The Positive Occupation Time of Brownian Motion with Two-Valued Drift and Asymptotic Dynamics of Sliding Motion with Noise

TL;DR

This paper derives the probability density function for the positive occupation time of one-dimensional Brownian motion with two-valued drift, analyzes its long-term behavior, and applies these results to stochastic sliding motion in higher-dimensional systems.

Contribution

It provides explicit formulas and asymptotic analysis for occupation times and extends these results to stochastic differential equations modeling sliding motion.

Findings

Explicit density function for occupation time derived

Asymptotic behavior of the density computed for long times

Application to stochastic sliding motion in multi-dimensional systems

Abstract

We derive the probability density function of the positive occupation time of one-dimensional Brownian motion with two-valued drift. Long time asymptotics of the density are also computed. We use the result to describe the transitional probability density function of a general N-dimensional system of stochastic differential equations representing stochastically perturbed sliding motion of a discontinuous, piecewise-smooth vector field on short time frames. A description of the density at larger times is obtained via an asymptotic expansion of the Fokker-Planck equation.