WIENER FUNCTIONALS OF SECOND ORDER AND THEIR LÉVY MEASURES

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.


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 L 2 -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 L 2 -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 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.

Lévy measures of Wiener functionals of second order 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 •, • and • H , respectively.Given a symmetric Hilbert-Schmidt operator A : where •, h n stands for the Itô integral of h n , and if there exists no a n with required property, the summation is defined to be equal to zero.It is possible to define f A, without using the eigenfunction decomposition of A. Indeed, if B : H → H is a symmetric non-negative definite Hilbert-Schmidt operator, then, for any N ∈ N, a bounded linear operator (B+εI) −N exp −{B+ εI} −1 on H converges strongly to a linear operator for any x = 0, k ≥ 2, and where 4) is an easy application of the monotone convergence theorem and an estimation for every m ∈ N.
Theorem 2. Let A : H → H be a symmetric Hilbert-Schmidt operator, ∈ H, and γ ∈ R.
Then, for any λ ∈ R, it holds that where i = √ −1 and 5) is guaranteed by Lemma 1.
(ii) The theorem asserts that the distribution of 1 2 Q A + •, + γ is infinitely divisible and the corresponding Lévy measure is f A, (x)dx.Moreover, the distribution of 1  2 Q A + γ is selfdecomposable.See [16, §8 and §15].
(iii) Let C n (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 ∈ C 2 (W ) ⊕ C 1 (W ) ⊕ C 0 (W ), and is of the form that F = (1/2)Q A + •, +γ for some symmetric Hilbert-Schmidt operator A, ∈ H, and γ ∈ R, , and γ ∈ C 0 (W ).Moreover, A, , and γ are determined so that A = ∇ 2 F , = W ∇F dµ, and γ = W F dµ, where ∇ stands for the Malliavin derivative.
In particular, the theorem is applicable to a quadratic form of the form 1 2 Q A (• − 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 a n 's, the eigenvalues of A.
1st step: the case where a n > 0 for all n ∈ N .Let λ > 0. Note that Since { •, h n : n ∈ N} is a family of independent Gaussian random variables of mean 0 and variance 1, we obtain the following well known identity: Applying the identities Plugging this into (7), we obtain Note that 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 f A, (x) = 0 for x ≤ 0.
2nd step: the case where a n < 0 for all n ∈ N .Since −Q A = Q −A , applying the result in the first step to −A, we obtain , this yields (5).
3rd step: the general case.Set Moreover, the random variables From the observations made in the 1st and 2nd steps, we obtain from which (5) follows.

Mellin transform
Proposition 5.The Mellin transform of f A, defined by (2) is given by for any s ≥ 2.

Proof.
Rewrite Then, by a change of variables x → −x on (−∞, 0), we obtain which completes the proof.

Remark 6.
A sufficient condition for the Mellin transform of f A, to have a meromorphic extension to C is given by Jorgenson and Lang ([10]).

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.

Exponential decay
Theorem 7. Let (W, H, µ) be an abstract Wiener space, A : H → H be a symmetric Hilbert-Schmidt operator, ∈ H, y ∈ R m , and η = {η 1 , . . ., η m } ⊂ W * be an orthonormal system in H. Suppose that A = 0 when µ is considered, and that (ii) Let for every λ ≥ λ a , γ ∈ R.Moreover, if both supremums a + , a − are attained as maximums, then the above assertion holds with a = max{a − , a H /2, which gives a much faster decay than the one discussed in Theorem 7. Similarly, if {P (η) (A(y • η) + )} A (η) = 0, then we obtain a much faster decay than the one discussed in the theorem.(ii) The integrability of x 2 f A, (x) and x 2 f A (η) ,P (η) (A(y•η)+ ) (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 = ∞ n=1 n −p h n ⊗ h n for some p > 1/2 and an orthonormal basis {h n } of H, then there exist C > 0 and λ 0 > 0 such that For details, see Lemma 10 and its proof.
Then, applying the monotone convergence theorem, we obtain the desired divergence.Similarly, if a n < 0 for some n, then lim λ→∞ Thus the assumption made on a ± is that only on the order of divergence.
from which it follows that a ± = 0.

Proof of Theorem 7. Assume that
By virtue of Theorems 2 and 4, we have Thus the estimations in (ii) and (iii) also follow.
where û is the Fourier transformation of u (cf.[6,Prop.8.4.2]).Since e −x ≤ α α x −α for α, x > 0, a sufficient condition for this to hold is that there exist We obtain the following from Theorem 7.
Corollary 9. Let a ± , b ± be as in Theorem 7.
(i) If A = 0 and max{a − , a + } > 0, then the distribution on R of 1 2 Q A + •, + γ under µ admits a density function, which is in G 1/a (R) for any a < max{a − , a + }, with respect to the Lebesgue measure.
We give a sufficient condition for a ± to be positive.It follows from (3) that f A, (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.
n=1 be eigenvalues of A. Suppose that there exist a subsequence {a n k }, C > 0, and p > 1/2 such that a n k ≥ C/k p for any k ∈ N.Then, for any δ > 0, holds for any x ∈ (0, δ).In particular, a + ≥ 1/p.
If there exist a subsequence {a n k }, C > 0, and p > 1/2 such that a n k ≤ −C/k p for any k ∈ N.
Then, for any δ > 0, Due to the first assumption, we have 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 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.

Density functions
Corollary 9 gives a sufficient condition for the distribution of 1 2 Q A + •, +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 Wiener space, A : H → H be a symmetric Hilbert-Schmidt operator, and decompose as
(ii) Suppose that a 2n−1 = a 2n for any n ∈ N and #{n : a n = 0} = ∞.Fix x ∈ R, and assume that there exists a family of simple where Res(f (ζ); z) denotes the residue of f at z.
. Then all assertions in (i) and (ii) hold, replacing Q A and det 2 (I − iζ A) by q A and det(I − iζ A), respectively.
(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 L 2 -norm on an interval of the one-dimensional Wiener process.[18]).Thus the assertion (i) follows as an fundamental application of the Malliavin calculus.By (7) and the assumption that a 2n−1 = a 2n , we have and hence 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 of q A /2, then R e iλx pA (x)dx = 1 det(I − iλ A) .

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 = 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.

The square of the L 2 -norm on an interval
Let T > 0 and consider the classical one-dimensional Wiener space (W T which is absolutely continuous and has a square integrable derivative dh/dt, and µ 1 T is the Wiener measure.The inner product in H 1  T is given by In this subsection we consider

3.1.1
We first compute the Lévy measure of T by applying Theorem 2, where ∈ H 1 T and γ ∈ R. Define a symmetric Hilbert-Schmidt operator A : , and so is h T .It is easily seen that ∇ 2 h T = 2A and that Then, by virtue of Remark 3 (iii), we observe that It is easy to see that where In particular, we have it is straightforward to see that By virtue of this and (20), applying Theorem 2, we arrive at: Proposition 13.It holds that where g H is defined by (22).

3.1.2
We next compute the Lévy measure of By a straightforward computation, we obtain and hence In particular, Note that where Hence, for x > 0, where gH (x; T, , y) Due to Theorem 4, we obtain Since η(w) = w(T )/ √ T , combined with (20), we conclude from this: Proposition 14.It holds that where and gH is given by (26).

3.1.3
We finally study the exponential decay of the characteristic function of As was seen in §3.1.1,the Hilbert-Schmidt operator A associated with h T has eigenvalues , each of them being of multiplicity one.By Theorem 7 and Lemma 10, there exist C 1 > 0 and λ 1 > 0 such that for any λ ≥ λ 1 , γ ∈ R.
Starting from these well-known expressions, and recalling the elementary formulae we can also show explicit expressions for the Lévy measures ν T (dx) and ν T,y (dx) of the distribution of h T /2 under µ 1 T and the conditional probability measure µ 1 T ( • |w(T ) = y) as described in Propositions 13 and 14 with = 0.Moreover, by using the Riemann zeta function ζ(s) = ∞ n=1 n −s , we can give explicit forms of the Mellin transforms of ν T and ν T,y .Namely, noting that and then plugging (21), (23), and (24) into (12), we obtain: Proposition 15.The Mellin transform of ν T (dx) and ν T,y (dx) are given by respectively, where Γ is the usual gamma function.

Lévy's stochastic area
Let T > 0 and consider the classical two-dimensional Wiener space T which is absolutely continuous and has a square integrable derivative dh/dt, and µ 2 T is the Wiener measure.The inner product in H 2  T is given by Define Lévy's stochastic area by where J = 0 −1 1 0 and dw(t) denotes the Itô integral.

3.2.1
We first compute the Lévy measure of s T + •, + γ under µ 2 T by applying Theorem 2, where ∈ H 2 T and γ ∈ R. Define a symmetric Hilbert-Schmidt operator B : 2), (2, 1)} and s < t, so is s T .It is easily seen that ∇ 2 s T = B, and hence, due to Remark 3 (iii), we have By a direct computation, we see that where In particular , and Then it is easily seen that By virtue of this and (30), applying Theorem 2, we arrive at; Proposition 16.It holds that where g L is defined by (32)

3.2.2
We next compute the Lévy measure of s T + •, + γ under the conditional probability For any h, g ∈ (H 2 T ) (η) 0 , the identity B (η) h, g = Bh, g holds, and hence Then it is straightforward to see that where and it holds that where , we obtain (35) Thus, if we put gL (x; T, , y) then, for x > 0, it holds that where we have used the identity ∞ n=1 ne −nx = 1/{4 sinh 2 (x/2)}.Similarly, for x < 0, we have 1 sinh 2 (πx/T ) + gL (x; T, , y).
We can give explicit expressions for the Mellin transforms of the Lévy measures σ T and σ T,y of the distributions of s T under µ 2 T and the conditional probability measure µ 2 T ( • |w(T ) = y), respectively.Namely, noting that and then plugging (31), (34), and (35) into (12), we obtain: Proposition 18.The Mellin transforms of σ T and σ T,y are given by

Sample variance
Let T > 0 and (W 1 T , H 1 T , µ 1 T ) be the two-dimensional classical Wiener space over [0, T ].In this subsection, we consider the sample variance v T (w) =

3.3.1
We first compute the Lévy measure of T by applying Theorem 2, where ∈ H 1  T and γ ∈ R. Define a symmetric Hilbert-Schmidt operator C : Since v T (w) = h T (w) − T w2 , due to the observation made at the beginning of §3.1.1,we have It is a straightforward computation to see that where {k A n } ∞ n=1 is the orthonormal basis of (H 1 T ) (η) 0 defined in (23).Comparing the above expansion of C with that of A (η) in (23), and recalling the definition of f A, , we obtain In conjunction with (25) and (39), Theorem 2 and Proposition 14 lead us to: where f H is the function defined in Proposition 14, and is given by (40).In particular, the distribution of T coincides with that of 1 2 h T + •, +γ under µ 1 T (•|w(T ) = 0).

3.3.2
We next compute the Lévy measure of 1 2 v T + •, + γ under the conditional probability By straightforward computations, we obtain and where In particular, Moreover it holds that Hence, for x > 0, where Recalling (39), and applying Theorem 4 and Proposition 14, we can conclude: Proposition 20.It holds that and g V is given by (41).Moreover, the distribution of v T /2 under the conditional probability µ 1 T (•|w(T ) = y) coincides with the one of {h T /2 + h T /2 }/2 under the product measure , where h T /2 denotes an independent copy of h T /2 .

Density functions
Let T > 0 and consider the classical two-dimensional Wiener space (W 2 T , H 2 T , µ 2 T ) 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 L 2 -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].

Lévy's stochastic area
By (30), (31), and Theorem 11(i), we see that the distribution of s T under µ 2 T admits a smooth density function p L with respect to the Lebesgue measure on R, and that the corresponding Hilbert-Schmidt operator B satisfies Let x < 0. By a straightforward computation, we can show that Γ n satisfies the conditions in Theorem 11(ii) and conclude that ) n e (2n+1)πx/T = 1 T cosh(πx/T ) .
For x > 0, the complex conjugate Γ n plays the same role as Γ n , and we obtain Using the same Γ n 's as above, this time with R n = (4n + 1)π/T , and then applying Theorem 11(ii), we can show that pL (x) = i  Since h (2)  T is a sum of two independent copies of h T coming from w T /2 ∈ dx = 1 2T 2 ϑ 1 (0|ixπ/T 2 )X (0,∞) (x)dx, µ 2 T h (2)   T /2 ∈ dx w(T ) = 0 = 1 4T 2 ϑ 4 (0|ixπ/T 2 )X (0,∞) (x)dx.Proof.We have already seen the first half.The last half is immediate consequence of the representations of p H and pH in the form of infinite sums.

2 T
34) and Remark 12(ii), the distribution of s T under µ (•|w(T ) = 0) admits a smooth density function pL with respect to the Lebesgue measure on R, and it holds that det 2