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

2012/04/26 by David J. W. Simpson, Simpson, David J. W., Rachel Kuske +1
Mathematics · #34F05 #37H10 #60H10 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:34F05 #msc:37H10 #msc:60H10

paper · pdf · doi:10.48550/arxiv.1204.5985

Submitted to: Stoch. Dyn

arxiv created 2013/06/05 · arxiv updated 2013/06/06

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.

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