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Global fluctuations for 1D log-gas dynamics

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  • Unterberger, Jérémie

Abstract

We study in this article the hydrodynamic limit in themacroscopic regime of the coupled system of stochastic differential equations, (0.1)dλti=1NdWti−V′(λti)dt+β2N∑j≠idtλti−λtj,i=1,…,N,with β>1, sometimes called generalized Dyson’s Brownian motion, describing the dissipative dynamics of a log-gas of N equal charges with equilibrium measure corresponding to a β-ensemble, with sufficiently regular convex potential V. The limit N→∞ is known to satisfy a mean-field Mc-Kean–Vlasov equation. We prove that, for suitable initial conditions, fluctuations around the limit are Gaussian and satisfy an explicit PDE.

Suggested Citation

  • Unterberger, Jérémie, 2018. "Global fluctuations for 1D log-gas dynamics," Stochastic Processes and their Applications, Elsevier, vol. 128(12), pages 4104-4153.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:12:p:4104-4153
    DOI: 10.1016/j.spa.2018.01.008
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    References listed on IDEAS

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    1. Israelsson, Stefan, 2001. "Asymptotic fluctuations of a particle system with singular interaction," Stochastic Processes and their Applications, Elsevier, vol. 93(1), pages 25-56, May.
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