Constrained Brownian motion: Fluctuations away from circular and parabolic barriers

Motivated 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=\tau T, \tau\in(-1,1), is only through the second derivative of c_T(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.
Comment: Published 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)