Takahashi, Hiroshi

写真a

Affiliation

Faculty of Business and Commerce (Hiyoshi)

Position

Professor
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Career 【 Display / hide 】

  • 1999.04
    -
    2005.03

    慶應義塾志木高等学校

  • 2005.04
    -
    2007.03

    松江工業高等専門学校　講師

  • 2007.04
    -
    2009.03

    理化学研究所　研究員

  • 2009.04
    -
    Present

    理化学研究所　客員研究員

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Academic Background 【 Display / hide 】

  • 1997.03

    Keio University, Faculty of Science and Engineering, 数理科学科

    University, Graduated

  • 1999.03

    Keio University, Graduate School, Division of Science and Engineering, 数理科学

    Graduate School, Completed, Master's course

  • 2004.03

    Keio University, Graduate School, Division of Science and Engineering, 基礎理工学

    Graduate School, Completed, Doctoral course

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Academic Degrees 【 Display / hide 】

  • 博士（理学）, Keio University, Coursework, 2004.03

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Research Areas 【 Display / hide 】

  • Natural Science / Basic mathematics (General Mathematics (includes Probability Theory/Statistical Mathematics))

  • Natural Science / Applied mathematics and statistics (General Mathematics (includes Probability Theory/Statistical Mathematics))

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Research Keywords 【 Display / hide 】

  • ランダム媒質

  • 確率過程論

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Papers 【 Display / hide 】

  • Diffusion Processes with One-sided Selfsimilar Random Potentials

    Suzuki Y., Takahashi H., Tamura Y.

    Potential Analysis  2024

    ISSN  09262601

    　View Summary

    Long-time behavior of diffusion processes with one-sided random potentials starting from the origin is studied. As random potentials, some strictly stable processes are given just on the negative side in the real line. This model is an extension of the diffusion process with a one-sided Brownian potential studied by Kawazu, Suzuki and Tanaka (Tokyo J. Math. 24, 211–229 2001) and Kawazu and Suzuki (J. Appl. Probab. 43, 997–1012 2006). In this paper, we analyze our model by different methods from theirs. We use the theory concerning the convergence of a sequence of bi-generalized diffusion processes studied by Ogura (J. Math. Soc. Japan 41, 213–242 1989) and Tanaka (Comm. Pure Appl. Math. 47, 755–766 1994). For diffusion processes with one-sided random potentials, the limit theorems introduced by them cannot be used. We improve their limit theorems and apply the improved limit theorem to examining the long-time behavior of our model. As a result, we show that limit distributions exist under the Brownian scaling with some probability, and under a sub-diffusive scaling with the remaining probability.

  • Diffusion processes in Brownian environments on disconnected selfsimilar fractal sets in R

    Takahashi H., Tamura Y.

    Statistics and Probability Letters (Statistics and Probability Letters)  193 2023.02

    ISSN  01677152

    　View Summary

    We investigate the limiting behavior of diffusion processes in Brownian environments on disconnected selfsimilar fractal sets in R. Due to the effect of Brownian environments, the diffusion processes exhibit ultra-slow diffusive behavior, which are called Brox-type diffusions. We show that the limiting distributions are given under suitable scalings determined by selfsimilar fractal sets and measures related to the sets. The scaling properties are different from that of the Brox-type diffusion on R.

  • On the rate of convergence of Euler–Maruyama approximate solutions of stochastic differential equations with multiple delays and their confidence interval estimations

    Hashimoto M., Takahashi H.

    AIMS Mathematics (AIMS Mathematics)  8 ( 6 ) 13747 - 13763 2023

    　View Summary

    In this paper, we investigate Euler–Maruyama approximate solutions of stochastic differential equations (SDEs) with multiple delay functions. Stochastic differential delay equations (SDDEs) are generalizations of SDEs. Solutions of SDDEs are influenced by both the present and past states. Because these solutions may include past information, they are not necessarily Markov processes. This makes representations of solutions complicated; therefore, approximate solutions are practical. We estimate the rate of convergence of approximate solutions of SDDEs to the exact solutions in the Lp-mean for p ≥ 2 and apply the result to obtain confidence interval estimations for the approximate solutions.

  • Limiting behavior of multi-dimensional diffusion process in stable Levy environment

    TAKAHASHI Hiroshi

    The Institute of Statistical Mathematics Cooperative Research Report 262   129-135 2011

    Research paper (bulletin of university, research institution), Joint Work

  • Recurrence of diffusion process in Gaussian field

    TAKAHASHI Hiroshi

    The Institute of Statistical Mathematics Cooperative Research Report 262   136-141 2011

    Research paper (bulletin of university, research institution), Joint Work

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Research Projects of Competitive Funds, etc. 【 Display / hide 】

  • Study of asymptotic behavior multi-dimensional diffusion processes in random environments with multiple inhomogeneities

    2024.04
    -
    2029.03

    基盤研究(C), Principal investigator

  • 劣拡散的なランダム媒質中の多次元拡散過程の漸近挙動と極限分布の研究

    2018.04
    -
    2023.03

    MEXT,JSPS, Grant-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Principal investigator

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Courses Taught 【 Display / hide 】

  • MATHEMATICS FOR MACHINE LEARNING

    2025

  • LINEAR ALGEBRA

    2025

  • GENERAL EDUCATION SEMINAR (S)

    2025

  • ECONOMIC MATHEMATICS 1

    2025

  • DIFFERENTIAL AND INTEGRAL CALCULUS

    2025

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