### 抄録

The equilibrium state of this dynamics is a determinantal random point field with the sine kernel. We prove for stochastic dynamics given by Dirichlet forms with determinantal random point fields as equilibrium states the particles never collide if the kernel of determining random point fields are locally Lipschitz continuous, and give examples of collision when H\"{o}lder continuous.

In addition we construct infinite volume dynamics (a kind of infinite dimensional diffusions) whose equilibrium states are determinantal random point fields. The last result is partial in the sense that we simply construct a diffusion associated with the {\em maximal closable part} of {\em canonical} pre Dirichlet forms for given determinantal random point fields as equilibrium states. To prove the closability of canonical pre Dirichlet forms for given determinantal random point fields is still an open problem. We prove these dynamics are the strong resolvent limit of finite volume dynamics.

元の言語 | 英語 |
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ホスト出版物のタイトル | Stochastic Analysis on Large Scale Interacting Systems |

編集者 | Tadahisa Funaki, Hirofumi Osada |

出版場所 | Mathematical Society of Japan |

ページ | 325-343 |

ページ数 | 29 |

巻 | 39 |

出版物ステータス | 出版済み - 2004 |

### 出版物シリーズ

名前 | Advanced Studies in Pure Mathematics |
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出版者 | Mathematical Society of Japan |

巻 | 39 |

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### これを引用

*Stochastic Analysis on Large Scale Interacting Systems*(巻 39, pp. 325-343). (Advanced Studies in Pure Mathematics ; 巻数 39). Mathematical Society of Japan.

**Non-collision and collision properties of Dyson's model in infinite dimension and other stochastic dynamics whose equilibrium states are determinantal random point fields.** / Osada, Hirofumi.

研究成果: 著書/レポートタイプへの貢献 › 会議での発言

*Stochastic Analysis on Large Scale Interacting Systems.*巻. 39, Advanced Studies in Pure Mathematics , 巻. 39, Mathematical Society of Japan, pp. 325-343.

}

TY - GEN

T1 - Non-collision and collision properties of Dyson's model in infinite dimension and other stochastic dynamics whose equilibrium states are determinantal random point fields

AU - Osada, Hirofumi

PY - 2004

Y1 - 2004

N2 - Dyson's model on interacting Brownian particles is a stochastic dynamics consisting of an infinite amount of particles moving in $ \R $ with a logarithmic pair interaction potential. For this model we will prove that each pair of particles never collide. The equilibrium state of this dynamics is a determinantal random point field with the sine kernel. We prove for stochastic dynamics given by Dirichlet forms with determinantal random point fields as equilibrium states the particles never collide if the kernel of determining random point fields are locally Lipschitz continuous, and give examples of collision when H\"{o}lder continuous. In addition we construct infinite volume dynamics (a kind of infinite dimensional diffusions) whose equilibrium states are determinantal random point fields. The last result is partial in the sense that we simply construct a diffusion associated with the {\em maximal closable part} of {\em canonical} pre Dirichlet forms for given determinantal random point fields as equilibrium states. To prove the closability of canonical pre Dirichlet forms for given determinantal random point fields is still an open problem. We prove these dynamics are the strong resolvent limit of finite volume dynamics.

AB - Dyson's model on interacting Brownian particles is a stochastic dynamics consisting of an infinite amount of particles moving in $ \R $ with a logarithmic pair interaction potential. For this model we will prove that each pair of particles never collide. The equilibrium state of this dynamics is a determinantal random point field with the sine kernel. We prove for stochastic dynamics given by Dirichlet forms with determinantal random point fields as equilibrium states the particles never collide if the kernel of determining random point fields are locally Lipschitz continuous, and give examples of collision when H\"{o}lder continuous. In addition we construct infinite volume dynamics (a kind of infinite dimensional diffusions) whose equilibrium states are determinantal random point fields. The last result is partial in the sense that we simply construct a diffusion associated with the {\em maximal closable part} of {\em canonical} pre Dirichlet forms for given determinantal random point fields as equilibrium states. To prove the closability of canonical pre Dirichlet forms for given determinantal random point fields is still an open problem. We prove these dynamics are the strong resolvent limit of finite volume dynamics.

M3 - Conference contribution

SN - 4-931469-24-8

VL - 39

T3 - Advanced Studies in Pure Mathematics

SP - 325

EP - 343

BT - Stochastic Analysis on Large Scale Interacting Systems

A2 - Funaki, Tadahisa

A2 - Osada, Hirofumi

CY - Mathematical Society of Japan

ER -