Probability and Stochastic Analysis - University of Bonn

Abstract of "Constrained Brownian motion: fluctuations away from circular and parabolic barriers"

with Herbert Spohn.

Motivated by the polynuclear growth model, we consider a Brownian bridge b(t) with b(T)=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=tau T, tau ∈ (-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.

Postscript file: [PS]

PDF file: [PDF]

arXiv: math.PR/0308242