Area fluctuations on a subinterval of Brownian excursion

TL;DR

This paper extends the Airy distribution to describe area fluctuations on subintervals of Brownian excursions, analyzing large deviations and revealing phase transitions in the conditioned distributions.

Contribution

It introduces a generalized distribution for subinterval areas of Brownian excursions and explores their large deviation properties and phase transitions.

Findings

Explicit Laplace transform for subinterval area distribution in Model 1

Identification of second and third order phase transitions in conditioned large areas

Analysis of tail behaviors and large deviation rate functions

Abstract

Area fluctuations of a Brownian excursion are described by the Airy distribution, which found applications in different areas of physics, mathematics and computer science. Here we generalize this distribution to describe the area fluctuations on a \emph{subinterval} of a Brownian excursion. In the first version of the problem (Model 1) no additional conditions are imposed. In the second version (Model 2) we study the distribution of the area fluctuations on a subinterval given the excursion area on the whole interval. Both versions admit convenient path-integral formulations. In Model 1 we obtain an explicit expression for the Laplace transform of the area distribution on the subinterval. In both models we focus on large deviations of the area by evaluating the tails of the area distributions, sometimes with account of pre-exponential factors. When conditioning on very large areas in Model 2, we uncover two singularities in the rate function of the subinterval area fraction. They can be interpreted as dynamical phase transitions of second and third order.