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Yuzuru Inahama

Publications and source records attributed to Yuzuru Inahama.

At least 19 recordsLinked to original sources
arXiv2026

Sub-Riemannian spectral distance

We study eigenvalues and eigenfunctions of the ``div-grad type" sub-Laplacian with respect to Popp's volume on a compact equiregular sub-Riemannian manifold $M$. Since Popp's volume is canonically determined by the sub-Riemannian structure of $M$, the spetra of the sub-Laplacian carry geometric meanings. In this paper, we first embed $M$ into the Hilbert space of square-summable sequences using eigenfunctions and then define a spectral distance between two compact equiregular sub-Riemannian manifolds. Our result is a sub-Riemannian analogue of Berard-Besson-Gallot's classical work in the Riemannian case.

math.DG↗
arXiv2025

Fractional Diffusion Bridges

Consider ``stochastic differential equations" driven by fractional Brownian motion with Hurst parameter H (1/4 <H< 1). Their solutions are sometimes called fractional diffusion processes. The main purpose of this paper is conditioning these processes to reach a given terminal point. We call the conditioned processes fractional diffusion bridges. Our main tool for mathematically rigorous conditioning is quasi-sure analysis, which is a potential theoretic part of Malliavin calculus. We also prove a small-noise large deviation principle of Freidlin-Wentzell type for scaled fractional diffusion bridges under a mild ellipticity assumption on the coefficient vector fields.

math.PR↗
arXiv2025

Wong-Zakai approximation of density functions

In this paper we prove the Wong-Zakai approximation of probability density functions of solutions at a fixed time of rough differential equations driven by fractional Brownian rough path with Hurst parameter $H$ $(1/4 <H \leq 1/2)$. Besides rough path theory, we use Hu-Watanabe's approximation theorem in the framework of Watanabe's distributional Malliavin calculus. When $H=1/2$, the random rough differential equations coincide with the corresponding Stratonovich-type stochastic differential equations. Even for that case, our main result seems new.

math.PR↗
arXiv2025

Averaging principle for slow-fast systems of rough differential equations via controlled paths

In this paper we prove the strong averaging principle for a slow-fast system of rough differential equations. The slow and the fast component of the system are driven by a rather general random rough path and Brownian rough path, respectively. These two driving noises are assumed to be independent. A prominent example of the driver of the slow component is fractional Brownian rough path with Hurst parameter between 1/3 and 1/2. We work in the framework of controlled path theory, which is one of the most widely-used frameworks in rough path theory. To prove our main theorem, we carry out Khas'minskii's time-discretizing method in this framework.

math.PR↗
arXiv2025

Averaging principle for rough slow-fast systems of level 3

The averaging principle for slow-fast systems of various kind of stochastic (partial) differential equations has been extensively studied. An analogous result was shown for slow-fast systems of rough differential equations driven by random rough paths a few years ago and the study of ``rough slow-fast systems" seems to be gaining momentum now. In all known results, however, the driving rough paths are of level 2. In this paper we formulate rough slow-fast systems driven by random rough paths of level 3 and prove the strong averaging principle of Khas'minskii-type.

math.PR↗
arXiv2024

Large deviation principle for slow-fast system with mixed fractional Brownian motion

This work focuses on a slow-fast system perturbed by mixed fractional Brownian motion with Hurst parameter $H\in(1/2,1)$. The integral with respect to fractional Brownian motion is the generalized Riemann-Stieltjes integral and the integral with respect to Brownian motion is the standard Itô integral. Our approach is based on the variational framework and the weak convergence criteria for mixed fractional Brownian motion. By combining the weak convergence method and Khasminskii's averaging principle, we show a large deviation principle for the slow component.

math.PR↗
arXiv2024

Moderate deviations for rough differential equations

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 H between 1/4 and 1/2. We prove a moderate deviation principle for this equation as the scale parameter tends to zero.

math.PR↗
arXiv2024

Moderate deviations for two-time scale systems with mixed fractional Brownian motion

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, 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.

math.DS↗
arXiv2023

Support theorem for pinned diffusion processes

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.

math.PR↗
arXiv2023

Averaging principle for fast-slow system driven by mixed fractional Brownian rough path

This paper is devoted to studying the averaging principle for fast-slow system of rough differential equations driven by mixed fractional Brownian rough path. The fast component is driven by Brownian motion, while the slow component is driven by fractional Brownian motion with Hurst index $H ~(1/3 < H\leq 1/2)$. Combining the fractional calculus approach to rough path theory and Khasminskii's classical time discretization method, we prove that the slow component strongly converges to the solution of the corresponding averaged equation in the $L^1$-sense. The averaging principle for a fast-slow system in the framework of rough path theory seems new.

math.PR↗
arXiv2021

Pathwise unique solutions and stochastic averaging for mixed stochastic partial differential equations driven by fractional Brownian motion and Brownian motion

This paper is devoted to a system of stochastic partial differential equations (SPDEs) that have a slow component driven by fractional Brownian motion (fBm) with the Hurst parameter $H >1/2$ and a fast component driven by fast-varying diffusion. It improves previous work in two aspects: Firstly, using a stopping time technique and an approximation of the fBm, we prove an existence and uniqueness theorem for a class of mixed SPDEs driven by both fBm and Brownian motion; Secondly, an averaging principle in the mean square sense for SPDEs driven by fBm subject to an additional fast-varying diffusion process is established. To carry out these improvements, we combine the pathwise approach based on the generalized Stieltjes integration theory with the Itô stochastic calculus. Then, we obtain a desired limit process of the slow component which strongly relies on an invariant measure of the fast-varying diffusion process.

math.PR↗
arXiv2021

Averaging Principles for Mixed Fast-Slow Systems Driven by Fractional Brownian Motion

We focus on fast-slow systems involving both fractional Brownian motion (fBm) and standard Brownian motion (Bm). The integral with respect to Bm is the standard Ito integral, and the integral with respect to fBm is the generalised Riemann-Stieltjes integral using the tools of fractional calculus. An averaging principle in which the fast-varying diffusion process of the fast-slow systems acts as a noise to be averaged out in the limit is established. It is shown that the slow process has a limit in the mean square sense, which is characterized by the solution of stochastic differential equations driven by fBm whose coefficients are averaged with respect to the stationary measure of the fast-varying diffusion. The implication is that one can ignore the complex original systems and concentrate on the averaged systems instead. This averaging principle paves the way for reduction of computational complexity.

math.DS↗
arXiv2020

Paracontrolled quasi-geostrophic equation with space-time white noise

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.

math.PR↗
arXiv2020

Positivity of the density for rough differential equations

Due to recent developments of Malliavin calculus for rough differential equations, it is now known that, under natural assumptions, the law of a unique solution at a fixed time has a smooth density function. Therefore, it is quite natural to ask whether or when the density is strictly positive. In this paper we study this problem from the viewpoint of Aida-Kusuoka-Stroock's general theory.

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↗
arXiv2020

Stochastic flows and rough differential equations on foliated spaces

Stochastic differential equations (SDEs) on compact foliated spaces were introduced a few years ago. As a corollary, a leafwise Brownian motion on a compact foliated space was obtained as a solution to an SDE. In this paper we construct stochastic flows associated with the SDEs by using rough path theory, which is something like a "deterministic version" of Itô's SDE theory.

math.DS↗
arXiv2019

Asymptotic expansion of the density for hypoelliptic rough differential equation

We study a rough differential equation driven by fractional Brownian motion with Hurst parameter $H$ $(1/4<H \le 1/2)$. Under Hörmander's condition on the coefficient vector fields, the solution has a smooth density for each fixed time. Using Watanabe's distributional Malliavin calculus, we obtain a short time full asymptotic expansion of the density under quite natural assumptions. Our main result can be regarded as a "fractional version" of Ben Arous' famous work on the off-diagonal asymptotics.

math.PR↗