Geometrical optics of constrained Brownian motion: Three short stories

Baruch Meerson, Naftali R. Smith

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23 Scopus citations


The optimal fluctuation method-essentially geometrical optics-gives a deep insight into large deviations of Brownian motion. Here we illustrate this point by telling three short stories about Brownian motions, 'pushed' into a large-deviation regime by constraints. In story 1 we compute the short-time large deviation function (LDF) of the winding angle of a Brownian particle wandering around a reflecting disk in the plane. Story 2 addresses a stretched Brownian motion above absorbing obstacles in the plane. We compute the short-time LDF of the position of the surviving Brownian particle at an intermediate point. Story 3 deals with survival of a Brownian particle in 1 + 1 dimension against absorption by a wall which advances according to a power law xw (t) ∼ tγ, where γ > 1/2. We also calculate the LDF of the particle position at an earlier time, conditional on the survival by a later time. In all three stories we uncover singularities of the LDFs which have a simple geometric origin and can be interpreted as dynamical phase transitions. We also use the small-deviation limit of the geometrical optics to reconstruct the distribution of typical fluctuations. We argue that, in stories 2 and 3, this is the Ferrari-Spohn distribution.

Original languageAmerican English
Article number415001
JournalJournal of Physics A: Mathematical and Theoretical
Issue number41
StatePublished - 18 Sep 2019

Bibliographical note

Publisher Copyright:
© 2019 IOP Publishing Ltd.


  • Brownian motion
  • dynamical phase transitions
  • large deviations
  • optimal fluctuation method


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