Constrained Brownian motion: Fluctuations away from circular and parabolic barriers

Abstract

Motivated by the polynuclear growth model, we consider a Brownian bridge b(t) with b(±T) = 0 conditioned to stay above the semicircle c T(t) = √T 2-t 2. In the limit of large T, the fluctuation scale of b(t) - c T(I) 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 r = τT, τ ∈ (-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. © Institute of Mathematical Statistics, 2005.