Delayed random walks: Investigating the interplay between delay and noise

A model for a 1–dimensional delayed random walk is developed by generalizing the Ehrenfest model of a discrete random walk evolving on a quadratic, or harmonic, potential to the case of non–zero delay. The Fokker–Planck equation derived from this delayed random walk (DRW) is identical to that obtained starting from the delayed Langevin equation, i.e. a first–order stochastic delay differential equation (SDDE). Thus this DRW and SDDE provide alternate, but complimentary 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 auto–correlation 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.