On the Quadratic Wiener Functional Associated with the Malliavin Derivative of the Square Norm of Brownian Sample Path on Interval

Exact expressions of the stochastic oscillatory integrals with the phase function, which is the quadratic Wiener functional obtained from the Malliavin derivative of the square norm of the Brownian sample path on interval, are given. As an application, the density function of the distribution of the half of the Wiener functional is given.


Introduction and statement of result
The study of quadratic Wiener functionals, i.e., elements in the space of Wiener chaos of order 2, goes back to Cameron-Martin [1,2] and Lévy [8]. While a stochastic oscillatory integral with quadratic Wiener functional as phase function has a general representation via Carleman-Fredholm determinant ( [3,6,10]), in our knowledge, a few examples, where the integrals are represented with more concrete functions like the ones used by Cameron-Martin and Lévy, are available. See [1,2,8,6,10] and references therein. In this paper, we study a new quadratic Wiener functional which admits a concrete expression of stochastic oscillatory integral, and apply the expression to compute the density function of the Wiener functional. Let T > 0, W be the space of all R-valued continuous functions w on [0, T ] with w(0) = 0, and P be the Wiener measure on W. The Wiener functional investigated in this paper is The functional q interests us because it is a key ingredient in the study of asymptotic theory on W. Namely, recall the Wiener functional which was studied first by 2,8]. As is well-known ( [15]), the stochastic oscillatory integral W exp(ζq 0 /2)δ y (w(T ))dP, where δ y (w(T )) is Watanabe's pull back of the Dirac measure δ y concentrated at y ∈ R via w(T ), relates to the fundamental solution to the heat equation associated with the Schrödinger operator (1/2){(d/dx) 2 +ζx 2 }, which describes the quantum mechanics of harmonic oscillator. If we denote by H the Cameron-Martin subspace of W (≡ the subspace of all absolutely continuous h ∈ W with square integrable derivativeḣ) and set h, where ∇ denotes the Malliavin gradient. Thus q determines the stationary points of q 0 . It should be noted that, in the context of the Malliavin calculus, the set of stationary points of q 0 , i.e. the set {∇q 0 = 0} = {q = 0} is determined uniquely up to equivalence of quasi-surely exceptional sets. On account of the stationary phase method on finite dimensional spaces (cf. [4]), q would play an important role in the study of asymptotic behavior of the stochastic oscillatory integral W exp(ζq 0 )ψdP with amplitude function ψ (cf. [9,11,12], in particular [13,14]). The aim of this paper is to show Theorem 1. (i) For sufficiently small λ > 0, the following identities hold.
The assertion (i) of Theorem 1 will be shown in Section 2 and (ii) will be proved in Section 3.

Proof of Theorem 1 (i)
In this section, we shall show the identities (1) and (2). The proof is broken into several steps, each being a lemma. We first show Lemma 1. Define the Hilbert-Schmidt operator A : H → H by Then it holds that where Q A = (∇ * ) 2 A, ∇ * being the adjoint operator of the Malliavin gradient ∇. Moreover, A is of trace class and tr A = T 4 /6. In particular, q = Q A + tr A.
Proof. Due to the integration by parts on [0, T ], it is easily seen that for h, k ∈ H, where H ⊗2 denotes the Hilbert space of all Hilbert-Schmidt operators on H, and ·, · H ⊗2 does its inner product. Hence Let C 2 be the space of Wiener chaos of order 2. Since From this and (6), we can conclude the identity (4). Let {h n } ∞ n=1 be an orthonormal basis of H, and define k t ∈ H, t ∈ [0, T ], by Thus A is of trace class and tr A = T 4 /6.
We next recall the following assertion achieved in [5,7]. (i) For sufficiently small λ ∈ R, it holds that Then, for sufficiently small λ ∈ R, it holds that Proof. The essential part of the proof can be found in [5,7]. For the completeness, we give the proof. Due to the splitting property of the Wiener measure, it holds that where det 2 denotes the Carleman-Fredholm determinant. For example, see [3,7]. Observe that, for Hilbert-Schmidt operators C, D : H → H such that C is of trace class, it holds that Since (9), we obtain that Thus (7) has been shown.
It is not known if, by just watching specific shape of quadratic Wiener functional, one can tell that the associated Hilbert-Schmidt operator admits a decomposition as a sum of a Volterra operator and a bounded operator with finite dimensional range. However, in our situation, we know a priori that the operator A admits such a decomposition. Namely, the Hilbert-Schmidt operator B associated with q 0 admits such a decomposition ( [7]). Being equal to the square of B (see Remark 1 below), so does A. The following lemma gives the concrete expression of the decomposition of A.
Proof. Let η j , j = 1, 2, be as in Lemma 3 (iii), and we have that Let f 1 , f 2 be as in the proof of Lemma 4. Then we see that Hence, if we put σ λ = sinh(λ 1/4 T ) and τ λ = sin(λ 1/4 T ), then Since R(A 1 ) ⊂ R(A F ), by Lemma 3 (ii), this yields that The identity (2) follows from this, because Lemmas 1, 2, and 3 imply that Remark 1. It may be interesting to see that (1) is also shown by using the infinite product expression. Namely, define B : H → H by Then there exists an orthonormal basis {h n } ∞ n=0 of H so that See [10]. Since A = B 2 , it holds that In conjunction with Lemma 1, this implies that Due to the splitting property of the Wiener measure, we then obtain that Due to the infinite product expressions of cosh x and cos x, this implies (1).