Year: 2019
Type: article
Abstract: We study large fluctuations of the area under a Brownian excursion on the time interval , constrained to stay away from a moving wall x 0 ( t ) such that and x 0 (| 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.
Author: Baruch Meerson
Institution: Hebrew University of Jerusalem
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2899868756