Large fluctuations of the area under a constrained Brownian excursion

Abstract

We study large fluctuations of the area under a Brownian excursion on the time interval , constrained to stay away from a moving wall x0(t) such that and x0(|t| < T) > 0. We focus on wall functions described by a family of generalized parabolas , where . 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 at a critical value of . 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 with T, with an exponent that depends continuously on and on k. We also consider the cosine wall 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.