Infinite-dimensional stochastic differential equations related to random matrices

Hirofumi Osada

    Research output: Contribution to journalArticlepeer-review

    34 Citations (Scopus)

    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 languageEnglish
    Pages (from-to)471-509
    Number of pages39
    JournalProbability Theory and Related Fields
    Volume153
    Issue number3-4
    DOIs
    Publication statusPublished - Aug 2012

    All Science Journal Classification (ASJC) codes

    • Analysis
    • Statistics and Probability
    • Statistics, Probability and Uncertainty

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