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July 2005 Constrained Brownian motion: Fluctuations away from circular and parabolic barriers
Patrik L. Ferrari, Herbert Spohn
Ann. Probab. 33(4): 1302-1325 (July 2005). DOI: 10.1214/009117905000000125

Abstract

Motivated by the polynuclear growth model, we consider a Brownian bridge b(t) with bT)=0 conditioned to stay above the semicircle $c_{T}(t)=\sqrt{T^{2}-t^{2}}$. In the limit of large T, the fluctuation scale of b(t)−cT(t) is T1/3 and its time-correlation scale is T2/3. We prove that, in the sense of weak convergence of path measures, the conditioned Brownian bridge, when properly rescaled, converges to a stationary diffusion process with a drift explicitly given in terms of Airy functions. The dependence on the reference point tT, τ∈(−1,1), is only through the second derivative of cT(t) at tT. We also prove a corresponding result where instead of the semicircle the barrier is a parabola of height Tγ, γ>1/2. The fluctuation scale is then T(2−γ)/3. More general conditioning shapes are briefly discussed.

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Patrik L. Ferrari. Herbert Spohn. "Constrained Brownian motion: Fluctuations away from circular and parabolic barriers." Ann. Probab. 33 (4) 1302 - 1325, July 2005. https://doi.org/10.1214/009117905000000125

Information

Published: July 2005
First available in Project Euclid: 1 July 2005

zbMATH: 1082.60071
MathSciNet: MR2150190
Digital Object Identifier: 10.1214/009117905000000125

Subjects:
Primary: 60J65
Secondary: 60J60

Keywords: Conditioned Brownian bridge , limiting diffusion process

Rights: Copyright © 2005 Institute of Mathematical Statistics

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Vol.33 • No. 4 • July 2005
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