# Geometrical optics of constrained Brownian motion: three short stories

## TL;DR

This paper uses geometrical optics to analyze large deviations in constrained Brownian motion, revealing singularities and phase transitions through three illustrative scenarios involving winding angles, obstacle avoidance, and survival against moving absorbing boundaries.

## Contribution

It introduces a geometrical optics approach to large deviations in Brownian motion, providing explicit calculations and insights into phase transitions and typical fluctuation distributions.

## Key findings

- Derived short-time large deviation functions for winding angles and positions.
- Identified singularities as dynamical phase transitions in the LDFs.
- Reconstructed the distribution of typical fluctuations, linking to Ferrari-Spohn distribution.

## 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.