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Vol. 7 (2002) Paper No. 14, pages 1–30.
Journal URL
http://www.math.washington.edu/~ejpecp/
Paper URL
http://www.math.washington.edu/~ejpecp/EjpVol7/paper14.abs.html
WIENER FUNCTIONALS OF SECOND ORDER
AND THEIR LÉVY MEASURES
Hiroyuki Matsumoto
School of Informatics and Sciences, Nagoya University
Chikusa-ku, Nagoya 464-8601, Japan
matsu@info.human.nagoya-u.ac.jp
Setsuo Taniguchi
Faculty of Mathematics, Kyushu University, Fukuoka 812-8581, Japan
taniguch@math.kyushu-u.ac.jp
Abstract The distributions of Wiener functionals of second order are infinitely divisible. An
explicit expression of the associated Lévy measures in terms of the eigenvalues of the corresponding Hilbert–Schmidt operators on the Cameron–Martin subspace is presented. In some
special cases, a formula for the densities of the distributions is given. As an application of the
explicit expression, an exponential decay property of the characteristic functions of the Wiener
functionals is discussed. In three typical examples, complete descriptions are given.
Keywords Wiener functional of second order, Lévy measure, Mellin transform, exponential
decay
AMS subject classification 60J65, 60E07
Submitted to EJP on July 19, 2001. Final version accepted on February 12, 2002.
Introduction
Let W be a classical Wiener space, and µ be the Wiener measure on it. A Wiener functional F
of second order is a measurable functional F : W → R with ∇3 F = 0, ∇ being the Malliavin
gradient. It is represented as a sum of Wiener chaos of order at most two. Widely known Wiener
functionals of second order are the square of the L2 -norm on an interval of the Wiener process,
Lévy’s stochastic area, and the sample variance of the Wiener process. The study of Wiener
functionals of second order has a history longer than a half century, and many contributions have
been made. Among them, pioneering works were made by Cameron-Martin and Lévy [2, 3, 12]
for the square of the L2 -norm on an interval of the Wiener process and Lévy’s stochastic area.
The sample variance plays an important role in the Malliavin calculus (cf. [8]), and it was
studied in detail. For example, see [5, 7].
There are a lot of reasons why one studies such Wiener functionals. One is that they are the
easiest functionals next to linear ones. This may sound rather nonsensical, but a wide gap
between Wiener functionals of first and second orders can be found in recent works by Lyons
(for example, see [13]). Recalling roles played by quadratic Lagrange functions in the theories
of Feynman path integrals and of semi-classical analysis for Schrödinger operators, one must
encounter another reason for studying Wiener functionals of second order. A Wiener functional
of second order is one of key ingredients in the asymptotic theories, the Laplace method, the
stationary phase method et al, on infinite dimensional spaces.
As was employed by Cameron-Martin and Lévy, a fundamental strategy to investigate Wiener
functionals of second order is computing their Laplace transforms or characteristic functions, and
then their Lévy measures. In this paper, we give explicit expressions of Lévy measures of Wiener
functionals of second order in terms of the eigenvalues and eigenfunctions of the corresponding
Hilbert-Schmidt operators. See Theorem 2. Moreover, we extend the result to the case where µ is
replaced by a conditional probability (Theorem 4). These explicit representations are essentially
based on the splitting property of the Wiener measure µ, in other words, a decomposition of
the Brownian motion via the eigenfunctions of the Hilbert-Schmidt operator. Wiener used a
decomposition of this kind, the Fourier series expansion, to construct a Brownian motion, and
a generalization we use is due to Itô-Nisio [9].
With the help of the explicit expression, we compute the Mellin transform of the Lévy measures.
See Proposition 5. Recently Biane, Pitman and Yor ([1, 15]) showed that certain probability
distributions corresponding to Wiener functionals of second order are closely related to special
functions like Riemann’s ζ-function. The general expression given in Proposition 5 will indicate
that the relations studied by them are very natural ones. As another application, we shall
investigate the order of decay of the characteristic function as the parameter of Fourier transform
tends to infinity. If the Lévy-Khintchine representation admits a Gaussian term, then the decay
is very fast, but if there is no Gaussian term, then the decay is determined by the behavior
of the Lévy measure at the origin. For details, see Theorem 7. A characteristic function of
a quadratic Wiener functional is a key object to investigate the principle of stationary phase
on the Wiener space, and its exponential decay is indispensable to achieve such a principle on
infinite dimensional spaces. The exponential decay also implies that the distribution of the
Wiener functional of second order has a density function of Gevrey class with respect to the
Lebesgue measure, which relates to the property called transversal analyticity by Malliavin [14].
Another criteria for the distribution to possess a smooth density function will also be given, and
2
a method to compute it by using the residue theorem is shown (Theorem 11).
In Section 3, all our general results are tested for three concrete Wiener functionals of second
order mentioned above. Comparisons with known results will be also discussed there.
1
1.1
Lévy measures of Wiener functionals of second order
General Scheme
Throughout this subsection, (W, H, µ) stands for an abstract Wiener space. For the definition,
see [11]. The inner product and the norm of H are denoted by h·, ·i and k · kH , respectively.
GivenPa symmetric Hilbert-Schmidt operator A : H → H and ℓ ∈ H, decomposing A as
2
∞
A
= ∞
n=1 an hn ⊗ hn with an ∈ R and an orthonormal basis {hn }n=1 of H such that kAk2 =
P∞
2
n=1 an < ∞, we define
QA =
∞
X
n=1
an h·, hn i2 − 1 ,
(1)
X
1
1 hℓ, hn i2
+
exp[−x/an ],
x > 0,
2
x
a3n
n;an >0
fA,ℓ (x) = 0,
x = 0,
2
X
1
hℓ,
h
i
1
n
exp[−x/an ], x < 0,
+
2
−x
−a3n
(2)
n;an 0,
x Tr T(1/x)A1 + x3 hℓ, T(1/x)A1 ℓi,
x = 0,
fA,ℓ (x) = 0,
1
1
(3)
Tr T (0)
(1/|x|)A2 + |x|3 hℓ, T(1/|x|)A2 ℓi, x < 0.
|x|
Lemma 1. It holds that
0 ≤ fA,ℓ (x) ≤
for any x 6= 0, k ≥ 2, and
Z
1
−1
k−2
kAk∞
2
2
k!kAk
+
(k
+
1)!kℓk
2
H
2|x|k+1
|x|2 fA,ℓ (x)dx ≤
1
kAk22 + kℓk2H ,
2
where kAk∞ = sup{|an | : n = 1, 2, . . . }.
3
(3)
(4)
Proof. Since exp[−|x|/|an |] ≤ k!(|an |/|x|)k , (3) follows. (4) is an easy application of the monotone convergence theorem and an estimation
Z
1
0
|x|m exp[−|x|/|an |]dx = |an |m+1
for every m ∈ N .
Z
1/|an |
0
y m exp[−y]dy ≤ m!|an |m+1 ,
Theorem 2. Let A : H → H be a symmetric Hilbert-Schmidt operator, ℓ ∈ H, and γ ∈ R .
Then, for any λ ∈ R, it holds that
Z
exp iλ
W
where i =
√
1
QA + h·, ℓi + γ
2
−1 and
dµ
2
Z
λ kℓA k2H
iλx
e − 1 − iλx fA,ℓ (x)dx , (5)
= exp −
+ iλγ +
2
R
ℓA =
X
hhn , ℓihn .
(6)
n;an =0
Remark 3. (i) The integrability of (eiλx − 1 − iλx)fA,ℓ (x) in (5) is guaranteed by Lemma 1.
(ii) The theorem asserts that the distribution of 12 QA + h·, ℓi + γ is infinitely divisible and the
corresponding Lévy measure is fA,ℓ (x)dx. Moreover, the distribution of 12 QA + γ is selfdecomposable. See [16, §8 and §15].
(iii) Let Cn (W ) be the space of Wiener chaos of order n. A Wiener functional F of second order
is a Wiener chaos of order at most two, i.e., F ∈ C2 (W ) ⊕ C1 (W ) ⊕ C0 (W ), and is of the form
that F = (1/2)QA +h·, ℓi+γ for some symmetric Hilbert-Schmidt operator A, ℓ ∈ H, and γ ∈ R,
and QA ∈ C2 (WR ), h·, ℓi ∈ C1 (W ), and
R γ ∈ C0 (W ). Moreover, A, ℓ, and γ are determined so that
2
A = ∇ F , ℓ = W ∇F dµ, and γ = W F dµ, where ∇ stands for the Malliavin derivative.
(iv) It should be noted that C2 (W ) ⊕ C1 (W ) ⊕ C0 (W ) is invariant under shifts in the direction
of H. Namely, let F = 12 QA + h·, ℓi + γ and h ∈ H. Then
1
F (· + h) = F + h·, Ahi + hh, Ahi + hh, ℓi ∈ C2 (W ) ⊕ C1 (W ) ⊕ C0 (W ).
2
In particular, the theorem is applicable to a quadratic form of the form 12 QA (· − h), which is
one of main ingredients in the study of the principle of stationary phase on W . See [17]
Proof of Theorem 2. The proof is divided into three steps according to the signs of an ’s, the
eigenvalues of A.
1st step: the case where an > 0 for all n ∈ N .
Let λ > 0. Note that
∞ h
i
X
an
1
QA + h·, ℓi =
h·, hn i2 − 1 − hℓ, hn ih·, hn i .
2
2
n=1
4
Since {h·, hn i : n ∈ N } is a family of independent Gaussian random variables of mean 0 and
variance 1, we obtain the following well known identity:
Z
1
QA + h·, ℓi + γ
dµ
exp −λ
2
W
2
∞
Y
λ hℓ, hn i2
−1/2 λan /2
= exp[−λγ]
(1 + λan )
. (7)
e
exp
2 1 + λan
n=1
Applying the identities
we obtain
Z ∞
Z ∞
e−ax
log(1 + (λ/a)) =
e−x/a dx = a,
(1 − e−λx )
dx,
x
0
0
Z ∞
3 λ2
a
, a, λ > 0,
e−λx − 1 + λx e−x/a dx =
1 + aλ
0
2
!
∞
2
Y
i
λ
hℓ,
h
n
log
(1 + an )−1/2 ean /2 exp
2
1
+
an
n=1
=−
=
Z
∞
∞
n=1
∞
−λx
n=1
1X
1 X λ2 hℓ, hn i2
{log(1 + λan ) − λan } +
2
2
1 + λan
0
(e
− 1 + λx)fA,ℓ (x)dx.
Plugging this into (7), we obtain
Z
1
QA + h·, ℓi + γ
dµ
exp −λ
2
W
Z ∞
−λx
= exp −λγ +
e
− 1 + λx fA,ℓ(x)dx .
(8)
0
Note that
|ζ|2 x3
d
ζx
2
dζ e − 1 − ζx ≤ 2 + 2|ζ|x
and
3|ζ|2 x2
ζx
e − 1 − ζx ≤
2
for ζ ∈ C with Reζ < 0 and x ≥ 0. Hence, continuing (8) holomorphically to Ω = {ζ ∈ C :
Reζ < 0}, and then letting Reζ → 0 in Ω, we we arrive at (5), because fA,ℓ (x) = 0 for x ≤ 0.
2nd step: the case where an < 0 for all n ∈ N .
Since −QA = Q−A , applying the result in the first step to −A, we obtain
Z
1
QA + h·, ℓi + γ
dµ
exp iλ
2
W
Z
1
exp i(−λ)
=
+ h·, −ℓi + (−γ)
dµ
Q
2 (−A)
X
Z
−iλx
= exp[iλγ] exp
e
− 1 + iλx f−A,−ℓ (x)dx .
R
5
Since f−A,−ℓ(−x) = fA,ℓ (x), this yields (5).
3rd step: the general case.
Set
X
A+ =
an hn ⊗ hn ,
A− =
n;an >0
ℓ+ =
X
hℓ, hn ihn ,
n;an >0
Then
ℓ− =
X
n;an 0 : lim sup λ
(
−a
λ→∞
a+ := sup a > 0 : lim sup λ
λ→∞
−a
Z
(−∞,0)
Z
(0,∞)
8
(cos(λx) − 1)fA,ℓ (x)dx < 0,
)
(cos(λx) − 1)fA,ℓ (x)dx < 0 ,
)
,
where a− = 0, a+ = 0 if {· · · } = ∅. If max{a− , a+ } > 0, then, for every a < max{a− , a+ },
there exist Ca > 0 and λa > 0 such that
Z
1
≤ exp[−Ca λa ],
Q
+
h·,
ℓi
+
γ
dµ
(15)
exp
iλ
A
2
W
for every λ ≥ λa , γ ∈ R . Moreover, if both supremums a+ , a− are attained as maximums, then
the above assertion holds with a = max{a− , a+ }.
(iii) Put
(
)
Z
−b
b− := sup b > 0 : lim sup λ
(cos(λx) − 1)fA(η) ,P (η) (A(y·η)+ℓ) (x)dx < 0 ,
(
λ→∞
(−∞,0)
−b
b+ := sup b > 0 : lim sup λ
λ→∞
Z
(0,∞)
)
(cos(λx) − 1)fA(η) ,P (η) (A(y·η)+ℓ) (x)dx < 0 ,
where b− = 0, b+ = 0 if {· · · } = ∅. If max{b− , b+ } > 0, then, for every b < max{b− , b+ }, there
exist Cb > 0 and λb > 0 such that
E µ exp iλ 1 QA + h·, ℓi + γ η(w) = y ≤ exp[−Cb λb ],
(16)
2
for any λ ≥ λb , γ ∈ R . Moreover, if both supremums b+ , b− are attained as maximums, then
the above assertion holds with b = max{b− , b+ }.
Remark 8. (i) If ℓA 6= 0, then we have an exponent −λ2 kℓA k2H /2, which gives a much faster
decay than the one discussed in Theorem 7. Similarly, if {P (η) (A(y · η) + ℓ)}A(η) 6= 0, then we
obtain a much faster decay than the one discussed in the theorem.
(ii) The integrability of x2 fA,ℓ (x) and x2 fA(η) ,P (η) (A(y·η)+ℓ) (x) is due to Lemma 1.
(iii) The lower estimates
(13) and (14) are sharp as we shall see in Lemma 10. For example,
P in −p
if we consider A = ∞
n
hn ⊗ hn for some p > 1/2 and an orthonormal basis {hn } of H,
n=1
then there exist C > 0 and λ0 > 0 such that
Z
1
QA + h·, ℓi + γ
dµ ≤ exp[−Cλ1/p ] for any λ > λ0 .
exp iλ
2
W
For details, see Lemma 10 and its proof.R
∞
(iv) If an > 0 for some n, then limλ→∞ 0 (cos(λx) − 1)fA,ℓ (x)dx = −∞. In fact, it holds that
fA,ℓ ≥ fA,0 and
Z ∞
Z ∞
1 − cos y X
(1 − cos(λx))fA,0 (x)dx =
exp[−y/(λan )]dy.
y
0
0
n;a >0
n
Then, applying the monotone convergence
theorem, we obtain the desired divergence. Similarly,
R0
if an < 0 for some n, then limλ→∞ −∞ (cos(λx) − 1)fA,ℓ (x)dx = −∞. Thus the assumption
made on a± is that only on the order of divergence.
If #{n; an 6= 0} < ∞, then a+ = a− = 0. Indeed, in this case, there are C, C ′ > 0 such that
fA,ℓ (x) ≤ C{(1/|x|) + 1} exp[−C ′ |x|] for every x ∈ R \ {0}. For each δ > 0, this implies the
9
existence of Cδ > 0 such that fA,ℓ (x) ≤ Cδ |x|−1−δ for any x ∈ R \ {0}. Hence, for every λ > 0,
0 ≤ max
Z
0
−∞
(1 − cos(λx))fA,ℓ (x)dx,
Z
∞
0
(1 − cos(λx))fA,ℓ (x)dx
≤ Cδ λ
δ
Z
∞
0
1 − cos y
dy,
y 1+δ
from which it follows that a± = 0.
Proof of Theorem 7. Assume that ℓA = 0 or {P (η) (A(y · η) + ℓ)}A(η) = 0 accordingly as µ or
µ(·|η(w) = y) is considered. By virtue of Theorems 2 and 4, we have
Z
Z
1
(cos(λx) − 1)fA,ℓ (x)dx
QA + h·, ℓi + γ
dµ = exp
exp iλ
2
R
W
E µ exp iλ 1 QA + h·, ℓi + γ η(w) = y
2
Z
= exp
(cos(λx) − 1)fA(η) ,P (η) (A(y·η)+ℓ) (x)dx
R
Since | cos x − 1| ≤ x2 /2 for any x ∈ R , the assertion (i) follows immediately.
Let f = fA,ℓ or = fA(η) ,P (η) (A(y·η)+ℓ) . Since (cos(λx) − 1)f (x) ≤ 0, we have
Z
R
(cos(λx) − 1)f (x)dx
≤ min
(Z
(−∞,0)
(cos(λx) − 1)f (x)dx,
Z
(0,∞)
)
(cos(λx) − 1)f (x)dx .
Thus the estimations in (ii) and (iii) also follow.
A function ϕ ∈ C ∞ (R ) is said to belong a Gevrey class of order a > 1 (ϕ ∈ Ga (R ) in notation)
if, for any compact subset K ⊂ R , there exists a constant CK > 0 such that
n
d ϕ
a n
for every x ∈ K, n ∈ N .
dxn (x) ≤ CK (CK (n + 1) )
A finite Radon measure u on R admits a density function of class Ga (R ) if there is a C > 0 such
that
C(n + 1)a n
|û(ξ)| ≤ C
for any ξ ∈ R \ {0}, n ∈ N ,
|ξ|
where û is the Fourier transformation of u (cf. [6, Prop.8.4.2]). Since e−x ≤ αα x−α for α, x > 0,
a sufficient condition for this to hold is that there exist C1 , C2 > 0 such that
|û(ξ)| ≤ C1 exp[−C2 |ξ|1/a ]
We obtain the following from Theorem 7.
10
for any ξ ∈ R .
Corollary 9. Let a± , b± be as in Theorem 7.
(i) If ℓA = 0 and max{a− , a+ } > 0, then the distribution on R of 12 QA + h·, ℓi + γ under µ admits
a density function, which is in G1/a (R ) for any a < max{a− , a+ }, with respect to the Lebesgue
measure.
(ii) If {P (η) (A(y · η) + ℓ)}A(η) = 0 and max{b− , b+ } > 0, then the distribution on R of 12 QA +
h·, ℓi + γ under the conditional probability µ(·|η(w) = y) admits a density function, which is in
G1/b (R ) for any b < max{b− , b+ }, with respect to the Lebesgue measure.
We give a sufficient condition for a± to be positive. It follows from (3) that fA,ℓ (x) diverges
at the order of at most |x|−3 as |x| → 0. If we assume a uniform order of divergence, then
max{a− , a+ } > 0.
Lemma 10. (i) If there exist δ, C > 0 and ε < 2 such that
fA,ℓ (x) ≥ Cxε−3
for any x ∈ (0, δ),
then a+ ≥ 2 − ε. If there exist δ, C > 0 and ε < 2 such that
fA,ℓ (x) ≥ C|x|ε−3 for any x ∈ (−δ, 0)
then a− ≥ 2 − ε.
(ii) Let {an }∞
n=1 be eigenvalues of A. Suppose that there exist a subsequence {ank }, C > 0, and
p > 1/2 such that ank ≥ C/kp for any k ∈ N . Then, for any δ > 0,
(
)
Z
C 1/p ∞ (1/p)−1 −z
fA,ℓ (x) ≥
z
e dz x−1−(1/p)
p
δ/C
holds for any x ∈ (0, δ). In particular, a+ ≥ 1/p.
If there exist a subsequence {ank }, C > 0, and p > 1/2 such that ank ≤ −C/kp for any k ∈ N .
Then, for any δ > 0,
(
)
Z
C 1/p ∞ (1/p)−1 −z
fA,ℓ (x) ≥
z
e dz x−1−(1/p)
p
δ/C
holds for any x ∈ (−δ, 0). In particular, a− ≥ 1/p.
Proof. (i) It follows from (3) that
Z ∞
δ
(cos(λx) − 1)fA,ℓ (x)dx ≤ 0.
Due to the first assumption, we have
Z
0
δ
Z
δ
(cos(λx) − 1)xε−3 dx
0
Z λδ
= Cλ2−ε
(cos x − 1)xε−3 dx.
(cos(λx) − 1)fA,ℓ (x)dx ≤ C
0
11
Hence
lim sup λ
−(2−ε)
λ→∞
Z
∞
0
(cos(λx) − 1)fA,ℓ (x)dx ≤ C
Z
∞
0
(cos x − 1)x3−ε dx < 0.
Thus the first half has been verified. The latter half can be seen in exactly the same way.
(ii) Suppose the first assumption. Then it holds that
fA,ℓ (x) ≥
∞ −xk p /C
X
e
k=1
x
≥
Z
∞
1
p /C
e−xy
x
dy = x
−1−(1/p) C
1/p
p
Z
∞
z (1/p)−1 e−z dz
x/C
for any x > 0. This yields the first estimation. Since −1 − (1/p) = {2 − (1/p)} − 3, by the
assertion (i), we have that a+ ≥ 1/p. Thus the first half has been verified.
The latter half can be seen similarly.
2.2
Density functions
Corollary 9 gives a sufficient condition for the distribution of 21 QA +h·, ℓi+γ under µ or µ(·|η(w) =
y) to have a smooth density function with respect to the Lebesgue measure. We now show
another condition for the distribution to possess a smooth density function, and also a method
to compute it.
Theorem 11. Let (W, H, µ) be an abstract
space, A : H → H be a symmetric HilbertPWiener
∞
Schmidt operator, and decompose as A = n=1 an hn ⊗ hn with an orthonormal basis {hn }∞
n=1
of H.
(i) Suppose that #{n : an 6= 0} = ∞. Then there exists a pA ∈ C ∞ (R ) such that µ QA /2 ∈
dx = pA (x)dx.
(ii) Suppose that a2n−1 = a2n for any n ∈ N and #{n : an 6= 0} = ∞. Fix x ∈ R, and
assume that there exists a family of simple C 1 curves Γn = {γn (t) : t ∈ [αn , βn ]} in C such that
(a1) Rγn (αn ) ∈ (−∞, 0), γn (βn ) ∈ (0, ∞), (a2) inf{|γn (t)| : t ∈
n , βn ]} → ∞ as n → ∞, and
P[α
∞
−iζx
b
b
/ det2 (I − iζ A)}dζ → 0 as n → ∞, where A = n=1 a2n h2n ⊗ h2n .
(a3) Γn {e
(ii-1) If Im γn (t) > 0 and −i/am ∈
/ Γn for any n ∈ N , t ∈ (αn , βn ), and m ∈ N with am < 0,
then
!
X
e−iζx
i
pA (x) = i
,
(17)
Res
;−
b
an
det2 (I − iζ A)
n:an 0,
then
!
X
e−iζx
i
.
(18)
pA (x) = −i
;−
Res
b
an
det2 (I − iζ A)
n:an >0
P
P∞
(iii) Suppose ∞
n=1 |an | < ∞, and set qA = QA +
n=1 an . Then all assertions in (i) and (ii)
b by qA and det(I − iζ A),
b respectively.
hold, replacing QA and det2 (I − iζ A)
12
b is holomorphic ([4]).
Remark 12. (i) The mapping C ∋ ζ 7→ det2 (I − iζ A)
(ii) Let η = {η1 , . . . , ηm } be an orthonormal system of H. By (11), it holds
m
o
n
X
(η) 1
µ QA ∈ dx η(w) = 0 = µ0
hAηi , ηi i ∈ dx ,
QA(η) −
2
2
n=1
1
(η) 1
µ qA ∈ dx η(w) = 0 = µ0
q (η) ∈ dx .
2
2 A
1
Thus, we can compute the density functions of the distributions of QA /2 and qA /2 under
µ(·|η(w) = 0), by applying Theorem 11 to QA(η) and qA(η) on W (η) .
(iii) The method to compute the density with the help of the residue theorem has been already
applied by Cameron-Martin ([2]) more than a half century ago to the square of the L2 -norm on
an interval of the one-dimensional Wiener process.
T
P
−1
p
2
2
Proof. Since k∇(QA /2)k2H = ∞
p>0 L (µ) if #{n : an 6= 0} =
n=1 an h·, hn i , k∇(QA /2)kH ∈
∞ (cf.[18]). Thus the assertion (i) follows as an fundamental application of the Malliavin calculus.
By (7) and the assumption that a2n−1 = a2n , we have
Z
1
,
eiλx pA (x)dx =
b
det2 (I − iλA)
R
and hence
(19)
Z
e−iλx
1
pA (x) =
dλ.
b
2π R det2 (I − iλA)
Then the assertions in (ii) are immediate consequences of the residue theorem.
To see the last assertion, it suffices to mention that (19) implies that, if we denote by p̃A the
density function of qA /2, then
Z
1
.
eiλx p̃A (x)dx =
b
det(I − iλA)
R
3
Typical quadratic Wiener functionals
In this section, we investigate how our results work for typical quadratic Wiener functionals.
Some of the computations below have been carried out in Ikeda-Manabe [7], but we give all the
results for convenience of the reader. Moreover, when we do not consider the first order terms
of the Wiener chaos, that is, when ℓ = 0 in (5), explicit expressions of the Fourier or Laplace
transforms of the distributions are well known for the examples considered in the following and,
from them, we can obtain the same results after some elementary calculations.
13
3.1
The square of the L2 -norm on an interval
Let T > 0 and consider the classical one-dimensional Wiener space (WT1 , HT1 , µ1T ) over [0, T ]; WT1
is the space of continuous functions w : [0, T ] → R with w(0) = 0, HT1 consists of h ∈ WT1 which
is absolutely continuous and has a square integrable derivative dh/dt, and µ1T is the Wiener
measure. The inner product in HT1 is given by
hh, ki =
Z
T
dh dk
(t) (t)dt,
dt
dt
0
h, k ∈ HT1 .
In this subsection we consider
Z
hT (w) =
T
w(t)2 dt,
0
w ∈ WT1 .
3.1.1
We first compute the Lévy measure of
ℓ ∈ HT1 and γ ∈ R .
h + h·, ℓi + γ under µ1T by applying Theorem 2, where
1
2 T
Define a symmetric Hilbert-Schmidt operator A : HT1 → HT1 by
d(Ah)
(t) =
dt
Z
T
h(s)ds,
t
h ∈ HT1 , t ∈ [0, T ].
Note
that w(t)2 is in C2 (WT1 ) ⊕ C0 (WT1 ), and so is hT . It is easily seen that ∇2 hT = 2A and that
R
1
2
W 1 hT dµT = T /2. Then, by virtue of Remark 3 (iii), we observe that
T
hT = Q A +
It is easy to see that
A=
∞
X
n=0
where
(20)
!2
(21)
A
hA
n ⊗ hn ,
1
n
+
2T
2 πt
sin
hA
.
n (t) =
1
T
n+ 2 π
√
In particular, we have
T
n + 12 π
T2
.
2
ℓA = 0 and
fA,ℓ (x) = 0,
x ≤ 0.
Set
!
n + 12 πt
ℓ(n) =
dt,
=
T
0
∞
(2n + 1)2 π 2 x
π6 X
6
2
(2n + 1) |ℓ(n) | exp −
.
gH (x; T, ℓ) = 7 6
2 T
4T 2
hℓ, hA
ni
r
2
T
Z
T
dℓ
(t) cos
dt
n=0
14
(22)
Since Jacobi’s theta function Θ(u) =
P
2
Z exp[−n u] enjoys the relation
n∈
Θ(u) − Θ(4u) = 2
∞
X
2
e−(2n+1) u ,
n=0
it is straightforward to see that
2
2
1
π x
π x
fA,ℓ (x) =
−
Θ
+ gH (x; T, ℓ),
Θ
4x
4T 2
T2
x > 0.
By virtue of this and (20), applying Theorem 2, we arrive at:
Proposition 13. It holds that
Z
1
dµ1T
hT + h·, ℓi + γ
exp iλ
2
WT1
"
2
Z ∞
1 π2x
π x
T2
iλx
e − 1 − iλx
−Θ
= exp iλ γ +
+
Θ
2
4
4x
4T
T2
0
#
+ gH (x; T, ℓ) dx ,
where gH is defined by (22).
3.1.2
We next compute the Lévy measure of 12 hT + h·, ℓi + γ under the conditional probability
µ1T (·|w(T ) = y) given w(T ) = y, where ℓ ∈ HT1 , γ ∈ R, and y ∈ R.
√
Set η1 (w) = w(T )/ T , w ∈ WT1 , and η = {η1 }. Note that
hA(y · η), (y · η)i − hAη1 , η1 i = (y 2 − 1)hAη1 , η1 i =
(y 2 − 1)T 2
,
3
By a straightforward computation, we obtain
Z T
Z Z T
d(A(η) h)
1 T
h(s)ds −
(t) =
h(u)du ds,
dt
T 0
t
s
and hence
(η)
A
In particular,
∞
X
T 2 A
=
kn ⊗ knA ,
nπ
n=1
{P (η) (A(y · η) + ℓ)}A(η) = 0
where
knA (t)
=
yℓ(T )
h(y · η), ℓi = √ .
T
(η)
h ∈ (HT1 )0 ,
√
nπt
2T
.
sin
nπ
T
and fA(η) ,P (η) (A(y·η)+ℓ) (x) = 0,
(23)
x ≤ 0.
Note that
hP (η) (A(y · η) + ℓ), knA i = yhAη, knA i + hℓ, knA i
√ T 2
= (−1)n+1 2y
+ ℓ̃(n) ,
nπ
15
(24)
where
ℓ̃(n) =
Hence, for x > 0,
hℓ, knA i
=
r
2
T
Z
0
T
nπt
dℓ
dt.
(t) cos
dt
T
fA(η) ,P (η) (A(y·η)+ℓ) (x)
∞
nπ 4
nπ 2
1 Xn 1
=
+ (−1)n+1 23/2 y
ℓ̃(n)
+ 2y 2
2
x
T
T
n=1
o
h n2 π 2 i
nπ 6
ℓ̃2(n) exp − 2 x
+
T
T
o π2 y2 π2 x
1 n π2 x
=
−1 −
+ g̃H (x; T, ℓ, y),
Θ
Θ′
4x
T2
2T 2
T2
where
g̃H (x; T, ℓ, y)
∞
nπ 6
h n2 π 2 x i
nπ 4
1X
2
n+1 3/2
ℓ̃(n) +
ℓ̃(n) exp −
.
(−1)
2 y
=
2 n=1
T
T
T2
(25)
(26)
Due to Theorem 4, we obtain
1
E µ1 exp iλ
QA + h·, ℓi + γ η(w) = y
T
2
2
1 n π2 x
Z ∞
o
(y − 1)T 2 yℓ(T )
= exp iλ
eiλx − 1 − iλx
Θ
+ √ +γ +
−
1
6
4x
T2
T
0
π2 y2 ′ π2 x
−
Θ
+ g̃H (x; T, ℓ, y) dx .
2T 2
T2
√
Since η(w) = w(T )/ T , combined with (20), we conclude from this:
Proposition 14. It holds that
E µ1
T
1
exp iλ
hT + h·, ℓi + γ w(T ) = y
2
n 2
o Z ∞
Ty
T 2 yℓ(T )
iλx
= exp iλ
e − 1 − iλx fH (x; T, ℓ, y)dx ,
+
+
+γ +
6
12
T
0
where
fH (x; T, ℓ, y) =
and g̃H is given by (26).
π2y2 π2x
√
1 π2 x
′
Θ
−
1
−
+
g̃
(x;
T,
ℓ,
y/
T ),
Θ
H
4x
T2
2T 3
T2
3.1.3
We finally study the exponential decay of the characteristic function of 12 hT + h·, ℓi + γ under
µ1T and µ1T (·|w(T ) = y).
16
As was seen in §3.1.1, the Hilbert-Schmidt operator A associated with hT has eigenvalues
{T 2 /[(n + 12 )π]2 ; n ∈ N ∪ {0}}, each of them being of multiplicity one. By Theorem 7 and
Lemma 10, there exist C1 > 0 and λ1 > 0 such that
Z
1
(27)
dµ ≤ exp[−C1 λ1/2 ]
hT + h·, ℓi + γ
exp iλ
W1
2
T
for any λ ≥ λ1 , γ ∈ R.
√
Let η1 (t) = t/ T and η = {η1 }. As was shown in §3.1.2, the Hilbert-Schmidt operator A(η)
2
possesses
√ eigenvalues {(T /(nπ)) ; n ∈ N }, each of them being of multiplicity 1. Since η(w) =
w(T )/ T , by Theorem 7 and Lemma 10, there exist C2 > 0 and λ2 > 0 such that
E 1 exp iλ 1 hT + h·, ℓi + γ w(T ) = y ≤ exp[−C2 λ1/2 ]
(28)
µ
T
2
for any λ ≥ λ2 , γ ∈ R.
When ℓ = 0 and γ = 0, it is well known ([3, 12] and [8, pp.470–473]) that
Z
λ
1
√
exp − hT dµ1T =
2
(cosh( λT ))1/2
WT1
and
λ
exp − hT w(T ) = y
T
2
!1/2
√
√
√ y 2
λT
√
=
exp 1 − λT coth λT
2T
sinh λT
E µ1
(29)
hold for λ > 0. Thus, continuing holomorphically, we see that our estimations (27) and (28)
coincide with the order obtained from these precise expressions.
Starting from these well-known expressions, and recalling the elementary formulae
∞
∞
Y
Y
4x2
x2
cosh(x) =
1+
,
sinh(x)
=
x
,
1
+
(2k + 1)2 π 2
k2 π2
k=0
coth(πx) =
k=1
∞
2x X
1
+
πx
π
k=1
x2
1
,
+ k2
we can also show explicit expressions for the Lévy measures νT (dx) and νT,y (dx) of the distribution of hT /2 under µ1T and the conditional probability measure µ1T ( · |w(T ) = y) as described
in Propositions 13 and 14 with ℓ = 0.
P
−s
Moreover, by using the Riemann zeta function ζ(s) = ∞
n=1 n , we can give explicit forms of
the Mellin transforms of νT and νT,y . Namely, noting that
νT (dx) = fA,0 (x)dx,
νT,y (dx) = fA(η) ,P (η) (A((y/√T )·η)) (x)dx,
and then plugging (21), (23), and (24) into (12), we obtain:
17
Proposition 15. The Mellin transform of νT (dx) and νT,y (dx) are given by
Z
∞
s
x νT (dx) =
0
4T 2
π2
s
22s − 1
Γ(s)ζ(2s)
22s+1
and
Z
0
∞
1
x νT,y (dx) =
2
s
T2
π2
s
y2
Γ(s)ζ(2s) +
T
T2
π2
s
Γ(s + 1)ζ(2s),
s ≥ 2,
respectively, where Γ is the usual gamma function.
Recently, Biane-Pitman-Yor [1] and Pitman-Yor [15] have discussed the related topics and shown
similar formulae.
3.2
Lévy’s stochastic area
Let T > 0 and consider the classical two-dimensional Wiener space (WT2 , HT2 , µ2T ) over [0, T ]; WT2
is the space of continuous functions w : [0, T ] → R 2 with w(0) = 0, HT2 consists of h ∈ WT2 which
is absolutely continuous and has a square integrable derivative dh/dt, and µ2T is the Wiener
measure. The inner product in HT2 is given by
hh, ki =
Z
0
T
dh
dk
dt,
(t), (t)
dt
dt
R2
h, k ∈ HT2 .
Define Lévy’s stochastic area by
Z
Z
1 T
1 T 1
2
hJw(t), dw(t)iR2 =
w (t)dw2 (t) − w2 (t)dw1 (t) ,
sT (w) =
2 0
2 0
0 −1
where J =
and dw(t) denotes the Itô integral.
1 0
3.2.1
We first compute the Lévy measure of
ℓ ∈ HT2 and γ ∈ R .
sT + h·, ℓi + γ under µ2T by applying Theorem 2, where
Define a symmetric Hilbert-Schmidt operator B : HT2 → HT2 by
d(Bh)
1
(t) = J h(t) − h(T ) , h ∈ HT2 , t ∈ [0, T ].
dt
2
Since wi (s){wj (t) − wj (s)} is in C2 (WT2 ) for (i, j) ∈ {(1, 2), (2, 1)} and s < t, so is sT . It is easily
seen that ∇2 sT = B, and hence, due to Remark 3 (iii), we have
1
2
sT = QB .
18
(30)
By a direct computation, we see that
B=
where
hB
n (t)
X
n∈Z
o
n
T
B
B
B
⊗
h
+
h̃
⊗
h̃
hB
n
n
n
n ,
(2n + 1)π
√
T
cos((2n + 1)πt/T ) − 1,
=
,
sin((2n + 1)πt/T )
(2n + 1)π
(31)
B
h̃B
n (t) = Jhn (t).
In particular
ℓB = 0.
Set
ℓ1(n)
ℓ2(n)
ℓ(n)
and
Z T
dℓ
1
− sin((2n + 1)πt/T )
√
=
=
dt,
(t),
cos((2n + 1)πt/T )
dt
T 0
2
R
Z T
dℓ
1
cos((2n + 1)πt/T )
B
(t),
= hℓ, h̃n i = − √
dt,
sin((2n + 1)πt/T )
dt
T 0
2
R
!
ℓ1(n)
,
= 2
ℓ(n)
hℓ, hB
ni
∞
π3 X
(2n + 1)3 |ℓ(n) |2 exp [−(2n + 1)πx/T ] ,
x>0
3
2T
n=0
gL (x; T, ℓ) = 0,
x = 0,
∞
3
X
π
(2n − 1)3 |ℓ(−n) |2 exp [−(2n − 1)πx/T ] , x < 0.
2T 3
(32)
n=1
Then it is easily seen that
fB,ℓ (x) =
1
+ gL (x; T, ℓ),
2x sinh(πx/T )
x ∈ R.
By virtue of this and (30), applying Theorem 2, we arrive at;
Proposition 16. It holds that
Z
WT2
exp [iλ (sT + h·, ℓi + γ)] dµ2T
"
Z
iλx
e − 1 − iλx
= exp iλγ +
R
where gL is defined by (32)
19
#
1
+ gL (x; T, ℓ) dx , (33)
2x sinh(πx/T )
3.2.2
We next compute the Lévy measure of sT + h·, ℓi + γ under the conditional probability
µ2T (·|w(T ) = y) given W (T ) = y, where ℓ ∈ HT2 , γ ∈ R, and y ∈ R2 .
√
√
Let η = {η1 , η2 } ⊂ W ∗ , where ηi (w) = wi (T )/ T . Since y · η = ty/ T and hJz, ziR2 = 0 for
any z ∈ R 2 ,
hB(y · η), (y · η)i −
2
X
hBηn , ηn i = 0,
n=1
h(y · η), ℓi =
hy, ℓ(T )iR2
√
.
T
(η)
For any h, g ∈ (HT2 )0 , the identity hB (η) h, gi = hBh, gi holds, and hence
d(B (η) h)
(t) = J(h(t) − h̄),
dt
1
where h̄ =
T
Z
T
h(t)dt.
0
Then it is straightforward to see that
X
B (η) =
n∈Z\{0}
where
knB (t)
√
T
=
2nπ
T B
kn ⊗ knB + k̃nB ⊗ k̃nB ,
2nπ
cos(2nπt/T ) − 1
,
sin(2nπt/T )
(34)
k̃nB (t) = JknB (t).
Hence
{P (η) (B(y · η) + ℓ)}B (η) = 0,
and it holds that
T y2
+ ℓ̃1(n) ,
2nπ
T y1
+ ℓ̃2(n) ,
=
2nπ
hP (η) (B(y · η) + ℓ), knB i = h(y · η), BknB i + ℓ̃1(n) = −
hP (η) (B(y · η) + ℓ), k̃nB i = h(y · η), B k̃nB i + ℓ̃2(n)
where
ℓ̃1(n) = hℓ, knB i =
ℓ̃2(n)
Setting ℓ̃(n) =
=
hℓ, k̃nB i
=
!
ℓ̃1(n)
, we obtain
ℓ̃2(n)
Z
Z
T
0
T
0
dℓ
1
(t), √
dt
T
dℓ
1
(t), √
dt
T
− sin(2nπt/T )
cos(2nπt/T )
R2
dt,
− cos(2nπt/T )
dt.
− sin(2nπt/T )
R2
hP (η) (B(y · η) + ℓ), knB i2 + hP (η) (B(y · η) + ℓ), k̃nB i2
T 2
T
|y|2 +
hℓ̃ , JyiR2 + |ℓ̃(n) |2 .
=
2nπ
nπ (n)
20
(35)
Thus, if we put
g̃L (x; T, ℓ, y)
∞n
2nπ 3
o
h 2nπx i
1X
2nπ 2
2
,
x > 0,
hℓ̃(n) , JyiR2 +
|ℓ̃(n) |2 exp −
T
T
T
2 n=1
= 0,
x = 0,
∞ n
o
h
i
X
2
3
1
2nπ
2nπx
2nπ
, x < 0,
−2
hℓ̃(−n) , JyiR2 +
|ℓ̃(−n) |2 exp
2
T
T
T
n=1
(36)
then, for x > 0, it holds that
fB (η) ,P (η) (B(y·η)+ℓ) (x)
∞
∞
1 X {T /(2nπ)}2 |y|2
1X2
exp[−2nπx/T ] +
exp[−2nπx/T ] + g̃L (x; T, ℓ, y)
=
2
x
2
{T /(2nπ)}3
n=1
=
1 exp[−2πx/T ]
+
x 1 − exp[−2πx/T ]
n=1
∞
2
π|y| X
T
n exp[−2nπx/T ] + g̃L (x; T, ℓ, y)
n=1
1
1
1
π|y|2
+ g̃L (x; T, ℓ, y),
+
2
x exp[2πx/T ] − 1
4T sinh (πx/T )
P
−nx = 1/{4 sinh 2 (x/2)}. Similarly, for x < 0, we have
where we have used the identity ∞
n=1 ne
=
fB (η) ,P (η) (B(y·η)+ℓ) (x) =
1
1
1
π|y|2
+ g̃L (x; T, ℓ, y).
+
2
|x| exp[2π|x|/T ] − 1
4T sinh (πx/T )
Applying Theorem 4, we get to
1
QB + h·, ℓi + γ η(w) = y
E µ2 exp iλ
T
2
1
Z
hy, ℓ(T )iR2
1
iλx
√
= exp iλ
e − 1 − iλx
+γ +
|x| exp[2π|x|/T ] − 1
T
R
2
π|y|
1
+
+
g̃
(x;
T,
ℓ,
y)
dx .
L
4T sinh2 (πx/T )
√
Since η(w) = w(T )/ T , combined with (30), this yields:
Proposition 17. It holds that
E µ2 exp [iλ (sT + h·, ℓi + γ)] w(T ) = y
T
n
o Z
hy, ℓ(T )iR2
iλx
= exp iλ
e − 1 − iλx fL (x; T, ℓ, y)dx ,
+γ +
T
R
where
fL (x; T, ℓ, y) =
√
1
1
π|y|2
+
+ g̃L (x; T, ℓ, y/ T ),
2
2
|x|{exp[2π|x|/T ] − 1}
4T sinh (πx/T )
and g̃L is given by (36)
21
3.2.3
We finally consider the exponential decay of the characteristic function of sT + h·, ℓi + γ under
µ2T and µ2T (·|w(T ) = y).
As was seen in §3.2.1, the corresponding Hilbert-Schmidt operator B possesses eigenvalues
{T /[(2n + 1)π]; n ∈ Z} and each of them is of multiplicity 2. By Theorem 7 and Lemma 10,
there exist C1 > 0 and λ1 > 0 such that
Z
s
exp
[iλ
(
(37)
+
h·,
ℓi
+
γ)]
dµ
≤ exp[−C1 λ]
T
W2
T
for every λ ≥ λ1 , γ ∈ R .
√
0
t/ T
√
Let η1 (t) =
, and η = {η1 , η2 }. As was shown in §3.2.2, the Hilbert, η2 (t) =
0
t/ T
Schmidt operator B (η) has eigenvalues
{(T /(2nπ)); n ∈ Z \ {0}}, each of them being of multi√
plicity 2. Since η(w) = w(T )/ T , by Theorem 7 and Lemma 10, there exist C2 > 0 and λ2 > 0
such that
(38)
E µ2 exp [iλ (sT + h·, ℓi + γ)] w(T ) = y ≤ exp[−C2 λ]
T
for every λ ≥ λ2 , γ ∈ R.
As in the previous subsection, when ℓ = 0 and γ = 0, it is well known ([12] and [8, pp.470–473])
that
Z
1
,
exp[iλsT ]dµ2T =
cosh(λT /2)
WT2
and
E µ2
T
exp[iλsT ] w(T ) = y =
λT /2
exp
sinh(λT /2)
λT
λT
1−
coth
2
2
|y|2
.
2T
Thus our estimations (37) and (38) coincide with the order obtained from these precise expressions.
We can give explicit expressions for the Mellin transforms of the Lévy measures σT and σT,y
of the distributions of sT under µ2T and the conditional probability measure µ2T ( · |w(T ) = y),
respectively. Namely, noting that
σT (dx) = fB,0 (x)dx,
σT,y (dx) = fB (η) ,P (η) (B((y/√T )·η)) (x)dx,
and then plugging (31), (34), and (35) into (12), we obtain:
Proposition 18. The Mellin transforms of σT and σT,y are given by
s s
Z
T
2 −1
s
|x| σT (dx) =
Γ(s)ζ(s)
π
2s−1
R
and
Z
R
s
|x| σT,y (dx) = 2
T
2π
s
|y|2
Γ(s)ζ(s) +
T
respectively.
See [1, 15] for the related topics.
22
T
2π
s
Γ(s + 1)ζ(s),
s ≥ 2,
3.3
Sample variance
Let T > 0 and (WT1 , HT1 , µ1T ) be the two-dimensional classical Wiener space over [0, T ]. In this
subsection, we consider the sample variance
Z T
Z
1 T
2
1
(w(t) − w̄) dt, w ∈ WT ,
where w̄ =
vT (w) =
w(t)dt.
T 0
0
3.3.1
We first compute the Lévy measure of
ℓ ∈ HT1 and γ ∈ R .
v + h·, ℓi + γ under µ1T by applying Theorem 2, where
1
2 T
Define a symmetric Hilbert-Schmidt operator C : HT1 → HT1 by
d(Ch)
(t) =
dt
Z
T
t
(h(s) − h̄)ds
h ∈ HT1 , t ∈ [0, T ].
Since vT (w) = hT (w) − T w̄2 , due to the observation made at the beginning
of §3.1.1, we have
R
that vT ∈ C2 (WT1 ) ⊕ C0 (WT1 ). It is easily seen that ∇2 vT = 2C and W 1 vT dµ1T = T 2 /6. By
T
Remark 3 (iii), we have
T2
vT = Q C + .
(39)
6
It is a straightforward computation to see that
√
∞
X
2T
T 2 C
nπt
C
C
C=
−1 .
hn ⊗ hn , where hn (t) =
cos
nπ
nπ
T
n=1
Hence ℓC = 0. Define
e =
ℓ(t)
∞
X
A
hℓ, hC
n ikn (t),
n=1
t ∈ [0, T ],
(40)
(η)
1
where {knA }∞
defined in (23). Comparing the above
n=1 is the orthonormal basis of (HT )0
(η)
expansion of C with that of A in (23), and recalling the definition of fA,ℓ , we obtain
fC,ℓ = fA(η) ,ℓe = fA(η) ,P (η) ℓe.
In conjunction with (25) and (39), Theorem 2 and Proposition 14 lead us to:
Proposition 19. It holds that
Z
1
exp iλ
vT + h·, ℓi + γ dµ1T
1
2
WT
Z ∞
T2
e 0)dx ,
eiλx − 1 − iλx fH (x; T, ℓ,
+
= exp iλ γ +
12
0
where fH is the function defined in Proposition 14, and ℓe is given by (40). In particular, the
e
distribution of 12 vT +h·, ℓi+γ under µ1T coincides with that of 21 hT +h·, ℓi+γ
under µ1T (·|w(T ) = 0).
23
3.3.2
We next compute the Lévy measure of 21 vT + h·, ℓi + γ under the conditional probability
µ2T (·|w(T ) = y) given W (T ) = y, where ℓ ∈ HT1 , γ ∈ R, and y ∈ R1 .
√
Set η1 (w) = w(T )/ T , w ∈ WT1 , and η = {η1 }. Observe that
(y 2 − 1)T 2
,
12
hC(y · η), (y · η)i − hCη1 , η1 i =
yℓ(T )
h(y · η), ℓi = √ .
T
By straightforward computations, we obtain
d(C (η) h)
(t) =
dt
Z
T
t
1
(h(s) − h̄)ds −
T
and
C (η) =
Z
T
0
Z
T
s
(η)
h ∈ (HT1 )0 ,
(h(u) − h̄)du ds,
∞
o
X
T 2 n C
kn ⊗ knC + k̃nC ⊗ k̃nC ,
2nπ
n=1
where
knC (t)
In particular,
√
2nπt
2T
=
,
sin
2nπ
T
k̃nC (t)
{P (η) (C(y · η) + ℓ)}C (η) = 0 and
√
2T
=
2nπ
cos
2nπt
T
−1 .
fC (η) ,P (η) (C(y·η)+ℓ) (x) = 0,
x ≤ 0.
Moreover it holds that
hP
(η)
(C(y · η) +
ℓ), knC i
√
= − 2y
T
2nπ
2
+ hℓ, knC i, hP (η) (C(y · η) + ℓ), k̃nC i = hℓ, k̃nC i.
Hence, for x > 0,
where
2
∞
2nπ 4
1X 2
2 2nπ
3/2
fC (η) ,P (η) (C(y·η)+ℓ) (x) =
+ 2y
−2 y
hℓ, knC i
2
x
T
T
n=1
2nπ 6
2
2
C 2
C 2
+
hℓ, kn i + hℓ, k̃n i
e−(2nπ) x/T
T
o 2π 2 y 2 4π 2 x
n
2
4π x
1
Θ′
−
1
−
+ gV (x; T, ℓ, y),
=
Θ
2x
T2
T2
T2
gV (x; T, ℓ, y) =
∞
1X
2nπ 4
−23/2 y
hℓ, knC i
2 n=1
T
2nπ 6
2
2
+
hℓ, knC i2 + hℓ, k̃nC i2 e−(2nπ) x/T . (41)
T
Recalling (39), and applying Theorem 4 and Proposition 14, we can conclude:
24
Proposition 20. It holds that
1
E µ1 exp iλ
vT + h·, ℓi + γ w(T ) = y
T
2
2
Z ∞
Ty
T 2 yℓ(T )
= exp iλ
eiλx − 1 − iλx fV (x; T, ℓ, y)dx ,
+
+
+γ +
24
24
T
0
where
o 2π 2 y 2 4π 2 x
√
1 n 4π 2 x
−
1
−
+ gV (x; T, ℓ, y/ T ),
Θ
Θ′
2
3
2
2x
T
T
T
and gV is given by (41). Moreover, the distribution of vT /2 under the conditional probability µ1T (·|w(T ) = y) coincides with the one of {hT /2 + h′T /2 }/2 under the product measure
√
µ1T /2 (·|w(T /2) = 0) ⊗ µ1T /2 (·|w(T /2) = y/ 2), where h′T /2 denotes an independent copy of hT /2 .
fV (x; T, ℓ, y) =
3.3.3
We finally consider the exponential decay of the characteristic function of 21 vT + h·, ℓi + γ under
µ1T and µ1T (·|w(T ) = y).
As was seen in §3.3.1, the corresponding Hilbert-Schmidt operator C possesses eigenvalues
{(T /[nπ])2 ; n ∈ N }, each of them being of multiplicity 1, and ℓC = 0. By Theorem 7, Lemma 10,
and (39), there exist C1 > 0 and λ1 > 0 such that
Z
1
exp iλ
(42)
dµ ≤ exp[−C1 λ1/2 ]
vT + h·, ℓi + γ
W1
2
T
for every λ ≥ λ1 , γ ∈ R .
√
Let η1 (t) = t/ T and η = {η1 }. As was shown in §3.3.2, the Hilbert-Schmidt operator C (η)
(η)
has eigenvalues {(T /(2nπ))2 ; n ∈ N
√}, each of them being of multiplicity 2, and {P (C(y · η) +
ℓ)}C (η) = 0. Since η(w) = w(T )/ T , by Theorem 7, Lemma 10, and (39), there exist C2 > 0
and λ2 > 0 such that
E 1 exp iλ 1 vT + h·, ℓi + γ w(T ) = y ≤ exp[−C2 λ1/2 ]
(43)
µT
2
for every λ ≥ λ2 , γ ∈ R.
When ℓ = 0 and γ = 0, combining the results in Propositions 19 and 20 with (29), we can show
the following explicit expressions of the Laplace transforms for the distributions of vT ; for λ > 0,
!1/2
√
Z
1
λT
√
exp − λvT dµ1T =
2
sinh( λT )
WT1
and
E µ1
T
1
exp − λvT w(T ) = y
2
"
√
√
λT /2
λT
√
=
exp
1−
coth
2
sinh( λT /2)
√
λT
2
!!
#
y2
.
2T
From these expressions, we see, in the same way as in §3.1.3, that our estimates (42) and (43)
coincide with the order obtained from the explicit expressions.
25
3.4
Density functions
Let T > 0 and consider the classical two-dimensional Wiener space (WT2 , HT2 , µ2T ) over [0, T ]. In
this subsection, as an application of Theorem 11, we show a way to obtain the explicit expressions
of the densities of the distributions of Lévy’s stochastic area and the square of the L2 -norm on
an interval of the two-dimensional Wiener process. We also compute the Mellin transforms of
the distributions. For the related topics, see [1, 15].
3.4.1
Lévy’s stochastic area
By (30), (31), and Theorem 11(i), we see that the distribution of sT under µ2T admits a smooth
density function pL with respect to the Lebesgue measure on R , and that the corresponding
Hilbert-Schmidt operator B satisfies
b =
det2 (I − iζ B)
∞
Y
1+
n=0
ζ 2T 2
(2n + 1)2 π 2
Define a simple curve Γn = {γn (t) : t ∈ [0, 4Rn ]} in C
−Rn + it,
γn (t) = t − 2Rn + iRn ,
Rn + i{4Rn − t},
= cos(iζT /2).
with Rn = 4nπ/T by
t ∈ [0, Rn ],
t ∈ [Rn , 3Rn ],
t ∈ [3Rn , 4Rn ].
Let x < 0. By a straightforward computation, we can show that Γn satisfies the conditions in
Theorem 11(ii) and conclude that
pL (x) = i
=
∞
X
Res
n=0
∞
X
2
T
e−iζx
(2n + 1)π
;i
cos(iζT /2)
T
(−1)n e(2n+1)πx/T =
n=0
1
.
T cosh(πx/T )
For x > 0, the complex conjugate Γn plays the same role as Γn , and we obtain
pL (x) = −i
∞
X
n=0
Res
(2n + 1)π
e−iζx
; −i
cos(iζT /2)
T
=
1
.
T cosh(πx/T )
√
√
0
t/ T
√ . Then η(w) = w(T )/ T . By
Let η = {η1 , η2 }, where η1 (t) =
and η2 (t) =
0
t/ T
(34) and Remark 12(ii), the distribution of sT under µ2T (·|w(T ) = 0) admits a smooth density
function p̃L with respect to the Lebesgue measure on R , and it holds that
d
(η) ) =
det2 (I − iζ B
∞
Y
sin(iζT /2)
ζ 2T 2
.
1+ 2 2 =
4n π
iζT /2
n=1
26
Using the same Γn ’s as above, this time with Rn = (4n + 1)π/T , and then applying Theorem 11(ii), we can show that
p̃L (x) = i
=
∞
X
Res
n=1
∞
X
2π
T
e−iζx (iζT /2) 2nπ
;i
sin(iζT /2)
T
(−1)n+1 ne−2nπ|x|/T =
n=1
π
,
2T cosh2 (πx/T )
x < 0.
For x > 0, the complex conjugate Γn plays the same role as Γn , and we obtain
p̃L (x) = −i
∞
X
n=1
Res
e−iζx (iζT /2)
2nπ
; −i
sin(iζT /2)
T
=
π
.
2T cosh2 (πx/T )
Thus we have:
Proposition 21. It holds that
µ2T (sT ∈ dx) =
1
dx,
T cosh(πx/T )
µ2T (sT ∈ dx|w(T ) = 0) =
π
dx.
2T cosh2 (πx/T )
Moreover, their Mellin transforms are given by
Z
1
4T s
|sT |s dµ2T = s+1 Γ(s + 1)Lχ4 (s + 1), s > −
2
π
2
WT
s s−1
T 2
1
−1
E µ2 [|sT |s |w(T ) = 0] =
Γ(s + 1)ζ(s), s > ,
2(s−1)
T
π
2
2
P
n
−s
where Lχ4 is Dirichlet’s L-function given by Lχ4 (s) = ∞
n=0 (−1) (2n + 1) .
Proof. We have already seen the first half. The last half is immediate consequence of the
representations of pL and p̃L in the form of infinite sums.
The densities of sT under µ2T and µ2T (·|w(T ) = 0) are first computed by Lévy ([12]).
3.4.2
L2 -norm on an interval of the two-dimensional Wiener process
Put
(2)
hT (w)
(2)
=
Z
T
0
|w(t)|2 dt,
w ∈ WT2 .
Since hT is a sum of two independent copies of hT coming from w1 and w2 , due to the observations made in §3.1.1, we see that the Hilbert-Schmidt operator D : HT2 → HT2 associated with
(2)
hT /2 has eigenvalues {(T /[(n + 21 )π])2 : n = 0, 1, . . . }, each being of multiplicity 2, and hence
(
2 )
∞
Y
p
T
b =
1 − iζ
det(I − iζ D)
= cos( iζ T ),
1
n+ 2 π
n=0
27
(2)
(2)
and that hT /2 = qD /2. By Theorem 11(iii), the distribution of hT /2 under µ2T admits a
smooth density function pH with respect to the Lebesgue measure on R . Consider a simple curve
Γn = {−i(Rn + it)2 : t ∈ [−Rn , Rn ]} in C with Rn = 2nπ/T . Let x > 0. By a straightforward
i
on
Electr
o
u
a
rn l
c
o
f
P
r
ob
abil
ity
Vol. 7 (2002) Paper No. 14, pages 1–30.
Journal URL
http://www.math.washington.edu/~ejpecp/
Paper URL
http://www.math.washington.edu/~ejpecp/EjpVol7/paper14.abs.html
WIENER FUNCTIONALS OF SECOND ORDER
AND THEIR LÉVY MEASURES
Hiroyuki Matsumoto
School of Informatics and Sciences, Nagoya University
Chikusa-ku, Nagoya 464-8601, Japan
matsu@info.human.nagoya-u.ac.jp
Setsuo Taniguchi
Faculty of Mathematics, Kyushu University, Fukuoka 812-8581, Japan
taniguch@math.kyushu-u.ac.jp
Abstract The distributions of Wiener functionals of second order are infinitely divisible. An
explicit expression of the associated Lévy measures in terms of the eigenvalues of the corresponding Hilbert–Schmidt operators on the Cameron–Martin subspace is presented. In some
special cases, a formula for the densities of the distributions is given. As an application of the
explicit expression, an exponential decay property of the characteristic functions of the Wiener
functionals is discussed. In three typical examples, complete descriptions are given.
Keywords Wiener functional of second order, Lévy measure, Mellin transform, exponential
decay
AMS subject classification 60J65, 60E07
Submitted to EJP on July 19, 2001. Final version accepted on February 12, 2002.
Introduction
Let W be a classical Wiener space, and µ be the Wiener measure on it. A Wiener functional F
of second order is a measurable functional F : W → R with ∇3 F = 0, ∇ being the Malliavin
gradient. It is represented as a sum of Wiener chaos of order at most two. Widely known Wiener
functionals of second order are the square of the L2 -norm on an interval of the Wiener process,
Lévy’s stochastic area, and the sample variance of the Wiener process. The study of Wiener
functionals of second order has a history longer than a half century, and many contributions have
been made. Among them, pioneering works were made by Cameron-Martin and Lévy [2, 3, 12]
for the square of the L2 -norm on an interval of the Wiener process and Lévy’s stochastic area.
The sample variance plays an important role in the Malliavin calculus (cf. [8]), and it was
studied in detail. For example, see [5, 7].
There are a lot of reasons why one studies such Wiener functionals. One is that they are the
easiest functionals next to linear ones. This may sound rather nonsensical, but a wide gap
between Wiener functionals of first and second orders can be found in recent works by Lyons
(for example, see [13]). Recalling roles played by quadratic Lagrange functions in the theories
of Feynman path integrals and of semi-classical analysis for Schrödinger operators, one must
encounter another reason for studying Wiener functionals of second order. A Wiener functional
of second order is one of key ingredients in the asymptotic theories, the Laplace method, the
stationary phase method et al, on infinite dimensional spaces.
As was employed by Cameron-Martin and Lévy, a fundamental strategy to investigate Wiener
functionals of second order is computing their Laplace transforms or characteristic functions, and
then their Lévy measures. In this paper, we give explicit expressions of Lévy measures of Wiener
functionals of second order in terms of the eigenvalues and eigenfunctions of the corresponding
Hilbert-Schmidt operators. See Theorem 2. Moreover, we extend the result to the case where µ is
replaced by a conditional probability (Theorem 4). These explicit representations are essentially
based on the splitting property of the Wiener measure µ, in other words, a decomposition of
the Brownian motion via the eigenfunctions of the Hilbert-Schmidt operator. Wiener used a
decomposition of this kind, the Fourier series expansion, to construct a Brownian motion, and
a generalization we use is due to Itô-Nisio [9].
With the help of the explicit expression, we compute the Mellin transform of the Lévy measures.
See Proposition 5. Recently Biane, Pitman and Yor ([1, 15]) showed that certain probability
distributions corresponding to Wiener functionals of second order are closely related to special
functions like Riemann’s ζ-function. The general expression given in Proposition 5 will indicate
that the relations studied by them are very natural ones. As another application, we shall
investigate the order of decay of the characteristic function as the parameter of Fourier transform
tends to infinity. If the Lévy-Khintchine representation admits a Gaussian term, then the decay
is very fast, but if there is no Gaussian term, then the decay is determined by the behavior
of the Lévy measure at the origin. For details, see Theorem 7. A characteristic function of
a quadratic Wiener functional is a key object to investigate the principle of stationary phase
on the Wiener space, and its exponential decay is indispensable to achieve such a principle on
infinite dimensional spaces. The exponential decay also implies that the distribution of the
Wiener functional of second order has a density function of Gevrey class with respect to the
Lebesgue measure, which relates to the property called transversal analyticity by Malliavin [14].
Another criteria for the distribution to possess a smooth density function will also be given, and
2
a method to compute it by using the residue theorem is shown (Theorem 11).
In Section 3, all our general results are tested for three concrete Wiener functionals of second
order mentioned above. Comparisons with known results will be also discussed there.
1
1.1
Lévy measures of Wiener functionals of second order
General Scheme
Throughout this subsection, (W, H, µ) stands for an abstract Wiener space. For the definition,
see [11]. The inner product and the norm of H are denoted by h·, ·i and k · kH , respectively.
GivenPa symmetric Hilbert-Schmidt operator A : H → H and ℓ ∈ H, decomposing A as
2
∞
A
= ∞
n=1 an hn ⊗ hn with an ∈ R and an orthonormal basis {hn }n=1 of H such that kAk2 =
P∞
2
n=1 an < ∞, we define
QA =
∞
X
n=1
an h·, hn i2 − 1 ,
(1)
X
1
1 hℓ, hn i2
+
exp[−x/an ],
x > 0,
2
x
a3n
n;an >0
fA,ℓ (x) = 0,
x = 0,
2
X
1
hℓ,
h
i
1
n
exp[−x/an ], x < 0,
+
2
−x
−a3n
(2)
n;an 0,
x Tr T(1/x)A1 + x3 hℓ, T(1/x)A1 ℓi,
x = 0,
fA,ℓ (x) = 0,
1
1
(3)
Tr T (0)
(1/|x|)A2 + |x|3 hℓ, T(1/|x|)A2 ℓi, x < 0.
|x|
Lemma 1. It holds that
0 ≤ fA,ℓ (x) ≤
for any x 6= 0, k ≥ 2, and
Z
1
−1
k−2
kAk∞
2
2
k!kAk
+
(k
+
1)!kℓk
2
H
2|x|k+1
|x|2 fA,ℓ (x)dx ≤
1
kAk22 + kℓk2H ,
2
where kAk∞ = sup{|an | : n = 1, 2, . . . }.
3
(3)
(4)
Proof. Since exp[−|x|/|an |] ≤ k!(|an |/|x|)k , (3) follows. (4) is an easy application of the monotone convergence theorem and an estimation
Z
1
0
|x|m exp[−|x|/|an |]dx = |an |m+1
for every m ∈ N .
Z
1/|an |
0
y m exp[−y]dy ≤ m!|an |m+1 ,
Theorem 2. Let A : H → H be a symmetric Hilbert-Schmidt operator, ℓ ∈ H, and γ ∈ R .
Then, for any λ ∈ R, it holds that
Z
exp iλ
W
where i =
√
1
QA + h·, ℓi + γ
2
−1 and
dµ
2
Z
λ kℓA k2H
iλx
e − 1 − iλx fA,ℓ (x)dx , (5)
= exp −
+ iλγ +
2
R
ℓA =
X
hhn , ℓihn .
(6)
n;an =0
Remark 3. (i) The integrability of (eiλx − 1 − iλx)fA,ℓ (x) in (5) is guaranteed by Lemma 1.
(ii) The theorem asserts that the distribution of 12 QA + h·, ℓi + γ is infinitely divisible and the
corresponding Lévy measure is fA,ℓ (x)dx. Moreover, the distribution of 12 QA + γ is selfdecomposable. See [16, §8 and §15].
(iii) Let Cn (W ) be the space of Wiener chaos of order n. A Wiener functional F of second order
is a Wiener chaos of order at most two, i.e., F ∈ C2 (W ) ⊕ C1 (W ) ⊕ C0 (W ), and is of the form
that F = (1/2)QA +h·, ℓi+γ for some symmetric Hilbert-Schmidt operator A, ℓ ∈ H, and γ ∈ R,
and QA ∈ C2 (WR ), h·, ℓi ∈ C1 (W ), and
R γ ∈ C0 (W ). Moreover, A, ℓ, and γ are determined so that
2
A = ∇ F , ℓ = W ∇F dµ, and γ = W F dµ, where ∇ stands for the Malliavin derivative.
(iv) It should be noted that C2 (W ) ⊕ C1 (W ) ⊕ C0 (W ) is invariant under shifts in the direction
of H. Namely, let F = 12 QA + h·, ℓi + γ and h ∈ H. Then
1
F (· + h) = F + h·, Ahi + hh, Ahi + hh, ℓi ∈ C2 (W ) ⊕ C1 (W ) ⊕ C0 (W ).
2
In particular, the theorem is applicable to a quadratic form of the form 12 QA (· − h), which is
one of main ingredients in the study of the principle of stationary phase on W . See [17]
Proof of Theorem 2. The proof is divided into three steps according to the signs of an ’s, the
eigenvalues of A.
1st step: the case where an > 0 for all n ∈ N .
Let λ > 0. Note that
∞ h
i
X
an
1
QA + h·, ℓi =
h·, hn i2 − 1 − hℓ, hn ih·, hn i .
2
2
n=1
4
Since {h·, hn i : n ∈ N } is a family of independent Gaussian random variables of mean 0 and
variance 1, we obtain the following well known identity:
Z
1
QA + h·, ℓi + γ
dµ
exp −λ
2
W
2
∞
Y
λ hℓ, hn i2
−1/2 λan /2
= exp[−λγ]
(1 + λan )
. (7)
e
exp
2 1 + λan
n=1
Applying the identities
we obtain
Z ∞
Z ∞
e−ax
log(1 + (λ/a)) =
e−x/a dx = a,
(1 − e−λx )
dx,
x
0
0
Z ∞
3 λ2
a
, a, λ > 0,
e−λx − 1 + λx e−x/a dx =
1 + aλ
0
2
!
∞
2
Y
i
λ
hℓ,
h
n
log
(1 + an )−1/2 ean /2 exp
2
1
+
an
n=1
=−
=
Z
∞
∞
n=1
∞
−λx
n=1
1X
1 X λ2 hℓ, hn i2
{log(1 + λan ) − λan } +
2
2
1 + λan
0
(e
− 1 + λx)fA,ℓ (x)dx.
Plugging this into (7), we obtain
Z
1
QA + h·, ℓi + γ
dµ
exp −λ
2
W
Z ∞
−λx
= exp −λγ +
e
− 1 + λx fA,ℓ(x)dx .
(8)
0
Note that
|ζ|2 x3
d
ζx
2
dζ e − 1 − ζx ≤ 2 + 2|ζ|x
and
3|ζ|2 x2
ζx
e − 1 − ζx ≤
2
for ζ ∈ C with Reζ < 0 and x ≥ 0. Hence, continuing (8) holomorphically to Ω = {ζ ∈ C :
Reζ < 0}, and then letting Reζ → 0 in Ω, we we arrive at (5), because fA,ℓ (x) = 0 for x ≤ 0.
2nd step: the case where an < 0 for all n ∈ N .
Since −QA = Q−A , applying the result in the first step to −A, we obtain
Z
1
QA + h·, ℓi + γ
dµ
exp iλ
2
W
Z
1
exp i(−λ)
=
+ h·, −ℓi + (−γ)
dµ
Q
2 (−A)
X
Z
−iλx
= exp[iλγ] exp
e
− 1 + iλx f−A,−ℓ (x)dx .
R
5
Since f−A,−ℓ(−x) = fA,ℓ (x), this yields (5).
3rd step: the general case.
Set
X
A+ =
an hn ⊗ hn ,
A− =
n;an >0
ℓ+ =
X
hℓ, hn ihn ,
n;an >0
Then
ℓ− =
X
n;an 0 : lim sup λ
(
−a
λ→∞
a+ := sup a > 0 : lim sup λ
λ→∞
−a
Z
(−∞,0)
Z
(0,∞)
8
(cos(λx) − 1)fA,ℓ (x)dx < 0,
)
(cos(λx) − 1)fA,ℓ (x)dx < 0 ,
)
,
where a− = 0, a+ = 0 if {· · · } = ∅. If max{a− , a+ } > 0, then, for every a < max{a− , a+ },
there exist Ca > 0 and λa > 0 such that
Z
1
≤ exp[−Ca λa ],
Q
+
h·,
ℓi
+
γ
dµ
(15)
exp
iλ
A
2
W
for every λ ≥ λa , γ ∈ R . Moreover, if both supremums a+ , a− are attained as maximums, then
the above assertion holds with a = max{a− , a+ }.
(iii) Put
(
)
Z
−b
b− := sup b > 0 : lim sup λ
(cos(λx) − 1)fA(η) ,P (η) (A(y·η)+ℓ) (x)dx < 0 ,
(
λ→∞
(−∞,0)
−b
b+ := sup b > 0 : lim sup λ
λ→∞
Z
(0,∞)
)
(cos(λx) − 1)fA(η) ,P (η) (A(y·η)+ℓ) (x)dx < 0 ,
where b− = 0, b+ = 0 if {· · · } = ∅. If max{b− , b+ } > 0, then, for every b < max{b− , b+ }, there
exist Cb > 0 and λb > 0 such that
E µ exp iλ 1 QA + h·, ℓi + γ η(w) = y ≤ exp[−Cb λb ],
(16)
2
for any λ ≥ λb , γ ∈ R . Moreover, if both supremums b+ , b− are attained as maximums, then
the above assertion holds with b = max{b− , b+ }.
Remark 8. (i) If ℓA 6= 0, then we have an exponent −λ2 kℓA k2H /2, which gives a much faster
decay than the one discussed in Theorem 7. Similarly, if {P (η) (A(y · η) + ℓ)}A(η) 6= 0, then we
obtain a much faster decay than the one discussed in the theorem.
(ii) The integrability of x2 fA,ℓ (x) and x2 fA(η) ,P (η) (A(y·η)+ℓ) (x) is due to Lemma 1.
(iii) The lower estimates
(13) and (14) are sharp as we shall see in Lemma 10. For example,
P in −p
if we consider A = ∞
n
hn ⊗ hn for some p > 1/2 and an orthonormal basis {hn } of H,
n=1
then there exist C > 0 and λ0 > 0 such that
Z
1
QA + h·, ℓi + γ
dµ ≤ exp[−Cλ1/p ] for any λ > λ0 .
exp iλ
2
W
For details, see Lemma 10 and its proof.R
∞
(iv) If an > 0 for some n, then limλ→∞ 0 (cos(λx) − 1)fA,ℓ (x)dx = −∞. In fact, it holds that
fA,ℓ ≥ fA,0 and
Z ∞
Z ∞
1 − cos y X
(1 − cos(λx))fA,0 (x)dx =
exp[−y/(λan )]dy.
y
0
0
n;a >0
n
Then, applying the monotone convergence
theorem, we obtain the desired divergence. Similarly,
R0
if an < 0 for some n, then limλ→∞ −∞ (cos(λx) − 1)fA,ℓ (x)dx = −∞. Thus the assumption
made on a± is that only on the order of divergence.
If #{n; an 6= 0} < ∞, then a+ = a− = 0. Indeed, in this case, there are C, C ′ > 0 such that
fA,ℓ (x) ≤ C{(1/|x|) + 1} exp[−C ′ |x|] for every x ∈ R \ {0}. For each δ > 0, this implies the
9
existence of Cδ > 0 such that fA,ℓ (x) ≤ Cδ |x|−1−δ for any x ∈ R \ {0}. Hence, for every λ > 0,
0 ≤ max
Z
0
−∞
(1 − cos(λx))fA,ℓ (x)dx,
Z
∞
0
(1 − cos(λx))fA,ℓ (x)dx
≤ Cδ λ
δ
Z
∞
0
1 − cos y
dy,
y 1+δ
from which it follows that a± = 0.
Proof of Theorem 7. Assume that ℓA = 0 or {P (η) (A(y · η) + ℓ)}A(η) = 0 accordingly as µ or
µ(·|η(w) = y) is considered. By virtue of Theorems 2 and 4, we have
Z
Z
1
(cos(λx) − 1)fA,ℓ (x)dx
QA + h·, ℓi + γ
dµ = exp
exp iλ
2
R
W
E µ exp iλ 1 QA + h·, ℓi + γ η(w) = y
2
Z
= exp
(cos(λx) − 1)fA(η) ,P (η) (A(y·η)+ℓ) (x)dx
R
Since | cos x − 1| ≤ x2 /2 for any x ∈ R , the assertion (i) follows immediately.
Let f = fA,ℓ or = fA(η) ,P (η) (A(y·η)+ℓ) . Since (cos(λx) − 1)f (x) ≤ 0, we have
Z
R
(cos(λx) − 1)f (x)dx
≤ min
(Z
(−∞,0)
(cos(λx) − 1)f (x)dx,
Z
(0,∞)
)
(cos(λx) − 1)f (x)dx .
Thus the estimations in (ii) and (iii) also follow.
A function ϕ ∈ C ∞ (R ) is said to belong a Gevrey class of order a > 1 (ϕ ∈ Ga (R ) in notation)
if, for any compact subset K ⊂ R , there exists a constant CK > 0 such that
n
d ϕ
a n
for every x ∈ K, n ∈ N .
dxn (x) ≤ CK (CK (n + 1) )
A finite Radon measure u on R admits a density function of class Ga (R ) if there is a C > 0 such
that
C(n + 1)a n
|û(ξ)| ≤ C
for any ξ ∈ R \ {0}, n ∈ N ,
|ξ|
where û is the Fourier transformation of u (cf. [6, Prop.8.4.2]). Since e−x ≤ αα x−α for α, x > 0,
a sufficient condition for this to hold is that there exist C1 , C2 > 0 such that
|û(ξ)| ≤ C1 exp[−C2 |ξ|1/a ]
We obtain the following from Theorem 7.
10
for any ξ ∈ R .
Corollary 9. Let a± , b± be as in Theorem 7.
(i) If ℓA = 0 and max{a− , a+ } > 0, then the distribution on R of 12 QA + h·, ℓi + γ under µ admits
a density function, which is in G1/a (R ) for any a < max{a− , a+ }, with respect to the Lebesgue
measure.
(ii) If {P (η) (A(y · η) + ℓ)}A(η) = 0 and max{b− , b+ } > 0, then the distribution on R of 12 QA +
h·, ℓi + γ under the conditional probability µ(·|η(w) = y) admits a density function, which is in
G1/b (R ) for any b < max{b− , b+ }, with respect to the Lebesgue measure.
We give a sufficient condition for a± to be positive. It follows from (3) that fA,ℓ (x) diverges
at the order of at most |x|−3 as |x| → 0. If we assume a uniform order of divergence, then
max{a− , a+ } > 0.
Lemma 10. (i) If there exist δ, C > 0 and ε < 2 such that
fA,ℓ (x) ≥ Cxε−3
for any x ∈ (0, δ),
then a+ ≥ 2 − ε. If there exist δ, C > 0 and ε < 2 such that
fA,ℓ (x) ≥ C|x|ε−3 for any x ∈ (−δ, 0)
then a− ≥ 2 − ε.
(ii) Let {an }∞
n=1 be eigenvalues of A. Suppose that there exist a subsequence {ank }, C > 0, and
p > 1/2 such that ank ≥ C/kp for any k ∈ N . Then, for any δ > 0,
(
)
Z
C 1/p ∞ (1/p)−1 −z
fA,ℓ (x) ≥
z
e dz x−1−(1/p)
p
δ/C
holds for any x ∈ (0, δ). In particular, a+ ≥ 1/p.
If there exist a subsequence {ank }, C > 0, and p > 1/2 such that ank ≤ −C/kp for any k ∈ N .
Then, for any δ > 0,
(
)
Z
C 1/p ∞ (1/p)−1 −z
fA,ℓ (x) ≥
z
e dz x−1−(1/p)
p
δ/C
holds for any x ∈ (−δ, 0). In particular, a− ≥ 1/p.
Proof. (i) It follows from (3) that
Z ∞
δ
(cos(λx) − 1)fA,ℓ (x)dx ≤ 0.
Due to the first assumption, we have
Z
0
δ
Z
δ
(cos(λx) − 1)xε−3 dx
0
Z λδ
= Cλ2−ε
(cos x − 1)xε−3 dx.
(cos(λx) − 1)fA,ℓ (x)dx ≤ C
0
11
Hence
lim sup λ
−(2−ε)
λ→∞
Z
∞
0
(cos(λx) − 1)fA,ℓ (x)dx ≤ C
Z
∞
0
(cos x − 1)x3−ε dx < 0.
Thus the first half has been verified. The latter half can be seen in exactly the same way.
(ii) Suppose the first assumption. Then it holds that
fA,ℓ (x) ≥
∞ −xk p /C
X
e
k=1
x
≥
Z
∞
1
p /C
e−xy
x
dy = x
−1−(1/p) C
1/p
p
Z
∞
z (1/p)−1 e−z dz
x/C
for any x > 0. This yields the first estimation. Since −1 − (1/p) = {2 − (1/p)} − 3, by the
assertion (i), we have that a+ ≥ 1/p. Thus the first half has been verified.
The latter half can be seen similarly.
2.2
Density functions
Corollary 9 gives a sufficient condition for the distribution of 21 QA +h·, ℓi+γ under µ or µ(·|η(w) =
y) to have a smooth density function with respect to the Lebesgue measure. We now show
another condition for the distribution to possess a smooth density function, and also a method
to compute it.
Theorem 11. Let (W, H, µ) be an abstract
space, A : H → H be a symmetric HilbertPWiener
∞
Schmidt operator, and decompose as A = n=1 an hn ⊗ hn with an orthonormal basis {hn }∞
n=1
of H.
(i) Suppose that #{n : an 6= 0} = ∞. Then there exists a pA ∈ C ∞ (R ) such that µ QA /2 ∈
dx = pA (x)dx.
(ii) Suppose that a2n−1 = a2n for any n ∈ N and #{n : an 6= 0} = ∞. Fix x ∈ R, and
assume that there exists a family of simple C 1 curves Γn = {γn (t) : t ∈ [αn , βn ]} in C such that
(a1) Rγn (αn ) ∈ (−∞, 0), γn (βn ) ∈ (0, ∞), (a2) inf{|γn (t)| : t ∈
n , βn ]} → ∞ as n → ∞, and
P[α
∞
−iζx
b
b
/ det2 (I − iζ A)}dζ → 0 as n → ∞, where A = n=1 a2n h2n ⊗ h2n .
(a3) Γn {e
(ii-1) If Im γn (t) > 0 and −i/am ∈
/ Γn for any n ∈ N , t ∈ (αn , βn ), and m ∈ N with am < 0,
then
!
X
e−iζx
i
pA (x) = i
,
(17)
Res
;−
b
an
det2 (I − iζ A)
n:an 0,
then
!
X
e−iζx
i
.
(18)
pA (x) = −i
;−
Res
b
an
det2 (I − iζ A)
n:an >0
P
P∞
(iii) Suppose ∞
n=1 |an | < ∞, and set qA = QA +
n=1 an . Then all assertions in (i) and (ii)
b by qA and det(I − iζ A),
b respectively.
hold, replacing QA and det2 (I − iζ A)
12
b is holomorphic ([4]).
Remark 12. (i) The mapping C ∋ ζ 7→ det2 (I − iζ A)
(ii) Let η = {η1 , . . . , ηm } be an orthonormal system of H. By (11), it holds
m
o
n
X
(η) 1
µ QA ∈ dx η(w) = 0 = µ0
hAηi , ηi i ∈ dx ,
QA(η) −
2
2
n=1
1
(η) 1
µ qA ∈ dx η(w) = 0 = µ0
q (η) ∈ dx .
2
2 A
1
Thus, we can compute the density functions of the distributions of QA /2 and qA /2 under
µ(·|η(w) = 0), by applying Theorem 11 to QA(η) and qA(η) on W (η) .
(iii) The method to compute the density with the help of the residue theorem has been already
applied by Cameron-Martin ([2]) more than a half century ago to the square of the L2 -norm on
an interval of the one-dimensional Wiener process.
T
P
−1
p
2
2
Proof. Since k∇(QA /2)k2H = ∞
p>0 L (µ) if #{n : an 6= 0} =
n=1 an h·, hn i , k∇(QA /2)kH ∈
∞ (cf.[18]). Thus the assertion (i) follows as an fundamental application of the Malliavin calculus.
By (7) and the assumption that a2n−1 = a2n , we have
Z
1
,
eiλx pA (x)dx =
b
det2 (I − iλA)
R
and hence
(19)
Z
e−iλx
1
pA (x) =
dλ.
b
2π R det2 (I − iλA)
Then the assertions in (ii) are immediate consequences of the residue theorem.
To see the last assertion, it suffices to mention that (19) implies that, if we denote by p̃A the
density function of qA /2, then
Z
1
.
eiλx p̃A (x)dx =
b
det(I − iλA)
R
3
Typical quadratic Wiener functionals
In this section, we investigate how our results work for typical quadratic Wiener functionals.
Some of the computations below have been carried out in Ikeda-Manabe [7], but we give all the
results for convenience of the reader. Moreover, when we do not consider the first order terms
of the Wiener chaos, that is, when ℓ = 0 in (5), explicit expressions of the Fourier or Laplace
transforms of the distributions are well known for the examples considered in the following and,
from them, we can obtain the same results after some elementary calculations.
13
3.1
The square of the L2 -norm on an interval
Let T > 0 and consider the classical one-dimensional Wiener space (WT1 , HT1 , µ1T ) over [0, T ]; WT1
is the space of continuous functions w : [0, T ] → R with w(0) = 0, HT1 consists of h ∈ WT1 which
is absolutely continuous and has a square integrable derivative dh/dt, and µ1T is the Wiener
measure. The inner product in HT1 is given by
hh, ki =
Z
T
dh dk
(t) (t)dt,
dt
dt
0
h, k ∈ HT1 .
In this subsection we consider
Z
hT (w) =
T
w(t)2 dt,
0
w ∈ WT1 .
3.1.1
We first compute the Lévy measure of
ℓ ∈ HT1 and γ ∈ R .
h + h·, ℓi + γ under µ1T by applying Theorem 2, where
1
2 T
Define a symmetric Hilbert-Schmidt operator A : HT1 → HT1 by
d(Ah)
(t) =
dt
Z
T
h(s)ds,
t
h ∈ HT1 , t ∈ [0, T ].
Note
that w(t)2 is in C2 (WT1 ) ⊕ C0 (WT1 ), and so is hT . It is easily seen that ∇2 hT = 2A and that
R
1
2
W 1 hT dµT = T /2. Then, by virtue of Remark 3 (iii), we observe that
T
hT = Q A +
It is easy to see that
A=
∞
X
n=0
where
(20)
!2
(21)
A
hA
n ⊗ hn ,
1
n
+
2T
2 πt
sin
hA
.
n (t) =
1
T
n+ 2 π
√
In particular, we have
T
n + 12 π
T2
.
2
ℓA = 0 and
fA,ℓ (x) = 0,
x ≤ 0.
Set
!
n + 12 πt
ℓ(n) =
dt,
=
T
0
∞
(2n + 1)2 π 2 x
π6 X
6
2
(2n + 1) |ℓ(n) | exp −
.
gH (x; T, ℓ) = 7 6
2 T
4T 2
hℓ, hA
ni
r
2
T
Z
T
dℓ
(t) cos
dt
n=0
14
(22)
Since Jacobi’s theta function Θ(u) =
P
2
Z exp[−n u] enjoys the relation
n∈
Θ(u) − Θ(4u) = 2
∞
X
2
e−(2n+1) u ,
n=0
it is straightforward to see that
2
2
1
π x
π x
fA,ℓ (x) =
−
Θ
+ gH (x; T, ℓ),
Θ
4x
4T 2
T2
x > 0.
By virtue of this and (20), applying Theorem 2, we arrive at:
Proposition 13. It holds that
Z
1
dµ1T
hT + h·, ℓi + γ
exp iλ
2
WT1
"
2
Z ∞
1 π2x
π x
T2
iλx
e − 1 − iλx
−Θ
= exp iλ γ +
+
Θ
2
4
4x
4T
T2
0
#
+ gH (x; T, ℓ) dx ,
where gH is defined by (22).
3.1.2
We next compute the Lévy measure of 12 hT + h·, ℓi + γ under the conditional probability
µ1T (·|w(T ) = y) given w(T ) = y, where ℓ ∈ HT1 , γ ∈ R, and y ∈ R.
√
Set η1 (w) = w(T )/ T , w ∈ WT1 , and η = {η1 }. Note that
hA(y · η), (y · η)i − hAη1 , η1 i = (y 2 − 1)hAη1 , η1 i =
(y 2 − 1)T 2
,
3
By a straightforward computation, we obtain
Z T
Z Z T
d(A(η) h)
1 T
h(s)ds −
(t) =
h(u)du ds,
dt
T 0
t
s
and hence
(η)
A
In particular,
∞
X
T 2 A
=
kn ⊗ knA ,
nπ
n=1
{P (η) (A(y · η) + ℓ)}A(η) = 0
where
knA (t)
=
yℓ(T )
h(y · η), ℓi = √ .
T
(η)
h ∈ (HT1 )0 ,
√
nπt
2T
.
sin
nπ
T
and fA(η) ,P (η) (A(y·η)+ℓ) (x) = 0,
(23)
x ≤ 0.
Note that
hP (η) (A(y · η) + ℓ), knA i = yhAη, knA i + hℓ, knA i
√ T 2
= (−1)n+1 2y
+ ℓ̃(n) ,
nπ
15
(24)
where
ℓ̃(n) =
Hence, for x > 0,
hℓ, knA i
=
r
2
T
Z
0
T
nπt
dℓ
dt.
(t) cos
dt
T
fA(η) ,P (η) (A(y·η)+ℓ) (x)
∞
nπ 4
nπ 2
1 Xn 1
=
+ (−1)n+1 23/2 y
ℓ̃(n)
+ 2y 2
2
x
T
T
n=1
o
h n2 π 2 i
nπ 6
ℓ̃2(n) exp − 2 x
+
T
T
o π2 y2 π2 x
1 n π2 x
=
−1 −
+ g̃H (x; T, ℓ, y),
Θ
Θ′
4x
T2
2T 2
T2
where
g̃H (x; T, ℓ, y)
∞
nπ 6
h n2 π 2 x i
nπ 4
1X
2
n+1 3/2
ℓ̃(n) +
ℓ̃(n) exp −
.
(−1)
2 y
=
2 n=1
T
T
T2
(25)
(26)
Due to Theorem 4, we obtain
1
E µ1 exp iλ
QA + h·, ℓi + γ η(w) = y
T
2
2
1 n π2 x
Z ∞
o
(y − 1)T 2 yℓ(T )
= exp iλ
eiλx − 1 − iλx
Θ
+ √ +γ +
−
1
6
4x
T2
T
0
π2 y2 ′ π2 x
−
Θ
+ g̃H (x; T, ℓ, y) dx .
2T 2
T2
√
Since η(w) = w(T )/ T , combined with (20), we conclude from this:
Proposition 14. It holds that
E µ1
T
1
exp iλ
hT + h·, ℓi + γ w(T ) = y
2
n 2
o Z ∞
Ty
T 2 yℓ(T )
iλx
= exp iλ
e − 1 − iλx fH (x; T, ℓ, y)dx ,
+
+
+γ +
6
12
T
0
where
fH (x; T, ℓ, y) =
and g̃H is given by (26).
π2y2 π2x
√
1 π2 x
′
Θ
−
1
−
+
g̃
(x;
T,
ℓ,
y/
T ),
Θ
H
4x
T2
2T 3
T2
3.1.3
We finally study the exponential decay of the characteristic function of 12 hT + h·, ℓi + γ under
µ1T and µ1T (·|w(T ) = y).
16
As was seen in §3.1.1, the Hilbert-Schmidt operator A associated with hT has eigenvalues
{T 2 /[(n + 12 )π]2 ; n ∈ N ∪ {0}}, each of them being of multiplicity one. By Theorem 7 and
Lemma 10, there exist C1 > 0 and λ1 > 0 such that
Z
1
(27)
dµ ≤ exp[−C1 λ1/2 ]
hT + h·, ℓi + γ
exp iλ
W1
2
T
for any λ ≥ λ1 , γ ∈ R.
√
Let η1 (t) = t/ T and η = {η1 }. As was shown in §3.1.2, the Hilbert-Schmidt operator A(η)
2
possesses
√ eigenvalues {(T /(nπ)) ; n ∈ N }, each of them being of multiplicity 1. Since η(w) =
w(T )/ T , by Theorem 7 and Lemma 10, there exist C2 > 0 and λ2 > 0 such that
E 1 exp iλ 1 hT + h·, ℓi + γ w(T ) = y ≤ exp[−C2 λ1/2 ]
(28)
µ
T
2
for any λ ≥ λ2 , γ ∈ R.
When ℓ = 0 and γ = 0, it is well known ([3, 12] and [8, pp.470–473]) that
Z
λ
1
√
exp − hT dµ1T =
2
(cosh( λT ))1/2
WT1
and
λ
exp − hT w(T ) = y
T
2
!1/2
√
√
√ y 2
λT
√
=
exp 1 − λT coth λT
2T
sinh λT
E µ1
(29)
hold for λ > 0. Thus, continuing holomorphically, we see that our estimations (27) and (28)
coincide with the order obtained from these precise expressions.
Starting from these well-known expressions, and recalling the elementary formulae
∞
∞
Y
Y
4x2
x2
cosh(x) =
1+
,
sinh(x)
=
x
,
1
+
(2k + 1)2 π 2
k2 π2
k=0
coth(πx) =
k=1
∞
2x X
1
+
πx
π
k=1
x2
1
,
+ k2
we can also show explicit expressions for the Lévy measures νT (dx) and νT,y (dx) of the distribution of hT /2 under µ1T and the conditional probability measure µ1T ( · |w(T ) = y) as described
in Propositions 13 and 14 with ℓ = 0.
P
−s
Moreover, by using the Riemann zeta function ζ(s) = ∞
n=1 n , we can give explicit forms of
the Mellin transforms of νT and νT,y . Namely, noting that
νT (dx) = fA,0 (x)dx,
νT,y (dx) = fA(η) ,P (η) (A((y/√T )·η)) (x)dx,
and then plugging (21), (23), and (24) into (12), we obtain:
17
Proposition 15. The Mellin transform of νT (dx) and νT,y (dx) are given by
Z
∞
s
x νT (dx) =
0
4T 2
π2
s
22s − 1
Γ(s)ζ(2s)
22s+1
and
Z
0
∞
1
x νT,y (dx) =
2
s
T2
π2
s
y2
Γ(s)ζ(2s) +
T
T2
π2
s
Γ(s + 1)ζ(2s),
s ≥ 2,
respectively, where Γ is the usual gamma function.
Recently, Biane-Pitman-Yor [1] and Pitman-Yor [15] have discussed the related topics and shown
similar formulae.
3.2
Lévy’s stochastic area
Let T > 0 and consider the classical two-dimensional Wiener space (WT2 , HT2 , µ2T ) over [0, T ]; WT2
is the space of continuous functions w : [0, T ] → R 2 with w(0) = 0, HT2 consists of h ∈ WT2 which
is absolutely continuous and has a square integrable derivative dh/dt, and µ2T is the Wiener
measure. The inner product in HT2 is given by
hh, ki =
Z
0
T
dh
dk
dt,
(t), (t)
dt
dt
R2
h, k ∈ HT2 .
Define Lévy’s stochastic area by
Z
Z
1 T
1 T 1
2
hJw(t), dw(t)iR2 =
w (t)dw2 (t) − w2 (t)dw1 (t) ,
sT (w) =
2 0
2 0
0 −1
where J =
and dw(t) denotes the Itô integral.
1 0
3.2.1
We first compute the Lévy measure of
ℓ ∈ HT2 and γ ∈ R .
sT + h·, ℓi + γ under µ2T by applying Theorem 2, where
Define a symmetric Hilbert-Schmidt operator B : HT2 → HT2 by
d(Bh)
1
(t) = J h(t) − h(T ) , h ∈ HT2 , t ∈ [0, T ].
dt
2
Since wi (s){wj (t) − wj (s)} is in C2 (WT2 ) for (i, j) ∈ {(1, 2), (2, 1)} and s < t, so is sT . It is easily
seen that ∇2 sT = B, and hence, due to Remark 3 (iii), we have
1
2
sT = QB .
18
(30)
By a direct computation, we see that
B=
where
hB
n (t)
X
n∈Z
o
n
T
B
B
B
⊗
h
+
h̃
⊗
h̃
hB
n
n
n
n ,
(2n + 1)π
√
T
cos((2n + 1)πt/T ) − 1,
=
,
sin((2n + 1)πt/T )
(2n + 1)π
(31)
B
h̃B
n (t) = Jhn (t).
In particular
ℓB = 0.
Set
ℓ1(n)
ℓ2(n)
ℓ(n)
and
Z T
dℓ
1
− sin((2n + 1)πt/T )
√
=
=
dt,
(t),
cos((2n + 1)πt/T )
dt
T 0
2
R
Z T
dℓ
1
cos((2n + 1)πt/T )
B
(t),
= hℓ, h̃n i = − √
dt,
sin((2n + 1)πt/T )
dt
T 0
2
R
!
ℓ1(n)
,
= 2
ℓ(n)
hℓ, hB
ni
∞
π3 X
(2n + 1)3 |ℓ(n) |2 exp [−(2n + 1)πx/T ] ,
x>0
3
2T
n=0
gL (x; T, ℓ) = 0,
x = 0,
∞
3
X
π
(2n − 1)3 |ℓ(−n) |2 exp [−(2n − 1)πx/T ] , x < 0.
2T 3
(32)
n=1
Then it is easily seen that
fB,ℓ (x) =
1
+ gL (x; T, ℓ),
2x sinh(πx/T )
x ∈ R.
By virtue of this and (30), applying Theorem 2, we arrive at;
Proposition 16. It holds that
Z
WT2
exp [iλ (sT + h·, ℓi + γ)] dµ2T
"
Z
iλx
e − 1 − iλx
= exp iλγ +
R
where gL is defined by (32)
19
#
1
+ gL (x; T, ℓ) dx , (33)
2x sinh(πx/T )
3.2.2
We next compute the Lévy measure of sT + h·, ℓi + γ under the conditional probability
µ2T (·|w(T ) = y) given W (T ) = y, where ℓ ∈ HT2 , γ ∈ R, and y ∈ R2 .
√
√
Let η = {η1 , η2 } ⊂ W ∗ , where ηi (w) = wi (T )/ T . Since y · η = ty/ T and hJz, ziR2 = 0 for
any z ∈ R 2 ,
hB(y · η), (y · η)i −
2
X
hBηn , ηn i = 0,
n=1
h(y · η), ℓi =
hy, ℓ(T )iR2
√
.
T
(η)
For any h, g ∈ (HT2 )0 , the identity hB (η) h, gi = hBh, gi holds, and hence
d(B (η) h)
(t) = J(h(t) − h̄),
dt
1
where h̄ =
T
Z
T
h(t)dt.
0
Then it is straightforward to see that
X
B (η) =
n∈Z\{0}
where
knB (t)
√
T
=
2nπ
T B
kn ⊗ knB + k̃nB ⊗ k̃nB ,
2nπ
cos(2nπt/T ) − 1
,
sin(2nπt/T )
(34)
k̃nB (t) = JknB (t).
Hence
{P (η) (B(y · η) + ℓ)}B (η) = 0,
and it holds that
T y2
+ ℓ̃1(n) ,
2nπ
T y1
+ ℓ̃2(n) ,
=
2nπ
hP (η) (B(y · η) + ℓ), knB i = h(y · η), BknB i + ℓ̃1(n) = −
hP (η) (B(y · η) + ℓ), k̃nB i = h(y · η), B k̃nB i + ℓ̃2(n)
where
ℓ̃1(n) = hℓ, knB i =
ℓ̃2(n)
Setting ℓ̃(n) =
=
hℓ, k̃nB i
=
!
ℓ̃1(n)
, we obtain
ℓ̃2(n)
Z
Z
T
0
T
0
dℓ
1
(t), √
dt
T
dℓ
1
(t), √
dt
T
− sin(2nπt/T )
cos(2nπt/T )
R2
dt,
− cos(2nπt/T )
dt.
− sin(2nπt/T )
R2
hP (η) (B(y · η) + ℓ), knB i2 + hP (η) (B(y · η) + ℓ), k̃nB i2
T 2
T
|y|2 +
hℓ̃ , JyiR2 + |ℓ̃(n) |2 .
=
2nπ
nπ (n)
20
(35)
Thus, if we put
g̃L (x; T, ℓ, y)
∞n
2nπ 3
o
h 2nπx i
1X
2nπ 2
2
,
x > 0,
hℓ̃(n) , JyiR2 +
|ℓ̃(n) |2 exp −
T
T
T
2 n=1
= 0,
x = 0,
∞ n
o
h
i
X
2
3
1
2nπ
2nπx
2nπ
, x < 0,
−2
hℓ̃(−n) , JyiR2 +
|ℓ̃(−n) |2 exp
2
T
T
T
n=1
(36)
then, for x > 0, it holds that
fB (η) ,P (η) (B(y·η)+ℓ) (x)
∞
∞
1 X {T /(2nπ)}2 |y|2
1X2
exp[−2nπx/T ] +
exp[−2nπx/T ] + g̃L (x; T, ℓ, y)
=
2
x
2
{T /(2nπ)}3
n=1
=
1 exp[−2πx/T ]
+
x 1 − exp[−2πx/T ]
n=1
∞
2
π|y| X
T
n exp[−2nπx/T ] + g̃L (x; T, ℓ, y)
n=1
1
1
1
π|y|2
+ g̃L (x; T, ℓ, y),
+
2
x exp[2πx/T ] − 1
4T sinh (πx/T )
P
−nx = 1/{4 sinh 2 (x/2)}. Similarly, for x < 0, we have
where we have used the identity ∞
n=1 ne
=
fB (η) ,P (η) (B(y·η)+ℓ) (x) =
1
1
1
π|y|2
+ g̃L (x; T, ℓ, y).
+
2
|x| exp[2π|x|/T ] − 1
4T sinh (πx/T )
Applying Theorem 4, we get to
1
QB + h·, ℓi + γ η(w) = y
E µ2 exp iλ
T
2
1
Z
hy, ℓ(T )iR2
1
iλx
√
= exp iλ
e − 1 − iλx
+γ +
|x| exp[2π|x|/T ] − 1
T
R
2
π|y|
1
+
+
g̃
(x;
T,
ℓ,
y)
dx .
L
4T sinh2 (πx/T )
√
Since η(w) = w(T )/ T , combined with (30), this yields:
Proposition 17. It holds that
E µ2 exp [iλ (sT + h·, ℓi + γ)] w(T ) = y
T
n
o Z
hy, ℓ(T )iR2
iλx
= exp iλ
e − 1 − iλx fL (x; T, ℓ, y)dx ,
+γ +
T
R
where
fL (x; T, ℓ, y) =
√
1
1
π|y|2
+
+ g̃L (x; T, ℓ, y/ T ),
2
2
|x|{exp[2π|x|/T ] − 1}
4T sinh (πx/T )
and g̃L is given by (36)
21
3.2.3
We finally consider the exponential decay of the characteristic function of sT + h·, ℓi + γ under
µ2T and µ2T (·|w(T ) = y).
As was seen in §3.2.1, the corresponding Hilbert-Schmidt operator B possesses eigenvalues
{T /[(2n + 1)π]; n ∈ Z} and each of them is of multiplicity 2. By Theorem 7 and Lemma 10,
there exist C1 > 0 and λ1 > 0 such that
Z
s
exp
[iλ
(
(37)
+
h·,
ℓi
+
γ)]
dµ
≤ exp[−C1 λ]
T
W2
T
for every λ ≥ λ1 , γ ∈ R .
√
0
t/ T
√
Let η1 (t) =
, and η = {η1 , η2 }. As was shown in §3.2.2, the Hilbert, η2 (t) =
0
t/ T
Schmidt operator B (η) has eigenvalues
{(T /(2nπ)); n ∈ Z \ {0}}, each of them being of multi√
plicity 2. Since η(w) = w(T )/ T , by Theorem 7 and Lemma 10, there exist C2 > 0 and λ2 > 0
such that
(38)
E µ2 exp [iλ (sT + h·, ℓi + γ)] w(T ) = y ≤ exp[−C2 λ]
T
for every λ ≥ λ2 , γ ∈ R.
As in the previous subsection, when ℓ = 0 and γ = 0, it is well known ([12] and [8, pp.470–473])
that
Z
1
,
exp[iλsT ]dµ2T =
cosh(λT /2)
WT2
and
E µ2
T
exp[iλsT ] w(T ) = y =
λT /2
exp
sinh(λT /2)
λT
λT
1−
coth
2
2
|y|2
.
2T
Thus our estimations (37) and (38) coincide with the order obtained from these precise expressions.
We can give explicit expressions for the Mellin transforms of the Lévy measures σT and σT,y
of the distributions of sT under µ2T and the conditional probability measure µ2T ( · |w(T ) = y),
respectively. Namely, noting that
σT (dx) = fB,0 (x)dx,
σT,y (dx) = fB (η) ,P (η) (B((y/√T )·η)) (x)dx,
and then plugging (31), (34), and (35) into (12), we obtain:
Proposition 18. The Mellin transforms of σT and σT,y are given by
s s
Z
T
2 −1
s
|x| σT (dx) =
Γ(s)ζ(s)
π
2s−1
R
and
Z
R
s
|x| σT,y (dx) = 2
T
2π
s
|y|2
Γ(s)ζ(s) +
T
respectively.
See [1, 15] for the related topics.
22
T
2π
s
Γ(s + 1)ζ(s),
s ≥ 2,
3.3
Sample variance
Let T > 0 and (WT1 , HT1 , µ1T ) be the two-dimensional classical Wiener space over [0, T ]. In this
subsection, we consider the sample variance
Z T
Z
1 T
2
1
(w(t) − w̄) dt, w ∈ WT ,
where w̄ =
vT (w) =
w(t)dt.
T 0
0
3.3.1
We first compute the Lévy measure of
ℓ ∈ HT1 and γ ∈ R .
v + h·, ℓi + γ under µ1T by applying Theorem 2, where
1
2 T
Define a symmetric Hilbert-Schmidt operator C : HT1 → HT1 by
d(Ch)
(t) =
dt
Z
T
t
(h(s) − h̄)ds
h ∈ HT1 , t ∈ [0, T ].
Since vT (w) = hT (w) − T w̄2 , due to the observation made at the beginning
of §3.1.1, we have
R
that vT ∈ C2 (WT1 ) ⊕ C0 (WT1 ). It is easily seen that ∇2 vT = 2C and W 1 vT dµ1T = T 2 /6. By
T
Remark 3 (iii), we have
T2
vT = Q C + .
(39)
6
It is a straightforward computation to see that
√
∞
X
2T
T 2 C
nπt
C
C
C=
−1 .
hn ⊗ hn , where hn (t) =
cos
nπ
nπ
T
n=1
Hence ℓC = 0. Define
e =
ℓ(t)
∞
X
A
hℓ, hC
n ikn (t),
n=1
t ∈ [0, T ],
(40)
(η)
1
where {knA }∞
defined in (23). Comparing the above
n=1 is the orthonormal basis of (HT )0
(η)
expansion of C with that of A in (23), and recalling the definition of fA,ℓ , we obtain
fC,ℓ = fA(η) ,ℓe = fA(η) ,P (η) ℓe.
In conjunction with (25) and (39), Theorem 2 and Proposition 14 lead us to:
Proposition 19. It holds that
Z
1
exp iλ
vT + h·, ℓi + γ dµ1T
1
2
WT
Z ∞
T2
e 0)dx ,
eiλx − 1 − iλx fH (x; T, ℓ,
+
= exp iλ γ +
12
0
where fH is the function defined in Proposition 14, and ℓe is given by (40). In particular, the
e
distribution of 12 vT +h·, ℓi+γ under µ1T coincides with that of 21 hT +h·, ℓi+γ
under µ1T (·|w(T ) = 0).
23
3.3.2
We next compute the Lévy measure of 21 vT + h·, ℓi + γ under the conditional probability
µ2T (·|w(T ) = y) given W (T ) = y, where ℓ ∈ HT1 , γ ∈ R, and y ∈ R1 .
√
Set η1 (w) = w(T )/ T , w ∈ WT1 , and η = {η1 }. Observe that
(y 2 − 1)T 2
,
12
hC(y · η), (y · η)i − hCη1 , η1 i =
yℓ(T )
h(y · η), ℓi = √ .
T
By straightforward computations, we obtain
d(C (η) h)
(t) =
dt
Z
T
t
1
(h(s) − h̄)ds −
T
and
C (η) =
Z
T
0
Z
T
s
(η)
h ∈ (HT1 )0 ,
(h(u) − h̄)du ds,
∞
o
X
T 2 n C
kn ⊗ knC + k̃nC ⊗ k̃nC ,
2nπ
n=1
where
knC (t)
In particular,
√
2nπt
2T
=
,
sin
2nπ
T
k̃nC (t)
{P (η) (C(y · η) + ℓ)}C (η) = 0 and
√
2T
=
2nπ
cos
2nπt
T
−1 .
fC (η) ,P (η) (C(y·η)+ℓ) (x) = 0,
x ≤ 0.
Moreover it holds that
hP
(η)
(C(y · η) +
ℓ), knC i
√
= − 2y
T
2nπ
2
+ hℓ, knC i, hP (η) (C(y · η) + ℓ), k̃nC i = hℓ, k̃nC i.
Hence, for x > 0,
where
2
∞
2nπ 4
1X 2
2 2nπ
3/2
fC (η) ,P (η) (C(y·η)+ℓ) (x) =
+ 2y
−2 y
hℓ, knC i
2
x
T
T
n=1
2nπ 6
2
2
C 2
C 2
+
hℓ, kn i + hℓ, k̃n i
e−(2nπ) x/T
T
o 2π 2 y 2 4π 2 x
n
2
4π x
1
Θ′
−
1
−
+ gV (x; T, ℓ, y),
=
Θ
2x
T2
T2
T2
gV (x; T, ℓ, y) =
∞
1X
2nπ 4
−23/2 y
hℓ, knC i
2 n=1
T
2nπ 6
2
2
+
hℓ, knC i2 + hℓ, k̃nC i2 e−(2nπ) x/T . (41)
T
Recalling (39), and applying Theorem 4 and Proposition 14, we can conclude:
24
Proposition 20. It holds that
1
E µ1 exp iλ
vT + h·, ℓi + γ w(T ) = y
T
2
2
Z ∞
Ty
T 2 yℓ(T )
= exp iλ
eiλx − 1 − iλx fV (x; T, ℓ, y)dx ,
+
+
+γ +
24
24
T
0
where
o 2π 2 y 2 4π 2 x
√
1 n 4π 2 x
−
1
−
+ gV (x; T, ℓ, y/ T ),
Θ
Θ′
2
3
2
2x
T
T
T
and gV is given by (41). Moreover, the distribution of vT /2 under the conditional probability µ1T (·|w(T ) = y) coincides with the one of {hT /2 + h′T /2 }/2 under the product measure
√
µ1T /2 (·|w(T /2) = 0) ⊗ µ1T /2 (·|w(T /2) = y/ 2), where h′T /2 denotes an independent copy of hT /2 .
fV (x; T, ℓ, y) =
3.3.3
We finally consider the exponential decay of the characteristic function of 21 vT + h·, ℓi + γ under
µ1T and µ1T (·|w(T ) = y).
As was seen in §3.3.1, the corresponding Hilbert-Schmidt operator C possesses eigenvalues
{(T /[nπ])2 ; n ∈ N }, each of them being of multiplicity 1, and ℓC = 0. By Theorem 7, Lemma 10,
and (39), there exist C1 > 0 and λ1 > 0 such that
Z
1
exp iλ
(42)
dµ ≤ exp[−C1 λ1/2 ]
vT + h·, ℓi + γ
W1
2
T
for every λ ≥ λ1 , γ ∈ R .
√
Let η1 (t) = t/ T and η = {η1 }. As was shown in §3.3.2, the Hilbert-Schmidt operator C (η)
(η)
has eigenvalues {(T /(2nπ))2 ; n ∈ N
√}, each of them being of multiplicity 2, and {P (C(y · η) +
ℓ)}C (η) = 0. Since η(w) = w(T )/ T , by Theorem 7, Lemma 10, and (39), there exist C2 > 0
and λ2 > 0 such that
E 1 exp iλ 1 vT + h·, ℓi + γ w(T ) = y ≤ exp[−C2 λ1/2 ]
(43)
µT
2
for every λ ≥ λ2 , γ ∈ R.
When ℓ = 0 and γ = 0, combining the results in Propositions 19 and 20 with (29), we can show
the following explicit expressions of the Laplace transforms for the distributions of vT ; for λ > 0,
!1/2
√
Z
1
λT
√
exp − λvT dµ1T =
2
sinh( λT )
WT1
and
E µ1
T
1
exp − λvT w(T ) = y
2
"
√
√
λT /2
λT
√
=
exp
1−
coth
2
sinh( λT /2)
√
λT
2
!!
#
y2
.
2T
From these expressions, we see, in the same way as in §3.1.3, that our estimates (42) and (43)
coincide with the order obtained from the explicit expressions.
25
3.4
Density functions
Let T > 0 and consider the classical two-dimensional Wiener space (WT2 , HT2 , µ2T ) over [0, T ]. In
this subsection, as an application of Theorem 11, we show a way to obtain the explicit expressions
of the densities of the distributions of Lévy’s stochastic area and the square of the L2 -norm on
an interval of the two-dimensional Wiener process. We also compute the Mellin transforms of
the distributions. For the related topics, see [1, 15].
3.4.1
Lévy’s stochastic area
By (30), (31), and Theorem 11(i), we see that the distribution of sT under µ2T admits a smooth
density function pL with respect to the Lebesgue measure on R , and that the corresponding
Hilbert-Schmidt operator B satisfies
b =
det2 (I − iζ B)
∞
Y
1+
n=0
ζ 2T 2
(2n + 1)2 π 2
Define a simple curve Γn = {γn (t) : t ∈ [0, 4Rn ]} in C
−Rn + it,
γn (t) = t − 2Rn + iRn ,
Rn + i{4Rn − t},
= cos(iζT /2).
with Rn = 4nπ/T by
t ∈ [0, Rn ],
t ∈ [Rn , 3Rn ],
t ∈ [3Rn , 4Rn ].
Let x < 0. By a straightforward computation, we can show that Γn satisfies the conditions in
Theorem 11(ii) and conclude that
pL (x) = i
=
∞
X
Res
n=0
∞
X
2
T
e−iζx
(2n + 1)π
;i
cos(iζT /2)
T
(−1)n e(2n+1)πx/T =
n=0
1
.
T cosh(πx/T )
For x > 0, the complex conjugate Γn plays the same role as Γn , and we obtain
pL (x) = −i
∞
X
n=0
Res
(2n + 1)π
e−iζx
; −i
cos(iζT /2)
T
=
1
.
T cosh(πx/T )
√
√
0
t/ T
√ . Then η(w) = w(T )/ T . By
Let η = {η1 , η2 }, where η1 (t) =
and η2 (t) =
0
t/ T
(34) and Remark 12(ii), the distribution of sT under µ2T (·|w(T ) = 0) admits a smooth density
function p̃L with respect to the Lebesgue measure on R , and it holds that
d
(η) ) =
det2 (I − iζ B
∞
Y
sin(iζT /2)
ζ 2T 2
.
1+ 2 2 =
4n π
iζT /2
n=1
26
Using the same Γn ’s as above, this time with Rn = (4n + 1)π/T , and then applying Theorem 11(ii), we can show that
p̃L (x) = i
=
∞
X
Res
n=1
∞
X
2π
T
e−iζx (iζT /2) 2nπ
;i
sin(iζT /2)
T
(−1)n+1 ne−2nπ|x|/T =
n=1
π
,
2T cosh2 (πx/T )
x < 0.
For x > 0, the complex conjugate Γn plays the same role as Γn , and we obtain
p̃L (x) = −i
∞
X
n=1
Res
e−iζx (iζT /2)
2nπ
; −i
sin(iζT /2)
T
=
π
.
2T cosh2 (πx/T )
Thus we have:
Proposition 21. It holds that
µ2T (sT ∈ dx) =
1
dx,
T cosh(πx/T )
µ2T (sT ∈ dx|w(T ) = 0) =
π
dx.
2T cosh2 (πx/T )
Moreover, their Mellin transforms are given by
Z
1
4T s
|sT |s dµ2T = s+1 Γ(s + 1)Lχ4 (s + 1), s > −
2
π
2
WT
s s−1
T 2
1
−1
E µ2 [|sT |s |w(T ) = 0] =
Γ(s + 1)ζ(s), s > ,
2(s−1)
T
π
2
2
P
n
−s
where Lχ4 is Dirichlet’s L-function given by Lχ4 (s) = ∞
n=0 (−1) (2n + 1) .
Proof. We have already seen the first half. The last half is immediate consequence of the
representations of pL and p̃L in the form of infinite sums.
The densities of sT under µ2T and µ2T (·|w(T ) = 0) are first computed by Lévy ([12]).
3.4.2
L2 -norm on an interval of the two-dimensional Wiener process
Put
(2)
hT (w)
(2)
=
Z
T
0
|w(t)|2 dt,
w ∈ WT2 .
Since hT is a sum of two independent copies of hT coming from w1 and w2 , due to the observations made in §3.1.1, we see that the Hilbert-Schmidt operator D : HT2 → HT2 associated with
(2)
hT /2 has eigenvalues {(T /[(n + 21 )π])2 : n = 0, 1, . . . }, each being of multiplicity 2, and hence
(
2 )
∞
Y
p
T
b =
1 − iζ
det(I − iζ D)
= cos( iζ T ),
1
n+ 2 π
n=0
27
(2)
(2)
and that hT /2 = qD /2. By Theorem 11(iii), the distribution of hT /2 under µ2T admits a
smooth density function pH with respect to the Lebesgue measure on R . Consider a simple curve
Γn = {−i(Rn + it)2 : t ∈ [−Rn , Rn ]} in C with Rn = 2nπ/T . Let x > 0. By a straightforward