Stiffness of random walks with reflecting boundary conditions

Abstract

We study the distribution of occupation times for a one-dimensional random walk restricted to a finite interval by reflecting boundary conditions. At short times the classical bimodal distribution due to Lévy is reproduced with walkers staying mostly either left or right to the initial point. With increasing time, however, the boundaries suppress large excursions from the starting point, and the distribution becomes unimodal converging to a $\delta$-distribution in the long time limit. An approximate spectral analysis of the underlying Fokker-Planck equation yields results in excellent agreement with numerical simulations.

arXiv comment6 pages, 9 figures, references added, small adjustments in the text

References in corpus (7)

• The Local and the Occupation Time of a Particle Diffusing in a Random Medium
• A minimal model of dynamical phase transition
• Dynamical phase transition in drifted Brownian motion
• Large-Deviation Functions for Nonlinear Functionals of a Gaussian Stationary Markov Process
• Residence Time Statistics for Normal and Fractional Diffusion in a Force Field
• Dynamical large deviations of reflected diffusions
• Renewal theory with fat tailed distributed sojourn times: typical versus rare

Cited by in corpus (1)

• Short-time large deviations of first-passage functionals for high-order stochastic processes