
Proposition 1. Let F ∈ L
2
(Ω, P). Then, F ∈ U if and only if
Φ
Θ
(1) < ∞.
Now consider two d-dimensional independent sub-fBms S
H
and

S
H
with indices H ∈ (0, 1). Let H
n
(x), x ∈ R be the Hermite polynomials
of degree n. That is,
H
n
(x) = (−1)
n
1
n!

for all t ∈ C and x ∈ R, which deduces
exp(iuξ, S
H
t
−

S
H
s
 +
1
2
u
2
|ξ|
2
Var(S
H,1
t
−

S
H,1
s
))
=
∞

n=0
(iu)

for ξ ∈ R
d
. Because of the
orthogonality of {H
n
(x), x ∈ R}
n∈Z
+
, we will get from (2.2) that
(iu)
n
σ
n
(t, s, ξ)H
n

ξ, S
H
t
−

S
H
s

σ(t, s, ξ)

is the n-th chaos of
exp


local time of S
H
and

S
H
, for an H =
1
2
and d ≥ 2. We have obtained
the following result.
Theorem 2. (i) If Hd < 2, then the 
ε,T
converges in L
2
(Ω). The
limit is denoted by 
T
(ii) If Hd ≥ 2, then
lim
ε→0
E(
ε,T
) = +∞,
and
lim
ε→0
Var(
ε,T
) = +∞.

(
ε
−
6
E
ε
) exists in L
2
(Ω). Condition (ii) implies that Varadhan renormal-
ization does not converge in this case.
For Hd ≥ 2, according to Theorem 2, 
ε,T
does not converge in
L
2
(Ω), and therefore, 
T
, the intersection local time of S
H
and

S
H
,
does not exist.
Using the following classical equality
p
ε
(x) =
1

T

0
p

(S
H
t
−

S
H
s
)dsdt
=
1
(2π)
d
T

0
T

0

R
d
e
iξ,S
H

)), so
E[e
iξ,S
H
t
−

S
H
s

] = e
−[(2−2
2H−1
)(t
2H
+s
2H
)]
|ξ|
2
2
.
Therefore,
E(
ε,T
) =
1
(2π)
d


0
T

0

R
d
e
−[ε+(2−2
2H−1
)(t
2H
+s
2H
)]
|ξ|
2
2
dξdsdt
=
1
(2π)
d
2
T

0
T


2π
ε + (2 −2
2H−1
)(t
2H
+ s
2H
)

d
2
.
7
We also have
E(
2
ε,T
) =
1
(2π)
2d

[0,T ]
4

R
2d
E

e

2
dξdηdsdtdudv.
(3.3)
Let we intro duce some notations that will be used throughout this
paper,
λ
s,t
= Var(S
H,1
t
− S
H,2
s
) = (2 − 2
2H−1
)(t
2H
+ s
2H
),
ρ
u,v
= Var(S
H,1
v
− S
H,2
u
) = (2 − 2
2H−1

2
H
+ v
2
H
−
1
2
[(t + v)
2
H
+ |t −v|
2
H
+ (s + u)
2
H
+ |s −u|
2
H
],
where S
H,1
and S
H,2
are independent one dimensional sub-fBms with
indices H. Using the above notations, we can write for any ε > 0
E(
2
ε,T

=
1
(2π)
d

[0,T ]
4

(λ
s,t
+ ε)(ρ
u,v
+ ε) −µ
2
s,t,u,v

−
d
2
dsdtdudv.
(3.4)
In order to prove the Theorem 2, we need some auxiliary lemmas.
Without loss of generality, we may assume v ≤ t, u ≤ s and v = xt, u =
ys with x, y ∈ [0, 1]. Then, we can rewrite ρ
u,v
and µ
s,t,u,v
as following.
ρ
u,v

1 + y
2H
−
1
2
[(1 + y)
2H
+ (1 −y)
2H
]

.
(3.5)
It follows that
λ
s,t
ρ
u,v
− µ
2
s,t,u,v
= t
4H
f(x) + s
4H
f(y) + t
2H
s
2H
g(x, y), (3.6)


x
2H
+ y
2H

− 2

1 + x
2H
−
1
2
(1 + x)
2H
−
1
2
(1 − x)
2H

×

1 + y
2H
−
1
2
(1 + y)
2H

g(x, y)  x
2H
(1 − y)
2H
+ y
2H
(1 − x)
2H
(3.9)
for all x, y ∈ [0, 1].
Lemma 4. Let
A
T
:=

[0,T ]
4

λ
s,t
ρ
u,v
− µ
2
s,t,u,v

−
d
2
dsdtdudv.









s = r cos ϕ
1
,
t = r sin ϕ
1
cos ϕ
2
,
u = r sin ϕ
1
sin ϕ
2
cos ϕ
3
,
v = r sin ϕ
1
sin ϕ
2
sin ϕ
3
.

u,v
−µ
2
s,t,u,v
is always positive, and λ
s,t
ρ
u,v
−µ
2
s,t,u,v
= r
4H
φ(θ),
we have
A
T
≥

D
ε
(λ
s,t
ρ
u,v
− µ
2
s,t,u,v
)
−

)
−
d
2
dsdtdudv,
where Υ = {(u, v, s, t) : 0 < u < s ≤ T, 0 < v < t ≤ T }.
By Lemma 3, we get
λ
s,t
ρ
u,v
− µ
2
s,t,u,v
= t
4H
f(x) + s
4H
f(y) + t
2H
s
2H
g(x, y)
 t
4H
x
2H
(1 − x)
2H
+ s

+ (1 −y)
2H
s
2H
]
= (v
2H
+ u
2H
)[(t − v)
2H
+ (s −u)
2H
].
(3.12)
These deduce for all H ∈ (0, 1) and T > 0,
Λ
T
≤ C
H
T

0
dt
t

0
(v
H
(t − v)

0
x
−
Hd
2
(1 − x)
−
Hd
2
dx


2
< ∞.

Proof of Theorem 2. Suppose Hd < 2, we have
E(
ε,T
· 
η,T
) =
1
(2π)
d

[0,T ]
4
((λ
s,t
+ ε)(ρ

dsdtdudv < ∞.
This is true due to Lemma 4.
If Hd ≥ 2, then from (3.2) and using monotone convergence theorem
lim
ε→0
E(
ε,T
) =
1
(2π(2 − 2
2H−1
))
d/2
T

0
T

0
(s
2H
+ t
2H
)
−
d
2
dsdt.
Making a p olar change of coordinates


1−Hd
(cos
2H
θ + sin
2H
θ)
−
d
2
drdθ,
and this integral is divergent if Hd ≥ 2. By the expression (3.2) and
(3.4), we have
lim
ε→0
Var(
ε,T
) = lim
ε→0
[E(
2
ε,T
) − (E
ε,T
)
2
]
=
1
(2π)
d

V ar(
ε,T
) = +∞.
11
In fact, as the integrand is always positive, we obtain

[0,T ]
4


λ
s,t
ρ
u,v
− µ
2
s,t,u,v

−
d
2
− (λ
s,t
ρ
u,v
)
−
d
2



0
r
3−2Hd
dr

Θ
ψ(θ)dθ,
where the integral in r is convergent if and only if Hd < 2, and the
angular integral is different from zero thanks to the positivity of the
integrand. Therefore, Hd ≥ 2 implies that
lim
ε→0
Var(
ε,T
) = +∞.
This completes the proof of Theorem 2. 
4. Smoothness of the intersection local time
In this section, we consider the smoothness of the intersection local
time. Our main object is to explain and prove the following theorem.
The idea is due to An and Yan [32] and Chen and Yan [7].
Theorem 5. Let 
T
be the intersection local time of two independent
d-dimensional sub-fBms S
H
and

S
H

= s
2H
+t
2H
+u
2H
+v
2H
−
1
2
[(t+v)
2H
+|t−v|
2H
+(s+u)
2H
+|s−u|
2H
],
for all s, t, u, v ≥ 0.
In order to prove Theorem 5, we need the following propositions.
Proposition 6. Under the assumptions above, the following statements
are equivalent:
(i) H <
2
d+2
;
(ii)
T

s,t
ρ
u,v
− µ
2
s,t,u,v
= t
4H
f(x) + s
4H
f(y) + t
2H
s
2H
g(x, y)
 t
4H
x
2H
(1 − x)
2H
+ s
4H
y
2H
(1 − y)
2H
+ t
2H
s

2H
≤ 1 + x
2H
−
1
2
(1 + x)
2H
−
1
2
(1 − x)
2H
≤ (2 − 2
2H−1
)x
2H
for all x, H ∈ (0, 1). By (3.5), we obtain
(t
2H
x
2H
+ s
2H
y
2H
)
2
≤ µ
2

λ
s,t
ρ
u,v
− µ
2
s,t,u,v

−
d
2
−1
µ
2
s,t,u,v
dsdtdudv
≥ C
H,T
T

0
1

0
T

0
1

0

1

0
1

0
(t
2H
x
2H
+ s
2H
y
2H
)st
((1 − x)
2H
t
2H
+ (1 −y)
2H
s
2H
)
1+
d
2
dydsdxdt
≥ C
H,T

y

0
x
4−H(d−2)
(1 − x)
2H(1+d/2)
dx = C
H,T
1

0
x
4−H(d−2)
(1 − x)
1−2H(1+d/2)
dx,
where C
H,T
> 0 is a constant depending only on H and T and its
value may differ from line to line, which implies that H <
2
d+2
if the
convergence (ii) holds.
13
On the other hand,
T

0

0
1

0
T

0
1

0
(t
2H
x
2H
+ s
2H
y
2H
)
2
st
[(x
2H
t
2H
+ y
2H
s
2H
)((1 − x)

y
2H
)
2
st
[(x
H
t
H
y
H
s
H
)((1 − x)
H
t
H
(1 − y)
H
s
H
)]
d/2+1
dydsdxdt
≤ C
H
T

0
1

dydsdxdt
< ∞
if H <
2
d+2
. Where C
H
> 0 is a constant depending only on H and its
value may differ from line to line. Thus, the proof is completed. 
Hence, Theorem 5 follows from the next proposition.
Proposition 7. Under the assumptions above, the following statements
are equivalent: 
T
∈ U if and only if
T

0
T

0
T

0
T

0
(λ
s,t
ρ
u,v

n
, m = n.
(4.4)
Moreover, elementary calculus can show that the following lemma holds.
Lemma 8 ( [7]). Suppose d ≥ 1. For any x ∈ [−1, 1) we have
∞

n=1
n

k
1
, ,k
d
=0
k
1
+···+k
d
=n
2n(2k
1
− 1)!! ·····(2k
d
− 1)!!
(2k
1
)!! · ··· · (2k
d
)!!

d
2
+1
=
µ
2
s,t,u,v
λ
s,t
ρ
u,v

1 −
µ
2
s,t,u,v
λ
s,t
ρ
u,v

−(
d
2
+1)

1
λ
s,t
ρ

µ
2n
s,t,u,v
(λ
s,t
ρ
u,v
)
n+
d
2
.
Proof of Proposition 7. For ε > 0, T ≥ 0, we denote
Φ
Θ
ε
(κ) := E(|Υ
√
κ

ε,T
|
2
)
and Φ
Θ
(κ) := E(|Υ
√
κ


d
T

0
T

0

R
d
e
iξ,S
H
t
−

S
H
s

e
−ε
|ξ|
2
2
dξdsdt
=
1
(2π)
d

−

S
H
s

σ(t, s, ξ)

dξdsdt
≡
∞

n=0
F
n
.
Thus, by (4.4) and Lemma 8, we have
Φ
Θ
ε
(1) =
∞

n=0
nE(|F
n
|
2
)
=

σ
n
(t, s, ξ)σ
n
(u, v, η)
H
n

ξ, S
H
t
−

S
H
s

σ(t, s, ξ)

H
n

η, S
H
u
−

S
H
v

s,t
+ε)|ξ|
2
+(ρ
u,v
+ε)|η|
2
)
ξ, η
n
dξdη
=
∞

n=1
1
(2π)
2d
(2n − 1)!

[0,T ]
4
µ
2n
s,t,u,v
dudvdsdt

R
2d
e

dudvdsdt
×

R
2d
e
−
1
2
(
(λ
s,t
+ε)(ξ
2
1
+···+ξ
2
d
)+(ρ
u,v
+ε)(η
2
1
+···+η
2
d
)
(ξ
1
η


R
2d
e
−
1
2
(
(λ
s,t
+ε)(ξ
2
1
+···+ξ
2
d
)+(ρ
u,v
+ε)(η
2
1
+···+η
2
d
)
)
n

k
1

1
. . . dξ
d
dη
1
. . . dη
d
=
1
(2π)
d
∞

n=1
n

k
1
, ,k
d
=0
k
1
+···+k
d
=n
2
n
(2
k


[0,T ]
4
µ
2
s,t,u,v
((λ
s,t
+ ε)(ρ
u,v
+ ε) −µ
2
s,t,u,v
)
−
d
2
−1
dudvdsdt,
where we have used the following fact:

R
ξ
2k
e
−
1
2
(λ
s,t

−(k+
1
2
)
=
√
2π(2k − 1)!!(λ
s,t
+ ε)
−(k+
1
2
)
.
It follows that
lim
ε→0
Φ
Θ
ε
(1) 

[0,T ]
4
µ
2
s,t,u,v
(λ
s,t
ρ

Proof. Let C > 0 be a constant depending only on H and d and its
value may differ from line to line. For any 0 ≤ r, l, u, v ≤ T , denote
σ
2
= Var

ξ

S
H
r
−

S
H
l

+ η

S
H
u
−

S
H
v

.
Then, the property of strong local nondeterminism (see Yan et al. [24]) :


S
H
t
j
− S
H
t
j−1

. (5.1)
holds for 0 ≤ t
1
< t
2
< ··· < t
n
≤ T and u
j
∈ R, j = 2, 3, . . . , n. and
(1.2) yield
σ
2
= Var

ξ

S
H
r

2
(|r −u|
2H
+ |l − v|
2H
) + (ξ + η)
2
(u
2H
+ v
2H
)

.
It follows from (3.1) that for 0 ≤ s ≤ t ≤ T
E|
ε,t
− 
ε,s
|
2
=
1
(2π)
2d
t

s
t



s
dr
t

s
dl
t

s
s

0
dudv

R
2d
e
−
1
2
(
σ
2
+ε|ξ|
2
+ε|η|
2
)
dξdη

2
+ε|η|
2
)
dξdη
≡
1
(2π)
2d
[A
1
(s, t) + 4A
2
(s, t) + 4A
3
(s, t)] .
17
We have
A
1
(s, t) =
t

s
t

s
drdl
t



s
t

s
dudv

(|r −u|
2H
+ |l − v|
2H
)(u
2H
+ v
2H
)

−
d
2
≤ C
t

s
t

s
t

s

−
Hd
2
u
−
Hd
2
drdu


2
≤ 4C


t

s
r

s
(r −u)
−
Hd
2
u
−
Hd
2
dudr


u
−
Hd
2
dudr =
t

s
r
1−Hd
dr
1

s/r
(1 − m)
−
Hd
2
m
−
Hd
2
dm
≤ C(t − s)
2−dH
,
which yields
A
1
(s, t) ≤ C(t −s)

−
1
2
(
σ
2
+ε|ξ|
2
+ε|η|
2
)
dξdη
≤ C
t

s
dr
t

s
dl
t

s
s

0
dudv

(|r −u|

s
dl
s

0
|l − v|
−
Hd
2
v
−
Hd
2
dv
≤ Ct
2−Hd
(t − s)
2−Hd
,
A
3
(s, t) =
t

s
dr
s

0
dl

drdl
t

s
s

0
dudv

(|r −u|
2H
+ |l − v|
2H
)(u
2H
+ v
2H
)

−
d
2
= C
t

s
t

s
|r −u|

t
− 
s
|
2
) = E(lim
ε→0
|
ε,t
− 
ε,s
|
2
) ≤ lim inf
ε→0
E(|
ε,t
− 
ε,s
|
2
) ≤ Ct
2−Hd
(t − s)
2−Hd
.
This completes the proof. 
Competing interests
The author declare that he has no competing interests.
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