Updated on 2024/10/18

写真a

 
KAWAMOTO Yosuke
 
Organization
Faculty of Environmental, Life, Natural Science and Technology Associate Professor
Position
Associate Professor
Homepage
http://www.mtds.okayama-u.ac.jp/faculty_members/kawamoto/japanese/index-j.html
External link

Degree

  • なし ( 2018.3   Kyushu University )

Research Areas

  • Natural Science / Mathematical analysis

Education

  • Kyushu University   数理学府  

    2015.4 - 2018.3

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  • Kyushu University   数理学府  

    2013.4 - 2015.3

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  • Kyoto University   理学部   理学科

    2008.4 - 2013.3

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Research History

  • Okayama University   環境生命科学学域   Associate Professor

    2022.10

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  • Fukuoka Dental College   口腔歯学部   Lecturer

    2018.10 - 2022.9

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  • Kyushu University   確率解析研究センター

    2018.4 - 2018.9

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  • Kyushu University   数理学府

    2015.4 - 2018.3

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Papers

  • Dynamical universality for random matrices Reviewed

    Yosuke Kawamoto, Hirofumi Osada

    Partial Differential Equations and Applications   3 ( 2 )   2022.4

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    Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

    Abstract

    We establish an invariance principle corresponding to the universality of random matrices. More precisely, we prove the dynamical universality of random matrices in the sense that, if the random point fields $$ \mu ^N $$ of N-particle systems describing the eigenvalues of random matrices or log-gases with general self-interaction potentials V converge to some random point field $$ \mu $$, then the associated natural $$ \mu ^N $$-reversible diffusions represented by solutions of stochastic differential equations (SDEs) converge to some $$ \mu $$-reversible diffusion given by the solution of an infinite-dimensional SDE (ISDE). Our results are general theorems that can be applied to various random point fields related to random matrices such as sine, Airy, Bessel, and Ginibre random point fields. In general, the representations of finite-dimensional SDEs describing N-particle systems are very complicated. Nevertheless, the limit ISDE has a simple and universal representation that depends on a class of random matrices appearing in the bulk, and at the soft- and at hard-edge positions. Thus, we prove that ISDEs such as the infinite-dimensional Dyson model and the Airy, Bessel, and Ginibre interacting Brownian motions are universal dynamical objects.

    DOI： 10.1007/s42985-022-00154-7

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    Other Link： https://link.springer.com/article/10.1007/s42985-022-00154-7/fulltext.html

  • Infinite-dimensional stochastic differential equations and tail $\sigma$-fields II: the IFC condition Reviewed

    Yosuke KAWAMOTO, Hirofumi OSADA, Hideki TANEMURA

    Journal of the Mathematical Society of Japan   74 ( -1 )   79 - 128   2022

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    Publishing type：Research paper (scientific journal)   Publisher：Mathematical Society of Japan (Project Euclid)  

    DOI： 10.2969/jmsj/85118511

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  • Interacting Brownian motions in infinite dimensions related to the origin of the spectrum of random matrices Reviewed

    Yosuke Kawamoto

    Modern Stochastics: Theory and Applications   9   89 - 122   2022

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    Authorship：Corresponding author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：VTeX  

    The generalised sine random point field arises from the scaling limit at the origin of the eigenvalues of the generalised Gaussian ensembles. We solve an infinite-dimensional stochastic differential equation (ISDE) describing an infinite number of interacting Brownian particles which is reversible with respect to the generalised sine random point field. Moreover, finite particle approximation of the ISDE is shown, that is, a solution to the ISDE is approximated by solutions to finite-dimensional SDEs describing finite-particle systems related to the generalised Gaussian ensembles.

    DOI： 10.15559/21-vmsta193

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  • Uniqueness of Dirichlet Forms Related to Infinite Systems of Interacting Brownian Motions Reviewed

    Yosuke Kawamoto, Hirofumi Osada, Hideki Tanemura

    Potential Analysis   55 ( 4 )   639 - 676   2021.12

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

    <title>Abstract</title>The Dirichlet forms related to various infinite systems of interacting Brownian motions are studied. For a given random point field <italic>μ</italic>, there exist two natural infinite-volume Dirichlet forms <inline-formula><alternatives><tex-math>$ (\mathcal {E}^{\mathsf {upr } },\mathcal {D}^{\mathsf {upr } })$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
    <mml:mo>(</mml:mo>
    <mml:msup>
    <mml:mrow>
    <mml:mi>E</mml:mi>
    </mml:mrow>
    <mml:mrow>
    <mml:mstyle>
    <mml:mi>u</mml:mi>
    <mml:mi>p</mml:mi>
    <mml:mi>r</mml:mi>
    </mml:mstyle>
    </mml:mrow>
    </mml:msup>
    <mml:mo>,</mml:mo>
    <mml:msup>
    <mml:mrow>
    <mml:mi>D</mml:mi>
    </mml:mrow>
    <mml:mrow>
    <mml:mstyle>
    <mml:mi>u</mml:mi>
    <mml:mi>p</mml:mi>
    <mml:mi>r</mml:mi>
    </mml:mstyle>
    </mml:mrow>
    </mml:msup>
    <mml:mo>)</mml:mo>
    </mml:math></alternatives></inline-formula> and <inline-formula><alternatives><tex-math>$(\mathcal {E}^{\mathsf {lwr } },\mathcal {D}^{\mathsf {lwr } })$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
    <mml:mo>(</mml:mo>
    <mml:msup>
    <mml:mrow>
    <mml:mi>E</mml:mi>
    </mml:mrow>
    <mml:mrow>
    <mml:mstyle>
    <mml:mi>l</mml:mi>
    <mml:mi>w</mml:mi>
    <mml:mi>r</mml:mi>
    </mml:mstyle>
    </mml:mrow>
    </mml:msup>
    <mml:mo>,</mml:mo>
    <mml:msup>
    <mml:mrow>
    <mml:mi>D</mml:mi>
    </mml:mrow>
    <mml:mrow>
    <mml:mstyle>
    <mml:mi>l</mml:mi>
    <mml:mi>w</mml:mi>
    <mml:mi>r</mml:mi>
    </mml:mstyle>
    </mml:mrow>
    </mml:msup>
    <mml:mo>)</mml:mo>
    </mml:math></alternatives></inline-formula> on <italic>L</italic>2(<italic>S</italic>,<italic>μ</italic>) describing interacting Brownian motions each with unlabeled equilibrium state <italic>μ</italic>. The former is a decreasing limit of a scheme of such finite-volume Dirichlet forms, and the latter is an increasing limit of another scheme of such finite-volume Dirichlet forms. Furthermore, the latter is an extension of the former. We present a sufficient condition such that these two Dirichlet forms are the same. In the first main theorem (Theorem 3.1) the Markovian semi-group given by <inline-formula><alternatives><tex-math>$(\mathcal {E}^{\mathsf {lwr } },\mathcal {D}^{\mathsf {lwr } })$</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
    <mml:mo>(</mml:mo>
    <mml:msup>
    <mml:mrow>
    <mml:mi>E</mml:mi>
    </mml:mrow>
    <mml:mrow>
    <mml:mstyle>
    <mml:mi>l</mml:mi>
    <mml:mi>w</mml:mi>
    <mml:mi>r</mml:mi>
    </mml:mstyle>
    </mml:mrow>
    </mml:msup>
    <mml:mo>,</mml:mo>
    <mml:msup>
    <mml:mrow>
    <mml:mi>D</mml:mi>
    </mml:mrow>
    <mml:mrow>
    <mml:mstyle>
    <mml:mi>l</mml:mi>
    <mml:mi>w</mml:mi>
    <mml:mi>r</mml:mi>
    </mml:mstyle>
    </mml:mrow>
    </mml:msup>
    <mml:mo>)</mml:mo>
    </mml:math></alternatives></inline-formula> is associated with a natural infinite-dimensional stochastic differential equation (ISDE). In the second main theorem (Theorem 3.2), we prove that these Dirichlet forms coincide with each other by using the uniqueness of weak solutions of ISDE. We apply Theorem 3.1 to stochastic dynamics arising from random matrix theory such as the sine, Bessel, and Ginibre interacting Brownian motions and interacting Brownian motions with Ruelle’s class interaction potentials, and Theorem 3.2 to the sine2 interacting Brownian motion and interacting Brownian motions with Ruelle’s class interaction potentials of <inline-formula><alternatives><tex-math>$ {C_{0}^{3 } } $</tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
    <mml:msubsup>
    <mml:mrow>
    <mml:mi>C</mml:mi>
    </mml:mrow>
    <mml:mrow>
    <mml:mn>0</mml:mn>
    </mml:mrow>
    <mml:mrow>
    <mml:mn>3</mml:mn>
    </mml:mrow>
    </mml:msubsup>
    </mml:math></alternatives></inline-formula>-class.

    DOI： 10.1007/s11118-020-09872-2

    arXiv

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    Other Link： https://link.springer.com/article/10.1007/s11118-020-09872-2/fulltext.html

  • Transitions of generalised Bessel kernels related to biorthogonal ensembles Reviewed

    KAWAMOTO Yosuke

    Stochastic Analysis on Large Scale Interacting Systems, RIMS Kôkyûroku Bessatsu   B79   19 - 31   2020

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    Authorship：Corresponding author   Language：English   Publishing type：Research paper (scientific journal)  

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  • Correction to: Dynamical Bulk Scaling Limit of Gaussian Unitary Ensembles and Stochastic Differential Equation Gaps

    Yosuke Kawamoto, Hirofumi Osada

    Journal of Theoretical Probability   32 ( 3 )   1613 - 1613   2019.9

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media {LLC}  

    DOI： 10.1007/s10959-019-00913-0

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  • Dynamical Bulk Scaling Limit of Gaussian Unitary Ensembles and Stochastic Differential Equation Gaps Reviewed

    Yosuke Kawamoto, Hirofumi Osada

    Journal of Theoretical Probability   32 ( 2 )   907 - 933   2019

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

    DOI： 10.1007/s10959-018-0816-2

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    Other Link： http://link.springer.com/content/pdf/10.1007/s10959-018-0816-2.pdf

  • Finite-particle approximations for interacting Brownian particles with logarithmic potentials Reviewed

    Yosuke KAWAMOTO, Hirofumi OSADA

    Journal of the Mathematical Society of Japan   70 ( 3 )   2018.7

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    Publishing type：Research paper (scientific journal)   Publisher：Mathematical Society of Japan (Project Euclid)  

    DOI： 10.2969/jmsj/75717571

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  • Density preservation of unlabeled diffusion in systems with infinitely many particles Reviewed

    KAWAMOTO YOSUKE

    Stochastic Analysis on Large Scale Interacting Systems, RIMS Kôkyûroku Bessatsu   B59   337 - 350   2016

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    Language：English   Publishing type：Research paper (scientific journal)  

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MISC

Presentations

  • The intertwining property for Laguerre processes with a fixed parameter

    2024.9.4 

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    Event date： 2024.9.3 - 2024.9.6

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  • ランダム行列への誘い（II） Invited

    河本陽介

    2023年度確率論ヤングサマーセミナー  2023.8.29 

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    Event date： 2023.8.28 - 2023.8.31

    Presentation type：Oral presentation (invited, special)  

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  • ランダム行列への誘い（I） Invited

    河本陽介

    2023年度確率論ヤングサマーセミナー  2023.8.28 

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    Event date： 2023.8.28 - 2023.8.31

    Presentation type：Oral presentation (invited, special)  

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  • 無限粒子系の確率力学の末尾事象保存性について Invited

    河本陽介

    関西大学 確率論研究会 2022  2022.11.12 

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    Event date： 2022.11.12 - 2022.11.13

    Presentation type：Oral presentation (invited, special)  

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  • Infinite-dimensional stochastic differential equations related to generalised sine random point fields Invited

    2022.1.14 

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    Event date： 2022.1.10 - 2022.10.14

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  • 一般化Sine点過程に関する無限次元確率微分方程式について Invited

    大規模相互作用系の確率解析  2021.12.9 

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    Event date： 2021.12.7 - 2021.12.9

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  • Transitions of generalised Bessel kernels related to biorthogonal ensembles

    KAWAMOTO Yosuke

    2020.3.16 

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    Event date： 2020.3.16 - 2020.3.19

    Presentation type：Oral presentation (general)  

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  • Laguerre過程が満たすintertwining関係式について

    河本陽介

    半田山微分方程式セミナー  2024.6.21 

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  • Interacting Brownian motions in infinite dimensions related to the origin of the spectrum of random matrices

    2021.10.8 

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    Presentation type：Public lecture, seminar, tutorial, course, or other speech  

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  • Biorthogonal ensembleについて Invited

    Workshop on “Random matrices, Determinantal point processes and Gaussian analytic functions”  2020.3.19 

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  • Uniqueness of Dirichlet forms associated with non-tail trivial random point field Invited

    KAWAMOTO YOSUKE

    The 18th Stochastic Analysis on Large Scale Interacting Systems  2019.11.7 

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    Language：Japanese   Presentation type：Oral presentation (invited, special)  

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  • Stochastic analysis on infinite particle systems related to random matrices International conference

    KAWAMOTO YOSUKE

    Seminar at KU Leuven  2019.10.11 

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    Language：English   Presentation type：Public lecture, seminar, tutorial, course, or other speech  

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  • Uniqueness of Dirichlet forms related to non-tail trivial random point fields Invited International conference

    KAWAMOTO YOSUKE

    Japanese-German Open Conference on Stochastic Analysis 2019  2019.9.2 

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    Language：English   Presentation type：Oral presentation (invited, special)  

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  • On infinite particle systems related to non-tail trivial random point fields Invited

    KAWAMOTO YOSUKE

    Stochastic analysis on infinite-particle systems II  2019.8.4 

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    Language：Japanese   Presentation type：Oral presentation (invited, special)  

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  • Stochastic analysis on infinite dimensional stochastic differential equations related to random matrices Invited

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    量子場の数理とその周辺  2019.6.26 

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    Language：Japanese   Presentation type：Oral presentation (invited, special)  

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  • Dynamical transitions between universal infinite particle systems related to random matrices

    河本陽介

    日本数学会 2019年度年会  2019.3.17 

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    Language：Japanese   Presentation type：Oral presentation (general)  

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  • Convergence theorems related to Airy interacting infinite dimensional stochastic differential equations Invited International conference

    KAWAMOTO YOSUKE

    Workshop on Random matrices, stochastic geometry and related topics  2019.3.15 

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    Language：English   Presentation type：Oral presentation (invited, special)  

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  • ランダム行列に関する普遍的な点過程間の遷移関係とその力学版,

    河本陽介

    東京確率論セミナー  2019.1.28 

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    Language：Japanese   Presentation type：Public lecture, seminar, tutorial, course, or other speech  

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  • Transitions between dynamics of infinite particle systems associated with universal random point fields related to random matrices Invited International conference

    KAWAMOTO YOSUKE

    17th workshop on Stochastic Analysis on Large Scale Interacting Systems  2018.11.5 

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    Language：English   Presentation type：Oral presentation (invited, special)  

    Venue：Kyoto university  

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  • Dynamical universality for the Airy random point fields Invited International conference

    KAWAMOTO YOSUKE

    Random matrices and their applications  2018.5.21 

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    Language：English   Presentation type：Oral presentation (invited, special)  

    Venue：Kyoto University  

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  • Universality of strong and weak non-Hermitian limit Invited

    KAWAMOTO YOSUKE

    Workshop on "Random matrices, determinantal processes and their related topics"  2018.3.9 

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    Language：Japanese   Presentation type：Oral presentation (invited, special)  

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  • 無限粒子系の確率力学におけるMosco収束について

    河本陽介

    マルコフ過程とその周辺  2018.3.8 

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    Language：Japanese   Presentation type：Oral presentation (general)  

    Venue：福岡大学  

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  • ランダム行列に関係する無限次元確率微分方程式について Invited

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    確率解析の諸相  2018.1.6 

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    Language：Japanese   Presentation type：Oral presentation (invited, special)  

    Venue：九州大学  

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  • Mosco convergence for infinite particle systems related to random matrices Invited International conference

    KAWAMOTO YOSUKE

    16th workshop on Stochastic Analysis on Large Scale Interacting Systems  2017.11.9 

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    Language：English   Presentation type：Oral presentation (invited, special)  

    Venue：The University of Tokyo  

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  • Dynamical universality for infinite-dimensional interacting Brownian motions related to random matrices Invited International conference

    KAWAMOTO YOSUKE

    Japanese-German Conference 2017 on Stochastic Analysis  2017.9.8 

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    Language：English   Presentation type：Oral presentation (invited, special)  

    Venue：TU Kaiserslautern (Germany)  

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  • 非Hermiteランダム行列の固有値について

    河本陽介

    2017年度確率論ヤングサマーセミナー  2017.8.10 

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    Language：Japanese   Presentation type：Oral presentation (general)  

    Venue：国民宿舎 良寛荘（岡山県）  

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  • Dynamical universality for infinite dimensional stochastic differential equations related to random matrices International conference

    KAWAMOTO YOSUKE

    The 39th Conference on Stochastic Processes and their Applications  2017.7.24 

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    Language：English   Presentation type：Oral presentation (general)  

    Venue：Moscow (Russia)  

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  • ランダム行列の普遍性と力学的対応物

    河本陽介

    無限粒子系の確率解析学キックオフミーティング  2017.1.25 

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    Language：Japanese   Presentation type：Oral presentation (invited, special)  

    Venue：九州大学  

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  • Dynamic Universality for Random Matrices

    KAWAMOTO YOSUKE

    2016.12.20 

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    Language：Japanese   Presentation type：Oral presentation (general)  

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  • 無限次元干渉Brown粒子系の密度保存性

    河本陽介

    阪大確率論セミナー  2016.12.6 

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    Language：Japanese   Presentation type：Public lecture, seminar, tutorial, course, or other speech  

    Venue：大阪大学  

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  • Dynamical bulk universality of Gaussian unitary ensembles International conference

    KAWAMOTO YOSUKE

    Forum "Math-for-Industry" 2016  2016.11.23 

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    Language：English   Presentation type：Poster presentation  

    Venue：Queensland University of Technology (Australia)  

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  • Some property of infinite-dimensional Dyson's model with multiple tails Invited International conference

    KAWAMOTO YOSUKE

    15th Stochastic Analysis on Large Scale Interacting Systems  2016.11.2 

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    Language：English   Presentation type：Oral presentation (invited, special)  

    Venue：The University of Tokyo  

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  • Density preservation of infinite-dimensional interacting Brownian motions

    KAWAMOTO YOSUKE

    2016.9.15 

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    Language：Japanese   Presentation type：Oral presentation (general)  

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  • 無限次元干渉Brown運動の密度保存性について

    河本陽介

    確率論サマースクール2016  2016.9.7 

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    Language：Japanese   Presentation type：Oral presentation (general)  

    Venue：信州大学  

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  • 無限次対称群に付随する拡散過程

    河本陽介

    2016年度確率論ヤングサマーセミナー  2016.8.10 

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    Language：Japanese   Presentation type：Oral presentation (general)  

    Venue：三重県伊勢市  

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  • Finite particle approximation and density preservation of infinite-dimensional interacting Brownian motions related to random matrices International conference

    KAWAMOTO YOSUKE

    The 46th Saint-Flour Probability Summer School  2016.7.4 

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    Language：English   Presentation type：Oral presentation (general)  

    Venue：Saint-Flour (France)  

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  • 無限粒子系の拡散過程の密度保存性について

    河本陽介

    東京確率論セミナー  2016.5.9 

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    Language：Japanese   Presentation type：Oral presentation (general)  

    Venue：東京大学  

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  • Density preservation of unlabeled diffusion in systems with infinitely many particles Invited

    KAWAMOTO YOSUKE

    Workshop on "Random matrices, determinantal processes and their related topics"  2016.3.8 

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    Language：Japanese   Presentation type：Oral presentation (invited, special)  

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  • Infinite-dimensional SDE related to random matrices -Finite particle approximation and the density preserving property- Invited International conference

    KAWAMOTO YOSUKE

    Statistics & Applied Probability Department Seminar  2016.2.24 

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    Language：English   Presentation type：Oral presentation (invited, special)  

    Venue：UC Santa Barbara (USA)  

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  • Finite particle approximation of infinite-dimensional SDE related to random matrices Invited International conference

    KAWAMOTO YOSUKE

    14th Stochastic Analysis on Large Scale Interacting Systems  2015.10.29 

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    Language：English   Presentation type：Oral presentation (invited, special)  

    Venue：Kyoto University  

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  • Finite Particle Approximation of Interacting Brownian Motion Invited International conference

    KAWAMOTO YOSUKE

    Stochastic Analysis and Applications  2015.8.31 

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    Language：English   Presentation type：Poster presentation  

    Venue：Tohoku University  

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  • Airy random point fieldに対応する無限次元SDEとその普遍性

    河本陽介

    2015年度確率論ヤングサマーセミナー  2015.8.20 

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    Language：Japanese   Presentation type：Oral presentation (general)  

    Venue：鈴岡旅館（愛知県)  

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  • ランダム行列に関する無限次元SDEの有限粒子系近似

    河本陽介

    関西確率論セミナー  2015.7.24 

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    Language：Japanese   Presentation type：Oral presentation (invited, special)  

    Venue：京都大学  

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  • Finite particle approximations of interacting Brownian motions related to random matrices International conference

    KAWAMOTO YOSUKE

    38th Conference on Stochastic Processes and their Applications  2015.7.16 

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    Language：English   Presentation type：Oral presentation (general)  

    Venue：Oxford University (UK)  

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  • 魅力的な申請書の書き方 Invited

    河本陽介

    第19回Quricon  2015.4.11 

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    Language：Japanese   Presentation type：Symposium, workshop panel (nominated)  

    Venue：九州大学  

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  • Ito calculusと粒子系 Invited

    河本陽介

    第19回Quricon  2015.4.11 

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    Language：Japanese   Presentation type：Oral presentation (invited, special)  

    Venue：九州大学  

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  • Finite Particle Approximation of Dyson’s Brownian Motion International conference

    KAWAMOTO YOSUKE

    Kick-off Meeting of IMI Australia Branch in La Trobe - Mathematics Bridge over the Pacific for Competitive Edge in Industry  2015.3.13 

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    Language：English   Presentation type：Poster presentation  

    Venue：La Trobe University (Australia)  

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  • Universality of random matrices Invited

    KAWAMOTO YOSUKE

    Workshop on "Random matrices, determinantal processes and integrable probability" in Beppu 2015  2015.3.6 

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    Language：Japanese   Presentation type：Oral presentation (invited, special)  

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  • Finite particle approximation of interacting Brownian motions corresponding to geometric universality

    KAWAMOTO YOSUKE

    2014.12.19 

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    Language：Japanese   Presentation type：Oral presentation (general)  

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  • 確率幾何の普遍性に対応する無限次元確率力学の普遍性について

    河本陽介

    無限粒子系、確率場の諸問題X  2014.11.29 

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    Language：Japanese   Presentation type：Oral presentation (general)  

    Venue：横浜情報文化センター  

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  • Dynamical convergence of infinite particle system related to random matrices Invited International conference

    KAWAMOTO YOSUKE

    13th workshop on Stochastic Analysis on Large Scale Interacting Systems  2014.11.5 

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    Language：English   Presentation type：Oral presentation (invited, special)  

    Venue：The University of Tokyo  

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  • 無限次元確率力学の有限系からの力学的収束

    河本陽介

    九州確率論セミナー  2014.10.31 

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    Language：Japanese   Presentation type：Oral presentation (general)  

    Venue：九州大学  

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  • On SDE gaps International conference

    KAWAMOTO YOSUKE

    Summer school on Probability 2014  2014.9.11 

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    Language：English   Presentation type：Oral presentation (general)  

    Venue：Shinshu University  

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  • 無限次元確率力学の力学的収束とSDEギャップ

    河本陽介

    2014年度確率論ヤングサマーセミナー  2014.8.18 

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    Language：Japanese   Presentation type：Oral presentation (general)  

    Venue：新潟県月岡温泉  

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Research Projects

  • 表現論に関する無限粒子系における，確率解析的手法の新研究と代数的手法との融合

    Grant number：21K13812  2021.04 - 2025.03

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Early-Career Scientists  Grant-in-Aid for Early-Career Scientists

    河本 陽介

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    Grant amount：\4680000 （ Direct expense: \3600000 、 Indirect expense：\1080000 ）

    本研究が対象とするのは，ランダム行列に関する無限粒子系の確率力学である．ランダム行列に関する無限粒子系は，強い相互作用を持つ系であり，数学的にも物理的にも興味深い．そこで，このような無限粒子系について，自然に対応する確率力学の構成・解析を行いたい．
    長距離相互作用する無限粒子系の確率力学における代表的な研究手法としては，特別な2次形式であるDirichlet形式を用いた確率解析的手法と，可積分構造を使った明示的な計算を用いる代数的手法の2つがある．本研究の目標は，無限粒子系の確率力学に対して確率解析的手法と代数的手法の両方からアプローチすることで，両者の長所を取り込み，様々な観点からの解析を可能にすることである．
    当該年度では確率解析的手法での研究をメインに行った．1つ目の成果は，粒子配置を表す確率測度である点過程の強収束から，対応する確率力学の収束を導く一般論を確立したことである．これは，Dirichlet形式を用いた一般性の高い定理であり，代数的手法では直接計算するのが困難な収束も示すことができる．
    また，ランダム行列の固有値から得られるある重要な無限粒子系の点過程について，対応する確率力学を構成した．この研究では確率解析的手法を用いたことにより，確率力学の（無限次元）確率微分方程式を具体的に記述することができた．さらにこの方程式が，全く振る舞いが異なる別の確率力学と同じものであるという興味深い性質を，厳密に確かめることに成功した．

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  • ランダム行列と無限次元確率力学の普遍性

    2015.04 - 2018.03

    日本学術振興会  特別研究員奨励費 

    河本陽介

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    Authorship：Principal investigator  Grant type：Competitive

    Grant amount：\2800000 （ Direct expense: \2800000 ）

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