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.