Equilibrium fluctuation of the Atlas model
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Amir Dembo [1] ; Li-Cheng Tsai [2]
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[1] Stanford University
Stanford University
Estados Unidos
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[2] Columbia University
Columbia University
Estados Unidos
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 6, 2, 2017, págs. 4529-4560
- Idioma: inglés
- Enlaces
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Resumen
We study the fluctuation of the Atlas model, where a unit drift is assigned to the lowest ranked particle among a semi-infinite (Z+Z+-indexed) system of otherwise independent Brownian particles, initiated according to a Poisson point process on R+R+. In this context, we show that the joint law of ranked particles, after being centered and scaled by t−14t−14, converges as t→∞t→∞ to the Gaussian field corresponding to the solution of the Additive Stochastic Heat Equation (ASHE) on R+R+ with the Neumann boundary condition at zero. This allows us to express the asymptotic fluctuation of the lowest ranked particle in terms of a fractional Brownian Motion (fBM). In particular, we prove a conjecture of Pal and Pitman [Ann. Appl. Probab. 18 (2008) 2179–2207] about the asymptotic Gaussian fluctuation of the ranked particles.
