九州大学 研究者情報
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長田 博文(おさだ ひろふみ) データ更新日:2018.10.09



主な研究テーマ
確率論


キーワード:確率論 確率解析 拡散過程の均質化 フラクタル上の拡散過程 無限粒子系 干渉ブラウン運動 ランダム行列
1980.04.
従事しているプロジェクト研究
日本学術振興会 日独二国間共同研究
2012.04~2014.03, 代表者:竹田雅好, 東北大学, 日本.
日本学術振興会 日独二国間共同研究
2007.04~2009.03, 代表者:重川一郎, 京都大学, 日本.
研究業績
主要著書
主要原著論文
1. 長田博文, 無限粒子系の確率解析学, 数学, 69, 3, 225-259, 30ページ, 2017.07, 対称性のある無限次元確率微分方程式を解くための新しい一般論を解説する。.
2. Hirofumi Osada, Tomoyuki Shirai, Absolute continuity and singularity of Palm measures of the Ginibre point process, Probability Theory and Related Fields, 10.1007/s00440-015-0644-6, 165, 3-4, 725-770, 2016.08, [URL], We prove a dichotomy between absolute continuity and singularity of the Ginibre point process G and its reduced Palm measures { G<sub>x</sub>, x∈ C<sup>ℓ</sup>, ℓ= 0 , 1 , 2 … } , namely, reduced Palm measures G<sub>x</sub> and G<sub>y</sub> for x∈ C<sup>ℓ</sup> and y∈ C<sup>n</sup> are mutually absolutely continuous if and only if ℓ= n; they are singular each other if and only if ℓ≠ n. Furthermore, we give an explicit expression of the Radon–Nikodym density dG<sub>x</sub>/ dG<sub>y</sub> for x, y∈ C<sup>ℓ</sup>..
3. Hirofumi Osada, Hideki Tanemura, Strong Markov property of determinantal processes with extended kernels, Stochastic Processes and their Applications, 10.1016/j.spa.2015.08.003, 126, 1, 186-208, 2016.01, [URL], Noncolliding Brownian motion (Dyson's Brownian motion model with parameter β=2) and noncolliding Bessel processes are determinantal processes; that is, their space-time correlation functions are represented by determinants. Under a proper scaling limit, such as the bulk, soft-edge and hard-edge scaling limits, these processes converge to determinantal processes describing systems with an infinite number of particles. The main purpose of this paper is to show the strong Markov property of these limit processes, which are determinantal processes with the extended sine kernel, extended Airy kernel and extended Bessel kernel, respectively. We also determine the quasi-regular Dirichlet forms and infinite-dimensional stochastic differential equations associated with the determinantal processes.
ランダム行列に関係する1次元空間の無限粒子系の運動を記述する確率力学の構成には、確率解析的な手法と、時空間相関関数の計算に基づく代数的な方法がある。この論文は、代数的な手法によって構成された確率力学の強マルコフ性を証明し、筆者達の他の結果と合わせて、これら二つの手法によって構成された確率力学が同一であることを証明した。その結果、解析的手法によって証明されている粒子の運動のパスとしての性質、また、代数的に手法によって証明されている確率力学の定量的な性質が、これらは、それぞれ別の手法ではとr手も証明されないものだが、これら二つの確率力学の同一性が示されたために、両方に対して成立することが判明した。.
4. Hirofumi Osada, Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials II
Airy random point field, Stochastic Processes and their Applications, 10.1016/j.spa.2012.11.002, 123, 3, 813-838, 2013.01, [URL], We give a new sufficient condition of the quasi-Gibbs property. This result is a refinement of one given in a previous paper (Osada (in press) [18]), and will be used in a forthcoming paper to prove the quasi-Gibbs property of Airy random point fields (RPFs) and other RPFs appearing under soft-edge scaling. The quasi-Gibbs property of RPFs is one of the key ingredients to solve the associated infinite-dimensional stochastic differential equation (ISDE). Because of the divergence of the free potentials and the interactions of the finite particle approximation under soft-edge scaling, the result of the previous paper excludes the Airy RPFs, although Airy RPFs are the most significant RPFs appearing in random matrix theory. We will use the result of the present paper to solve the ISDE for which the unlabeled equilibrium state is the Airy β RPF with β=1,2,4..
5. Hirofumi Osada, Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials, Annals of Probability, 10.1214/11-AOP736, 41, 1, 1-49, 2013.01, [URL], We investigate the construction of diffusions consisting of infinitely numerous Brownian particles moving in Rd and interacting via logarithmic functions (two-dimensional Coulomb potentials). These potentials are very strong and act over a long range in nature. The associated equilibrium states are no longer Gibbs measures. We present general results for the construction of such diffusions and, as applications thereof, construct two typical interacting Brownian motions with logarithmic interaction potentials, namely the Dyson model in infinite dimensions and Ginibre interacting Brownian motions. The former is a particle system in R, while the latter is in R2. Both models are translation and rotation invariant in space, and as such, are prototypes of dimensions d = 1, 2, respectively. The equilibrium states of the former diffusion model are determinantal or Pfaffian random point fields with sine kernels. They appear in the thermodynamical limits of the spectrum of the ensembles of Gaussian random matrices such as GOE, GUE and GSE. The equilibrium states of the latter diffusion model are the thermodynamical limits of the spectrum of the ensemble of complex non-Hermitian Gaussian random matrices known as the Ginibre ensemble..
6. Hirofumi Osada, Infinite-dimensional stochastic differential equations related to random matrices, Probability Theory and Related Fields, 10.1007/s00440-011-0352-9, 153, 3-4, 471-509, 2012.08, [URL], We solve infinite-dimensional stochastic differential equations (ISDEs) describing an infinite number of Brownian particles interacting via two-dimensional Coulomb potentials. The equilibrium states of the associated unlabeled stochastic dynamics are the Ginibre random point field and Dyson's measures, which appear in random matrix theory. To solve the ISDEs we establish an integration by parts formula for these measures. Because the long-range effect of two-dimensional Coulomb potentials is quite strong, the properties of Brownian particles interacting with two-dimensional Coulomb potentials are remarkably different from those of Brownian particles interacting with Ruelle's class interaction potentials. As an example, we prove that the interacting Brownian particles associated with the Ginibre random point field satisfy plural ISDEs..
7. Hirofumi Osada, Tagged particle processes and their non-explosion criteria, Journal of the mathematical society of Japan, 10.2969/jmsj/06230867, 62, 3, 867-894, 2010.06.
主要総説, 論評, 解説, 書評, 報告書等
主要学会発表等
1. Hirofumi Osada, Infinite-dimensional stochastic differential equations related to random matrices, 2012.03, We give general theorems to construct natural stochastic dynamicswhose unlabeled processes are reversible with respect to random point fields(RPF) related to random matrices. We solve the infinite-dimensional stochasticequations describing the labeled diffusion process of above mentioned dynamics.Typical examples of applications of our results are Dyson, Airy, and Besselrandom point fields, and the Ginibre random point field. All canonical Gibbs measures with Ruelle’sclass interaction potentials satisfying suitable marginal assumptions arecovered by our theorems as trivial applications. We focus the talk on the Ginibre RPF. As well as Ginibre interactingBrownian motions, we discuss quite interesting properties of Ginibre RPF thatare very different from usual translation invariant Gibbs measures, and theirapplication to homogenization problem of particles in 2D Coulomb environment..
2. Hirofumi Osada, Interacting Brownian motions related to random matrices, Stochastic Processes and their applications, 2007.08.
学会活動
所属学会名
日本数学会
学協会役員等への就任
2012.01~2013.12, ベルヌイ協会, the Committee for Conferences on Stochastic Processesの委員.
2007.04, 日本数学会, 受賞候補者推薦委員.
学会大会・会議・シンポジウム等における役割
2011.12.05~2011.12.07, Stochastic Analysis on Large Scale Interacting Systems, 座長(Chairmanship).
2011.07.11~2011.07.15, the Conference in Honor of the 70th Birthday of S. R. Srinivasa Varadhan., 司会(Moderator).
2009.07.27~2009.07.31, SPA Berlin 2009: 33rd Conference on Stochastic Processes and their applications, 座長(Chairmanship).
2011.12.05~2011.12.07, Stochastic Analysis on large Scale interacting Systems, オーガナイザー.
2010.09.06~2010.09.10, SPA 2010 (Stochastic Processes and its Applicatins 2010), Scientific Program Committee 委員長.
2008.09.08~2008.09.12, Stochastic Analysis and Applications 2008 (日独シンポジウム), Organizing Committee 委員.
2006.10, Stochastic Analysis on large Scale interacting Systems, オーガナイザー.
2006.08, 確率論サマースクール2006, organizer.
2006.07, 大規模相互作用系の確率解析, organizer.
2005.10, 大規模相互作用系の確率解析, organizer.
2004.10, 大規模相互作用系の確率解析, organizer.
学会誌・雑誌・著書の編集への参加状況
2012.01~2017.12, Electric Journal of Probability, 国際, 編集委員.
2012.01~2017.12, Electric Communications of Probability, 国際, 編集委員.
2002.07~2003.07, Advanced Studies in Pure Mathematics 39, Stochastic Analysis on Large Scale Interacting Systems, 国際, 編集者.
学術論文等の審査
年度 外国語雑誌査読論文数 日本語雑誌査読論文数 国際会議録査読論文数 国内会議録査読論文数 合計
2015年度 21        21 
2014年度 14        14 
2013年度 16        16 
2012年度      
2011年度      
2010年度      
2009年度      
2008年度      
2007年度    
2006年度      
2005年度      
2004年度 13        13 
その他の研究活動
海外渡航状況, 海外での教育研究歴
ポアンカレ研究所, France, 2009.01~2009.01.
ベルリン工科大学, Germany, 2009.08~2009.08.
Cornell大学, UnitedStatesofAmerica, 2008.06~2008.06.
イリノイ大学, UnitedStatesofAmerica, 2007.08~2007.08.
台湾大学, Taiwan, 2006.06~2006.06.
外国人研究者等の受入れ状況
2012.05~2012.07, ENSTA ParisTech, , .
受賞
日本数学会秋季賞, 日本数学会, 2018.09.
解析学賞, 日本数学会, 2014.09.
Itô Prize , Elsevier, 2013.08.
研究資金
科学研究費補助金の採択状況(文部科学省、日本学術振興会)
2009年度~2011年度, 基盤研究(B), 代表, ランダム行列、統計物理に動機づけられた無限次元確率力学系.
2005年度~2008年度, 基盤研究(A), 代表, 統計力学に動機付けを持つ諸問題の確率解析による総合的かつ統合的研究.

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