Constrained Brownian motion: Fluctuations away from circular and parabolic barriers

• 成果类型: Article
• 署名作者: Ferrari, PL; Spohn, H
• 署名单位: Technical University of Munich
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117905000000125
• 发表日期: 2005
• 页码: 1302-1325
• 关键词: airy

摘要

Motivated by the polynuclear growth model, we consider a Brownian bridge b(t) with b(+/- T) = Oconditioned to stay above the semicircle C-T(t) = root T-2 - t(2). In the limit of large T, the fluctuation scale of b(t) - CT(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 resealed, converges to a stationary diffusion process with a drift explicitly given in terms of Airy functions. The dependence on the reference point t = tau T, tau is an element of (-1, 1), is only through the second derivative Of CT(t) at t = tau T. We also prove a corresponding result where instead of the semicircle the barrier is a parabola of height T-gamma, gamma > 1/2. The fluctuation scale is then T(2-gamma)/(3). More general conditioning shapes are briefly discussed.