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.

Publisher：Applied Mathematics and Optimization  

This work focuses on moderate deviations for two-time scale systems with mixed fractional Brownian motion. Our proof uses the weak convergence method which is based on the variational representation formula for mixed fractional Brownian motion. Throughout this paper, the Hurst parameter of fractional Brownian motion is larger than 1/2 and the integral along the fractional Brownian motion is understood as the generalized Riemann-Stieltjes integral. First, we consider single-time scale systems with fractional Brownian motion. The key of our proof is showing the weak convergence of the controlled system. Next, we extend our method to show moderate deviations for two-time scale systems. To this goal, we combine the Khasminskii-type averaging principle and the weak convergence approach.

DOI： 10.1007/s00245-024-10159-w

Web of Science

Scopus

Publisher：Bulletin of the London Mathematical Society  

Small noise problems are quite important for all types of stochastic differential equations. In this paper, we focus on rough differential equations driven by scaled fractional Brownian rough path with Hurst parameter (Formula presented.). We prove a moderate deviation principle for this equation as the scale parameter tends to zero.

DOI： 10.1112/blms.13097

Web of Science

Scopus

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Nagoya Mathematical 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

Web of Science

Scopus

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

Publisher：Publications of the Research Institute for Mathematical Sciences  

In this paper we study a large deviation principle of Freidlin–Wentzell type for pinned hypoelliptic diffusion measures associated with a natural sub-Laplacian on a compact sub-Riemannian manifold. To prove this large deviation principle, we use rough path theory and manifold-valued Malliavin calculus.

DOI： 10.4171/PRIMS/60-1-4

Web of Science

Scopus

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。

Language：Japanese  

CiNii Books

Event date： 2024.11

Language：English   Presentation type：Oral presentation (invited, special)  

Venue：Mathematisches Forshungsinstitut Oberwolfach   Country：Germany  

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.