### Abstract

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.

Original language | English |
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Pages (from-to) | 471-509 |

Number of pages | 39 |

Journal | Probability Theory and Related Fields |

Volume | 153 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - Aug 1 2012 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

**Infinite-dimensional stochastic differential equations related to random matrices.** / Osada, Hirofumi.

Research output: Contribution to journal › Article

*Probability Theory and Related Fields*, vol. 153, no. 3-4, pp. 471-509. https://doi.org/10.1007/s00440-011-0352-9

}

TY - JOUR

T1 - Infinite-dimensional stochastic differential equations related to random matrices

AU - Osada, Hirofumi

PY - 2012/8/1

Y1 - 2012/8/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84864387490&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84864387490&partnerID=8YFLogxK

U2 - 10.1007/s00440-011-0352-9

DO - 10.1007/s00440-011-0352-9

M3 - Article

VL - 153

SP - 471

EP - 509

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 3-4

ER -