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KAKEN_15K04916seika.pdf
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Title |
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長距離相互作用系の確率解析
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Kana |
チョウキョリ ソウゴ サヨウケイ ノ カクリツ カイセキ
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Chōkyori sōgo sayōkei no kakuritsu kaiseki
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Stochastic calculus for systems with long range interaction
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種村, 秀紀
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タネムラ, ヒデキ
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Tanemura, Hideki
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慶應義塾大学・理工学部 (矢上) ・教授
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Research team head
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科研費研究者番号 : 40217162
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2019
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科学研究費補助金研究成果報告書
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2018
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近年の長田氏との共同研究により、長距離相互作用をもつ拡散型無限粒子系に対して、無限次元確率微分方程式に対する解の存在と一意性を導いた。この拡散系の結果を飛躍型無限粒子系(江崎氏との共同研究)、および無限剛体球系に対して一般化することに成功した。無限次元確率微分方程式の一意性の応用として、対応するディリクレ形式の一意性を導いた(長田氏、河本氏との共同研究)Rahul Roy 氏との共同研究により、ユークリッド空間上に中心がポアッソン配置されたstick の繋がりから定まる浸透模型を導入し、stick の方向分布とクラスターの形状の関連を調べ、ある種の相転移現象を導いた。
We studied infinite dimensional stochastic differential equations (ISDEs) representing infinite particles systems with long range interaction. We proved the existence and uniqueness of solutions of ISDEs for systems of interacting diffusion processes (joint work with H. Osada ). We generalized the result to systems of jump type process including Levy process (joint work with S. Esaki) and those of hard core balls. As an application of these results, we showed the uniqueness of Dirichlet forms associated the systems (joint work with Y. Kawamoto and H. Osada ).
We also studied on percolation models composed by sticks whose centers are arranged by a Poisson point process and directions are i. i. d. We examined the relation between the distribution of direction of sticks and the shape of cluster, and obtained a kind of phase transition (joint work with R. Roy).
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研究種目 : 基盤研究 (C) (一般)
研究期間 : 2015~2018
課題番号 : 15K04916
研究分野 : 確率論
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