On large deviation properties of Brownian motion with dry friction

TL;DR

This paper analyzes the large deviation properties of Brownian motion with dry friction, deriving explicit expressions for the distribution and moments of various functionals, revealing deviations from Gaussian behavior in long-time limits.

Contribution

It extends the backward Fokker-Planck technique to arbitrary support functionals and provides explicit solutions for the distribution of local time, occupation time, and displacement in Brownian motion with dry friction.

Findings

Explicit expressions for moments and distributions of functionals.

Quantitative measures of deviation from Gaussian behavior.

Extension of the backward Fokker-Planck technique.

Abstract

We investigate piecewise-linear stochastic models as with regards to the probability distribution of functionals of the stochastic processes, a question which occurs frequently in large deviation theory. The functionals that we are looking into in detail are related to the time a stochastic process spends at a phase space point or in a phase space region, as well as to the motion with inertia. For a Langevin equation with discontinuous drift, we extend the so-called backward Fokker-Planck technique for nonnegative support functionals to arbitrary support functionals, to derive explicit expressions for the moments of the functional. Explicit solutions for the moments and for the distribution of the so-called local time, the occupation time and the displacement are derived for the Brownian motion with dry friction, including quantitative measures to characterize deviation from Gaussian behaviour in the asymptotic long time limit.