[论文解读] Quadratic Wiener functionals -- transformations and quadratic forms

Setsuo Taniguchi|arXiv (Cornell University)|Mar 1, 2026

Algebraic and Geometric Analysis被引用 0

一句话总结

该论文为在 Wiener 空间上使用 Malliavin 计算、一次变换以及基于行列式的变量变换，建立一个统一的框架来评估二次 Wiener 泛函，重新考察 Lévy 与 Kac 的历史结果并推导新表述及应用。

ABSTRACT

Quadratic Wiener functionals are investigated systematically through transformations of order one on the Wiener space with the help of Malliavin calculus. The bi-directional relationship between quadratic Wiener functionals and transformations of order one is established via change of variables formulas on the Wiener space. The relationship is applied to the investigation of Laplace transformations of quadratic Wiener functionals. This note is made due to establishing a systematic framework to study quadratic Wiener functionals and revisiting the past works by the author with the framework.

研究动机与目标

• Motivate the study of integrals of the form ∫_W f e^{q} dμ for quadratic Wiener functionals and their role in stochastic analysis.
• Provide a unified framework using Wiener space transformations to handle exponential integrals of quadratic forms.
• Revisit and extend classic results by Lévy and Kac through Malliavin calculus and operator-theoretic methods.
• Derive change-of-variables formulas and invertibility conditions for order-one transformations and their connections to Girsanov’s theorem.

提出的方法

• Introduce and characterize quadratic forms q_η via a Hilbert-Schmidt operator B_η and its kernel η in S_2.
• Give a series expansion of q_η as (1/2) (D^*)^2 B_η and as a sum over an ONB {h_n} of H with D^*h_n terms (converging in L^p for p in (1,∞)).
• Define transformations of order one ι + F_κ with κ ∈ L_2 and derive η(κ) from κ using explicit formula.
• Prove change-of-variables formulas involving det_2(I + B_κ) and conditions ensuring exponential integrability of q_η.
• Explore the inverse transformation ι + F_{reve{κ}} and establish density-transform relations and Girsanov-type results.
• Apply the framework to linear adapted transformations, Feynman-Kac densities, heat kernels on nilpotent Lie groups, and connections to Euler/Bernoulli/Eulerian polynomials and KdV.

实验结果

研究问题

• RQ1 How can quadratic Wiener functionals be represented and manipulated using Malliavin calculus and Hilbert-Schmidt operators?
• RQ2 What are the necessary and sufficient conditions for exponential integrability of q_η and for the existence/invertibility of associated transformations?
• RQ3 How can change-of-variables formulas be formulated for order-one transformations and what are their implications (e.g., Girsanov theorem)?
• RQ4 What explicit representations arise for specific transformations (e.g., harmonic-oscillator type, Lévy’s stochastic area) and their applications in analysis and PDEs?
• RQ5 How does the framework connect with classical results (Lévy, Kac) and extend to applications like Feynman-Kac densities and KdV-related structures?

主要发现

• A series expansion of quadratic forms q_η is established as (1/2) Σ_{n,m} ⟨B_η h_n, h_m⟩ (D^* h_n)(D^* h_m) − δ_{nm}, converging in L^p for p ∈ (1,∞).
• The transformation ι + F_κ yields a quadratic form q_{η(κ)} with a change-of-variables formula involving det_2(I + B_κ) and a normalization factor e^{(1/2)‖κ‖_2^2} when certain operator norms are small.
• Equivalent conditions are given for e^{q_η} ∈ L^1(μ), including Λ(B_η) < 1 and representations of η via τ or ρ in appropriate function spaces, with explicit density transformations.
• The inverse transformation ι + F_{reve{κ}} exists under suitable conditions, enabling dual change-of-variables formulas and connections to Girsanov’s theorem.
• Applications include explicit Laplace transforms, Feynman-Kac densities, heat kernels on two-step nilpotent Lie groups, and stochastic representations of Euler, Bernoulli, Eulerian polynomials and KdV-related structures.
• The framework recovers classical results (Lévy’s stochastic area, Kac’s oscillator) as special cases and clarifies the role of harmonic-oscillator type functionals.