TY - JOUR

T1 - Geometrical optics of constrained Brownian motion

T2 - Three short stories

AU - Meerson, Baruch

AU - Smith, Naftali R.

N1 - Publisher Copyright:
© 2019 IOP Publishing Ltd.

PY - 2019/9/18

Y1 - 2019/9/18

N2 - 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.

AB - 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.

KW - Brownian motion

KW - dynamical phase transitions

KW - large deviations

KW - optimal fluctuation method

UR - http://www.scopus.com/inward/record.url?scp=85073103646&partnerID=8YFLogxK

U2 - 10.1088/1751-8121/ab3f0f

DO - 10.1088/1751-8121/ab3f0f

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AN - SCOPUS:85073103646

SN - 1751-8113

VL - 52

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

IS - 41

M1 - 415001

ER -