E l e c t ro n ic
Jo u r n
a l o
f P
r o b
a b i l i t y
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 L´EVY 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 L´evy measures in terms of the eigenvalues of the corre-sponding 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, L´evy measure, Mellin transform, exponential decay
AMS subject classification 60J65, 60E07
Introduction
LetW be a classical Wiener space, andµbe the Wiener measure on it. A Wiener functionalF
of second order is a measurable functional F :W → R with ∇
3F = 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 theL2-norm on an interval of the Wiener process, L´evy’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 L´evy [2, 3, 12] for the square of the L2-norm on an interval of the Wiener process and L´evy’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 Schr¨odinger 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 L´evy, a fundamental strategy to investigate Wiener functionals of second order is computing their Laplace transforms or characteristic functions, and then their L´evy measures. In this paper, we give explicit expressions of L´evy 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 whereµis 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 Itˆo-Nisio [9].
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
L´
evy measures of Wiener functionals of second order
1.1 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.
Given a symmetric Hilbert-Schmidt operator A : H → H and ℓ ∈ H, decomposing A as
A =P∞n=1anhn⊗hn with an ∈ R and an orthonormal basis {hn}∞
n=1 of H such that kAk22 =
P∞
n=1a2n<∞, we define QA=
∞
X
n=1
an
h·, hni2−1 , (1)
fA,ℓ(x) =
1 2
X
n;an>0
1
x +
hℓ, hni2 a3
n
exp[−x/an], x >0,
0, x= 0,
1 2
X
n;an<0
1 −x +
hℓ, hni2
−a3
n
exp[−x/an], x <0,
(2)
whereh·, hnistands for the Itˆo integral ofhn, and if there exists no an with required property, the summation is defined to be equal to zero. It is possible to define fA,ℓ without using the
eigenfunction decomposition ofA. Indeed, if B :H →H is a symmetric non-negative definite Hilbert-Schmidt operator, then, for anyN ∈N, a bounded linear operator (B+εI)
−Nexp−{B+ εI}−1onHconverges strongly to a linear operatorT(N)
B of trace class. DecomposingA:H→H
asA=A1−A2 with symmetric non-negative definite Hilbert-Schmidt operatorsA1 and A2 on
H such that A1A2 =A2A1= 0, we obtain
fA,ℓ(x) =
1
xTrT
(0) (1/x)A1 +
1
x3hℓ, T (3)
(1/x)A1ℓi, x >0,
0, x= 0,
1 |x|TrT
(0)
(1/|x|)A2+ 1 |x|3hℓ, T
(3)
(1/|x|)A2ℓi, x <0. Lemma 1. It holds that
0≤fA,ℓ(x)≤ k Akk−2
∞ 2|x|k+1
k!kAk22+ (k+ 1)!kℓk2H (3)
for anyx6= 0, k≥2, and
Z 1
−1|
x|2fA,ℓ(x)dx≤
1 2kAk
2
2+kℓk2H, (4)
Proof. Since exp[−|x|/|an|]≤k!(|an|/|x|)k, (3) follows. (4) is an easy application of the mono-tone convergence theorem and an estimation
Z 1
0 |
x|mexp[−|x|/|an|]dx=|an|m+1
Z 1/|an|
0
ymexp[−y]dy ≤m!|an|m+1,
for everym∈N.
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
W
exp
iλ
1
2QA+h·, ℓi+γ
dµ
= exp
−λ 2kℓ
Ak2H
2 +iλγ+
Z
R
eiλx−1−iλxfA,ℓ(x)dx
, (5)
where i=√−1 and
ℓA= X
n;an=0
hhn, ℓihn. (6)
Remark 3. (i) The integrability of (eiλx−1−iλx)f
A,ℓ(x) in (5) is guaranteed by Lemma 1.
(ii) The theorem asserts that the distribution of 12QA+h·, ℓi+γ is infinitely divisible and the
corresponding L´evy measure is fA,ℓ(x)dx. Moreover, the distribution of 12QA+γ is
selfdecom-posable. See [16,§8 and §15].
(iii) LetCn(W) be the space of Wiener chaos of ordern. A Wiener functionalF 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
thatF = (1/2)QA+h·, ℓi+γ for some symmetric Hilbert-Schmidt operatorA,ℓ∈H, andγ∈R,
andQA∈C2(W),h·, ℓi ∈C1(W), andγ ∈C0(W). Moreover,A,ℓ, andγ are determined so that A=∇2F,ℓ=R
W∇F dµ, and γ=
R
W F dµ, where∇stands for the Malliavin derivative.
(iv) It should be noted thatC2(W)⊕C1(W)⊕C0(W) is invariant under shifts in the direction
of H. Namely, letF = 12QA+h·, ℓi+γ and h∈H. Then F(·+h) =F+h·, Ahi+1
2hh, Ahi+hh, ℓi ∈C2(W)⊕C1(W)⊕C0(W).
In particular, the theorem is applicable to a quadratic form of the form 12QA(· −h), which is
one of main ingredients in the study of the principle of stationary phase onW. See [17]
Proof of Theorem 2. The proof is divided into three steps according to the signs of an’s, the
eigenvalues ofA.
1st step: the case where an >0 for all n∈N.
Letλ >0. Note that 1
2QA+h·, ℓi= ∞
X
n=1
han
2
h·, hni2−1 − hℓ, hnih·, hni
i
Since {h·, hni : n ∈ N} is a family of independent Gaussian random variables of mean 0 and
variance 1, we obtain the following well known identity:
Z
Applying the identities
log(1 + (λ/a)) =
Plugging this into (7), we obtain
Since f−A,−ℓ(−x) =fA,ℓ(x), this yields (5).
3rd step: the general case. Set
A+=
X
n;an>0
anhn⊗hn, A−= X
n;an<0
anhn⊗hn,
ℓ+=
X
n;an>0
hℓ, hnihn, ℓ−=
X
n;an<0
hℓ, hnihn.
Then
1
2QA+h·, ℓi =h·, ℓAi+ 1
2QA++h·, ℓ+i+ 1
2QA−+h·, ℓ−i.
Moreover, the random variablesh·, ℓAi, 12QA++h·, ℓ+i, and 1
2QA−+h·, ℓ−iare independent under
µ, and
fA,ℓ=fA+,ℓ+ +fA−,ℓ−.
From the observations made in the 1st and 2nd steps, we obtain
Z
W
exp
iλ
1
2QA+h·, ℓi+γ
dµ
=
Z
W
exp [iλh·, ℓAi]dµ×
Z
W
exp
iλ
1
2QA++h·, ℓ+i+γ
dµ
×
Z
W
exp
iλ
1
2QA−+h·, ℓ−i
dµ
= exp
−λ 2kℓ
Ak2H
2
exp
iλγ+
Z
R
eiλx−1−iλxfA+,ℓ+(x)dx
×exp
Z
R
eiλx−1−iλxfA−,ℓ−(x)dx
,
from which (5) follows.
1.2 Conditional expectation
Let η ={η1, . . . , ηm} ⊂W∗,W∗ being the dual space of W, be an orthonormal system in H ;
hηi, ηji=δij. Setting
W0(η)={w∈W;η(w) = 0}, H0(η)=H∩W0(η),
where η(w) = (η1(w), . . . , ηm(w)) ∈R
m, we define a projection P(η) :W → W(η)
0 by P(η)w=
w−Pmn=1ηn(w)ηn, and denote byµ(0η) the induced measure ofµ on W (η)
0 via P(η). Then the triplet (W0(η), H0(η), µ(0η)) is an abstract Wiener space.
For a symmetric Hilbert-Schmidt operatorA:H →H, we define a symmetric Hilbert-Schmidt operator A(η) on H0(η) by A(η) =P(η)A. We denote by Eµ[F|η(w) =y] the conditional
expec-tation of a Wiener functional F : W → R given η(w) = y. For y = (y
1, . . . , ym) ∈ R
m, put y·η =Pmn=1ynη
Theorem 4. Let A : H → H be a symmetric Hilbert-Schmidt operator, ℓ ∈ H, and γ ∈ R.
Hence we have
Eµ[F|η(w) =y] =
Z
W0(η)
F(w0+y·η)µ0(η)(dw0). (10) Using a finite dimensional approximation argument, we can easily show that
1
1.3 Mellin transform
Proposition 5. The Mellin transform offA,ℓ defined by (2) is given by
Then, by a change of variablesx→ −xon (−∞,0), we obtain
which completes the proof.
Remark 6. A sufficient condition for the Mellin transform of fA,ℓ to have a meromorphic
extension toC is given by Jorgenson and Lang ([10]).
2
Exponential decay and density functions
In this section, as an application of Theorems 2 and 4, we first study how fast a stochastic oscillatory integral decays when its phase function is a Wiener chaos of order at most two. Moreover we also show how to compute the density function of its distribution.
2.1 Exponential decay
Hilbert-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
W
exp
iλ
1
2QA+h·, ℓi+γ
dµ
≤exp[−Caλa], (15)
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
b− := sup
(
b >0 : lim sup
λ→∞
λ−b
Z
(−∞,0)
(cos(λx)−1)fA(η),P(η)(A(y·η)+ℓ)(x)dx <0
)
,
b+ := sup
(
b >0 : lim sup
λ→∞
λ−b
Z
(0,∞)
(cos(λx)−1)fA(η),P(η)(A(y·η)+ℓ)(x)dx <0
)
,
where b− = 0, b+ = 0if {· · · }=∅. If max{b−, b+}>0, then, for every b <max{b−, b+}, there
exist Cb >0 andλb >0 such that
Eµ
exp
iλ
1
2QA+h·, ℓi+γ η(w) =y
≤exp[−Cbλb], (16)
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 −λ2kℓAk2
H/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 x2f
A,ℓ(x) and x2fA(η),P(η)(A(y·η)+ℓ)(x) is due to Lemma 1.
(iii) The lower estimates in (13) and (14) are sharp as we shall see in Lemma 10. For example, if we consider A = P∞n=1n−ph
n⊗hn for some p > 1/2 and an orthonormal basis {hn} of H,
then there exist C >0 and λ0 >0 such that
Z
W
exp
iλ
1
2QA+h·, ℓi+γ
dµ
≤exp[−Cλ1/p] for any λ > λ0. For details, see Lemma 10 and its proof.
(iv) If an>0 for somen, then limλ→∞
R∞
0 (cos(λx)−1)fA,ℓ(x)dx=−∞. In fact, it holds that
fA,ℓ≥fA,0 and
Z ∞
0
(1−cos(λx))fA,0(x)dx=
Z ∞
0
1−cosy y
X
n;an>0
exp[−y/(λan)]dy.
Then, applying the monotone convergence theorem, we obtain the desired divergence. Similarly, if an < 0 for some n, then limλ→∞
R0
−∞(cos(λx)−1)fA,ℓ(x)dx = −∞. Thus the assumption made ona± 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
existence ofCδ>0 such thatfA,ℓ(x)≤Cδ|x|−1−δ for any x∈R \ {0}. Hence, for everyλ >0,
0≤max
Z 0
−∞
(1−cos(λx))fA,ℓ(x)dx,
Z ∞
0
(1−cos(λx))fA,ℓ(x)dx
≤Cδλδ
Z ∞
0
1−cosy y1+δ dy,
from which it follows thata±= 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
W
exp
iλ
1
2QA+h·, ℓi+γ
dµ
= exp
Z
R
(cos(λx)−1)fA,ℓ(x)dx
Eµ
exp
iλ
1
2QA+h·, ℓi+γ η(w) =y
= exp
Z
R
(cos(λx)−1)fA(η),P(η)(A(y·η)+ℓ)(x)dx
Since |cosx−1| ≤x2/2 for any x∈
R, the assertion (i) follows immediately.
Letf =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 ordera >1 (ϕ∈G a(
R) in notation)
if, for any compact subsetK ⊂R, there exists a constantCK >0 such that
dnϕ dxn(x)
≤CK(CK(n+ 1)a)n for every x∈K, n∈N.
A finite Radon measureuonR admits a density function of classG a(
R) if there is aC >0 such
that
|uˆ(ξ)| ≤C
C(n+ 1)a |ξ|
n
for any ξ ∈R \ {0}, n∈N,
where ˆuis the Fourier transformation ofu(cf. [6, Prop.8.4.2]). Sincee−x≤ααx−α forα, x >0, a sufficient condition for this to hold is that there existC1, C2>0 such that
|uˆ(ξ)| ≤C1exp[−C2|ξ|1/a] for any ξ∈R.
Corollary 9. Let a±, b± be as in Theorem 7.
(i)If ℓA= 0 andmax{a−, a+}>0, then the distribution on R of
1
2QA+h·, ℓi+γ underµadmits
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
1 2QA+ 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 anyx∈(0, δ),
then a+≥2−ε. If there exist δ, C >0 andε <2 such that
fA,ℓ(x)≥C|x|ε−3for 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/k
p for any k∈
N. Then, for anyδ >0, fA,ℓ(x)≥
(
C1/p p
Z ∞
δ/C
z(1/p)−1e−zdz
)
x−1−(1/p)
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/k
p for any k∈ N.
Then, for any δ >0,
fA,ℓ(x)≥
(
C1/p p
Z ∞
δ/C
z(1/p)−1e−zdz
)
x−1−(1/p)
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 δ
0
(cos(λx)−1)fA,ℓ(x)dx≤C
Z δ
0
(cos(λx)−1)xε−3dx
=Cλ2−ε
Z λδ
0
Hence
lim sup
λ→∞
λ−(2−ε)
Z ∞
0
(cos(λx)−1)fA,ℓ(x)dx≤C
Z ∞
0
(cosx−1)x3−εdx <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)≥
∞
X
k=1
e−xkp/C
x ≥
Z ∞
1
e−xyp/C
x dy =x
−1−(1/p)C1/p
p
Z ∞
x/C
z(1/p)−1e−zdz
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 12QA+h·, ℓi+γunderµorµ(·|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 Wiener space, A:H →H be a symmetric Hilbert-Schmidt operator, and decompose as A =P∞n=1anhn⊗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 C1 curves Γ
n={γn(t) :t∈[αn, βn]} in C such that
(a1) γn(αn) ∈ (−∞,0), γn(βn) ∈ (0,∞), (a2) inf{|γn(t)|:t ∈ [αn, βn]} → ∞ as n → ∞, and
(a3) RΓ
n{e
−iζx/det
2(I−iζAb)}dζ →0 as n→ ∞, where Ab=P∞n=1a2nh2n⊗h2n.
(ii-1) If Imγn(t) >0 and −i/am ∈/ Γn for any n ∈ N, t ∈ (αn, βn), and m ∈ N with am < 0,
then
pA(x) =i
X
n:an<0
Res e−
iζx
det2(I −iζAb) ;− i
an
!
, (17)
where Res(f(ζ);z) denotes the residue of f atz.
(ii-2) If Imγn(t) <0 and −i/am ∈/ Γn for any n ∈ N, t ∈ (αn, βn), and m ∈ N with am > 0,
then
pA(x) =−i X n:an>0
Res e−
iζx
det2(I−iζAb);−
i an
!
. (18)
(iii) Suppose P∞n=1|an|<∞, and set qA= QA+P∞n=1an. Then all assertions in (i) and (ii)
Remark 12. (i) The mapping C ∋ζ 7→det2(I −iζAb) is holomorphic ([4]).
(ii) Let η={η1, . . . , ηm} be an orthonormal system ofH. By (11), it holds
µ1
2QA∈dx
η(w) = 0=µ(0η)1
2
n
QA(η) −
m
X
n=1
hAηi, ηii
o
∈dx,
µ1
2qA∈dx
η(w) = 0=µ(0η)1
2qA(η) ∈dx
.
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 theL2-norm on an interval of the one-dimensional Wiener process.
Proof. Since k∇(QA/2)k2H =
P∞
n=1a2nh·, hni2,k∇(QA/2)k−H1 ∈
T
p>0Lp(µ) if #{n:an 6= 0}=
∞(cf.[18]). Thus the assertion (i) follows as an fundamental application of the Malliavin calcu-lus.
By (7) and the assumption thata2n−1=a2n, we have
Z
R
eiλxpA(x)dx=
1 det2(I−iλAb)
, (19)
and hence
pA(x) =
1 2π
Z
R
e−iλx
det2(I−iλAb)dλ.
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 ˜pA the
density function ofqA/2, then
Z
R
eiλxpA˜ (x)dx= 1 det(I −iλAb).
3
Typical quadratic Wiener functionals
3.1 The square of the L2-norm on an interval
LetT >0 and consider the classical one-dimensional Wiener space (W1
T, HT1, µ1T) over [0, T];WT1
is the space of continuous functionsw: [0, T]→R withw(0) = 0, H
1
T consists ofh∈WT1 which
is absolutely continuous and has a square integrable derivative dh/dt, and µ1T is the Wiener measure. The inner product inH1
T is given by
hh, ki=
Z T
0
dh dt(t)
dk
dt(t)dt, h, k∈H
1
T.
In this subsection we consider
hT(w) =
Z T
0
w(t)2dt, w∈WT1.
3.1.1
We first compute the L´evy measure of 12hT +h·, ℓi+γ underµ
1
T by applying Theorem 2, where ℓ∈HT1 and γ∈R.
Define a symmetric Hilbert-Schmidt operatorA:HT1 →HT1 by
d(Ah)
dt (t) =
Z T
t
h(s)ds, h∈HT1, t∈[0, T].
Note thatw(t)2 is in C2(W
1
T)⊕C0(W
1
T), and so ishT. It is easily seen that∇
2
hT = 2Aand that
R
W1
T
hTdµ
1
T =T2/2. Then, by virtue of Remark 3 (iii), we observe that
hT =QA+ T2
2 . (20)
It is easy to see that
A= ∞
X
n=0
T n+12π
!2
hAn ⊗hAn, (21)
where
hAn(t) = √
2T n+12πsin
n+12πt T
.
In particular, we have
ℓA= 0 and fA,ℓ(x) = 0, x≤0.
Set
ℓ(n)=hℓ, hAni=
r
2
T
Z T
0
dℓ dt(t) cos
n+12πt T
!
dt,
gH(x;T, ℓ) = π6
27T6 ∞
X
n=0
(2n+ 1)6|ℓ(n)|2exp
−(2n+ 1) 2π2x 4T2
Since Jacobi’s theta function Θ(u) =Pn∈
Z
exp[−n2u] enjoys the relation Θ(u)−Θ(4u) = 2
∞
X
n=0
e−(2n+1)2u,
it is straightforward to see that
fA,ℓ(x) =
1 4x
Θ
π2x 4T2
−Θ
π2x
T2
+gH(x;T, ℓ), x >0.
By virtue of this and (20), applying Theorem 2, we arrive at: Proposition 13. It holds that
Z
W1
T
exp
iλ
1
2hT +h·, ℓi+γ
dµ1T
= exp
"
iλ
γ+T 2 4
+
Z ∞
0
eiλx−1−iλx 1
4x
Θ
π2x
4T2
−Θ
π2x T2
+gH(x;T, ℓ)
dx
#
,
where gH is defined by (22).
3.1.2
We next compute the L´evy measure of 12hT + h·, ℓi +γ under the conditional probability µ1
T(·|w(T) =y) given w(T) =y, whereℓ∈HT1,γ ∈R, andy∈R.
Setη1(w) =w(T)/ √
T,w∈WT1, and η={η1}. Note that hA(y·η),(y·η)i − hAη1, η1i= (y2−1)hAη1, η1i=
(y2−1)T2
3 , h(y·η), ℓi=
yℓ(T) √
T .
By a straightforward computation, we obtain
d(A(η)h)
dt (t) =
Z T
t
h(s)ds− 1
T
Z T
0
Z T
s
h(u)du
ds, h∈(HT1)(0η),
and hence
A(η) = ∞
X
n=1
T
nπ
2
knA⊗kAn, where kAn(t) = √
2T nπ sin
nπt
T
. (23)
In particular,
{P(η)(A(y·η) +ℓ)}A(η) = 0 and fA(η),P(η)(A(y·η)+ℓ)(x) = 0, x≤0. Note that
hP(η)(A(y·η) +ℓ), knAi=yhAη, kAni+hℓ, kAni = (−1)n+1√2y T
nπ
2
where
Due to Theorem 4, we obtain
E Proposition 14. It holds that
E
We finally study the exponential decay of the characteristic function of 12hT +h·, ℓi+γ under µ1
As was seen in §3.1.1, the Hilbert-Schmidt operator A associated with hT has eigenvalues
{T2/[(n+ 12)π]2;n ∈ N ∪ {0}}, each of them being of multiplicity one. By Theorem 7 and
Lemma 10, there existC1 >0 andλ1 >0 such that
Z
W1
T
exp
iλ
1
2hT +h·, ℓi+γ
dµ
≤exp[−C1λ
1/2] (27)
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)
possesses eigenvalues {(T /(nπ))2;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 andλ2 >0 such that
E
µ1
T
exp
iλ
1
2hT +h·, ℓi+γ
w(T) =y
≤exp[−C2λ1/2] (28) 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
W1
T
exp
−λ 2hT
dµ1T = 1 (cosh(√λT))1/2 and
E µ1
T
exp
−λ 2hT
w(T) =y
=
√
λT
sinh √λT
!1/2
exp1−√λTcoth √λT y
2 2T
(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
cosh(x) = ∞
Y
k=0
1 + 4x 2 (2k+ 1)2π2
, sinh(x) =x
∞
Y
k=1
1 + x 2
k2π2
,
coth(πx) = 1
πx+
2x π
∞
X
k=1 1
x2+k2,
we can also show explicit expressions for the L´evy measures νT(dx) and νT,y(dx) of the distri-bution ofhT/2 underµ
1
T and the conditional probability measureµ1T(· |w(T) =y) as described
in Propositions 13 and 14 withℓ= 0.
Moreover, by using the Riemann zeta function ζ(s) = P∞n=1n−s, we can give explicit forms of the Mellin transforms ofνT and νT,y. Namely, noting that
Proposition 15. The Mellin transform of νT(dx) and νT,y(dx) are given by
Z ∞
0
xsνT(dx) =
4T2 π2
s
22s−1
22s+1 Γ(s)ζ(2s)
and
Z ∞
0
xsνT,y(dx) =
1 2
T2
π2
s
Γ(s)ζ(2s) +y 2
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 L´evy’s stochastic area
LetT >0 and consider the classical two-dimensional Wiener space (WT2, HT2, µ2T) over [0, T];WT2
is the space of continuous functionsw: [0, T]→R
2 withw(0) = 0,H2
T consists ofh∈WT2 which
is absolutely continuous and has a square integrable derivative dh/dt, and µ2T is the Wiener measure. The inner product inHT2 is given by
hh, ki=
Z T
0
dh dt(t),
dk dt(t)
R
2
dt, h, k∈HT2.
Define L´evy’s stochastic area by
sT(w) =
1 2
Z T
0 h
Jw(t), dw(t)i2
R
2 = 1 2
Z T
0
w1(t)dw2(t)−w2(t)dw1(t) ,
whereJ =
0 −1 1 0
and dw(t) denotes the Itˆo integral.
3.2.1
We first compute the L´evy measure of sT +h·, ℓi+γ under µ
2
T by applying Theorem 2, where ℓ∈HT2 and γ∈R.
Define a symmetric Hilbert-Schmidt operatorB :HT2 →HT2 by
d(Bh)
dt (t) =J
h(t)−1 2h(T)
, h∈HT2, t∈[0, T].
Sincewi(s){wj(t)−wj(s)} is in C2(W
2
T) for (i, j)∈ {(1,2),(2,1)} ands < t, so issT. It is easily
seen that∇2s
T =B, and hence, due to Remark 3 (iii), we have
sT =
1
By a direct computation, we see that
B =X
n∈Z T
(2n+ 1)π
n
hBn ⊗hBn + ˜hBn ⊗˜hBno, (31)
where
hBn(t) = √
T
(2n+ 1)π
cos((2n+ 1)πt/T)−1,
sin((2n+ 1)πt/T)
, ˜hBn(t) =JhBn(t).
In particular
ℓB= 0.
Set
ℓ1(n)=hℓ, hBni= √1
T
Z T
0
dℓ dt(t),
−sin((2n+ 1)πt/T) cos((2n+ 1)πt/T)
R
2
dt,
ℓ2(n)=hℓ,˜hBni=−√1
T
Z T
0
dℓ dt(t),
cos((2n+ 1)πt/T) sin((2n+ 1)πt/T)
R
2
dt,
ℓ(n)= ℓ 1 (n)
ℓ2(n)
!
,
and
gL(x;T, ℓ) =
π3
2T3 ∞
X
n=0
(2n+ 1)3|ℓ(n)|2exp [−(2n+ 1)πx/T], x >0
0, x= 0,
π3
2T3 ∞
X
n=1
(2n−1)3|ℓ(−n)|2exp [−(2n−1)πx/T], x <0.
(32)
Then it is easily seen that
fB,ℓ(x) =
1
2xsinh(πx/T) +gL(x;T, ℓ), x∈R.
By virtue of this and (30), applying Theorem 2, we arrive at; Proposition 16. It holds that
Z
W2
T
exp [iλ(sT +h·, ℓi+γ)] dµ
2
T
= exp
"
iλγ+
Z
R
eiλx−1−iλx 1
2xsinh(πx/T)+gL(x;T, ℓ)
dx
#
, (33)
3.2.2
We next compute the L´evy measure of sT + h·, ℓi + γ under the conditional probability µ2T(·|w(T) =y) given W(T) =y, whereℓ∈HT2,γ ∈R, andy∈R
2. Let η ={η1, η2} ⊂ W∗, where ηi(w) =wi(T)/
√
T. Since y·η = ty/√T and hJz, ziR
2 = 0 for any z∈R
2,
hB(y·η),(y·η)i − 2
X
n=1
hBηn, ηni= 0, h(y·η), ℓi= h
y, ℓ(T)iR
2 √
T .
For anyh, g ∈(H2
T)
(η)
0 , the identity hB(η)h, gi =hBh, gi holds, and hence
d(B(η)h)
dt (t) =J(h(t)−¯h), where ¯h=
1
T
Z T
0
h(t)dt.
Then it is straightforward to see that
B(η) = X
n∈Z\{0} T
2nπ
knB⊗knB+ ˜knB⊗˜knB, (34)
where
kBn(t) = √
T
2nπ
cos(2nπt/T)−1 sin(2nπt/T)
, ˜knB(t) =JknB(t).
Hence
{P(η)(B(y·η) +ℓ)}B(η) = 0, and it holds that
hP(η)(B(y·η) +ℓ), kBni=h(y·η), BkBni+ ˜ℓ1(n)=−T y 2 2nπ + ˜ℓ
1 (n), hP(η)(B(y·η) +ℓ),k˜Bni=h(y·η), B˜knBi+ ˜ℓ2(n)= T y
1 2nπ + ˜ℓ
2 (n), where
˜
ℓ1(n)=hℓ, knBi=
Z T
0
dℓ dt(t),
1 √
T
−sin(2nπt/T) cos(2nπt/T)
R
2
dt,
˜
ℓ2(n)=hℓ,˜knBi=
Z T
0
dℓ dt(t),
1 √
T
−cos(2nπt/T) −sin(2nπt/T)
R
2
dt.
Setting ˜ℓ(n)= ˜
ℓ1(n)
˜
ℓ2(n)
!
, we obtain
hP(η)(B(y·η) +ℓ), knBi2+hP(η)(B(y·η) +ℓ),˜knBi2 = T
2nπ
2
|y|2+ T
nπhℓ˜(n), JyiR
Thus, if we put Applying Theorem 4, we get to
E Proposition 17. It holds that
3.2.3
We finally consider the exponential decay of the characteristic function ofsT +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 existC1 >0 and λ1 >0 such that
Z
W2
T
exp [iλ(sT +h·, ℓi+γ)]dµ
≤exp[−C1λ] (37)
for everyλ≥λ1, γ ∈R.
Letη1(t) =
t/√T
0
, η2(t) =
0
t/√T
, and η={η1, η2}. As was shown in§3.2.2, the Hilbert-Schmidt operator B(η) has eigenvalues {(T /(2nπ));n ∈
Z\ {0}}, each of them being of
multi-plicity 2. Sinceη(w) =w(T)/√T, by Theorem 7 and Lemma 10, there existC2>0 andλ2 >0
such that
E
µ2
T
exp [iλ(sT +h·, ℓi+γ)]
w(T) =y≤exp[−C2λ] (38) for everyλ≥λ2, γ∈R.
As in the previous subsection, whenℓ= 0 andγ = 0, it is well known ([12] and [8, pp.470–473])
that Z
W2
T
exp[iλsT]dµ
2
T =
1 cosh(λT /2), and
E µ2
T
exp[iλsT]
w(T) =y= λT /2
sinh(λT /2)exp
1−λT 2 coth
λT
2
|y|2 2T
.
Thus our estimations (37) and (38) coincide with the order obtained from these precise expres-sions.
We can give explicit expressions for the Mellin transforms of the L´evy measures σT and σT,y
of the distributions of sT under µ
2
T and the conditional probability measureµ2T( · |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
Z
R
|x|sσT(dx) =
T π
s
2s−1
2s−1 Γ(s)ζ(s)
and
Z
R
|x|sσT,y(dx) = 2
T
2π
s
Γ(s)ζ(s) +|y| 2
T
T
2π
s
Γ(s+ 1)ζ(s), s≥2,
respectively.
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
vT(w) =
Z T
0
(w(t)−w¯)2dt, w∈WT1, where w¯= 1
T
Z T
0
w(t)dt.
3.3.1
We first compute the L´evy measure of 12vT +h·, ℓi+γ underµ
1
T by applying Theorem 2, where ℓ∈HT1 and γ∈R.
Define a symmetric Hilbert-Schmidt operatorC :H1
T →HT1 by d(Ch)
dt (t) =
Z T
t
(h(s)−h¯)ds h∈HT1, t∈[0, T].
Since vT(w) =hT(w)−Tw¯
2, due to the observation made at the beginning of §3.1.1, we have that vT ∈ C2(W
1
T)⊕C0(W
1
T). It is easily seen that ∇2vT = 2C and
R
W1
T
vTdµ
1
T = T2/6. By
Remark 3 (iii), we have
vT =QC+ T2
6 . (39)
It is a straightforward computation to see that
C= ∞
X
n=1
T nπ
2
hCn ⊗hCn, wherehCn(t) = √
2T nπ
cosnπt
T
−1
.
HenceℓC = 0. Define
e
ℓ(t) = ∞
X
n=1
hℓ, hCniknA(t), t∈[0, T], (40)
where {kA
n}∞n=1 is the orthonormal basis of (HT1)
(η)
0 defined in (23). Comparing the above expansion of C with that ofA(η) in (23), and recalling the definition offA,ℓ, we obtain
fC,ℓ=fA(η),eℓ=fA(η),P(η)eℓ.
In conjunction with (25) and (39), Theorem 2 and Proposition 14 lead us to: Proposition 19. It holds that
Z
W1
T
exp
iλ
1
2vT +h·, ℓi+γ
dµ1T
= exp
iλ
γ+T 2 12
+
Z ∞
0
eiλx−1−iλxfH(x;T,ℓ,e0)dx
,
where fH is the function defined in Proposition 14, and ℓeis given by (40). In particular, the
distribution of 12vT+h·, ℓi+γ underµ
1
T coincides with that of 12hT+h·,ℓei+γ underµ
1
3.3.2
By straightforward computations, we obtain
d(C(η)h) Moreover it holds that
Proposition 20. It holds that
and gV is given by (41). Moreover, the distribution of vT/2 under the conditional
proba-bility µ1
T /2 denotes an independent copy ofh T /2.
3.3.3
We finally consider the exponential decay of the characteristic function of 12vT+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, andℓC = 0. By Theorem 7, Lemma 10,
Whenℓ= 0 andγ = 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 ofvT; forλ >0,
Z
3.4 Density functions
LetT >0 and consider the classical two-dimensional Wiener space (W2
T, 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 L´evy’s stochastic area and the square of theL2-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 L´evy’s stochastic area
By (30), (31), and Theorem 11(i), we see that the distribution ofsT underµ
2
T admits a smooth
density function pL with respect to the Lebesgue measure on R, and that the corresponding
Hilbert-Schmidt operatorB satisfies
det2(I−iζBb) = ∞
Y
n=0
1 + ζ 2T2 (2n+ 1)2π2
= cos(iζT /2).
Define a simple curve Γn={γn(t) :t∈[0,4Rn]} inC with Rn= 4nπ/T by
γn(t) =
−Rn+it, t∈[0, Rn], t−2Rn+iRn, t∈[Rn,3Rn], Rn+i{4Rn−t}, 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
n=0 Res
e−iζx
cos(iζT /2);i
(2n+ 1)π T
= 2
T
∞
X
n=0
(−1)ne(2n+1)πx/T = 1
Tcosh(πx/T). Forx >0, the complex conjugate Γn plays the same role as Γn, and we obtain
pL(x) =−i
∞
X
n=0 Res
e−iζx
cos(iζT /2);−i
(2n+ 1)π T
= 1
Tcosh(πx/T).
Let η ={η1, η2}, where η1(t) =
t/√T
0
and η2(t) =
0
t/√T
. Then η(w) = w(T)/√T. By (34) and Remark 12(ii), the distribution of sT under µ
2
T(·|w(T) = 0) admits a smooth density
function ˜pL with respect to the Lebesgue measure on R, and it holds that
det2(I−iζBd(η)) = ∞
Y
n=1
1 + ζ 2T2 4n2π2
= sin(iζT /2)
Using the same Γn’s as above, this time with Rn = (4n+ 1)π/T, and then applying
Theo-rem 11(ii), we can show that
˜
pL(x) =i
∞
X
n=1 Res
e−iζx(iζT /2)
sin(iζT /2) ;i 2nπ
T
= 2π
T
∞
X
n=1
(−1)n+1ne−2nπ|x|/T = π
2Tcosh2(πx/T), x <0. Forx >0, the complex conjugate Γn plays the same role as Γn, and we obtain
˜
pL(x) =−i
∞
X
n=1 Res
e−iζx(iζT /2)
sin(iζT /2) ;−i 2nπ
T
= π
2Tcosh2(πx/T). Thus we have:
Proposition 21. It holds that
µ2T(sT ∈dx) =
1
Tcosh(πx/T)dx, µ 2
T(sT ∈dx|w(T) = 0) =
π
2Tcosh2(πx/T)dx.
Moreover, their Mellin transforms are given by
Z
W2
T
|sT| sdµ2
T =
4Ts
πs+1Γ(s+ 1)Lχ4(s+ 1), s >− 1 2
E µ2
T[|
sT|
s|w(T) = 0] =
T π
s
2s−1−1
22(s−1) Γ(s+ 1)ζ(s), s > 1 2,
where Lχ4 is Dirichlet’sL-function given by Lχ4(s) =
P∞
n=0(−1)n(2n+ 1)−s.
Proof. We have already seen the first half. The last half is immediate consequence of the representations of pL and ˜pL in the form of infinite sums.
The densities of sT under µ
2
T and µ2T(·|w(T) = 0) are first computed by L´evy ([12]).
3.4.2 L2-norm on an interval of the two-dimensional Wiener process Put
h
(2)
T (w) =
Z T
0 |
w(t)|2dt, w∈WT2.
Since h
(2)
T is a sum of two independent copies of hT coming from w
1 and w2, due to the obser-vations made in§3.1.1, we see that the Hilbert-Schmidt operator D:H2
T →HT2 associated with h
(2)
T /2 has eigenvalues{(T /[(n+ 12)π])2 :n= 0,1, . . .}, each being of multiplicity 2, and hence det(I−iζDb) =
∞
Y
n=0
(
1−iζ
T n+12π
2)
and that h
(2)
T /2 = qD/2. By Theorem 11(iii), the distribution of h
(2)
T /2 under µ2T admits a
smooth density functionpH with respect to the Lebesgue measure onR. Consider a simple curve
Γn ={−i(Rn+it)2 :t∈[−Rn, Rn]} inC withRn= 2nπ/T. Let x >0. By a straightforward
computation, we can show that Γnsatisfies the conditions in Theorem 11(ii) and conclude that pH(x) =−i
∞
X
n=0
Res e−
iζx
cos(√iζ T);−i
n+1 2
π T
2!
= π
T2 ∞
X
n=0
(−1)n(2n+ 1)e−(2n+1)2π2x/(4T2)= 1
2T2ϑ′1(0|ixπ/T2), whereϑ1(u|τ) denotes the theta function of the first kind with parameterτ,
ϑ1(u|τ) = 2 ∞
X
n=0
(−1)neτ πi(n+1/2)2sin[(2n+ 1)πu],
and ϑ′1 stands for the first derivative inu. Obviously,pH(x) = 0 if x <0.
Let η={η1, η2}, where η1(t) =
t/√T
0
and η2(t) =
0
t/√T
. Due to the observations made in§3.1.2, D(η) :HT2 →HT2 has eigenvalues{(T /[nπ])2 :n= 1,2, . . .}, each being of multiplicity 2, and hence
det(I−iζDd(η)) = ∞
Y
n=1
(
1−iζ
T nπ
2) = sin(
√
iζ T) √
iζ T .
By Remark 12(iii), the distribution of h
(2)
T /2 under µ2T(·|w(T) = 0) admits a smooth density
function ˜pH with respect to the Lebesgue measure on R. Let Rn= 2n+
1 2
π/T, and consider the same curves {Γn;n ∈ N} as above and x > 0. By a straightforward computation, we can
show that Γn satisfies the conditions in Theorem 11(ii) and conclude that
˜
pH(x) =−i
∞
X
n=1 Res
√
iζ T e−iζx
sin(√iζ T);−i
nπ T
2!
= 2
π T
2X∞
n=1
(−1)n+1n2e−(nπ/T)2x= 1
4T2ϑ′′4(0|ixπ/T 2),
whereϑ4(u|τ) denotes the theta function of the fourth kind with parameterτ,
ϑ4(u|τ) = 1 + 2 ∞
X
n=1
(−1)neτ πin2cos(2nπu),
and ϑ′′4 stands for the second derivative in u. Obviously, ˜pH(x) = 0 if x <0.
Thus we have:
Proposition 22. It holds that
µ2T h
(2)
T /2∈dx
= 1
2T2ϑ′1(0|ixπ/T2)X(0,∞)(x)dx,
µ2T h
(2)
T /2∈dx
w(T) = 0 = 1
4T2ϑ′′4(0|ixπ/T 2)X(0
Moreover, their Mellin transforms are given by
Z
W2
T
h
(2)
T /2
s
dµ= 2
2s+2T2s
π2s+1 Γ(s+ 1)Lχ4(2s+ 1), s >− 1 4,
E µ2
T
h
h
(2)
T /2
s
w(T) = 0i= 2
T π
2s
Γ(s+ 1)(1−21−2s)ζ(2s), s > 1
4.
Proof. We have already seen the first half. The last half is immediate consequence of the representations of pH and ˜pH in the form of infinite sums.
Acknowledgements. The authors are grateful to Professor N. Ikeda who opened their eyes to the study of the L´evy measures of quadratic Wiener functionals. The paper has grown out of stimulative discussions with him.
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