arXiv subjects

Setsuo Taniguchi

Publications and source records attributed to Setsuo Taniguchi.

8 recordsLinked to original sources

arXiv2026

Quadratic Wiener functionals -- transformations and quadratic forms

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.

math.PR↗

arXiv2025

Transformations of order one and quadratic forms on Wiener spaces

It will be shown that transformations of order one on the Wiener space give rise to quadratic forms as exponents of change of variables formulas, and conversely every exponentially integrable quadratic form has a transformation of order one realizing the form in such a manner. Several expressions of corresponding change of variables formulas are also discussed.

math.PR↗

arXiv2024

Transformations and quadratic forms on Wiener spaces

Two-way relationships between transformations and quadratic forms on Wiener spaces are investigated with the help of change of variables formulas on Wiener spaces. Further the evaluation of Laplace transforms of quadratic forms via Riccati or linear second order ODEs will be shown.

math.PR↗

arXiv2021

On the transition density function of the diffusion process generated by the Grushin operator

The short-time asymptotic behavior of the transition density function of the diffusion process generated by the general Grushin operator will be investigated, by using its explicit expression in terms of expectation. Further the dependence on of the asymptotics will be seen.

math.PR↗

arXiv2020

Heat trace asymptotics on equiregular sub-Riemannian manifolds

We study a "div-grad type" sub-Laplacian with respect to a smooth measure and its associated heat semigroup on a compact equiregular sub-Riemannian manifold. We prove a short time asymptotic expansion of the heat trace up to any order. Our main result holds true for any smooth measure on the manifold, but it has a spectral geometric meaning when Popp's measure is considered. Our proof is probabilistic. In particular, we use S. Watanabe's distributional Malliavin calculus.

math.DG↗

arXiv2017

Short time full asymptotic expansion of hypoelliptic heat kernel at the cut locus

In this paper we prove a short time asymptotic expansion of a hypoelliptic heat kernel on an Euclidean space and a compact manifold. We study the "cut locus" case, namely, the case where energy-minimizing paths which join the two points under consideration form not a finite set, but a compact manifold. Under mild assumptions we obtain an asymptotic expansion of the heat kernel up to any order. Our approach is probabilistic and the heat kernel is regarded as the density of the law of a hypoelliptic diffusion process, which is realized as a unique solution of the corresponding stochastic differential equation. Our main tools are S. Watanabe's distributional Malliavin calculus and T. Lyons' rough path theory.

math.PR↗

arXiv2015

A construction of diffusion processes associated with sub-Laplacian on CR manifolds and its applications

A diffusion process associated with the real sub-Laplacian $Δ_b$, the real part of the complex Kohn-Spencer laplacian $\square_b$, on a strictly pseudoconvex CR manifold is constructed via the Eells-Elworthy-Malliavin method by taking advantage of the metric connection due to Tanaka-Webster. Using the diffusion process and the Malliavin calculus, the heat kernel and the Dirichlet problem for $Δ_b$ are studied in a probabilistic manner. Moreover, distributions of stochastic line integrals along the diffusion process will be investigated.

math.PR↗

arXiv2005

Brownian sheet and reflectionless potentials

The bijectivity of the mapping, which is represented as expectation, from a family of Gaussian measures parametrized by linear combinations of Dirac measures to the space of classical reflectionless potentials is shown. It is also shown that the bijectivity extends to the space of generalized reflectionless potentials, which was used by V. Marchenko to study the Cauchy problem for the KdV equation. In the extension, the stochastic calculus based on the Brownian sheet plays a key role.

math.PR↗