Large deviations of currents in diffusions with reflective boundaries

TL;DR

This paper investigates the probabilities of current fluctuations in diffusions within bounded regions with reflections, deriving boundary conditions for the spectral problem that characterizes these large deviations, and illustrating results with particle diffusion models.

Contribution

It introduces boundary conditions for the spectral problem of current large deviations in reflected diffusions, extending previous density-focused results and providing two derivation methods.

Findings

Derived boundary conditions for the spectral problem in reflected diffusions.

Presented two methods: diffusive limit of random walks and Feynman--Kac equation.

Applied results to an N-particle diffusion model on a ring.

Abstract

We study the large deviations of current-type observables defined for Markov diffusion processes evolving in smooth bounded regions of $\mathbb{R}^d$ with reflections at the boundaries. We derive for these the correct boundary conditions that must be imposed on the spectral problem associated with the scaled cumulant generating function, which gives, by Legendre transform, the rate function characterizing the likelihood of current fluctuations. Two methods for obtaining the boundary conditions are presented, based on the diffusive limit of random walks and on the Feynman--Kac equation underlying the evolution of generating functions. Our results generalize recent works on density-type observables, and are illustrated for an $N$-particle single-file diffusion on a ring, which can be mapped to a reflected $N$-dimensional diffusion.