Large fluctuations of the area under a constrained Brownian excursion

Baruch Meerson*

*Corresponding author for this work

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11 Scopus citations


We study large fluctuations of the area A under a Brownian excursion x(t) on the time interval |t| ≤ T , constrained to stay away from a moving wall x0(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) = Tγ[1 - (t/T )2k], where k ≥ 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 A at a critical value of 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 A with T, with an exponent that depends continuously on γ and on k. We also consider the cosinewall x 0 (t) = Tγ cos[πt/(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.

Original languageAmerican English
Article number013210
JournalJournal of Statistical Mechanics: Theory and Experiment
Issue number1
StatePublished - 30 Jan 2019

Bibliographical note

Funding Information:
I am very grateful to Naftali Smith for valuable advice and for producing figure 1. I acknowledge a useful discussion with Tal Agranov. I am also grateful to the Center of Mathematical Research (Centro di Ricerca Matematica) Ennio De Giorgi in Pisa, where this work started, for hospitality. This research was supported by the Israel Science Foundation (grant No. 807/16).

Publisher Copyright:
© 2019 IOP Publishing Ltd.


  • Brownian motion
  • diffusion
  • fluctuation phenomena
  • large deviations in non-equilibrium systems


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