## Advanced Studies in Pure Mathematics

### Non-Collision and Collision Properties of Dyson's Model in Infinite Dimension and Other Stochastic Dynamics Whose Equilibrium States are Determinantal Random Point Fields

#### Abstract

Dyson's model on interacting Brownian particles is a stochastic dynamics consisting of an infinite amount of particles moving in $\mathbb{R}$ 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 Hölder 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 maximal closable part of 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.

#### Article information

Dates
Revised: 31 March 2003
First available in Project Euclid: 1 January 2019

https://projecteuclid.org/ euclid.aspm/1546369043

Digital Object Identifier
doi:10.2969/aspm/03910325

Mathematical Reviews number (MathSciNet)
MR2073339

Zentralblatt MATH identifier
1061.60109

#### Citation

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, 325--343, Mathematical Society of Japan, Tokyo, Japan, 2004. doi:10.2969/aspm/03910325. https://projecteuclid.org/euclid.aspm/1546369043