Constrained Brownian motion: Fluctuations away from circular and parabolic barriers

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dc.creatorFerrari, Patrik L.
dc.creatorSpohn, Herbert
dc.date2003-08-26
dc.date2005-08-25
dc.date.accessioned2026-07-07T05:00:37Z
dc.date.available2026-07-07T05:00:37Z
dc.descriptionMotivated by the polynuclear growth model, we consider a Brownian bridge b(t) with b(\pm T)=0 conditioned to stay above the semicircle c_T(t)=\sqrtT^2-t^2. In the limit of large T, the fluctuation scale of b(t)-c_T(t) is T^{1/3} and its time-correlation scale is T^{2/3}. 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 t=τT, τ\in(-1,1), is only through the second derivative of c_T(t) at t=τT. We also prove a corresponding result where instead of the semicircle the barrier is a parabola of height T^γ, γ>1/2. The fluctuation scale is then T^{(2-γ)/3}. More general conditioning shapes are briefly discussed.
dc.descriptionPublished at http://dx.doi.org/10.1214/009117905000000125 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
dc.identifierhttps://arxiv.org/abs/math/0308242
dc.identifierhttp://arxiv.org/abs/math/0308242
dc.identifierAnnals of Probability 2005, Vol. 33, No. 4, 1302-1325
dc.identifierdoi:10.1214/009117905000000125
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/68385
dc.subjectProbability
dc.subjectMathematical Physics
dc.subject60J65 (Primary) 60J60 (Secondary)
dc.titleConstrained Brownian motion: Fluctuations away from circular and parabolic barriers
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