Large fluctuations of the area under a constrained Brownian excursion

Abstract

We study large fluctuations of the area $\mathcal{A}$ under a Brownian excursion $x(t)$ on the time interval $|t|\leq T$, constrained to stay away from a moving wall $x_0(t)$ such that $x_0(-T)=x_0(T)=0$ and $x_0(|t|<T)>0$. We focus on wall functions described by a family of generalized parabolas $x_0(t)=T^{\gamma} [1-(t/T)^{2k}]$, where $k\geq 1$. Using the optimal fluctuation method (OFM), we calculate the large deviation function (LDF) of the area at long times. The OFM provides a simple description of the area fluctuations in terms of optimal paths, or rays, of the Brownian motion. We show that the LDF has a jump in the third derivative with respect to $\mathcal{A}$ at a critical value of $\mathcal{A}$. This singularity results from a qualitative change of the optimal path, and it can be interpreted as a third-order dynamical phase transition. Although the OFM is not applicable for typical (small) area fluctuations, we argue that it correctly captures their power-law scaling of $\mathcal{A}$ with $T$ with an exponent that depends continuously on $\gamma$ and on $k$. We also consider the cosine wall $x_0(t)=T^{\gamma} \cos[\pi t/(2T)]$ to illustrate a different possible behavior of the optimal path and of the scaling of typical fluctuations. For some wall functions additional phase transitions, which result from a coexistence of multiple OFM solutions, should be possible.