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 $c_{T}(t)=\sqrt{T^{2}-t^{2}}$. 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∈(−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.