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

### 抄録

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.
元の言語 英語 Stochastic Analysis on Large Scale Interacting Systems Tadahisa Funaki, Hirofumi Osada Mathematical Society of Japan 325-343 29 39 出版済み - 2004

### 出版物シリーズ

名前 Advanced Studies in Pure Mathematics Mathematical Society of Japan 39

### Fingerprint

Infinite Dimensions
Stochastic Dynamics
Equilibrium State
Collision
Dirichlet Form
Model
kernel
Resolvent
Finite Volume
Lipschitz
Open Problems
Logarithmic
Partial
Interaction

### これを引用

Osada, H. (2004). Non-collision and collision properties of Dyson's model in infinite dimension and other stochastic dynamics whose equilibrium states are determinantal random point fields. ： T. Funaki, & H. Osada (版), Stochastic Analysis on Large Scale Interacting Systems (巻 39, pp. 325-343). (Advanced Studies in Pure Mathematics ; 巻数 39). Mathematical Society of Japan.
Stochastic Analysis on Large Scale Interacting Systems. 版 / Tadahisa Funaki; Hirofumi Osada. 巻 39 Mathematical Society of Japan, 2004. p. 325-343 (Advanced Studies in Pure Mathematics ; 巻 39).

Osada, H 2004, Non-collision and collision properties of Dyson's model in infinite dimension and other stochastic dynamics whose equilibrium states are determinantal random point fields. ： T Funaki & H Osada (版), Stochastic Analysis on Large Scale Interacting Systems. 巻. 39, Advanced Studies in Pure Mathematics , 巻. 39, Mathematical Society of Japan, pp. 325-343.
Osada H. Non-collision and collision properties of Dyson's model in infinite dimension and other stochastic dynamics whose equilibrium states are determinantal random point fields. ： Funaki T, Osada H, 編集者, Stochastic Analysis on Large Scale Interacting Systems. 巻 39. Mathematical Society of Japan. 2004. p. 325-343. (Advanced Studies in Pure Mathematics ).
Osada, Hirofumi. / Non-collision and collision properties of Dyson's model in infinite dimension and other stochastic dynamics whose equilibrium states are determinantal random point fields. Stochastic Analysis on Large Scale Interacting Systems. 編集者 / Tadahisa Funaki ; Hirofumi Osada. 巻 39 Mathematical Society of Japan, 2004. pp. 325-343 (Advanced Studies in Pure Mathematics ).
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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.

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T3 - Advanced Studies in Pure Mathematics

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BT - Stochastic Analysis on Large Scale Interacting Systems