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Preprints and accepted papers

5 papers have appeared. (1 papers accepted.) Last modified: 12 Dec 2023.

2023

  • Toyomu Matsuda and Avi Mayorcas, Pathwise Uniqueness for Multiplicative Young and Rough Differential Equations Driven by Fractional Brownian Motion, arXiv:2312.06473, 2023 [arxiv].
    abstractWe show \emph{pathwise uniqueness} of multiplicative SDEs, in arbitrary dimensions, driven by fractional Brownian motion with Hurst parameter $H\in (1/3,1)$ with volatility coefficient $\sigma$ that is at least $\gamma$-H\"older continuous for $\gamma > \frac{1}{2H} \vee \frac{1-H}{H}$. This improves upon the long-standing results of Lyons (94, 98) and Davie (08) which cover the same regime but require $\sigma$ to be at least $\frac{1}{H}$-H\"older continuous. Our central innovation is to combine stochastic averaging estimates with refined versions of the stochastic sewing lemma, due to Lê (20), Gerencsér (22) and Matsuda and Perkowski (22).
  • Purba Das, Rafał Łochowski, Toyomu Matsuda and Nicolas Perkowski, Level crossings of fractional Brownian motion, arXiv:2308.08274, 2023 [arxiv].
    abstractSince 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.
  • Toyomu Matsuda and Willem van Zuijlen, Anderson Hamiltonians with singular potentials, arXiv:2211.01199, 2023 [arxiv].
    abstractWe construct random Schrödinger 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.

2022

  • Toyomu Matsuda and Nicolas Perkowski, An extension of the stochastic sewing lemma and applications to fractional stochastic calculus, arXiv:2206.01686, 2022 [arxiv].
    abstractWe 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 $\sum_{[s,t] \in \pi} A_{s,t}$ 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.
  • Matsuda, Toyomu, Integrated density of states of the Anderson Hamiltonian with two-dimensional white noise, Stochastic Processes and their Applications 153, 91-127, 2022 [html] [doi] [arxiv].
    abstractWe 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.

Master’s work

Not for publications.