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.
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03 April 2022
A Correction to this paper has been published: https://doi.org/10.1007/s00440-022-01124-1
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Osada, H. Infinite-dimensional stochastic differential equations related to random matrices. Probab. Theory Relat. Fields 153, 471–509 (2012). https://doi.org/10.1007/s00440-011-0352-9
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DOI: https://doi.org/10.1007/s00440-011-0352-9
Keywords
- Interacting Brownian particles
- Coulomb potentials
- Random matrices
- Ginibre random point field
- Dyson’s model
- Infinite-dimensional stochastic differential equations