Journal Article

Constrained Brownian Motion: Fluctuations Away from Circular and Parabolic Barriers

Spohn, Herbert, Ferrari, Patrik L.

2005

Overview

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)-cT(t) is T1/3and its time-correlation scale is T2/3. 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 cT(t) at t = τT. We also prove a corresponding result where instead of the semicircle the barrier is a parabola of height Tγ, γ > 1/2. The fluctuation scale is then T(2-γ)/3. More general conditioning shapes are briefly discussed.

• Publisher: Institute of Mathematical Statistics,The Institute of Mathematical Statistics
• Subject: 60J60 / 60J65 / Airy function / Approximation / Brownian bridge / Brownian motion / Circles / Conditioned Brownian bridge / Continuous functions / Density / Exact sciences and technology / limiting diffusion process / Markov processes / Mathematics / Parabolas / Probability and statistics / Probability theory and stochastic processes / Sciences and techniques of general use / Semicircles / Transition probabilities