11 research outputs found
Delayed Random Walks: Investigating the Interplay Between Delay and Noise
Summary. A model for a 1âdimensional delayed random walk is developed by gen-eralizing the Ehrenfest model of a discrete random walk evolving on a quadratic, or harmonic, potential to the case of nonâzero delay. The FokkerâPlanck equation derived from this delayed random walk (DRW) is identical to that obtained starting from the delayed Langevin equation, i.e. a firstâorder stochastic delay differential equation (SDDE). Thus this DRW and SDDE provide alternate, but complimen-tary ways for describing the interplay between noise and delay in the vicinity of a fixed point. The DRW representation lends itself to determinations of the joint probability function and, in particular, to the autoâcorrelation function for both the stationary and transient states. Thus the effects of delay are manisfested through experimentally measurable quantities such as the variance, correlation time, and the power spectrum. Our findings are illustrated through applications to the analysis of the fluctuations in the center of pressure that occur during quiet standing. Key words: delay, random walk, stochastic delay differential equation, Fokker-Planck equation, autoâcorrelation function, postural sway Feedback control mechanisms are ubiquitous in physiology [2, 8, 17, 22