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
Let
c
δ
be the Dirac delta function at
c ∈
. Donsker’s delta function for
2
f L
∈ is defined by
,
: ,
, .
c f c
f f
c
δ δ
δ
= ⋅
= ⋅
− In order to “pin” Brownian motion at a point c
∈ we consider the composition of the Dirac delta function with Brownian motion
[0,
,1 .
t
c
δ
⋅ −
This can be given a rigorous meaning as a Hida distribution, see for example Kuo 1996. Donsker’s
delta function has some important applications, e.g. partial differential equations, quantum mechanical propagator in Feynman path integral and self-intersection
local time of a stochastic process. We will present an application of the characterization theorem in the previous section to show that Donsker’s delta
function for mixed fractional Brownian motion is a Hida distribution. This result will be a useful tool, particularly in the three areas mentioned above.
From distribution theory the following integral representation of Dirac delta is well known
1 exp
2
c
x i
x c d
δ λ
λ π
= −
∫
. We want to realize the Donsker’s delta function of mixed fractional Brownian
motion
, , , ,
1 exp
2
H a b H a b
c t
t
M i
M c d
δ λ
λ π
= −
∫
12
as Bochner integral in S
∗
. The following estimation was proved in Drumond et al 2008 and will be very useful for our purpose.
Theorem 4.1.
Let 0,1
H ∈
and f
S ∈
. Then there exist constant
H
C
, independent of
f
, such that
[0,
1 sup
sup
H t
H x
x
f x N x dx
C t f x
f x f
± ∈
∈
⎛ ⎞
′ ≤
+ +
⎜ ⎟
⎝ ⎠
∫
.
Let S
η
∈ and define
, ,
: exp
H a b t
F S
i M
c
λ
η λ
η
= −
. Then we have
2 2
2 2 1
[0, 2
exp exp
1 .
t H
H t
F a t
b t i
c a
x dx b
x N x dx
λ
η λ
λ η
η
−
⎛ ⎞
⎛ ⎞
⎜ ⎟
= −
+ −
− −
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎝ ⎠
⎝ ⎠
∫ ∫
Thus the mapping
F
λ
λ η is measurable for all
S
η
∈
. For the boundedness, let
z ∈
2 2
2 2
1 2
[0,
exp .
exp 1
.
H t
H t
F a t
b t z a
x dx z b
x N x dx
λ
η λ
λ η
λ η
−
≤ −
+ ⎛
⎞ ⎜
⎟ +
⎜ ⎟
⎝ ⎠
∫ ∫
2 2
2 2
1 2
exp exp
sup sup
sup .
H H
x x
x
a t b t
z a t x
z b C t x
x
λ λ
η λ
η η
η
∈ ∈
∈
≤ −
+ ⎛
⎞ ⎛
⎞ ′
+ +
+ ⎜
⎟ ⎜
⎟ ⎝
⎠ ⎝
⎠ Define a continuous norm on
S as follow
: sup sup
x x
x x
η η
η η
∗ ∈
∈
′ =
+ +
,
S η ∈
. Hence
2 2
2 2
1 2
exp exp
.
H H
F a t
b t z k
λ
η λ
λ η
∗
≤ −
+ For simplicity let
2 2 2
:
Ht
a t b t
β = +
and :
H
k γ
η
∗
= .
Then
13
2 2
2 2
1 4
2 2
2 1
4
exp exp
2 exp
exp .
z z
F z
λ
λ η
λ β β
γ γ
β β
λ β γ
β
⎛ ⎞
⎛ ⎞
⎜ ⎟
≤ −
− −
+ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎝
⎠ ⎝
⎠ ⎛
⎞ ⎜
⎟ ≤
− ⎜
⎟ ⎝
⎠ where
2 1
1 4
exp ,
L dx
λ β
− ∈
and
2
,
H
k L
dx
β
∞
∈ .
Hence F
λ
η
satisfies both conditions in Corollary 3.8, and by integrating over we obtain an explicit formula for S-transform of Donsker’s delta function of mixed
fractional Brownian motion. In other words we have proved the following result.
Theorem 4.2.
The Bochner integral
, , , ,
1 exp
2
H a b H a b
c t
t
M i
M c d
δ λ
λ π
= −
∫
is a Hida distribution with S-transform given by
, , 2
[0, 2
2 2 2
2 2
1 1
exp 1
, 2
2
H a b c
t t
H t
H H
S M
c a
x dx b
x N x dx
a t b t
a t b t
δ η
η η
π
−
⎛ ⎞
⎛ ⎞
⎜ ⎟
= −
− −
⎜ ⎟
⎜ ⎟
⎜ ⎟
+ +
⎝ ⎠
⎝ ⎠
∫ ∫
for every
S η ∈
.
5. Conclusion
