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The Wigner semi-circle law and eigenvalues of matrix-valued diffusions. (English) Zbl 0767.60050
Summary: A system of stochastic differential equations for the eigenvalues of a symmetric matrix whose components are independent Ornstein-Uhlenbeck processes is derived. This corresponds to a diffusion model of an interacting particles system with linear drift towards the origin and electrostatic inter-particle repulsion. The associated empirical distribution of particles is shown to converge weakly (as the number of particles tends to infinity) to a limiting measure-valued process which may be characterized as the weak solution of a deterministic ODE. The Wigner semi-circle density is found to be one of the equilibrium points of this limiting equation.

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
Full Text: DOI
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