Geometrical optics of large deviations of Brownian motion in inhomogeneous media

Abstract

Geometrical optics provides an instructive insight into Brownian motion, ``pushed" into a large-deviations regime by imposed constraints. Here we extend geometrical optics of Brownian motion by accounting for diffusion inhomogeneity in space. We consider three simple model problems of Brownian motion on the line or in the plane in situations where the diffusivity of the Brownian particle depends on one spatial coordinate. One of our results describes ``Brownian refraction": an analog of refraction of light passing through a boundary between two media with different refraction indices.