Language：English   Publishing type：Research paper (scientific journal)  

In this paper we study rough differential equations
driven by Gaussian rough paths from the viewpoint of Malliavin calculus.
Under mild assumptions on coefficient vector fields and underlying Gaussian processes,
we prove that solutions at a fixed time is smooth in the sense of Malliavin calculus.
Examples of Gaussian processes include fractional Brownian motion with Hurst parameter larger than 1/4.

Language：English   Publishing type：Research paper (scientific journal)  

In this paper we prove a support theorem of Stroock-Varadhan type for pinned diffusion processes.To this end we use two powerful results from stochastic analysis.One is quasi-sure analysis for Brownian rough path. The other is Aida-Kusuoka-Stroock's positivity theorem for the densities of weighted laws of non-degenerate Wiener functionals.

DOI： 10.1017/nmj.2023.25

Repository Public URL： https://hdl.handle.net/2324/7173583

Language：English   Publishing type：Research paper (scientific journal)  

DOI： https://doi.org/10.1007/s10959-021-01116-2

Language：English   Publishing type：Research paper (scientific journal)  

DOI： https://doi.org/10.1016/j.jde.2021.08.006

Language：English   Publishing type：Research paper (scientific journal)  

We study the stochastic dissipative quasi-geostrophic equation with space-time white noise on the two-dimensional torus.
This equation is highly singular and basically ill-posed in its original form.
The main objective of the present paper is to formulate and solve this equation locally in time in the framework of paracontrolled calculus
when the differential order of the main term, the fractional Laplacian, is larger than $7/4$. No renormalization has to be done for this model.

Language：Japanese   Book type：Scholarly book

伊藤流の確率微分方程式論をまったく別の角度から見る「ラフパス理論」は、性質の悪い連続なパスに沿った線積分を、確率論を使わずに定式化することを可能にする。理論の基礎的理解を目指し、ラフパスで駆動される常微分方程式など非測度論的な部分に焦点を当て解説。Brownラフパス理論に関する一連の確率論的な結果についても触れる。

Event date： 2016.8

Language：English   Presentation type：Oral presentation (general)  

Venue：King Monkut's Institute of Technology, Latkrabang, Bangkok, Thailand.   Country：Thailand  

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.

Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (scientific journal)  

Type：Competition, symposium, etc.