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Yuzuru Inahama

Publications and source records attributed to Yuzuru Inahama.

30 records · Page 2Linked to original sources
arXiv2017

Stochastic complex Ginzburg-Landau equation with space-time white noise

We study the stochastic cubic complex Ginzburg-Landau equation with complex-valued space-time white noise on the three dimensional torus. This nonlinear equation is so singular that it can only be under- stood in a renormalized sense. In the first half of this paper we prove local well-posedness of this equation in the framework of regularity structure theory. In the latter half we prove local well-posedness in the framework of paracontrolled distribution theory.

math.PR↗
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↗
arXiv2016

Short time kernel asymptotics for rough differential equation driven by fractional Brownian motion

We study a stochastic differential equation in the sense of rough path theory driven by fractional Brownian rough path with Hurst parameter H (1/3 < H <= 1/2) under the ellipticity assumption at the starting point. In such a case, the law of the solution at a fixed time has a kernel, i.e., a density function with respect to Lebesgue measure. In this paper we prove a short time off-diagonal asymptotic expansion of the kernel under mild additional assumptions. Our main tool is Watanabe's distributional Malliavin calculus.

math.PR↗
arXiv2016

Rough path theory and stochastic calculus

T. Lyons' rough path theory is something like a deterministic version of K. Ito's theory of stochastic differential equations, combined with ideas from K. T. Chen's theory of iterated path integrals. In this article we survey rough path theory, in particular, its probabilistic aspects.

math.PR↗
arXiv2015

Large deviations for rough path lifts of Watanabe's pullbacks of delta functions

We study Donsker-Watanabe's delta functions associated with strongly hypoelliptic diffusion processes indexed by a small parameter. They are finite Borel measures on the Wiener space and admit a rough path lift. Our main result is a large deviation principle of Schilder type for the lifted measures on the geometric rough path space as the scale parameter tends to zero. As a corollary, we obtain a large deviation principle conjectured by Takanobu and Watanabe, which is a generalization of a large deviation principle of Freidlin-Wentzell type for pinned diffusion processes.

math.PR↗
arXiv2014

Malliavin differentiability of solutions of rough differential equations

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 $1/4$.

math.PR↗
arXiv2014

Short time kernel asymptotics for Young SDE by means of Watanabe distribution theory

In this paper we study short time asymptotics of a density function of the solution of a stochastic differential equation driven by fractional Brownian motion with Hurst parameter $H \in (1/2, 1)$ when the coefficient vector fields satisfy an ellipticity condition at the starting point. We prove both on-diagonal and off-diagonal asymptotics under mild additional assumptions. Our main tool is Malliavin calculus, in particular, Watanabe's theory of generalized Wiener functionals.

math.PR↗
arXiv2014

Large deviation principle for certain spatially lifted Gaussian rough path

In rough stochastic PDE theory of Hairer type, rough path lifts with respect to the space variable of two-parameter continuous Gaussian processes play a main role. A prominent example of such processes is the solution of the stochastic heat equation under the periodic condition. The main objective of this paper is to show that the law of the spatial lift of this process satisfies a Schilder type large deviation principle on the continuous path space over a geometric rough path space.

math.PR↗
arXiv2013

Large deviation principle of Freidlin-Wentzell type for pinned diffusion processes

Since T. Lyons invented rough path theory, one of its most successful applications is a new proof of Freidlin-Wentzell's large deviation principle for diffusion processes. In this paper we extend this method to the case of pinned diffusion processes under a mild ellipticity assumption. Besides rough path theory, our main tool is quasi-sure analysis, which is a kind of potential theory in Malliavin calculus.

math.PR↗
arXiv2010

A moment estimate of the derivative process in rough path theory

In this paper we prove the derivative process of a rough differential equation driven by Brownian rough path has finite $L^r$-moment for any $r /ge 1$. Thanks to Burkholder-Davis-Gundy's inequality, this kind of problem is easy in the usual SDE theory. In the context of rough path theory, however, it does not seem so obvious.

math.PR↗
arXiv2010

A stochastic Taylor-like expansion in the rough path theory

In this paper we establish a Taylor-like expansion in the context of the rough path theory for a family of It ^{o} maps indexed by a small parameter. We treat not only the case that the roughness $p$ satisfies $[p]=2$, but also the case that $[p] ge 3$. As an application, we discuss the Laplace asymptotics for It^{o} functionals of Brownian rough paths.

math.PR↗