Geometrical optics of constrained Brownian motion: three short stories

Abstract

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 $x_{\text{w}}\left(t\right)\sim t^{\gamma }$, where $\gamma >1/2$. 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 \emph{typical} fluctuations. We argue that, in stories 2 and 3, this is the Ferrari-Spohn distribution.

13 pages, 7 figures

References in corpus (6)

• Large Deviations of Extreme Eigenvalues of Random Matrices
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
• Mortality, Redundancy, and Diversity in Stochastic Search
• Constrained Brownian motion: Fluctuations away from circular and parabolic barriers
• Large fluctuations of the area under a constrained Brownian excursion

Cited by in corpus (27)

• The Airy distribution: experiment, large deviations and additional statistics
• Statistics of first-passage Brownian functionals
• Toward the full short-time statistics of an active Brownian particle on the plane
• Nonequilibirum steady state for harmonically-confined active particles
• Condensation transition in large deviations of self-similar Gaussian processes with stochastic resetting
• Search efficiency in the Adam-Delbrück reduction-of-dimensionality scenario versus direct diffusive search
• Geometrical optics of large deviations of fractional Brownian motion
• Geometrical optics of first-passage functionals of random acceleration
• Brownian flights over a circle
• Anomalous scalings of fluctuations of the area swept by a Brownian particle trapped in a $|x|$ potential
• Area fluctuations on a subinterval of Brownian excursion
• Geometrical optics of large deviations of Brownian motion in inhomogeneous media
• Large deviations in statistics of the convex hull of passive and active particles: A theoretical study
• Cosmic-ray transport in inhomogeneous media
• Nonequilibrium steady state of Brownian motion in an intermittent potential
• Stretching of a fractal polymer around a disc reveals KPZ-like statistics
• Fixman problem revisited: When fluctuations of inflated ideal polymer loop are non-Gaussian?
• Constrained non-crossing Brownian motions, fermions and the Ferrari-Spohn distribution
• Thermodynamic uncertainty relations for many-body systems with fast jump rates and large occupancies
• Exact short-time height distribution and dynamical phase transition in the relaxation of a Kardar-Parisi-Zhang interface with random initial condition
• Short-time large deviation of constrained random acceleration process
• Orientational Distribution of an Active Brownian Particle: an analytical study
• Mean square winding angle of Brownian motion around an impenetrable cylinder
• Non-algebraic first return probability of a stretched random walk near a convex boundary and its effect on adsorption
• Dimensionality-induced dynamical phase transition in the large deviation of local time density for Brownian motion
• Short-time large deviations of first-passage functionals for high-order stochastic processes
• A Tale of Two (and More) Altruists