arXiv subjects

Toyomu Matsuda

Publications and source records attributed to Toyomu Matsuda.

7 recordsLinked to original sources

arXiv2025

Pathwise Uniqueness for Multiplicative Young and Rough Differential Equations Driven by Fractional Brownian Motion

We show pathwise uniqueness of multiplicative SDEs, in arbitrary dimensions, driven by fractional Brownian motion with Hurst parameter $H\in (1/3,1)$ with volatility coefficient $σ$ that is at least $γ$-Hölder continuous for $γ> \frac{1}{2H} \vee \frac{1-H}{H}$. This improves upon the long-standing results of [Lyo94 , Lyo98 , Dav08] which cover the same regime but require $σ$ to be at least $\frac{1}{H}$-Hölder continuous. Our central innovation is to combine stochastic averaging estimates with refined versions of the stochastic sewing lemma, due to [Lê20, Ger22, MP22].

math.PR↗

arXiv2023

An extension of the stochastic sewing lemma and applications to fractional stochastic calculus

We give an extension of Lê's stochastic sewing lemma [Electron. J. Probab. 25: 1 - 55, 2020]. The stochastic sewing lemma proves convergence in $L_m$ of Riemann type sums $\sum _{[s,t] \in π} A_{s,t}$ for an adapted two-parameter stochastic process $A$, under certain conditions on the moments of $A_{s,t}$ and of conditional expectations of $A_{s,t}$ given $\mathcal {F}_s$. Our extension replaces the conditional expectation given $\mathcal F_s$ by that given $\mathcal F_v$ for $v < s$, and it allows to make use of asymptotic decorrelation properties between $A_{s,t}$ and $\mathcal {F}_v$ by including a singularity in $(s-v)$. We provide three applications for which Lê's stochastic sewing lemma seems to be insufficient.The first is to prove the convergence of Itô or Stratonovich approximations of stochastic integrals along fractional Brownian motions under low regularity assumptions. The second is to obtain new representations of local times of fractional Brownian motions via discretization. The third is to improve a regularity assumption on the diffusion coefficient of a stochastic differential equation driven by a fractional Brownian motion for pathwise uniqueness and strong existence.

math.PR↗

arXiv2023

Level crossings of fractional Brownian motion

Since the classical work of Lévy, it is known that the local time of Brownian motion can be characterized through the limit of level crossings. While subsequent extensions of this characterization have primarily focused on Markovian or martingale settings, this work presents a highly anticipated extension to fractional Brownian motion -- a prominent non-Markovian and non-martingale process. Our result is viewed as a fractional analogue of Chacon et al. (1981). Consequently, it provides a global path-by-path construction of fractional Brownian local time. Due to the absence of conventional probabilistic tools in the fractional setting, our approach utilizes completely different argument with a flavor of the subadditive ergodic theorem, combined with the shifted stochastic sewing lemma recently obtained in Matsuda and Perkowski (22, arXiv:2206.01686). Furthermore, we prove an almost-sure convergence of the (1/H)-th variation of fractional Brownian motion with the Hurst parameter H, along random partitions defined by level crossings, called Lebesgue partitions. This result raises an interesting conjecture on the limit, which seems to capture non-Markovian nature of fractional Brownian motion.

math.PR↗

arXiv2022

Anderson Hamiltonians with singular potentials

We construct random Schr\"odinger operators, called Anderson Hamiltonians, with Dirichlet and Neumann boundary conditions for a fairly general class of singular random potentials on bounded domains. Furthermore, we construct the integrated density of states of these Anderson Hamiltonians, and we relate the Lifschitz tails (the asymptotics of the left tails of the integrated density of states) to the left tails of the principal eigenvalues.

math.PR↗

arXiv2022

Integrated density of states of the Anderson Hamiltonian with two-dimensional white noise

We construct the integrated density of states of the Anderson Hamiltonian with two-dimensional white noise by proving the convergence of the Dirichlet eigenvalue counting measures associated with the Anderson Hamiltonians on the boxes. We also determine the logarithmic asymptotics of the left tail of the integrated density of states. Furthermore, we apply our result to a moment explosion of the parabolic Anderson model in the plane.

math.PR↗

arXiv2020

Global well-posedness of the two-dimensional stochastic complex Ginzburg-Landau equation with cubic nonlinearity

The aim of this paper is to prove, under minimum assumptions, the global well-posedness of the two-dimensional stochastic complex Ginzburg-Landau equation on the torus driven by the additive space-time white noise. In addition to the global well-posedness, we prove an estimate of the solution which is uniform with respect to the initial condition and the strong Feller property of the dynamics.

math.PR↗

arXiv2020

Characterization of the support for Wick powers of the additive stochastic heat equation

Let $Z$ be the stationary solution of the additive stochastic heat equation $\partial_t Z = (Δ- 1) Z + ξ$ on the two-dimensional torus, where $ξ$ is the space-time white noise. The aim of this paper is to determine the support of Wick powers $\{Z^{:k:}\}_{k=1}^{\infty}$. This leads to an elementary proof of a support theorem for the dynamic $P(Φ)_2$ equation. In addition, we show that the approach can be used to determine the support of the law of the Gaussian multiplicative chaos in the $L^2$-phase.

math.PR↗