Exponential decay of stochastic oscillatory integrals and its application

Setsuo Taniguchi (Faculty of Math., Kyushu Univ.) ^{∗} ^{†}

1. Outline

Let (W, H, μ) be an abstract Wiener space. A stochastic oscillatory integral with phase function q : W → ℝ and amplitude function ψ : W → ℝ is given by

I_{q,ψ}(λ) = ∫_{W} e^{iλq(w)}ψ(w)μ(dw), λ ∈ ℝ, where i^{2} = −1.

We report about its asymptotic behaviour as λ → ∞ when q is a Wiener chaos of order at most 2 and ψ ≡ 1, and discuss on examples and an application.

2. Lévy measure and exponential decay

Let q be a Wiener chaos of order at most 2, i.e. ∇^{3}q = 0, where ∇ stands for the Malliavin gradient. Setting γ = ∫_{W} qdμ, ℓ = ∫_{W} ∇qdμ ∈ H, A = ∇^{2}q ∈ H^{(2)}(the space of all symmetric Hilbert-Schmidt operator on H), and Q_{A} ≡ (∇^{∗})^{2}A, we have

q = \frac{1}{2}Q_{A} + ⟨·, ℓ⟩ + γ.

Decomposing A as A = ∑_{n=1}^{∞} a_{n}h_{n} ⊗ h_{n} with an ONB {h_{n}}_{n=1}^{∞} of H and real numbers {a_{n}} such that ∑_{n=1}^{∞} a_{n}^{2} < ∞, we define f_{A,ℓ} : ℝ → [0, ∞) by

f_{A,ℓ}(x) = \begin{cases}\frac{1}{2} ∑_{n:a_{n}>0}\{\frac{1}{x} + \frac{⟨ℓ, h_{n}⟩^{2}}{a_{n}^{3}} \} exp[−x/a_{n}], & if x > 0, \\ 0, & if x = 0, \\ \frac{1}{2} ∑_{n:a_{n}<0}\{\frac{1}{|x|} + \frac{⟨ℓ, h_{n}⟩^{2}}{|a_{n}|^{3}} \} exp[−|x|/|a_{n}|], & if x < 0. \end{cases}

Theorem 1 (Matsumoto-T.[1]). For any λ ∈ ℝ, it holds

∫_{W} e^{iλq}dμ = exp [−\frac{λ^{2}∥ℓ_{A}∥_{H}^{2}}{2} + iλγ + ∫_{ℝ} (e^{iλx} − 1 − iλx) f_{A,ℓ}(x)dx] , (1)

where ℓ_{A} = ∑_{n;a_{n}=0}⟨h_{n}, ℓ⟩h_{n}．

Remark 1. (i) It holds that 0 ≤ f_{A,ℓ}(x) ≤ \frac{∥A∥_{∞}^{k−2}}{2|x|^{k+1}} \{k!∥A∥_{2}^{2} + (k + 1)!∥ℓ∥_{H}^{2}\} and ∫_{−1}^{1} |x|^{2}f_{A,ℓ}(x)dx ≤ \frac{1}{2}∥A∥_{2}^{2}+∥ℓ∥_{H}^{2} (∀x ≠ 0, k ≥ 2), where ∥A∥_{2} is the Hilbert-Schmidt norm and ∥A∥_{∞} = sup\{|a_{n}| : n = 1, 2, … \}. Hence (e^{iλx} − 1 − iλx)f_{A,ℓ}(x) is integrable on ℝ. (ii) A key ingredient to lead us to (1) is the identity log(1 − iλa_{n}) = ∫_{0}^{∞}(1 − e^{iλx})(e^{−x/a_{n}}/x)dx. (iii) Let η_{1}, …, η_{n} ∈ W^{∗}, and define a Wiener functional η : W → ℝ^{n} by η(w) = (η_{1}(w), …, η_{n}(w)). The identity (1) extends to the conditional probability μ( · |η = y). Theorem 2 (Matsumoto-T.[1]). Suppose ℓ_{A} = 0.

(i) (lower estimation) It holds

|∫_{W} e^{iλq}dμ| ≥ exp[−\frac{λ^{2}}{2} ∫_{ℝ} x^{2}f_{A,ℓ}(x)dx], ∀λ ∈ ℝ. (2)

∗Abstract for the meeting “Stochastic analysis and the related topics” during Oct. 25 – 27, 2001, at Kyoto Univ.

†The talk is based on the joint work [1] with H. Matsumoto (Nagoya Univ.) and the recent development[2]

(ii) (upper estimation) Suppose that there exists a > 0 such that either of the following estimations holds

lim inf_{λ→∞} λ^{−a} ∫_{(−∞,0)}{1 − cos(λx)}f_{A,ℓ}(x)dx > 0, lim inf_{λ→∞} λ^{−a} ∫_{(0,∞)}{1 − cos(λx)}f_{A,ℓ}(x)dx > 0.

Then there are C_{0} > 0 and λ_{0} > 0 such that

|∫_{W} e^{iλq}dμ| ≤ exp[−C_{0}λ^{a}], ∀λ ≥ λ_{0}. (3)

Moreover, the distribution of q admits a density function of Gevrey class of order 1/a. Remark 2. (i) Because of the splitting property of μ, the assumption ℓ_{A} = 0 does no harm. (ii) Since |e^{z}| = e^{Rez}, Theorem 2 is an immediate consequence of Theorem 1. (iii) It holds that ∫_{ℝ_{±}}{1−cos(λx)}f_{A,ℓ}(x)dx ≥ ∫_{ℝ_{±}}{1−cos(λx)}f_{A,0}(x)dx → ∞(λ → ∞), where ℝ_{+} = {x > 0} and ℝ_{−} = {x < 0}. Thus the assumption in Theorem 2 (ii) is made just on the order of divergence. (iv) If #{n : a_{n} > 0} < ∞, lim_{λ→∞} λ^{−a} ∫_{0}^{∞}{1 − cos(λ)}f_{A,ℓ}(x)dx = 0. If #{n : a_{n} > 0} = ∞ and N_{A}(y) ≡ #{n : a_{n} > 1/y} satisfies that (a) lim_{y→∞} e^{−cy}N_{A}(y) = 0 (∀c > 0) and (b) lim inf_{y→∞} y^{−a}N_{A}(y) > 0 (∃a > 0), then lim inf_{λ→∞} λ^{−a} ∫_{0}^{∞}{1 − cos(λx)}f_{A,ℓ}(x)dx > 0.

3. Examples and an application

Example 1. Let W = C_{0}([0, T]; ℝ) and q(w) = \frac{1}{2} ∫_{0}^{T} w(t)^{2}dt. Then ∫_{W} e^{iλq}dμ = \{cos(\sqrt{iλ} T)\}^{−1/2}. The eigenvalues of the corresponding Hilbert-Schmidt operators are \{[T/(n + \frac{1}{2})π]^{2}; n = 0, 1, … \}(multiplicity 1). By Theorem 2, we have | ∫_{W} e^{iλq}dμ| ≤ exp[−Cλ^{1/2}] for ∀λ ≥ λ_{0} (∃C, λ_{0} > 0). This order coincides with the one coming from the above exact expression.

Example 2. Let W = C_{0}([0, T]; ℝ^{2}) and q(w) = \frac{1}{2} ∫_{0}^{T} \{w^{1}(t)dw(t)^{2} − w^{2}(t)dw^{1}(t)\}. Then ∫_{W} e^{iλq}dμ = 1/ cosh(λT/2). The eigenvalues of the corresponding Hilbert-Schmidt operators are {T/[(2n + 1)π]; n = 0, ±1, ±2, … }(multiplicity 2). By Theorem 2, we have | ∫_{W} e^{iλq}dμ| ≤ exp[−Cλ] for ∀λ ≥ λ_{0} (∃C, λ_{0} > 0). This order coincides with the one coming from the above exact expression.

Example 3. Let W = C_{0}([0, T]; ℝ) and q(w) = \frac{1}{2} ∫_{0}^{T} \{w(t) − \frac{1}{T} ∫_{0}^{T} w(s)ds\}dt.

Then ∫_{W} e^{iλq}dμ = \{\sqrt{iλ} T/ sin(\sqrt{iλ} T)\}^{1/2}. The eigenvalues of the corresponding Hilbert-Schmidt operators are {[T/nπ]^{2}; n = 1, 2, … }(multiplicity 1). By Theorem 2, we have | ∫_{W} e^{iλq}dμ| ≤ exp[−Cλ^{1/2}] for ∀λ ≥ λ_{0} (∃C, λ_{0} > 0). This order coincides with the one coming from the above exact expression.

Application to the Fourier transform on the hypersurface {Q_{A}/2 = x} Theorem 3 (T.[2]). Let h ∈ R(A), the range of A. Suppose that dim R(A) = ∞, and that there exist α, β > 0 such that

(4)  lim inf_{|ξ|→∞} |ξ|^{α} ∫_{ℝ}(1 − cos(ξy))f_{A,0}(y)dy > 0, lim inf_{|ξ|→∞} |ξ|^{β}⟨h, (I + ξ^{2}A^{2})^{−1}h⟩ > 0.

Then, Q_{A}/2 is smooth and nondegenerate in the sense of the Malliavin calculus. Moreover, for every 0 < α′ < α, there exist C_{1}, C_{2} > 0 so that

|∫_{X} e^{\sqrt{−1} λ⟨·,h⟩}δ_{x}(Q_{A}/2)dμ| ≤ C_{1} exp[−C_{2}|λ|^{2α′/(α′+β)}], ∀λ ∈ ℝ. (5)

Remark 3. (i) If A is singular, we have a faster decay for h ∉ R(A). Namely, defining h_{A} as ℓ_{A} in Theorem 1, we have ∫_{X} e^{\sqrt{−1} λ⟨·,h⟩}δ_{x}(Q_{A}/2)dμ = (∫_{X} δ_{x}(Q_{A}/2)dμ) exp[−λ^{2}∥h_{A}∥_{H}^{2}/2](∀λ ∈ ℝ).

(ii) If dim R(A) = ∞, for any β > 0, we can find h ∈ R(A) such that lim inf_{|ξ|→∞} |ξ|^{β}⟨h, (I + ξ^{2}A^{2})^{−1}h⟩ > 0. This reflects the fact the surface {Q_{A}/2 = x} gets flatter and flatter at infinity.

References

[1] H. Matsumoto and S. Taniguchi, Wiener functionals of second order and their Lévy measures, preprint, June 2001.

[2] S. Taniguchi, Oscillatory integrals on submanifolds in an abstract Wiener space, in preparation.