6
Definition 2.2.
For 0,1
H
∈
, ,
a b
∈
, , 0,0
a b
≠
a mixed fractional Brownian motion of parameter
, H a
, and b in the white noise space is given by the continuous
version of
[0, [0,
., 1 1
H t
t
a bN
−
+ .
3. Mixed Fractional White Noise
As we know a mixed fractional Brownian motion
H
M is nowhere differentiable on almost every path. However, we are going to show that
H
M is differentiable as a mapping from
into a space of stochastic generalized functions, the so-called Hida distributions. The distributional derivative of mixed
fractional Brownian motion is called mixed fractional white noise. According to Wiener-Ito decomposition theorem every
2 2
: , ,
L L S
B ϕ
μ ′
∈ =
can be decomposed uniquely as
2
ˆ :
:, ,
n n
n n
c n
f f
L ϕ
∞ ⊗
=
= ⋅
∈
∑
4 where
2
ˆ
n c
L denotes the space of symmetric complex-valued
2
L -functions on
n
, and n
⊗ denotes the n times tensor product. The above decomposition is called the Wiener-Ito expansion of
ϕ . Moreover the
2
L -norm ϕ of ϕ is given
by
2 2
2
:
n n
E n f
μ
ϕ ϕ
∞ =
= =
∑
5 Now consider Hamiltonian of harmonic oscillator
2 2
2
: 1
d A
x dx
= − +
+ and we define its second quantization operator
A Τ
in terms of the Wiener-Ito expansion. The domain of
A Τ
, denoted by D
A Τ
, is the space of function
ϕ
of the form 4 such that
n n
f D A
⊗
∈
and
2
n n
n
n A f
∞ ⊗
=
∞
∑
. Then, we define :
: :,
,
n n
n n
A A f
D A
ϕ ϕ
∞ ⊗
⊗ =
Τ =
⋅ ∈
Τ
∑
.
7
Both operators, A and A
Τ
, are densely defined on
2
L and
2
L , respectively. Furthermore they are invertible and the inverse operators are
bounded. For
p ∈
and
p
D A
ϕ
∈ Τ
we define a more general norm as follow :
p p
A
ϕ ϕ
= Τ .
Now define
{ }
2 2
: :
p p
p
S L
A exists and
A L
ϕ ϕ
= ∈
Τ Τ
∈ and endow
p
S with the norm
p
⋅ . If we define
: S
=
projective limit of
{ }
:
p
S p
∈ then
S is a nuclear space and it is called the space of Hida test function. The topological dual
S
∗
of S is called the space of Hida distribution. It can be
shown
p p
S S
∗ ∗
≥
=
∪
and the norms on the dual space
p
S
∗
of
p
S is given by
: ,
p p
A p
ϕ ϕ
− −
= Τ ∈
. Hence we arrive at the Gelfand triple
2
S L
S
∗
⊂ ⊂
. Dual pairing of
S ϕ
∗
∈ and
S η ∈
is denoted by ,
ϕ η . If
2
L ϕ ∈
then ,
. E
μ
ϕ η ϕ η
= .
For a complete discussion about S and
S
∗
see Hida et al 1993 or Kuo 1996.
Definition 3.1.
1. Let I ⊂ be an interval. A mapping
: X I
S
∗
→ is called a stochastic
distribution process. 2. A stochastic distribution process X is said to be differentiable if the limit
lim
t h t
h
X X
h
+ →
−
exists in S
∗
.
8
Note that convergence in S
∗
means convergence in the inductive limit topology.
Now we are in the position to show that mixed fractional Brownian motion
H
M is a differentiable stochastic distribution process. For n ∈ let
n
ξ be the n th Hermite function. First, recall that tempered distribution space
S′ can be
reconstructed as an inductive limit as follow. Define a family of norms on
2
L by
2 2
2 2
: 2
2 ,
,
p p
k p
k
f A f
k f
p ξ
∞ −
− −
=
= =
+ ∈
∑
. The last equation follows from the fact that
k
ξ is an eigenfunction of A with eigenvalue 2
2 k
+ . Then S′
is the inductive limit of
p
S
−
, p ∈ . Note that
convergence in the inductive limit topology coincides with both the convergence in the strong and the weak-
∗ topology of S′
.
Lemma 3.2.
Let 0,1 H
∈ . Then
[0,
1 :
H
N S
− ⋅
′ →
is differentiable and
[0,
1
H H
t k
k k
d N
N t
dt ξ
ξ
∞ −
+ =
=
∑
. See Bender 2003 for proof.
From the representation
[0,
, 1
H H
t t
B N
−
= ⋅ the preceding lemma might
suggest that
,
H H
t k
k k
d B
N t
dt
ξ ξ
∞ +
=
= ⋅
∑
. Now the integrand in this Wiener integral is no longer an element of
2
L but a
tempered distribution. From 3 and isometry 2 we obtain the following isometry ,
,
p p
p p
f A f
A f f
− −
− −
⋅ = ⋅
= =
6 Thus the Wiener integral can be extended to
p
f S
−
∈ such that the isometry
6 holds, and consequently to
f S′
∈
. Note that this extended Wiener integral
9
is a Hida distribution and need not to be a random variable. The following theorem of Bender 2003 enables us to calculate the derivative of
H
B .
Theorem 3.3.
Let I ⊂ be an interval and let
: F I
S′ →
be differentiable. Then , F t
⋅ is a
differentiable stochastic distribution process and ,
, d
d F t
F t dt
dt ⋅
= ⋅ .
Combining this theorem with Lemma 3.2 we see that
H
B is differentiable for 0,1
H ∈
and ,
H H
t k
k k
d B
N t
dt
ξ ξ
∞ +
=
= ⋅
∑
. Now for
t ∈
we define the distribution ,
:
H H
t
N f
N f t
δ
+ +
= ,
where
t
δ
is Dirac delta function at t . Then
2 2
2 1
2 2
, ,
0.
H H
H H
k k
t k
k n
t n
k k
k
N t
N n
N t
N ξ
ξ δ
ξ ξ ξ
δ ξ
∞ ∞
∞ −
+ +
+ +
= =
= −
⎛ ⎞
− =
+ −
⎜ ⎟
⎝ ⎠
=
∑ ∑
∑
Hence
,
H H
t t
d B
N dt
δ
+
= ⋅
.
Definition 3.4.
Let 0,1
H ∈
. Then the derivative of
H t
M in S
∗
: ,
H H
t t
t
W a
b N
δ δ
+
= ⋅ +
is called mixed fractional white noise. Note that this is really a generalization of the classical white noise ,
t
δ ⋅
. One of the fundamental tools in white noise analysis is the S-transform.
Definition 3.5.
For S
∗
Φ ∈ the S-transform is defined by
: ,: exp
, : ,
S S
η η
η Φ
= Φ ⋅
∈ .
10
The S-transform is well defined, because the Wick exponential
2 1
2
: exp ,
: : exp ,
η η
η ⋅
= ⋅
− of the Wiener integral of a smooth rapidly decreasing function is a Hida test
function. The S-transform also gives a convenient way to characterize an element in
S
∗
.
Definition 3.6.
A mapping :
F S → is called U-functional if it satisfies the two conditions:
i. F is ray analytic, i.e. for all ,
S η ζ ∈
the mapping F
λ η λζ
+ is
analytic on , and
ii.
there exist
1 2
, K K
such that
2 2
1 2
exp F z
K K z
η η
≤ for all z
∈ , S
η ∈ and some continuous norm
⋅ on S
.
Theorem 3.7.
The S-transform defines a bijection between the Hida distribution space S
∗
and the space of U-functional.
An important consequence of the above characterization theorem is the following corollary concerns the Bochner integration of family of the same type of
distributions which depend on an additional parameter.
Corollary 3.8.
Let , ,
F m Ω
be a measure space and
λ
λ Φ be a mapping from Ω to
S
∗
.If the S-transform of
λ
Φ fulfils the following two conditions: i. the mapping
S
λ
λ η
Φ is measurable for every
S η ∈
, ii. there exist
1 1
, C
L m
λ ∈ Ω
,
2
, C
L m
λ
∞
∈ Ω
, and a continuous norm ⋅ on
S such that
2 2
1 2
exp ,
S z
C C
z
λ
η λ
λ η
Φ ≤
11
for all z ∈ and
S η ∈
, then
λ
Φ is Bochner integrable with respect to some Hilbertian norm which topologizing
S
∗
. Hence dm
S
λ
λ
∗ Ω
Φ ∈
∫
and furthermore
. S
dm S
dm
λ λ
λ η η
λ
Ω Ω
⎛ ⎞
Φ =
Φ ⎜
⎟ ⎜
⎟ ⎝
⎠
∫ ∫
We refer to Hida et al 1993 or Kuo 1996 for details and proofs.
4. Donsker’s Delta Function
